# Equal temperament

Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). The best known example of such a system is twelve-tone equal temperament, sometimes abbreviated to 12-TET, which is nowadays used in most Western music. Other equal temperaments do exist (some music has been written in 19-TET and 31-TET for example), but they are so rare that when people use the term equal temperament without qualification, it is usually understood that they are talking about the twelve tone variety.

The distance between each step and the next is aurally the same for any two adjacent steps, though, because steps form a geometric sequence, the difference in frequency increases from one to the next. A linear sequence of one frequency difference would create ever smaller intervals (ratios), such as the harmonic series. See also logarithmic scale.

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## History

The first person known to introduce a mathematically accurate specification for equal temperament is Chu Tsai-Yu in the Ming Dynasty, who published a theory of the temperament in 1584. Soon after, European mathematicians Simon Stevin (1585) and Marin Mersenne (1636) accurately described equal temperament, independently from China.

Twelve tone equal temperament was introduced in the West to permit the playing of music in all keys with an equal amount of mis-tuning in each, without having to provide more than 12 pitches per octave on instruments, while still roughly approximating just intonation intervals. This allows much more facile harmonic motion, while losing some subtlety of intonation. True equal temperament was not available to musicians before about 1870 because scientific tuning and measurement was not available. And in fact, from about 1450 to about 1800 musicians tolerated even less mistuning in the most common keys, like C major. Instead, they used approximations that emphasized the tuning of thirds or fifths in these keys, such as Meantone temperament.

At the time equal temperament was beginning to take hold in the West, many people perceived the much-increased mis-tuning of the music, relative to meantone temperament, as a disgrace. Those in opposition to equal temperament worried that the temperament, by degrading the purity of each chord, would degrade the purity of music. The composers against equal temperament included Giuseppe Tartini.

Equal temperament does have a weak point in tonal music. Group of musicians such as string ensemble or a capella, where tuning by microtones can be possible simultaneously during concerts, often prefer to tune the parts comprising each chord in just tuning relative to one another, in order to maximize the effect of consonance. Other instruments, such as wind, keyboard, and fretted-instruments, use equal temperament or quasi-equal temperament, when the instruments have technical limitations to be tuned exactly equal. The dissonance of such temperaments is known to be noticed by an average audience. Some claim that this especially troubling in the lower register, and that this had somewhat constrained composers in the classical and romantic eras from writing chords narrower than octave for the left hand in keyboard music, while such examples in cello parts of string quartets are more common. Others hear the dissonance as most troubling in the higher register, where beating between harmonics of mistuned consonances is faster, and where combinational tones, often an entire semitone out-of-tune in equal temperament, are louder.

On the other hand, J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament, where in some keys the consonances are even more degraded than in equal temperament. There is some reason to believe that when composers and theoreticians of this era wrote of the "colors" of the keys, they described the subtly different dissonances of particular tuning methods, though it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. Well temperaments were gradually supplanted by equal temperament over the course of the 19th century, and it is in the environment of equal temperament that the new styles of symmetrical tonality and polytonality, atonal music such as that written with the twelve tone technique or serialism, and jazz (at least its piano component) developed and flourished.

## Twelve-tone equal temperament

The ratio between two adjacent semitones can be found with a few steps:

1. Let an be the frequency of a semitone n, with a12 an octave above a0. This creates twelve tones for each octave.
2. Since the frequency ratio of a tone from one octave to the next is 2:1, the ratio of the frequency of one tone (a12) to the frequency of a tone an octave lower (a0) is 2:1 as well, so
[itex]\frac{a_{12}}{a_0} = 2[itex]
3. Since the tones are in a geometric sequence, the frequency for a tone k (relative to the tone designated zero) will be equal to ska0 where s is the constant ratio between adjacent frequencies. This gives for k = 12,
[itex]a_{12} = s^{12} a_0[itex]
[itex]\frac{a_{12}}{a_0} = s^{12}[itex]
4. Since a12 / a0 was found to be two, the formula with constant ratio s is
[itex]2 = s^{12}[itex]
[itex]s = \sqrt{2}[itex]

Therefore, the ratio between two adjacent frequencies is equal to the twelfth root of two or approximately 1.05946309 to one.

