Course:PHYS341/Archive/2016wTerm2/EffectivenessOfEqualTemperament

Equal temperament is a method of tuning in which adjacent notes have the equal frequency ratios. It has become the standard tuning system of modern pianos and keyboard instruments.

Equal Temperament

History

Several factors influenced the standardization of equal temperament. Many scholars at the time were convinced that use of equal temperament was promoted by Bach, as stated in notes of the Well-Tempered Clavier. In truth, Bach and his contemporaries used a modified version of meantone temperament, and there is in fact more evidence that suggest Bach writing for unequal temperament. In terms of the progression of music composition, thirds in early polyphony were little used. However since the renaissance and going into the Baroque period, the major triad, which involves the third and fifth, became arguably the most important chord in composition. Enharmonic modulation became prominent in 18th century compositions, making equal temperament a great fit for keyboard music. Most composers from 18th century and onward presumed equal temperament would be used, and if not, would specify their tuning method, such as in the music of Ligeti.

In Practice

Grand Piano Tuning

Although other instruments like winds and strings can easily alter pitch and tuning with a number of accessible factors, the situation for pianos and other keyboard instruments are more peculiar. The absolute frequency of the note is fixed after the entire instrument is tuned; therefore the temperament must be decided in advance. Furthermore, the tools and knowledge required to tune a piano are so particular that piano tuning is a whole discipline in and of itself.

In a scenario where the piano was playing major thirds, and the instrumentalist being accompanied were to play in tune, we would likely hear the phenomenon known as beating, due to the fifth and fourth harmonics not being exactly identical. This is why it has become common practice for the instrumentalist to tune to each different piano before they perform.

Due to these complexities, most keyboard players rarely get to explore tuning methods outside of equal temperament, nor get a chance to hear for themselves which type of sound they prefer over another.

Advantages and Disadvantages

As defined in pure intonation, how pure or how consonant an interval sounds is based entire on how precisely the intervals correlate with harmonic series. In the case of equal temperament, only octaves are maximally consonant, therefore remaining pure in any temperament. Additionally, the tone of intervals are consistent across all the keys.

The problem is within the intervals, notably the disproportion of major thirds and fifths. For example, the interval from F4 to A4 is a major third with a frequency ratio of 5/4, but when we compare the ratio of A6/F4 it becomes 81/16, and in the same octave 81/64. When compared to a pure third, 80/64, it is larger than it by a factor of the syntonic comma. The major thirds are typically sharp by 14 cents, causing beating.

Comparison between meantone temperaments and standard 12-tone equal temperament. The Y-Axis indicates cents and X-Axis is the pitch relative to C

The relevancy of the disadvantage is notable when compared. One can attempt to hear the difference in this sample: "A pair of major thirds, followed by a pair of full major chords. The first in each pair is in equal temperament; the second is in just intonation." The lack of pure consonance makes the beating more audible in the chords tuned to equal temperament.

Alternatives

Meantone temperament

A Harpsichord with a Split Sharp

Meantone temperament is a method of tuning constructed by a series of perfect fifths : F-C-G-D-A. Most common is the quarter-comma meantone, which flattens the fifths by 1/4 of the syntonic comma, but achieves pure thirds in exchange. Most can accept the loss of the pure fifth, as a difference of 5 cent is hard to notice. However, the major downfall of meantone temperament is the range in which it works. For example, in the series of fifths E♭–B♭–F–C–G–D–A–E–B–F#–C#–G#, the thirds will only remain pure up from E♭ to E, thirds from B and onward will be tuned too sharp to be pure thirds. In other words, meantone temperament only works for 7 keys, which would be difficult to use especially in pieces where keys are changing constantly. The concept of a split sharps have been introduced in order to increase the amount of keys accessible with meantone temperament, but unfortunately never took off in modern pianos.

Closed cycle temperament

In the series of pure fifths, E♭–B♭–F–C–G–D–A–E–B–F#-C#-G#-D#, the frequency gap is a ratio of (3/2)¹² / 2⁷, known as the Pythagorean comma. Closed cycle temperament is achieved by tuning using fractions of the pythagorean comma, the syntonic comma or other positions of pure fifths. If distributed uniformly using 1/12 of the Pythagorean comma on the circle of fifths, this becomes the standard 12-tone equal temperament on modern pianos.

Kirnberger Temperament

If the distribution is unequal, we have a less extreme case of closed cycle temperament. Kirnberger temperament is a prime example of one. His idea is to tune the third C-E as a pure third, and the series, C-G-D-A-E, 1/4 of a syntonic comma flat, similar to meantone temperament. What changes is that the B a fifth above the E will be a pure fifth, and the F# above the B will also be a pure fifth. This causes deviations with the thirds, for example the D to that F# compared to the meantone will deviate by 10.8 cent, or 2/4 of the syntonic comma. In essence, the farther it is away from C major in the circle of fifths, the less pure the intervals sound. Regardless, his tuning method ensures that intervals will never exceed his acceptable margin of error, 1 syntonic comma, or 21.51 cent.