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Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-04-06T20:48:56Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

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===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are fastened to fixed points. If excited, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set. Along with the fundamental frequency, the string produces [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. Strings can vary in thickness, and length, which change the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, represented in a numerical value, the string should be either tightened or loosened, via the string's tuning pin. This process usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. Certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it is noticeable. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

A notable feature of well temperament, is that each interval can have more than one type. There can be: 5 kinds of minor 2ⁿᵈˢ and major 7ᵗʰˢ, 7 kinds of major 3ʳᵈˢ and minor 6ᵗʰˢ, 5 kinds of major 2ⁿᵈˢ and minor 7ᵗʰˢ, 6 kinds of minor 3ʳᵈˢ and major 6ᵗʰˢ, 5 kinds of tritons, and 3 kinds of perfect 5ᵗʰˢ and 4ᵗʰˢ.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

In the black keys for example, there are 5ᵗʰˢ of 702 cents,<ref name=Gann1997>{{Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref name=Gann1997 /> The middle keys, such as Eb and A are considered more neutral. <ref name=Gann1997 />
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major 3ʳᵈ interval is created, (391.7 cents) so it reverberates for a long period of time. Repeating the same experiment, but with F# and A#, the major 3ʳᵈ does not sing as richly, and [https://en.wikipedia.org/wiki/Damping_(music) dampens] at an accelerated rate. The 3ʳᵈ created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the 3ʳᵈ comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# 3ʳᵈ is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here shows how certain keys can vary in excitement to produce varied sound colour in well temperament.

The most stable major 3ʳᵈˢ are thought to be C-E, followed by G-B and F-A. The most stable minor 3ʳᵈˢ are those of the keys of A and E, while the most stable major 2ⁿᵈˢ fall on the black keys.<ref name=Rubinstein2000 /> The stable sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2ⁿᵈˢ, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3ʳᵈ, minor 3ʳᵈ, and perfect 5ᵗʰ intervals can be seen in figure 2.

In well temperament, because there are several types of perfect 5ᵗʰˢ and 4ᵗʰˢ, the piano resonates in different ways, depending on which key is being played, when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref name=Gann1997 /> The key of Db has been described to be quite harsh<ref name=Gann1997 />, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T02:32:46Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are fastened to fixed points. If excited, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set. Along with the fundamental frequency, the string produces [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. Strings can vary in thickness, and length, which change the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, represented in a numerical value, the string should be either tightened or loosened, via the string's tuning pin. This process usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. Certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it is noticeable. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

A notable feature of well temperament, is that each interval can have more than one type. There can be: 5 kinds of minor 2ⁿᵈˢ and major 7ᵗʰˢ, 7 kinds of major 3ʳᵈˢ and minor 6ᵗʰˢ, 5 kinds of major 2ⁿᵈˢ and minor 7ᵗʰˢ, 6 kinds of minor 3ʳᵈˢ and major 6ᵗʰˢ, 5 kinds of tritons, and 3 kinds of perfect 5ᵗʰˢ and 4ᵗʰˢ.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

In the black keys for example, there are 5ᵗʰˢ of 702 cents,<ref name=Gann1997>{{Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref name=Gann1997 /> The middle keys, such as Eb and A are considered more neutral. <ref name=Gann1997 />
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major 3ʳᵈ interval is created, (391.7 cents) so it reverberates for a long period of time. Repeating the same experiment, but with F# and A#, the major 3ʳᵈ does not sing as richly, and [https://en.wikipedia.org/wiki/Damping_(music) dampens] at an accelerated rate. The 3ʳᵈ created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the 3ʳᵈ comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# 3ʳᵈ is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here shows how certain keys can vary in excitement to produce varied sound colour in well temperament.

The most stable major 3ʳᵈˢ are thought to be C-E, followed by G-B and F-A. The most stable minor 3ʳᵈˢ are those of the keys of A and E, while the most stable major 2ⁿᵈˢ fall on the black keys.<ref name=Rubinstein2000 /> The stable sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2ⁿᵈˢ, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3ʳᵈ, minor 3ʳᵈ, and perfect 5ᵗʰ intervals can be seen in figure 2.

