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# equal temperament

equal temperament: see tuning systems.
Equal temperament is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings an interval — usually the octave — is divided into a series of equal steps (equal frequency ratios). For modern Western music, the most common tuning system is twelve-tone equal temperament, sometimes abbreviated as 12-TET, which divides the octave into 12 (logarithmically) equal parts. It is usually tuned relative to a standard pitch of 440 Hz.

Other equal temperaments exist (some music has been written in 19-TET and 31-TET for example, and Arabian music is based on 24-TET), but in western countries when people use the term equal temperament without qualification, it is usually understood that they are talking about 12-TET.

Equal temperaments may also divide some interval other than the octave, a pseudo-octave, into a whole number of equal steps. An example is an equally-tempered Bohlen-Pierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is sometimes preferred. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on; however, when composers and music-theorists use "EDO" their intention is generally that a temperament (i.e., a reference to just intonation intervals) is not implied.

## History

Vincenzo Galilei (father of Galileo Galilei) has been one of the first advocates of twelve-tone equal temperament in a 1581 treatise, along two sets of dance suites on each of the 12 notes of the chromatic scale, and 24 ricercars in all the "major/minor keys.", along with his countryman and fellow lutenist Giacomo Gorzanis who had written music based on this temperament by 1567. Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino wrote a "Recercare de tutti li Toni" as early as 1507. John Wilson (the English 17th-century composer) wrote 26 preludes including 24 in all the major/minor keys.

Historically, there was Seven-equal temperament or Hepta-equal temperament practice in ancient Chinese tradition. The first person known to have attempted a numerical specification for 12-TET is probably Zhu Zaiyu (朱載堉) a prince of the Ming court, who published a theory of the temperament in 1584. It is possible that this idea was spread to Europe by way of trade, which intensified just at the moment when Zhu Zaiyu published his calculations. Within fifty-two years of Zhu's publication, the same ideas had been published by Marin Mersenne and Simon Stevin.

From 1450 to about 1800 there is evidence that musicians expected much less mistuning (than that of Equal Temperament) in the most common keys, such as C major. Instead, they used approximations that emphasized the tuning of thirds or fifths in these keys, such as meantone temperament. Some theorists, such as Giuseppe Tartini, were opposed to the adoption of Equal Temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music.

String ensembles and vocal groups, who have no mechanical tuning limitations, often use a tuning much closer to just intonation, as it is naturally more consonant. Other instruments, such as some wind, keyboard, and fretted-instruments, often only approximate equal temperament, where technical limitations prevent exact tunings, other wind instruments, who can easily and spontaneously bend their tone, most notably double-reeds, use tuning similar to string ensembles and vocal groups.

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. It is reasonable to believe that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they each described the subtly different dissonances made available within a particular tuning method. However, it is difficult to determine with any exactness the actual tunings used in different places at different times by any composer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about the moods and colors of particular keys.)

Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and was a better approximation to just intonation than the nearby alternative equal temperaments. It permitted total harmonic freedom at the expense of just a little purity in every interval. This allowed greater expression through modulation, which became extremely important in the 19th century music of composers such as Chopin, Schumann, Liszt, and others.

A precise equal temperament was not attainable until Johann Heinrich Scheibler developed a tuning fork tonometer in 1834 to accurately measure pitches. The use of this device was not widespread, and it was not until 1917 that William Braid White published a practical aural method of tuning the piano to equal temperament.

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.

## General properties

In an equal temperament, the distance between each step of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly-spaced, and would not permit transposition to different keys.) Specifically, the smallest interval in an equal tempered scale is the ratio:

$r^n_\left\{\right\}=p$
$r=sqrt\left[n\right]\left\{p\right\}$

where the ratio r divides the ratio p (often the octave, which is 2/1) into n equal parts. (See Twelve-tone equal temperament below.)

Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in Ethnomusicology. The basic step in cents for any equal temperament can be found by taking the width of p above in cents (usually the octave, which is 1200 cents wide), called below w, and dividing it into n parts:

$c = frac\left\{w\right\}\left\{n\right\}$

In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulo is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g. 'C' is 0 regardless of octave register. The MIDI encoding standard uses integer note designations.

