Wikipedia

# Equal temperament

A comparison of some "EDO" equal temperament scales.[1] The graph spans one octave horizontally (open the image to view the full width), and each shaded rectangle is the width of one step in a scale. The just interval ratios are separated in rows by their prime limits.
12-tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps.

An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In other words, the ratios of the frequencies of any adjacent pair of notes is the same, and, as pitch is perceived roughly as the logarithm of frequency,[2] equal perceived "distance" from every note to its nearest neighbor.

In equal temperament tunings, the generating interval is often found by dividing some larger desired interval, often the octave (ratio 2:1), into a number of smaller equal steps (equal frequency ratios between successive notes).

In classical music and Western music in general, the most common tuning system since the 18th century has been twelve-tone equal temperament (also known as 12 equal temperament, 12-TET or 12-ET), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2 (122 ≈ 1.05946). That resulting smallest interval, ​112 the width of an octave, is called a semitone or half step. In modern times, 12-TET is usually tuned relative to a standard pitch of 440 Hz, called A440, meaning one note, A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones apart from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz. It has varied and generally risen over the past few hundred years.[3]

Other equal temperaments divide the octave differently. For example, some music has been written in 19-TET and 31-TET. Arabic music uses 24-TET as a notational convention. In Western countries the term equal temperament, without qualification, generally means 12-TET. To avoid ambiguity between equal temperaments that divide the octave and those that divide some other interval (or that use an arbitrary generator without first dividing a larger interval), the term equal division of the octave, or EDO is preferred for the former. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, and so on.

An example of an equal temperament that finds its smallest interval by dividing an interval other than the octave into equal parts is the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts.

Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.[citation needed] Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.

A comparison of equal temperaments between 10-TET and 60-TET on each main interval of small prime limits (red: 3/2, green: 5/4, blue: 7/4, yellow: 11/8, cyan: 13/8). Each colored graph shows how much error occurs (in cents) when each EDO approximates the corresponding just interval (the black line on the center). Two black curves surrounding the graph on both sides represent the maximum possible error, while the gray ones inside of them indicate the half of it.

## History

The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and Simon Stevin in 1585. According to Fritz A. Kuttner, a critic of the theory,[4] it is known that "Chu-Tsaiyu presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that "Simon Stevin offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently.[5]

Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu[6] and provides textual quotations as evidence.[7] Zhu Zaiyu is quoted as saying that, in a text dating from 1584, "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."[7] Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications."[4] Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors.[8]

### China

#### Early history

The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun. Allegedly his writings discussed the equal division of the scale in the 27th century BC.[9] However, evidence of the origins of writing in this period (the early Longshan) in China is limited to rudimentary inscriptions on oracle bones and pottery.[10]

A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng (early Warring States, c. 5th century BCE in the Chinese Bronze Age), covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.[11]

An approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history: 900 849 802 758 715 677 638 601 570 536 509.5 479 450.[12]

Prince Zhu Zaiyu constructed 12 string equal temperament tuning instrument, front and back view

Historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition.[13][14]

Zhu Zaiyu (朱載堉), a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 律暦融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-TET in his 5,000-page work Complete Compendium of Music and Pitch (Yuelü quan shu 樂律全書) in 1584.[15] An extended account is also given by Joseph Needham.[16] Zhu obtained his result mathematically by dividing the length of string and pipe successively by 122 ≈ 1.059463, and for pipe length by 242,[17] such that after twelve divisions (an octave) the length was divided by a factor of 2:

${\displaystyle \left({\sqrt[{12}]{2}}\right)^{12}=2}$

Similarly, after 84 divisions (7 octaves) the length was divided by a factor of 128:

${\displaystyle \left({\sqrt[{12}]{2}}\right)^{84}=2^{7}=128}$

#### Zhu Zaiyu

Zhu Zaiyu has been credited as the first person to solve the equal temperament problem mathematically.[18] At least one researcher has proposed that Matteo Ricci, a Jesuit in China recorded this work in his personal journal[18][19] and may have transmitted the work back to Europe. (Standard resources on the topic make no mention of any such transfer.[20]) In 1620, Zhu's work was referenced by a European mathematician[who?].[19] Murray Barbour said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyü's brilliant solution remains an enigma."[21] The 19th-century German physicist Hermann von Helmholtz wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.[22]

Zhu Zaiyu's equal temperament pitch pipes

Zhu Zaiyu illustrated his equal temperament theory by the construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the length of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory.[23]

### Europe

Simon Stevin's Van de Spiegheling der singconst c. 1605.