[itex]s = \sqrt{2} \approx 1.05946309[itex]

The half tone interval:

[itex] 1 : 2^{1/12}[itex]

is also known as 100 cent. 1 cent is therefore the ratio between two tone frequencies with an interval of one hundredth of an equal-tempered semitone.

The distance between two notes whose frequencies are f1 and f2 is 12log2(f1/f2) half tones, that is 1200log2(f1/f2) cents.

### Cent values of equal temperament

 Tone Cents C1 C# D Eb E F F# G G# A Bb B C2 0 100 200 300 400 500 600 700 800 900 1000 1100 1200

12-TET allows the use of integer notation and modulo 12, and this allows for proofs concerning pitch.

The following table shows the values of the intervals of 12 TET, along with one interval from just intonation that each approximates, and the percentage by which they differ:

Name Exact value in 12-TET Decimal value Just intonation interval Percent difference
Unison   1 1.000000 1 = 1.000000 0.00%
Minor second [itex]\sqrt{2^1} = \sqrt{2}[itex] 1.059463 16/15 = 1.066667 -0.68%
Major second [itex]\sqrt{2^2} = \sqrt{2}[itex] 1.122462 9/8 = 1.125000 -0.23%
Minor third [itex]\sqrt{2^3} = \sqrt{8}[itex] 1.189207 6/5 = 1.200000 -0.91%
Major third [itex]\sqrt{2^4} = \sqrt{2}[itex] 1.259921 5/4 = 1.250000 +0.79%
Perfect fourth [itex]\sqrt{2^5} = \sqrt{32}[itex] 1.334840 4/3 = 1.333333 +0.11%
Diminished fifth [itex]\sqrt{2^6} = \sqrt{2}[itex] 1.414214 7/5 = 1.400000 +1.02%
Perfect fifth [itex]\sqrt{2^7} = \sqrt{128}[itex] 1.498307 3/2 = 1.500000 -0.11%
Minor sixth [itex]\sqrt{2^8} = \sqrt{4}[itex] 1.587401 8/5 = 1.600000 -0.79%
Major sixth [itex]\sqrt{2^9} = \sqrt{8}[itex] 1.681793 5/3 = 1.666667 +0.90%
Minor seventh [itex]\sqrt{2^{10}} = \sqrt{32}[itex] 1.781797 16/9 = 1.777778 +0.23%
Major seventh [itex]\sqrt{2^{11}} = \sqrt{2048}[itex] 1.887749 15/8 = 1.875000 +0.68%
Octave [itex]\sqrt{2^{12}} = {2}[itex] 2.000000 2/1 = 2.000000 0.00%

These mappings from equal temperament to just intonation are by no means unique. The minor seventh, for example, can be meaningfully said to approximate both 16/9 and 9/5, depending on context or simultaneously in a chord -- and probably even 7/4.

## Non-12 TET

Five and seven tone equal temperament, with 240 and 171 cent steps relatively, seem the most common outside of 12-tET. A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-tET. A Ugandan Chop xylophone measured by Haddon (1952) also tuned to 171 cent steps. Gamelans are tuned to 5-tET according to Kunst (1949), but according to Hood (1966) and McPhee (1966) their tuning varies widely and according to Tenzer (2000) contain stretched octaves. It is now well-accepted that of the two primary tuning systems in Gamelan music, Slendro and Pelog, only Slendro somewhat resembles 5-tone equal temperament while Pelog is highly unequal. However, Wachsmann (1950) used a Stroboconn to measure a Ugandan harp and women singing unaccompanied, finding variations of 15 and 5 cents respectively. ( ← check accuracy of fragment repair) A South American Indian scale from a preinstrumental culture measured by Boiles (1969) featured 175 cent equal temperament which stretches the octave slightly as with instrumental gamelan music.

