12-tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music.
History The two figures frequently credited with the achievement of exact calculation of equal temperament are
Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: ) in 1584 and
Simon Stevin in 1585. According to F. A. Kuttner, a critic of giving credit to Zhu, Kenneth Robinson credits the invention of equal temperament to Zhu and provides textual quotations as evidence. In 1584 Zhu wrote: : I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.
China 's equal temperament pitch pipes Chinese theorists had previously come up with approximations for , but Zhu was the first person to mathematically solve 12-tone equal temperament, which he described in two books, published in 1580 and 1584. Needham also gives an extended account. Zhu obtained his result by dividing the length of string and pipe successively by {{nobr|\sqrt[12]{2} ≈ 1.059463}}, and for pipe length by {{nobr|\sqrt[24]{2} ≈ 1.029302}}, such that after 12 divisions (an octave), the length was halved. Zhu created several instruments tuned to his system, including bamboo pipes.
Europe Some of the first Europeans to advocate equal temperament were lutenists
Vincenzo Galilei,
Giacomo Gorzanis, and
Francesco Spinacino, all of whom wrote music in it.
Simon Stevin was the first to develop 12 based on the
twelfth root of two, which he described in
van de Spiegheling der singconst (), published posthumously in 1884. Plucked instrument players (lutenists and guitarists) generally favored equal temperament, while others were more divided. In the end, 12-tone equal temperament won out. This allowed
enharmonic modulation, new styles of symmetrical tonality and
polytonality,
atonal music such as that written with the
12-tone technique or
serialism, and
jazz (at least its piano component) to develop and flourish.
Mathematics In 12-tone equal temperament, which divides the octave into 12 equal parts, the width of a
semitone, i.e. the
frequency ratio of the interval between two adjacent notes, is the
twelfth root of two: : \sqrt[12]{2\ } = 2^{\tfrac{1}{12}} \approx 1.059463 This interval is divided into 100 cents.
Calculating absolute frequencies To find the frequency, , of a note in 12 , the following formula may be used: :\ P_n = P_a\ \cdot\ \Bigl(\ \sqrt[12]{2\ }\ \Bigr)^{ n-a }\ In this formula represents the pitch, or frequency (usually in
hertz), to be calculated. is the frequency of a reference pitch. The index numbers and are the labels assigned to the desired pitch () and the reference pitch (). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A (the reference pitch) is the 49th key from the left end of a piano (tuned to
440 Hz), and C (
middle C), and F are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of C and F: :P_{40} = 440\ \text{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(40-49)} \approx 261.626\ \text{Hz}\ :P_{46} = 440\ \text{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(46-49)} \approx 369.994\ \text{Hz}\
Converting frequencies to their equal temperament counterparts To convert a frequency (in Hz) to its equal 12 counterpart, the following formula can be used: :\ E_n = E_a\ \cdot\ 2^{\ x}\ \quad where in general \quad\ x\ \equiv\ \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12\log_{2} \left(\frac{\ n\ }{ a }\right) \Biggr) ~. is the frequency of a pitch in equal temperament, and is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that and have the following frequencies, respectively: : E_{660} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 7 }{\ 12\ }\right)}\ \approx\ 659.255\ \mathsf{Hz}\ \quad where in this case \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl(\ 12 \log_{2}\left(\frac{\ 660\ }{ 440 }\right)\ \Biggr) = \frac{ 7 }{\ 12\ } ~. : E_{550} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 1 }{\ 3\ }\right)}\ \approx\ 554.365\ \mathsf{Hz}\ \quad where in this case \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12 \log_{2}\left(\frac{\ 550\ }{ 440 }\right)\Biggr) = \frac{ 4 }{\ 12\ } = \frac{ 1 }{\ 3\ } ~.
Comparison with just intonation The intervals of 12 closely approximate some intervals in
just intonation. The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away. In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents. :
Seven-tone equal division of the fifth Violins, violas, and cellos are tuned in perfect fifths ( for violins and for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12-tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of \sqrt[7]{3/2} to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves. During actual play, however, violinists, violists, and cellists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio. ==Other equal temperaments==
Five-, seven-, and nine-tone temperaments in ethnomusicology Five- and seven-tone equal temperament (''
and 7 '' ), with 240-cent and 171-cent steps, respectively, are fairly common. and mark the endpoints of the
syntonic temperament's valid tuning range, as shown in
Figure 1. • In the tempered perfect fifth is 720 cents wide (at the top of the tuning continuum), and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0 cents. • In the tempered perfect fifth is 686 cents wide (at the bottom of the tuning continuum), and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second (at 171 cents each.