In well temperament, because there are several types of perfect 5ᵗʰˢ and 4ᵗʰˢ, the piano resonates in different ways, depending on which key is being played, when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref name=Gann1997 /> The key of Db has been described to be quite harsh<ref name=Gann1997 />, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T02:08:56Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are fastened to fixed points. If excited, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways. In the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string produces [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. Strings can vary in thickness, and length, which change the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, represented in a numerical value, the string should be either tightened or loosened, via the string's tuning pin. This process usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. Certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it is noticeable. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

A notable feature of well temperament, is that each interval can have more than one type. There can be: 5 kinds of minor 2ⁿᵈˢ and major 7ᵗʰˢ, 7 kinds of major 3ʳᵈˢ and minor 6ᵗʰˢ, 5 kinds of major 2ⁿᵈˢ and minor 7ᵗʰˢ, 6 kinds of minor 3ʳᵈˢ and major 6ᵗʰˢ, 5 kinds of tritons, and 3 kinds of perfect 5ᵗʰˢ and 4ᵗʰˢ.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament. In the black keys for example, there are 5ᵗʰˢ of 702 cents,<ref name=Gann1997>{{Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref name=Gann1997 /> The middle keys, such as Eb and A are considered more neutral. <ref name=Gann1997 />
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major 3ʳᵈ interval is created, (391.7 cents) so it reverberates for a long period of time. Repeating the same experiment, but with F# and A#, the major 3ʳᵈ does not sing as richly, and [https://en.wikipedia.org/wiki/Damping_(music) dampens] at an accelerated rate. The 3ʳᵈ created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the 3ʳᵈ comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# 3ʳᵈ is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here shows how certain keys can vary in excitement to produce varied sound colour in well temperament.

The most stable major 3ʳᵈˢ are thought to be C-E, followed by G-B and F-A. The most stable minor 3ʳᵈˢ are those of the keys of A and E, while the most stable major 2ⁿᵈˢ fall on the black keys.<ref name=Rubinstein2000 /> The stable sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2ⁿᵈˢ, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3ʳᵈ, minor 3ʳᵈ, and perfect 5ᵗʰ intervals can be seen in figure 2.

In well temperament, because there are several types of perfect 5ᵗʰˢ and 4ᵗʰˢ, the piano resonates in different ways, depending on which key is being played, when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref name=Gann1997 /> The key of Db has been described to be quite harsh<ref name=Gann1997 />, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:49:32Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

A notable feature of well temperament, is that each interval can have more than one different type. For example, there can be: 5 kinds of minor 2ⁿᵈˢ and major 7ᵗʰˢ, 7 kinds of major 3ʳᵈˢ and minor 6ᵗʰˢ, 5 kinds of major 2ⁿᵈˢ and minor 7ᵗʰˢ, 6 kinds of minor 3ʳᵈˢ and major 6ᵗʰˢ, 5 kinds of tritons, and 3 kinds of perfect 5ᵗʰˢ and 4ᵗʰˢ.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament. In the black keys for example, there are 5ᵗʰˢ of 702 cents,<ref name=Gann1997>{{Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref name=Gann1997 /> The middle keys, such as Eb and A are considered more neutral. <ref name=Gann1997 />
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major 3ʳᵈ interval is created, (391.7 cents) and so it reverberates for a long period of time. Repeating the same experiment, but with F# and A#, the major 3ʳᵈ does not sing as richly, and dampens at an accelerated rate. The 3ʳᵈ created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the 3ʳᵈ comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# 3ʳᵈ is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major 3ʳᵈˢ are thought to be C-E, followed by G-B and F-A. The most stable minor 3ʳᵈˢ are those of the keys of A and E, while the most stable major 2ⁿᵈˢ fall on the black keys.<ref name=Rubinstein2000 /> The stable sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2ⁿᵈˢ, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3ʳᵈ, minor 3ʳᵈ, and perfect 5ᵗʰ intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ᵗʰˢ and 4ᵗʰˢ, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref name=Gann1997 /> The key of Db has been described to be quite harsh<ref name=Gann1997 />, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:15:40Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref name=Gann1997>{{Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref name=Gann1997 /> The middle keys, such as Eb and A are considered more neutral. <ref name=Gann1997 />
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref name=Rubinstein2000 /> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref name=Gann1997 /> The key of Db has been described to be quite harsh<ref name=Gann1997 />, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:13:41Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref name=Gann1997>{{Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref name=Rubinstein2000 /> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:12:50Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref name=Gann1997>{{<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>}}</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref name=Rubinstein2000 /> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:11:09Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref name=Rubinstein2000 />The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref name=Rubinstein2000 /> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref name=Rubinstein2000 /> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref name=Rubinstein2000 /> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref name=Rubinstein2000 /> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref name=Rubinstein2000 /> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref name=Rubinstein2000 />]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref name=Rubinstein2000 />

