Course:PHYS341/Archive/2016wTerm2/WellTempermentAndItsEffectOnMusicalColour

Well Temperament and Its Effect On Musical Colour

Well Temperament is a type of tempered tuning; a system used to tune strings in an instrument, (in most cases the piano,) to certain frequency integer ratios. The term appears in the title of J.S. Bach’s composition, The Well-Tempered Clavier. The origins of well temperament are hard to pinpoint, although one of its earliest mentions is by Werckmeister, in "Orgelprobe, 1681.” It was later described in 1799 by Thomas Young 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.

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.

How Strings Can Be Tuned

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 stringed instrument will contain a set of strings, that are fastened to fixed points. If excited, these strings vibrate with a fundamental frequency, depending on how tight or loose the string is set. Along with the fundamental frequency, the string produces 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. 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 tuning instruments and a good ear. A piano's tuning pin can be seen in figure 1.1.

Well Temperament

Unlike equal temperament, well temperament was not designed to divide the octave into equal semitones, or 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, 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.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, which gives the keys related to C a gentle and sweet quality, while the black keys are described as being more austere, and noble. The middle keys, such as Eb and A are considered more neutral. The major chords in the keys of C or F have been described as sounding much more stable 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 tune.

Fig.2.A cent is a unit of measurement for pitch. In equal temperament 1 semitone is 100 cents. 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 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. This is due to the phenomenon of 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. 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. The stable sounding chromatics, i.e. minor 2ⁿᵈˢ, are E-F and B-C. 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 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.

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. The key of Db has been described to be quite harsh, 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.

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.

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.

See Also

• musical temperament
• piano
• strings
• musical acoustics