5-tone and 9-tone equal temperament According to
Jaap Kunst (1949), Indonesian
gamelans are tuned to but according to
Mantle Hood (1966) and
Colin McPhee (1966) their tuning varies widely, and according to
Michael Tenzer (2000) they contain
stretched octaves. It is now 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, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to (133-cent steps ).
7-tone equal temperament A
Thai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from . According to Morton, : "Thai instruments of fixed pitch are tuned to an equidistant system of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning system are not used in one mode (often referred to as 'scale'); in the Thai system five of the seven are used in principal pitches in any mode, thus establishing a pattern of nonequidistant intervals for the mode." A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175-cent seven-tone equal temperament, which stretches the octave slightly, as in instrumental gamelan music.
Chinese music has traditionally used .
Various equal temperaments 's notation system for 16 equal temperament: Intervals are notated similarly to those they approximate and there are fewer
enharmonic equivalents. ;
19 EDO: Many instruments have been built using
19 EDO tuning. Equivalent to meantone, it has a slightly flatter perfect fifth (at 695 cents), but its minor third and major sixth are less than one-fifth of a cent away from just, with the lowest EDO that produces a better minor third and major sixth than 19 EDO being 232 EDO. Its
perfect fourth (at 505 cents), is seven cents sharper than
just intonation's and five cents sharper than 12-EDO's. ;
22 EDO:
22 EDO is one of the most accurate EDOs to represent "superpythagorean" temperament (where 7:4 and 16:9 are the same interval). The perfect fifth is tuned sharp, resulting in four fifths and three fourths reaching supermajor thirds (9/7) and subminor thirds (7/6). One step closer to each other are the classical major and minor thirds (5/4 and 6/5). ;
23 EDO:
23 EDO is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics (3:2, 5:4, 7:4, 11:8) within 20 cents, but it does approximate some ratios between them (such as the 6:5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic territory. ;
24 EDO:
24 EDO, the
quarter-tone scale, is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who are also interested in microtonality. Because 24 EDO contains all the pitches of 12 EDO, musicians employ the additional colors without losing any tactics available in 12-tone harmony. That 24 is a multiple of 12 also makes 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments tuned a quarter-tone apart, such as two pianos, which also allows each performer (or one performer playing a different piano with each hand) to read familiar 12-tone notation. Various composers, including
Charles Ives, experimented with music for quarter-tone pianos. 24 EDO also approximates the 11th and 13th harmonics very well, unlike 12 EDO. ; 26 EDO: 26 is the denominator of a convergent to log2(7), tuning the
7th harmonic (7:4) with less than half a cent of error. Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a major third 17 cents flat (equated with the 11:9 neutral third). 26 EDO has two minor thirds and two minor sixths and could be an alternate temperament for
barbershop harmony. ; 27 EDO: 27 is the lowest number of equal divisions of the octave that uniquely represents all intervals involving the first eight harmonics. It tempers out the
septimal comma but not the
syntonic comma. ;
29 EDO:
29 is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12 EDO, in which the fifth is 1.5 cents sharp instead of 2 cents flat. Its classic major third is roughly as inaccurate as 12 EDO, but is tuned 14 cents flat rather than 14 cents sharp. It also tunes the 7th, 11th, and 13th harmonics flat by roughly the same amount, allowing 29 EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces
58 EDO, which allows for lower errors for some just tones. ;
31 EDO:
31 EDO was advocated by
Christiaan Huygens and
Adriaan Fokker and represents a rectification of
quarter-comma meantone into an equal temperament. 31 EDO does not have as accurate a perfect fifth as 12 EDO (like 19 EDO), but its major thirds and minor sixths are less than 1 cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate. ;
34 EDO:
34 EDO gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31 EDO does, despite having a slightly less accurate fit for 5:4. 34 EDO does not accurately approximate the seventh harmonic or ratios involving 7, and is not meantone since its fifth is sharp instead of flat. It enables the 600-cent tritone, since 34 is an even number. ;
41 EDO:
41 is the next EDO with a better perfect fifth than 29 EDO and 12 EDO. Its classical major third is also more accurate, at only six cents flat. It is not a meantone temperament, so it distinguishes 10:9 and 9:8, along with the classic and Pythagorean major thirds, unlike 31 EDO. It is more accurate in the 13 limit than 31 EDO. ; 46 EDO: 46 EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many say that this gives major triads a characteristic bright sound. The prime harmonics up to 17 are all within 6 cents of accuracy, with 10:9 and 9:5 a fifth of a cent away from pure. As it is not a meantone system, it distinguishes 10:9 and 9:8. ;
53 EDO:
53 EDO has only had occasional use, but is better at approximating the traditional
just consonances than 12, 19 or 31 EDO. Its extremely accurate
perfect fifths make it equivalent to an extended
Pythagorean tuning, as 53 is the denominator of a convergent to log2(3). With its accurate cycle of fifths and multi-purpose comma step, 53 EDO has been used in