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:09:04Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref name=Rubinstein2000 /> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-30T01:07:43Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref name=Rubinstein2000>{{Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html}}</ref>, or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

[Very interesting; I didn't know any of this until you mentioned it. Simplify your text somewhat; it is already over the word limit. You don't need to cite any one source more than once or twice, and you don't have to write out the reference each time. - CEW]

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course talk:PHYS341/Archive/2016wTerm2/AcousticsoftheGuitar

2017-03-23T21:44:14Z

AlexanderFonarev: Talk page autocreated when first thread was posted

Thread:Course talk:PHYS341/AcousticsoftheGuitar/Peer Review of your work

2017-03-23T21:44:14Z

AlexanderFonarev: New thread: Peer Review of your work

Hi,

There have been things done right here, but also a few things that I have caught.
Firstly, I think that the level of description is at about where it should be, It was detailed enough for me to understand, but not too dense, which I think is good.
You have a table of contents, an introduction paragraph, and a title, and everything seems to be structured as it should be.

That being said, it looks like you have a few grammar and spelling mistakes scattered throughout. Some that I can point out for you are:
"The mass of the string effects the pitch" -> it should be "affects," not effects.
"The strings provide the initial energy to vibrating top plate." -> Is missing a "the" in between "to" and "vibrating"
"therefore the sounds wave is thin and insignificant." -> there is an extra "s" at the end of "sound" (Also the word insignificant could be seen as too strong of an adjective, it maybe makes the writing feel more persuasive than objective, so be careful in your use of adjectives!)

Also, be careful about being too colloquial, for example "plays less of a role" could perhaps be worded differently.

One part of the prose I felt was worded a little awkwardly is: "While the mode of vibration of the string is plucked as to produce sound, the musician must pluck them." I think "while" can just be omitted here.

I think for some concepts, torque for example, could benefit from being hyperlinked, so that if the reader doesn't understand a concept they can read about it on another wiki.

It's good that you've made mental notes to add diagrams and pictures to the article, those are important! Also, make sure you add to the "see also" section at the bottom of the page, otherwise it would look rather awkward to leave it empty!

Overall, I think that this is a good draft, the references are consistent, and the word count has been met. Your style is for the most part consistent and digestible for the reader, except in a few places that I pointed out, but make sure to scan through your draft thoroughly to tighten things up. Good work.

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:40:32Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:34:29Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point, which allows for the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:32:52Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point. In doing so it allows the sound wave to travel back and forth through the string, as well as allow the string tightness to be adjusted to change the frequency it vibrates at.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

File:Tuningpin.png

2017-03-17T02:32:00Z

AlexanderFonarev: User created page with UploadWizard

=={{int:filedesc}}==
{{Information
|description={{en|1=Tuning pins fix the string in place and provide the sound wave something to reflect from. They are used to tighten or loosen the string to change its pitch}}
|date=2017-03-16 19:31:08
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Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:30:44Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

[[File:tuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point. In doing so it allows the sound wave to travel back and forth through the string, as well as allow the string tightness to be adjusted to change the frequency it vibrates at.]]

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:29:11Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

[[File:Pianotuningpin.jpg|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point. In doing so it allows the sound wave to travel back and forth through the string, as well as allow the string tightness to be adjusted to change the frequency it vibrates at.]]