## Twelve-tone equal temperament

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the ratio of frequencies between two adjacent semitones is the twelfth root of two:

$r = 100 cents = sqrt\left[12\right]\left\{2\right\} approx 1.0594630943593$

This interval is equal to 100 cents. (The cent is sometimes for this reason defined as one hundredth of a semitone.)

### Calculating absolute frequencies

To find the frequency, $P^\left\{\right\}_n$, of a note in 12-TET, the following definition may be used:

$P_n=P_a times 2^frac\left\{n-a\right\}\left\{12\right\}$

In this formula $P^\left\{\right\}_n$ refers to the pitch, or frequency (usually in hertz), you are trying to find. $P^\left\{\right\}_a$ refers to the frequency of a reference pitch (usually 440Hz). n and a refer to numbers assigned to the desired pitch and the reference pitch, respectively. These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A4 (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and C4 (middle C) is the 40th key. These numbers can be used to find the frequency of C4:

$P_\left\{40\right\} = 440~mathrm\left\{Hz\right\} times 2^frac\left\{40-49\right\}\left\{12\right\} approx 261.626~mathrm\left\{Hz\right\}$

### Comparison to just intonation

The intervals of 12-TET closely approximate some intervals in Just intonation. In particular, it approximates just fourths, fifths, thirds, and sixths better than any equal temperament with fewer divisions of the octave. Its fifths and fourths in particular are almost indistinguishably close to just. In general the next lowest viable equal temperament (as an approximation to just) is 19-TET, which has better thirds and sixths, but weaker fourths and fifths than 12-TET.

In the following table the sizes of various just intervals are compared against their equal tempered counterparts, given as a ratio as well as cents.

Name Exact value in 12-TET Decimal value in 12-TET Cents Just intonation interval Cents in just intonation Difference
Unison (C) $2^frac\left\{0\right\}\left\{12\right\} = 1$ 1.000000 0 $begin\left\{matrix\right\} frac\left\{1\right\}\left\{1\right\} end\left\{matrix\right\}$ = 1.000000 0.0000 0
Minor second (C♯) $2^frac\left\{1\right\}\left\{12\right\} = sqrt\left[12\right]\left\{2\right\}$ 1.059463 100 $begin\left\{matrix\right\} frac\left\{16\right\}\left\{15\right\} end\left\{matrix\right\}$ = 1.066667 111.73 11.73
Major second (D) $2^frac\left\{2\right\}\left\{12\right\} = sqrt\left[6\right]\left\{2\right\}$ 1.122462 200 $begin\left\{matrix\right\} frac\left\{9\right\}\left\{8\right\} end\left\{matrix\right\}$ = 1.125000 203.91 3.91
Minor third (D♯) $2^frac\left\{3\right\}\left\{12\right\} = sqrt\left[4\right]\left\{2\right\}$ 1.189207 300 $begin\left\{matrix\right\} frac\left\{6\right\}\left\{5\right\} end\left\{matrix\right\}$ = 1.200000 315.64 15.64
Major third (E) $2^frac\left\{4\right\}\left\{12\right\} = sqrt\left[3\right]\left\{2\right\}$ 1.259921 400 $begin\left\{matrix\right\} frac\left\{5\right\}\left\{4\right\} end\left\{matrix\right\}$ = 1.250000 386.31 -13.69
Perfect fourth (F) $2^frac\left\{5\right\}\left\{12\right\} = sqrt\left[12\right]\left\{32\right\}$ 1.334840 500 $begin\left\{matrix\right\} frac\left\{4\right\}\left\{3\right\} end\left\{matrix\right\}$ = 1.333333 498.04 -1.96
Diminished fifth (F♯) $2^frac\left\{6\right\}\left\{12\right\} = sqrt\left\{2\right\}$ 1.414214 600 $begin\left\{matrix\right\} frac\left\{7\right\}\left\{5\right\} end\left\{matrix\right\}$ = 1.400000 582.51 -17.49
Perfect fifth (G) $2^frac\left\{7\right\}\left\{12\right\} = sqrt\left[12\right]\left\{128\right\}$ 1.498307 700 $begin\left\{matrix\right\} frac\left\{3\right\}\left\{2\right\} end\left\{matrix\right\}$ = 1.500000 701.96 1.96
Minor sixth (G♯) $2^frac\left\{8\right\}\left\{12\right\} = sqrt\left[3\right]\left\{4\right\}$ 1.587401 800 $begin\left\{matrix\right\} frac\left\{8\right\}\left\{5\right\} end\left\{matrix\right\}$ = 1.600000 813.69 13.69
Major sixth (A) $2^frac\left\{9\right\}\left\{12\right\} = sqrt\left[4\right]\left\{8\right\}$ 1.681793 900 $begin\left\{matrix\right\} frac\left\{5\right\}\left\{3\right\} end\left\{matrix\right\}$ = 1.666667 884.36 -15.64
Minor seventh (A♯) $2^frac\left\{10\right\}\left\{12\right\} = sqrt\left[6\right]\left\{32\right\}$ 1.781797 1000 $begin\left\{matrix\right\} frac\left\{7\right\}\left\{4\right\} end\left\{matrix\right\}$ = 1.750000 968.826 -31.91
Major seventh (B) $2^frac\left\{11\right\}\left\{12\right\} = sqrt\left[12\right]\left\{2048\right\}$ 1.887749 1100 $begin\left\{matrix\right\} frac\left\{15\right\}\left\{8\right\} end\left\{matrix\right\}$ = 1.875000 1088.27 -11.73
Octave (C) $2^frac\left\{12\right\}\left\{12\right\} = \left\{2\right\}$ 2.000000 1200 $begin\left\{matrix\right\} frac\left\{2\right\}\left\{1\right\} end\left\{matrix\right\}$ = 2.000000 1200.0 0