#### Early history

One of the earliest discussions of equal temperament occurs in the writing of Aristoxenus in the 4th century BC.[24]

Vincenzo Galilei (father of Galileo Galilei) was one of the first practical advocates of twelve-tone equal temperament. He composed a set of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "Fronimo", 24 + 1 ricercars.[25] He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).[26]

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567.[27][28] Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino wrote a "Recercare de tutti li Toni" (Ricercar in all the Tones) as early as 1507.[29] In the 17th century lutenist-composer John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.[30][31]

Henricus Grammateus drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".[32] Zarlino in his polemic with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute in his Sopplimenti musicali in 1588.

#### Simon Stevin

The first mention of equal temperament related to the twelfth root of two in the West appeared in Simon Stevin's manuscript Van De Spiegheling der singconst (ca. 1605), published posthumously nearly three centuries later in 1884.[33] However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values.[20] As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is claimed by Gene Cho as incorrect.[34]

The following were Simon Stevin's chord length from Van de Spiegheling der singconst:[35]

Tone Chord 10000 from Simon Stevin Ratio Corrected chord
semitone 9438 1.0595465 9438.7
whole tone 8909 1.0593781
tone and a half 8404 1.0600904 8409
ditone 7936 1.0594758 7937
ditone and a half 7491 1.0594046 7491.5
tritone 7071 1.0593975 7071.1
tritone and a half 6674 1.0594845 6674.2
four-tone 6298 1.0597014 6299
four-tone and a half 5944 1.0595558 5946
five-tone 5611 1.0593477 5612.3
five-tone and a half 5296 1.0594788 5297.2
full tone 1.0592000

A generation later, French mathematician Marin Mersenne presented several equal tempered chord lengths obtained by Jean Beaugrand, Ismael Bouillaud, and Jean Galle.[36]

In 1630 Johann Faulhaber published a 100-cent monochord table, which contained several errors due to his use of logarithmic tables. He did not explain how he obtained his results.[37]

#### Baroque era

From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament,[38] and the Brossard lute Manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Bocquet written in all keys, including the last prelude, entitled Prelude sur tous les tons, which enharmonically modulates through all keys.[39] Angelo Michele Bartolotti published a series of passacaglias in all keys, with connecting enharmonically modulating passages. Among the 17th-century keyboard composers Girolamo Frescobaldi advocated equal 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, although Andreas Werckmeister emphatically advocated equal temperament in his 1707 treatise published posthumously.[40]

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 possible 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.)[citation needed]

Twelve-tone equal temperament took hold for a variety of reasons. It was a convenient fit for the existing keyboard design, and permitted total harmonic freedom at the expense of just a little impurity in every interval. This allowed greater expression through enharmonic modulation, which became extremely important in the 18th century in music of such composers as Francesco Geminiani, Wilhelm Friedemann Bach, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel.[citation needed]

The progress of equal temperament from the mid-18th century on is described with detail in quite a few modern scholarly publications: it was already the temperament of choice during the Classical era (second half of the 18th century),[citation needed] and it became standard during the Early Romantic era (first decade of the 19th century),[citation needed] except for organs that switched to it more gradually, completing only in the second decade of the 19th century. (In England, some cathedral organists and choirmasters held out against it even after that date; Samuel Sebastian Wesley, for instance, opposed it all along. He died in 1876.)[citation needed]

A precise equal temperament is possible using the 17th-century Sabbatini method of splitting the octave first into three tempered major thirds.[41] This was also proposed by several writers during the Classical era. Tuning without beat rates but employing several checks, achieving virtually modern accuracy, was already done in the first decades of the 19th century.[42] Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.[43] The ultimate precision was available with 2-decimal tables published by White in 1917.[44]

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 two adjacent steps 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:

${\displaystyle r^{n}=p}$
${\displaystyle r={\sqrt[{n}]{p}}}$

where the ratio r divides the ratio p (typically 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:

${\displaystyle c={\frac {w}{n}}}$

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 modulus 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

One octave of 12-tet on a monochord

In twelve-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:

${\displaystyle {\sqrt[{12}]{2}}=2^{\frac {1}{12}}\approx 1.059463}$

This is equivalent to:

${\displaystyle e^{{\frac {1}{12}}\ln 2}\approx 1.059463}$

This interval is divided into 100 cents.