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:27:35Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

[[File:Pianotuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point. In doing so it allows the sound wave to travel back and forth through the string, as well as allow the string tightness to be adjusted to change the frequency it vibrates at.]]

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

File:Pianotuningpin.jpg

2017-03-17T02:25:31Z

AlexanderFonarev: User created page with UploadWizard

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{{Information
|description={{en|1=A piano's tuning pins hold the string in place to provide the sound wave something to reflect off of. They can also be tightened or loosened to change the pitch of the string.}}
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Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-17T02:20:46Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

[[File:pianotuningpin.png|thumbnail|Fig.1.1.A tuning pin holds the string in place and provides for it a fixed point. In doing so it allows the sound wave to travel back and forth through the string, as well as allow the string tightness to be adjusted to change the frequency it vibrates at.]]

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened, via the string's tuning pin (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>. A piano's tuning pin can be seen in figure 1.1.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:22:26Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:21:44Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key is being played, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds. The perceived musical colour can vary depending on which intervals are being played.

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:17:21Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament with what is being played. In the black keys for example, there are 5ths of 702 cents,<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:13:58Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
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===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch of musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:08:35Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning]; a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:08:01Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table however, illustrates how well temperament has intervals that vary in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:06:25Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
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__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tableofcentintervals.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

File:Tableofcentintervals.png

2017-03-16T22:05:39Z

AlexanderFonarev: User created page with UploadWizard

=={{int:filedesc}}==
{{Information
|description={{en|1=A table displaying the cent values of various intervals}}
|date=2017-03-16 15:04:19
|source=<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
|author=Kyle Gann
|permission=
|other_versions=
}}

=={{int:license-header}}==
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[[Category:Uploaded with UploadWizard]]

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T22:02:24Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:tablecents.png|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which have the ratios of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

File:Wtvseqratio.png

2017-03-16T05:44:23Z

AlexanderFonarev: User created page with UploadWizard

=={{int:filedesc}}==
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|description={{en|1=A table that shows the differences between well temperament and equal temperament ratios.}}
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Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:40:33Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
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===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:wtvseqratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:39:18Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:tableratio.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

File:Ratiotable.png

2017-03-16T05:37:36Z

AlexanderFonarev: User created page with UploadWizard

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Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:36:20Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:ratiotable.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:33:48Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:Ratiograph.png|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:32:49Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:Ratiograph.jpg|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

File:Ratiograph.png

2017-03-16T05:32:09Z

AlexanderFonarev: User created page with UploadWizard

=={{int:filedesc}}==
{{Information
|description={{en|1=a graph displaying the differences of the musical ratios between well and equal temperament}}
|date=2017-03-15 22:30:11
|source=Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html
|author=Michael Rubinstein
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Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:29:40Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
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===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

[[File:ratiograph.jpg|thumbnail|Fig.2.1.A table to compare the tuning ratios between well temperament and equal temperament. The ratios of well temperate are close to those of equal temperament but vary slightly.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>]]

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:05:39Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it, a set of strings, that are securely fastened to fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:04:43Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T05:04:17Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of [https://en.wikipedia.org/wiki/Musical_tuning tempered tuning], a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T04:07:07Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Piano piano]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T04:06:24Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[https://en.wikipedia.org/wiki/Musical_temperament musical temperament]
*[https://en.wikipedia.org/wiki/Equal_temperament equal temperament]
*[https://en.wikipedia.org/wiki/String_(music) strings]
*[https://en.wikipedia.org/wiki/Musical_acoustics musical acoustics]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T04:04:41Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[[[https://en.wikipedia.org/wiki/String_(music) strings]]]
*[[[https://en.wikipedia.org/wiki/Equal_temperament equal temperament]]]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T04:02:32Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

===See Also===
*[[strings]]
*[[equal temperament]]