Turkish music theory. It is not a meantone temperament, which put good thirds within easy reach by stacking fifths; instead, like all
schismatic temperaments, the very consonant thirds are represented by a Pythagorean diminished fourth (C-F), reached by stacking eight perfect fourths. It also tempers out the
kleisma, allowing its fifth to be reached by a stack of six minor thirds (6:5). ;
58 EDO:
58 equal temperament is a duplication of 29 EDO, which it contains as an embedded temperament. Like 29 EDO it can match intervals such as 7:4, 7:5, 11:7, and 13:11 very accurately, as well as better approximating just thirds and sixths. ;
72 EDO:
72 EDO approximates many
just intonation intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72 EDO has been taught, written and performed in practice by
Joe Maneri and his students (whose atonal inclinations typically avoid any reference to
just intonation whatsoever). As it is a multiple of 12, 72 EDO can be considered an extension of 12 EDO, containing six copies of 12 EDO starting on different pitches, three copies of 24 EDO, and two copies of 36 EDO. ;
96 EDO:
96 EDO approximates all intervals within 6.25 cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12 EDO. It has been advocated by several composers, especially
Julián Carrillo. Other equal divisions of the octave that have found occasional use include
13 EDO,
15 EDO,
17 EDO, and 55 EDO. 2, 5, 12, 41, 53, 306, 665 and 15601 are
denominators of first
convergents of log(3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601
just twelfths/fifths than in any equal temperament with fewer tones. 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote.
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 plus an
octave (that is, a perfect twelfth), called in this theory a
tritave (), and split into 13 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[13]{3}.
Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120 cents. These were called
alpha,
beta, and
gamma. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes: •
alpha: \sqrt[9]{\frac{3}{2}} (78.0 cents) •
beta: \sqrt[11]{\frac{3}{2}} (63.8 cents) •
gamma: \sqrt[20]{\frac{3}{2}} (35.1 cents) Alpha and beta may be heard on the title track of Carlos's 1986 album
Beauty in the Beast.
Equal temperament with a non-integral number of notes per octave While traditional equal temperaments—such as 12‑TET, 19‑TET, or 31‑TET—divide the octave into an integral number of equal parts, it is also possible to explore systems that divide the octave into a non-integral (often irrational) number. In such temperaments, the interval between successive pitches is defined by the ratio 2^(1/N), where N is not an integer. This results in irrational step sizes, meaning their multiples never exactly equal an octave. Such tunings are of interest because, by deliberately sacrificing the octave (i.e., the second harmonic), they can yield a system that offers an improved overall approximation of other intervals in the harmonic series. For example, in a tuning system based on 18.911‑EDO, the step size is 1200⁄18.911 ≈ 63.45 cents. Approximating the just perfect fifth (with a ratio of 3:2, or about 701.96 cents) requires about 11 steps: •
11 steps × 63.45 cents ≈ 698.95 cents, yielding an error of roughly 3 cents. Similarly, for the just major third (with a ratio of 5:4, or about 386.31 cents), 6 steps are used: •
6 steps × 63.45 cents ≈ 380.70 cents, resulting in an error of approximately 5.61 cents. Thus, although a perfect octave is absent, the consonance of many other intervals in these systems can be significantly higher than in integer-based equal temperaments.
Proportions between semitone and whole tone In this section,
semitone and
whole tone may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be , and the number of steps in a tone be . There is exactly one family of equal temperaments that fixes the semitone to any
proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example, , , , , and are in ascending order if they preserve their usual relationships to ). That is, fixing to a proper fraction in the relationship also defines a unique family of one equal temperament and its multiples that fulfil this relationship. For example, where is an integer, sets sets and sets The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the
circle of fifths. (This is not true in general; in 24 , the half-sharps and half-flats are not in the circle of fifths generated starting from .) The extreme cases are where and the semitone becomes a unison, and , where and the semitone and tone are the same interval. Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into and the perfect fifth into If there are notes outside the circle of fifths, one must then multiply these results by , the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24 , six in 72 ). (One must take the small semitone for this purpose: 19 has two semitones, one being tone and the other being . Similarly, 31 has two semitones, one being tone and the other being .) The smallest of these families is and in particular, 12 is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12 has become the most commonly used equal temperament. (Another reason is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.) Each choice of fraction for the relationship results in exactly one equal temperament family, but the converse is not true: 47 has two different semitones, where one is tone and the other is , which are not complements of each other like in 19 ( and ). Taking each semitone results in a different choice of perfect fifth. == Related tuning systems ==