### Calculating absolute frequencies

To find the frequency, Pn, of a note in 12-TET, the following definition may be used:

${\displaystyle P_{n}=P_{a}\left({\sqrt[{12}]{2}}\right)^{(n-a)}}$

In this formula Pn refers to the pitch, or frequency (usually in hertz), you are trying to find. Pa refers to the frequency of a reference pitch. 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 440Hz), and C4 (middle C),and F#4 are the 40th and 46th key respectively. These numbers can be used to find the frequency of C4 and F#4 :

${\displaystyle P_{40}=440\left({\sqrt[{12}]{2}}\right)^{(40-49)}\approx 261.626\ \mathrm {Hz} }$
${\displaystyle P_{46}=440\left({\sqrt[{12}]{2}}\right)^{(46-49)}\approx 369.994\ \mathrm {Hz} }$

### Historical comparison

Year Name Ratio Cents
400 He Chengtian 1.060070671 101.0
1580 Vincenzo Galilei 18:17 [1.058823529] 99.0
1581 Zhu Zaiyu 1.059463094 100.0
1585 Simon Stevin 1.059546514 100.1
1630 Marin Mersenne 1.059322034 99.8
1630 Johann Faulhaber 1.059490385 100.0

Reference: Date, name, ratio, cents: from equal temperament monochord tables p55-p78; J. Murray Barbour Tuning and Temperament, Michigan State University Press 1951

### Comparison with Just Intonation

The intervals of 12-TET closely approximate some intervals in just intonation.[45] 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 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) 2012 = 1 1 0 11 = 1 0 0
Minor second (C/D) 2112 = 122 1.059463 100 1615 = 1.06666… 111.73 −11.73
Major second (D) 2212 = 62 1.122462 200 98 = 1.125 203.91 −3.91
Minor third (D/E) 2312 = 42 1.189207 300 65 = 1.2 315.64 −15.64
Major third (E) 2412 = 32 1.259921 400 54 = 1.25 386.31 +13.69
Perfect fourth (F) 2512 = 1232 1.334840 500 43 = 1.33333… 498.04 +1.96
Tritone (F/G) 2612 = 2 1.414214 600 75 = 1.4

107 = 1.42857...
582.51

617.49
+17.49

−17.49
Perfect fifth (G) 2712 = 12128 1.498307 700 32 = 1.5 701.96 −1.96
Minor sixth (G/A) 2812 = 34 1.587401 800 85 = 1.6 813.69 −13.69
Major sixth (A) 2912 = 48 1.681793 900 53 = 1.66666… 884.36 +15.64
Minor seventh (A/B) 21012 = 632 1.781797 1000 169 = 1.77777… 996.09 +3.91
Major seventh (B) 21112 = 122048 1.887749 1100 158 = 1.875 1088.27 +11.73
Octave (C) 21212 = 2 2 1200 21 = 2 1200.00 0

### Seven-tone equal division of the fifth

Violins, violas and cellos are tuned in perfect fifths (G – D – A – E, for violins, and C – G – D – A, for violas and cellos), which suggests that their semi-tone ratio is slightly higher than in the conventional twelve-tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval is covered in 7 steps, each tone is in the ratio of 732 to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2 but a slightly widened octave with a ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1 ratio, because twelve perfect fifths do not equal seven octaves.[46] During actual play, however, the violinist chooses pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio.

## Other equal temperaments

### 5 and 7 tone temperaments in ethnomusicology

Approximation of 7-tet

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. 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."[47]

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 (133-cent steps  ).

A South American Indian scale from a pre-instrumental culture measured by Boiles (1969) featured 175-cent seven-tone equal temperament, which stretches the octave slightly as with instrumental gamelan music.

5-TET and 7-TET mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure 1.

• In 5-TET 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 7-TET 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).

### Various Western equal temperaments

Easley Blackwood's notation system for 16 equal temperament: intervals are notated similarly to those they approximate and there are fewer enharmonic equivalents.[48]
Comparison of equal temperaments from 9 to 25 (after Sethares (2005), p.58).[1]

24 EDO, the quarter tone scale (or 24-TET), was a popular microtonal tuning in the 20th century probably because it represented a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who were also interested in microtonality. Because 24 EDO contains all of the pitches of 12 EDO, plus new pitches halfway between each adjacent pair of 12 EDO pitches, they could employ the additional colors without losing any tactics available in 12-tone harmony. The fact that 24 is a multiple of 12 also made 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments purposely tuned a quarter-tone apart, such as two pianos, which also allowed 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 approximates the 11th harmonic very well, unlike 12 EDO.