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T04:00:00Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate<ref>White, H. E., & White, D. H. (2014). Physics and music: the science of musical sound. Mineola, NY: Dover Publications, Inc.</ref>. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear<ref>Piano tuning. (2017, March 15). Retrieved March 15, 2017, from https://en.wikipedia.org/wiki/Piano_tuning</ref>.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones]<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> , or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T03:54:12Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young]<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones], or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T03:53:24Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young<ref>Thomas Young (1800). Outlines of Experiments and Inquiries Respecting Sound and Light. By Thomas Young, M. D. F. R. S. In a Letter to Edward Whitaker Gray, M. D. Sec. R. S. Retrieved March 15, 2017, from https://archive.org/stream/philtrans09850747/09850747#page/n37/mode/1up, The material on temperaments appears on pages 143-47</ref> Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
{{Help Nav}}
__NOEDITSECTION__
__TOC__

===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones], or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

==References==
{{Reflist}}

Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

2017-03-16T03:48:58Z

AlexanderFonarev:

[[category:PHYS341-2017|WellTempermentAndItsEffectOnMusicalColour]]

<big><big><big>Well Temperament and Its Effect On Musical Colour</big></big></big>

[https://en.wikipedia.org/wiki/Well_temperament '''Well Temperament'''] is a type of tempered tuning, a system used to tune strings in an instrument, (in most cases predominately the [https://en.wikipedia.org/wiki/Piano piano],) to certain [https://en.wikipedia.org/wiki/Interval_ratio frequency integer ratios.] The term appears in the title of [https://en.wikipedia.org/wiki/Johann_Sebastian_Bach J.S. Bach]’s composition, The Well-Tempered Clavier<ref>Bach, Johann Sebastian; Novack, Saul (1983). "The Well-Tempered Clavier: Books I and II, complete". ISBN 978-0-486-24532-4.</ref>. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by [https://en.wikipedia.org/wiki/Andreas_Werckmeister Werckmeister], in "Orgelprobe, 1681<ref>Well temperament. (2017, March 14). Retrieved March 13, 2017, from https://en.wikipedia.org/wiki/Well_temperament</ref>.” It was later described in 1799 by [https://en.wikipedia.org/wiki/Thomas_Young_(scientist) Thomas Young<ref>Young (1800). The material on temperaments appears on pages 143-47</ref>.] Well temperament can be characterized as having a distinct “sound colour” that is similar to that of equal temperament, but with a few subtle differences.
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===How Strings Can Be Tuned===

[[File:pianostrings.jpg|thumbnail|Fig.1.A piano's strings are fixed at the ends so that the sound wave can reflect back and forth. The strings themselves can vary in thickness and length to alter how the wave travels and thus resonates.]]

A [https://en.wikipedia.org/wiki/String_instrument stringed instrument] will contain within it a set of strings, that are securely fastened fixed points. If excited into action, these strings vibrate with a [https://en.wikipedia.org/wiki/Fundamental_frequency fundamental frequency], depending on how tight or loose the string is set to be. A string can be excited through a multitude of ways, and in the case of the piano, the string is struck with a hammer. Along with the fundamental frequency, the string will also produce [https://en.wikipedia.org/wiki/Harmonic harmonics], which produce sound in integer multiples of the fundamental frequency. These strings can vary in thickness, and length, which changes the properties of the sound they emit as they vibrate. A picture of a piano’s strings can be seen in figure 1.

In order to tune a string to a particular frequency ratio, which is represented in a numerical value. The string should be either tightened or loosened (depending on its initial state) in a way that usually requires the use of certain [https://en.wikipedia.org/wiki/Tuning_wrench tuning instruments] and a good ear.

===Well Temperament===

Unlike [https://en.wikipedia.org/wiki/Equal_temperament equal temperament], well temperament was not designed to divide the octave into [https://en.wikipedia.org/wiki/Semitone equal semitones], or [https://en.wikipedia.org/wiki/Interval_(music) intervals]. (Another way to refer to the pitch between musical intervals is called '''cents'''.) Well temperament prioritizes certain keys and harmonies to sound more pure and some to sound less so. That is to say, certain semitones have larger gaps between the frequencies, and some have smaller gaps. Although the difference is very subtle, at most '''9% of a semitone''',<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> it can be heard by most ears. Through this slight difference in semitones, some keys thus sound more harmonious, and some sound more dissonant. In making the division of the octaves so invariable, each key is given its own sense of “character”.