(These mappings from equal temperament to just intonation are by no means unique. The minor seventh, for example, can be meaningfully said to approximate 9/5, 7/4, or 16/9 depending on context. The 7/4 ratio is used to emphasize this tuning's poor fit to the 7th partial in the harmonic series.)

## Other equal temperaments

### 5 and 7 tone temperaments in ethnomusicology

Five and seven tone equal temperament (5-TET and 7-TET), with 240 and 171 cent steps respectively, are fairly common. A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system. Indonesian 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) they 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 five-tone equal temperament while pelog is highly unequal; however, Surjodiningrat et al. (1972) has analyzed pelog as a seven-note subset of nine-tone equal temperament. 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.

### Various Western equal temperaments

Many systems that divide the octave equally can be considered relative to other systems of temperament. 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 the Netherlands, there advocated by Christiaan Huygens and Adriaan Fokker. 31-TET, like most Meantone temperaments, has a less accurate fifth than 12-TET.

There are in fact five numbers by which the octave can be equally divided to give progressively smaller total mistuning of thirds, fifths and sixths (and hence minor sixths, fourths and minor thirds): 12, 19, 31, 34 and 53. The sequence continues with 118, 441, 612..., but these finer divisions produce improvements that are not audible.

In the 20th century, standardized Western pitch and notation practices having been placed on a 12-TET foundation made the quarter tone scale (or 24-TET) a popular microtonal tuning. Though it only improved non-traditional consonances, such as 11/4, 24-TET can be easily constructed by superimposing two 12-TET systems tuned half a semitone apart. It is based on steps of 50 cents, or $sqrt\left[24\right]\left\{2\right\}$.

29-TET is the lowest number of equal divisions of the octave which produces a better perfect fifth than 12-TET; however, it does not contain a good approximation of the pure major third, and so it is not widely used.

41-TET is the second lowest number of equal divisions which produces a better perfect fifth than 12-TET. It is not often used, however. (One of the reasons 12-TET is so widely favoured among the equal temperaments is that it is very practical in that with an economical number of keys it achieves better consonance than the other systems with a comparable number of tones.)