The quarter tone scale or 24-tET is, similarly, based on steps of 50 cents or powers of [itex]\sqrt{2}[itex]. Other equal divisions of the octave, though, can be better considered temperaments; 24 is usually best considered simply as an equal division (e.g., a bisection of 12-tET). 19-tET and especially 31-tET are extended varieties of Meantone temperament and approximate most just intonation intervals considerably better than 12-tET. They have been used sporadically since the 16th century, with 31-tET particularly popular in Holland, there advocated by Christiaan Huygens and Adriaan Fokker. 53-tET is much better still at approximating the traditional just intonation consonances, but has had very little use. It doesn't fit the Meantone mold that shaped the development of Western harmony and tonality since the Rennaissance, though it does fit schismatic temperament and the Pythagorean tuning of medieval music, and is sometimes used in Turkish music theory. In 53-tET, most traditional compositions would necessitate subtle microtonal pitch shifts or a drifting pitch level in order to make use of the tuning's excellent just intonation triads. 55-tET, not as close to just intonation, was a bit closer to common practice. As an excellent representative of the variety of meantone temperament popular in the 18th century, 55-tET it was considered ideal by Georg Philipp Telemann and other prominent musicians. Wolfgang Amadeus Mozart's surviving violin lessons conform closely to such a model.

In the 20th century, standardized Western pitch and notation practices having been placed on a 12-tET foundation made the quarter tone scale a much more popular microtonal tuning. A further extension of 12-tET is 72-tET, which though not a Meantone tuning, approximates most just intonation intervals, including non-traditional ones like 7/4, 9/7, 11/5, 11/6 and 11/7, much better. 72-tET has been taught, written and performed in practice, for example by Joe Maneri and his students -- whose atonal inclinations typically avoid any reference to just intonation intervals whatsoever. Still other equal temperaments occupying more than a few musicians include 5-tET, 7-tET, 15-tET, 22-tET, and 48-tET. Theoretically interesting temperaments which have found occasional use include division of the octave into 34, 41, 46, 99 or 171 parts.

More generally, every step in n tone equal temperament is 1200/n cents. However, if one wishes to create an equal tempered scale that does not repeat at the octave, a scale with n equal steps in a pseudo-octave p is based on the ratio r

[itex] r = \sqrt[n]{p} [itex].

This still may be easier to calculate in cents, for instance the pseudo-octave of ratio 2.1:1 is an interval of 1284 cents. Equal tempered scales can also be generated simply by picking the number of cents that each step will consist of.

Wendy Carlos created two equal tempered scales for the title track of her album Beauty In The Beast, the Alpha and Beta scales. Beta splits a perfect fourth into two equal parts, which creates a scale where each step is almost 64 cents. Alpha does the same to a minor third to create a scale of 78 cent steps.

The equal tempered version of the Bohlen-Pierce scale consists of the ratio 3:1, 1902 cents, conventionally an octave and a just fifth, used as a tritave, and split into a thirteen tone equal temperament where each step is

[itex] \sqrt{3} [itex]

or 146.3 cents. This provides a very close match to justly tuned ratios consisting only of odd numbers.

Australian aboriginal music extensively measured by Ellis (1965) was based on arithmetic scales (the harmonic series is an arithmetic scale, though presumably the Australian scales began an interval smaller than an octave) with an equal separation in hertz.

## Sources

• Burns, Edward M. (1999). "Intervals, Scales, and Tuning", The Psychology of Music second edition. Deutsch, Diana, ed. San Diego: Academic Press. ISBN 0122135644. Cited:
• Ellis, C. (1965). "Pre-instrumental scales", Journal of the Acoustical Society of America, 9, 126-144.
• Morton, D. (1974). "Vocal tones in traditional Thai music", Selected reports in ethnomusicology (Vol. 2, p.88-99). Los Angeles: Institute for Ethnomusicology, UCLA.
• Haddon, E. (1952). "Possible origin of the Chopi Timbila xylophone", African Music Society Newsletter, 1, 61-67.
• Kunzt, J. (1949). Music in Java (Vol. II). The Hague: Marinus Nijhoff.
• Hood, M. (1966). "Slendro and Pelog redefined", Selected Reports in Ethnomusicology, Institute of Ethnomusicology, UCLA, 1, 36-48.
• Tenzer, (2000). Gamelan Gong Kebyar: The Art of Twentieth-Century Balinese Music. ISBN 0226792811 and ISBN 0226792838
• Boiles, J. (1969). "Terpehua though-song", Ethnomusicology, 13, 42-47.
• Wachsmann, K. (1950). "An equal-stepped tuning in a Ganda harp", Nature (Longdon), 165, 40.

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