19 EDO is famous and some instruments are tuned in 19 EDO. It has slightly flatter perfect fifth (at 694 cents), but its major sixth are less than a single cent away from just intonation's major sixth (at 884 cents). Its minor third is also less than a cent from just intonation's. Its perfect fourth (at 503 cents), is only 5 cents sharp than just intonation's and 3 cents sharp from 12-tet's.

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, making it attractive to microtonalists looking for unusual microtonal harmonic territory.

27 EDO is the smallest EDO that uniquely represents all intervals involving the first eight harmonics. It tempers out the septimal comma but not the syntonic comma.

29 EDO is the lowest number of equal divisions of the octave that produces a better perfect fifth than 12 EDO. Its major third is roughly as inaccurate as 12-TET; however, it is tuned 14 cents flat rather than 14 cents sharp. It tunes the 7th, 11th, and 13th harmonics flat as well, by roughly the same amount. This means intervals such as 7:5, 11:7, 13:11, etc., are all matched extremely well in 29-TET.

31 EDO was advocated by Christiaan Huygens and Adriaan Fokker. 31 EDO has a slightly less accurate fifth than 12 EDO, but provides near-just major thirds, and provides decent matches for harmonics up to at least 13, of which the seventh harmonic is particularly accurate.

34 EDO gives slightly less total combined errors of approximation to the 5-limit just ratios 3:2, 5:4, 6:5, and their inversions than 31 EDO does, although the approximation of 5:4 is worse. 34 EDO doesn't approximate ratios involving prime 7 well. It contains a 600-cent tritone, since it is an even-numbered EDO.

41 EDO is the second lowest number of equal divisions that produces a better perfect fifth than 12 EDO. Its major third is more accurate than 12 EDO and 29 EDO, about 6 cents flat. It's not meantone, so it distinguishes 10:9 and 9:8, unlike 31edo. It is more accurate in 13-limit than 31edo.

53 EDO is better at approximating the traditional just consonances than 12, 19 or 31 EDO, 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 EDO, the very consonant thirds would be reached instead by using a Pythagorean diminished fourth (C-F), as it is an example of schismatic temperament, just like 41 EDO.

72 EDO approximates many just intonation intervals well, even into the 7-limit and 11-limit, such as 7:4, 9:7, 11:5, 11:6 and 11:7. 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). It can be considered an extension of 12 EDO because 72 is a multiple of 12. 72 EDO has a smallest interval that is six times smaller than the smallest interval of 12 EDO and therefore contains six copies of 12 EDO starting on different pitches. It also contains three copies of 24 EDO and two copies of 36 EDO, which are themselves multiples of 12 EDO. 72 EDO has also been criticized for its redundancy by retaining the poor approximations contained in 12 EDO, despite not needing them for any lower limits of just intonation (e.g. 5-limit).

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 from 1924 to the 1940s.[49]

Other equal divisions of the octave that have found occasional use include, 15 EDO, 17 EDO, 19 EDO and 22 EDO.

2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of log2(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 approximation of 2, 5, 12, 41, 53, 306, 665 and 15601 just twelfths/fifths than for any equal temperaments with less tones.[50][51]

1, 2, 3, 5, 7, 12, 29, 41, 53, 200... (sequence A060528 in the OEIS) is the sequence of divisions of octave that provide better and better approximations of the perfect fifth. Related sequences contain divisions approximating other just intervals.[52]

This application: [2] calculates the frequencies, approximate cents, and MIDI pitch bend values for any systems of equal division of the octave. Note that 'rounded' and 'floored' produce the same MIDI pitch bend value.

### 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 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 133.

Wendy Carlos created 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.[53] Their step sizes:

• alpha: 932 (78.0 cents)
• beta: 1132 (63.8 cents)
• gamma: 2032 (35.1 cents)

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

### 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 s, and the number of steps in a tone be t.

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, C, D, E, F, and F are in ascending order if they preserve their usual relationships to C). That is, fixing q to a proper fraction in the relationship qt = s also defines a unique family of one equal temperament and its multiples that fulfil this relationship.

