Equal temperament

Equal temperament is a scheme of musical tuning in which an interval, usually the octave, is divided into a series of equal steps (equal frequency ratios), forming an equal division of the octave. 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, and Arabian and eastern styled music is based on a twenty four tone equal temperament), but in the Western world when people use the term equal temperament without qualification, it is usually understood that they are talking about the twelve tone variety.

Other equal temperaments also exist, which 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, an equal temperament which divides the octave is sometimes called an Equal Division of the Octave, or EDO. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on.

Explanation

The distance between each step and the next is aurally the same for any two adjacent steps. However, the frequencies exhibit exponential growth, that is, the steps form a geometric sequence meaning that the difference in frequency increases dramatically from one tone to the next. Naively, one may think that the frequency of the notes should grow linearly like the terms in the Harmonic Series. However, any linear sequence of frequencies would create ever smaller intervals, at least aurally speaking, and to the ear would seem like a logarithmic scale.

In theoretical writings, 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. 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.

History

Vincenzo Galilei (father of Galileo Galilei) may have been the first person to advocate equal temperament (in a 1581 treatise). The first person known to have attempted a numerical specification for equal temperament is probably Chu Tsai-Yu (朱載堉) in the Ming Dynasty, who published a theory of the temperament in 1584. It is possible that this idea was spread to Europe by way of trade, which had been intensified just at the moment when Chu Tsai-Yu went into print with his new theory. Within fifty-two years of Chu'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. Some listeners claim that the equal-tempered difference is especially troubling in the lower register, and 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 take issue with dissonance 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.

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 wind, keyboard, and fretted-instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.

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 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. (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 the early 20th century that a practical aural method of tuning the piano to equal temperament with precision was developed and disseminated.

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 a_n be the frequency of a tone 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

\frac{a_{12}}{a_0} = 2

3. Since the frequencies of the tones are in a geometric sequence, the frequency for a tone k (relative to the tone designated zero) will be equal to s^ka₀ where s is the constant ratio between adjacent frequencies. This gives for k = 12,

a_{12} = s^{12} a_0

\frac{a_{12}}{a_0} = s^{12}

4. Since a₁₂ / a₀ was found to be two, the formula with constant ratio s is

2 = s^{12}

s = \sqrt

^*{2}

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

s = \sqrt

^*{2} \approx 1.05946309

The half tone interval:

1 : 2^{1/12}

is also known as 100 cents. 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 12 log₂(f₁/f₂) half tones, that is 1200 log₂(f₁/f₂) cents.

Cent values of equal temperament

Tone	C¹	D♭	D	E♭	E	F	F♯	G	A♭	A	B♭	B	C²
Cents		100	200	300	400	500	600	700	800	900	1000	1100	1200

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	$\sqrt = \sqrt[12 {2}$	1.059463	16/15 = 1.066667	-0.68%
Major second	$\sqrt = \sqrt[6 {2}$	1.122462	9/8 = 1.125000	-0.23%
Minor third	$\sqrt = \sqrt[4 {2}$	1.189207	6/5 = 1.200000	-0.91%
Major third	$\sqrt = \sqrt[3 {2}$	1.259921	5/4 = 1.250000	+0.79%
Perfect fourth	$\sqrt = \sqrt[12 {32}$	1.334840	4/3 = 1.333333	+0.11%
Diminished fifth	$\sqrt$ ^*{2^6} = \sqrt{2}	1.414214	7/5 = 1.400000	+1.02%
Perfect fifth	$\sqrt = \sqrt[12 {128}$	1.498307	3/2 = 1.500000	-0.11%
Minor sixth	$\sqrt = \sqrt[3 {4}$	1.587401	8/5 = 1.600000	-0.79%
Major sixth	$\sqrt = \sqrt[4 {8}$	1.681793	5/3 = 1.666667	+0.90%
Minor seventh	$\sqrt = \sqrt[6 {32}$	1.781797	16/9 = 1.777778	+0.23%
Major seventh	$\sqrt = \sqrt[12 {2048}$	1.887749	15/8 = 1.875000	+0.68%
Octave	$\sqrt$ ^*{2^{12}} = {2}	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 9/5, 7/4, or 16/9 depending on context.)

Simple formula to find the frequency of any note, in equal temperament

Here is a simple, single formula to find the frequency in hertz of any pitch in the 12-tone per octave equally tempered scale, assuming you want to find the frequency of 88 notes (such as there are on a standard piano) or more, where P49 equals the assumed value of concert-tuned A (sometimes called A4, where the first A on a piano is A0 and the last A is A7), P49 being the 49th key on a standard 88-key piano, and the A above middle C (where middle C is P40, or C4). Generally P49 is 440 hz. Pn (P stands for pitch) = the freq of the note you want to calculate, tuned in equal temprament, relative to P49, with n being the note number from 1 to 88 (or greater in either the negative or positive direction) in the sequence of 88 notes:

$P_{n}=2^\frac{n-49}{12}P_{49}$

That's it. That, and the assumed value of P49 (it is usually assumed to be 440), is all you need to know, to calculate the value of any note in the equally tempered scale.

If you assume P49 equals 440 hz (the international standard value of A4, sometimes called A440), then the formula becomes

$P_{n}=2^\frac{n-49}{12}440$

Translating from mathematics to English, that would be: to find the frequency (pitch) in hertz, of any note from 1 to 88 (or 0 or less, or greater than 88) subtract 49 from the number of the note you want to find, and divide the difference by 12, and raise 2 to that power. Then multiply this by the frequency of A49 (440)

As an example, here is how we find middle C, note 40 on the piano, alternatively named C4 or P40. First subtract 49 from 40. You get -9. Divide -9 by 12. You get -.75. Raise 2 to the -.75 power. You get .59460. Multipy 440 (or whatever value A has been assumed to be) by .59460. You end up with P40 (C4) as being 261.63

It is really incredibly simple and with a calculator you can find any note in a few seconds. You can make a whole electronic spreadsheet in a few more seconds. If you know the frequency of, say, P49, and you want to figure out the pitch of P40, you don't have to divide P49 by 2^(1/12), to find P48, then divide that by 2^(1/12) to find P47, etcetera, until you reach P40. With the formula, you can find P40 the fast way; you just substitute 40 for the value of n in the formula, and solve the formula.

Nor are you limited to 88 notes. You can find notes where n is above 88 using the same formula. For the next lower note below P1, simply decrease the value of n by 1; give the n in Pn a zero value, P₀, and for the n in the formula, plug in zero; for the next lower note, use a minus value, -1, you will have P_-1, and you will substitute -1 for the n in the formula; then for the next lower note, use -2, et cetera.

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. 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 5-tone equal temperament while pelog is highly unequal; however, Surjodiningrat et al. (1972) has analyzed pelog as a seven-note subset of 9-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.

The quarter tone scale or 24-tET is, similarly, based on steps of 50 cents or powers of $\sqrt$ ^*{2}. 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. Another tuning which has seen some use in practice and is not a meantone system is 22-tET. 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

r = \sqrt

^*{p} .

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

\sqrt

^*{3}

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 with 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.
- Kunst, 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.
- Temple, Robert K. G. (1986)."The Genius of China". ISBN 0-671-62028-2
- 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.
- 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
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.

External links

Equal temperaments

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