53-TET is better at approximating the traditional just consonances than 12, 19 or 31-TET, but has had only occasional use. Its extremely good perfect fifths make it interchangeable with an extended Pythagorean tuning, but it also accommodates schismatic temperament, and is sometimes used in Turkish music theory. It does not, however, fit the requirements of meantone temperaments which put good thirds within easy reach via the cycle of fifths. In 53-TET the very consonant thirds would be reached instead by strange enharmonic relationships. (Another tuning which has seen some use in practice and is not a meantone system is 22-TET.)

Another extension of 12-TET is 72-TET (dividing the semitone into 6 equal parts), which though not a meantone tuning, approximates well most just intonation intervals, even less traditional ones such as 7/4, 9/7, 11/5, 11/6 and 11/7. 72-TET has been taught, written and performed in practice by Joe Maneri and his students (whose atonal inclinations interestingly typically avoid any reference to just intonation whatsoever).

Other equal divisions of the octave that have found occasional use include 15-TET, 34-TET, 41-TET, 46-TET, 48-TET, 99-TET, and 171-TET.

### Equal temperaments of non-octave intervals

The equal tempered version of the Bohlen-Pierce scale consists of the ratio 3:1, 1902 cents, conventionally a perfect fifth wider than an octave, called in this theory a tritave and split into a thirteen equal parts. This provides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents or $sqrt\left[13\right]\left\{3\right\}$.

Wendy Carlos discovered three unusual equal temperaments after a thorough study of the properties of possible temperaments having a step size between 30 and 120 cents. These were called alpha, beta, and gamma. They can be considered as equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes:

• alpha: $sqrt\left[9\right]\left\{3/2\right\}$ (78.0 cents)
• beta: $sqrt\left[11\right]\left\{3/2\right\}$ (63.8 cents)
• gamma: $sqrt\left[20\right]\left\{3/2\right\}$ (35.1 cents)

Alpha and Beta may be heard on the title track of her 1986 album Beauty in the Beast.

## References

• Burns, Edward M. (1999). "Intervals, Scales, and Tuning", The Psychology of Music second edition. Deutsch, Diana, ed. San Diego: Academic Press. ISBN 0-12-213564-4. Cited:

*Ellis, C. (1965). "Pre-instrumental scales", Journal of the Acoustical Society of America, 9, 126-144. Cited in Burns (1999).
*Morton, D. (1974). "Vocal tones in traditional Thai music", Selected reports in ethnomusicology (Vol. 2, p.88-99). Los Angeles: Institute for Ethnomusicology, UCLA. Cited in Burns (1999).
*Haddon, E. (1952). "Possible origin of the Chopi Timbila xylophone", African Music Society Newsletter, 1, 61-67. Cited in Burns (1999).
*Kunst, J. (1949). Music in Java (Vol. II). The Hague: Marinus Nijhoff. Cited in Burns (1999).
*Hood, M. (1966). "Slendro and Pelog redefined", Selected Reports in Ethnomusicology, Institute of Ethnomusicology, UCLA, 1, 36-48. Cited in Burns (1999).
*Temple, Robert K. G. (1986)."The Genius of China". ISBN 0-671-62028-2. Cited in Burns (1999).
*Tenzer, (2000). Gamelan Gong Kebyar: The Art of Twentieth-Century Balinese Music. ISBN 0-226-79281-1 and ISBN 0-226-79283-8. Cited in Burns (1999).
*Boiles, J. (1969). "Terpehua though-song", Ethnomusicology, 13, 42-47. Cited in Burns (1999).
*Wachsmann, K. (1950). "An equal-stepped tuning in a Ganda harp", Nature (Longdon), 165, 40. Cited in Burns (1999).

• Cho, Gene Jinsiong. (2003). The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston, NY: The Edwin Mellen Press.
• Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3
• Sethares, William A. (2005). Timbre, Spectrum, Scale. 2nd ed., London: Springer-Verlag.
• Surjodiningrat, W., Sudarjana, P.J., and Susanto, A. (1972) Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta, Gadjah Mada University Press, Jogjakarta 1972. Cited on http://web.telia.com/~u57011259/pelog_main.htm, accessed May 19, 2006.
• Stewart, P. J. (2006) "From Galaxy to Galaxy: Music of the Spheres"