Unlike in equal temperament, a notable feature of well temperament, is that each interval can have more than one different type, or kind. For example, there can be: 5 kinds of minor 2nds and major 7ths, 7 kinds of major 3rds and minor 6ths, 5 kinds of major 2nds and minor 7ths, 6 kinds of minor 3rds and major 6ths, 5 kinds of tritons, and 3 kinds of perfect 5ths and 4ths.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> The variance among the intervals is in part what gives well temperament its musical qualities.

====The Effect of Well Temperament on Musical Colour====

Hearing the intricacies of this tuning are more apparent to the pianist playing the instrument than the listener, although the listener can also hear the effects of the temperament on what is being played. In the black keys for example, there are 5ths of 702 cents, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The middle keys, such as Eb and A are considered more neutral. <ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>
The major chords in the keys of C or F have been described as sounding much more stable<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> compared to in the keys just a semitone above them. (C# or F#.) This stability is a result of the keys of C and F actually being more '''in [https://en.wikipedia.org/wiki/Musical_tuning tune'''].

To illustrate the stability described, one might play a C and an E. As a result, a very resonant major third interval is created, (391.7 cents) and so it reverberates for a long period of time. If one were to repeat the same experiment but with F# and A#, the major third does not sing as richly, and dampens at an accelerated rate. The third created by the F# and A# notes (407.9 cents) has been described as a having a wavy, and unstable, perhaps excited texture. The feeling of excitement and instability produced by the third comes from a pulsating that can be heard.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> This is due to the phenomenon of [https://en.wikipedia.org/wiki/Beat_(acoustics) '''beating''']. In other words, because the F# - A# third is slightly out of tune, the frequencies of the notes are partially out of sync, which causes the tones to increase and decrease in volume as they interfere with one another.<ref>Beat (acoustics). (2017, March 15). Retrieved March 12, 2017, from https://en.wikipedia.org/wiki/Beat_(acoustics)</ref> The example illustrated here is but one that shows how certain keys can create the sound colour of more excitement through the effect of well temperament on musical tone.

[[File:centstable.jpg|thumbnail|Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents<ref>R. Nave (n.d.). Cents. Retrieved March 15, 2017, from http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html</ref>. This table illustrates how well temperament does not have perfect ratios in pitch.]]

The most stable major thirds are thought to be C-E, followed by G-B and F-A. The most stable minor 3rds are those of the keys of A and E, while the most stable major seconds fall on the black keys. The best sounding [https://en.wikipedia.org/wiki/Diatonic_and_chromatic chromatics], i.e. minor 2nds, are E-F and B-C.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref> A table illustrating the cents between the major 3rd, minor 3rd, and perfect 5th intervals can be seen in figure 2.

Furthermore, in well temperament, because there are several types of perfect 5ths and 4ths, depending on which key the pianist is playing in, the piano resonates in different ways when the [https://en.wikipedia.org/wiki/Sustain_pedal sustain pedal] is pressed. Pressing the sustain pedal lifts the dampers off of the piano strings, which frees the other strings to resonate with other sounds.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

Some keys have been described to feel warmer than others, F# minor for example, has been noted to produce a very lush sound, especially when playing the adagio movement of the Hammerklavier sonata, as some pianists report.<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref> The key of Db has been described to be quite harsh<ref>Kyle Gann (1997). An Introduction to Historical Tunings. Retrieved March 12, 2017, from http://www.kylegann.com/histune.html#hist4</ref>, particularly the interval of Db and F (406 cents) has been noted to be surprising in its harshness, as it has a particularly strong beating.

The contrast in musical colour in well temperament versus that of equal temperament can be seen visually when comparing the frequency ratios of the two temperaments side by side. Although very close, each note has its own distinct ratio, which can be seen when graphed side by side. A figure showing the differences between well temperament and equal temperament can be seen in figure 2.1 The note in well temperament with the closest resemblance to equal temperament is F, which has the ratio of 1.334745462, and 1.334839854 respectively.<ref>Michael Rubinstein (2000). Well v.s. equal temperament. Retrieved March 12, 2017, from http://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html</ref>

==References==
{{Reflist}}