For example, where k is an integer, 12k-EDO sets q = ​12, and 19k-EDO sets q = ​13. The smallest multiples in these families (e.g. 12 and 19 above) has the additional property of having no notes outside the circle of fifths. (This is not true in general; in 24-EDO, the half-sharps and half-flats are not in the circle of fifths generated starting from C.) The extreme cases are 5k-EDO, where q = 0 and the semitone becomes a unison, and 7k-EDO, where q = 1 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 fulfilling the above properties (including having no notes outside the circle of fifths) divides the octave into 7t − 2s steps, and the perfect fifth into 4ts steps. If there are notes outside the circle of fifths, one must then multiply these results by n, which is the number of nonoverlapping circles of fifths required to generate all the notes (e.g. two in 24-EDO, six in 72-EDO). (One must take the small semitone for this purpose: 19-EDO has two semitones, one being ​13 tone and the other being ​23.)

The smallest of these families is 12k-EDO, and in particular, 12-EDO is the smallest equal temperament that has the above properties. Additionally, it also makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons why 12-EDO 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 q for the relationship results in exactly one equal temperament family, but the converse is not true: 47-EDO has two different semitones, where one is ​17 tone and the other is ​89, which are not complements of each other like in 19-EDO (​13 and ​23). Taking each semitone results in a different choice of perfect fifth.

## Related tuning systems

### Regular diatonic tunings

Figure 1: The regular diatonic tunings continuum, which include many notable "equal temperament" tunings (Milne 2007).[54]

The diatonic tuning in twelve equal can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps TTSTTTS (or a rotation of it) with all the T's and all the S's the same size and the S's smaller than the T's. In twelve equal the S is the semitone and is exactly half the size of the tone T. When the S's reduce to zero the result is TTTTT or a five-tone equal temperament, As the semitones get larger, eventually the steps are all the same size, and the result is in seven tone equal temperament. These two endpoints are not included as regular diatonic tunings.

The notes in a regular diatonic tuning are connected together by a cycle of seven tempered fifths. The twelve-tone system similarly generalizes to a sequence CDCDDCDCDCDD (or a rotation of it) of chromatic and diatonic semitones connected together in a cycle of twelve fifths. In this case, seven equal is obtained in the limit as the size of C tends to zero and five equal is the limit as D tends to zero while twelve equal is of course the case C = D.

Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems. For instance if the diatonic semitone is double the size of the chromatic semitone, i.e. D = 2*C the result is nineteen equal with one step for the chromatic semitone, two steps for the diatonic semitone and three steps for the tone and the total number of steps 5*T + 2*S = 15 + 4 = 19 steps. The resulting twelve-tone system closely approximates to the historically important 1/3 comma meantone.

If the chromatic semitone is two-thirds of the size of the diatonic semitone, i.e. C = (2/3)*D, the result is thirty one equal, with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone where 5*T + 2*S = 25 + 6 = 31 steps. The resulting twelve-tone system closely approximates to the historically important 1/4 comma meantone.

## References

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6. Kenneth Robinson: A critical study of Chu Tsai-yü's contribution to the theory of equal temperament in Chinese music. (Sinologica Coloniensia, Bd. 9.) x, 136 pp. Wiesbaden: Franz Steiner Verlag GmbH, 1980. DM 36. p.vii "Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world
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51. "convergents(log2(3), 10)". WolframAlpha. Retrieved 2014-06-18.
• 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3 (sequence A054540 in the OEIS)
• 3:2 and 4:3, 5:4 and 8:5 (sequence A060525 in the OEIS)
• 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7 (sequence A060526 in the OEIS)
• 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7, 16:11 and 11:8 (sequence A060527 in the OEIS)
• 4:3 and 3:2, 5:4 and 8:5, 6:5 and 5:3, 7:4 and 8:7, 16:11 and 11:8, 16:13 and 13:8 (sequence A060233 in the OEIS)
• 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45 (sequence A061920 in the OEIS)
• 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45, 27:20 and 40:27, 32:27 and 27:16, 81:64 and 128:81, 256:243 and 243:128 (sequence A061921 in the OEIS)
• 5:4 and 8:5 (sequence A061918 in the OEIS)
• 6:5 and 5:3 (sequence A061919 in the OEIS)
• 6:5 and 5:3, 7:5 and 10:7, 7:6 and 12:7 (sequence A060529 in the OEIS)
• 11:8 and 16:11 (sequence A061416 in the OEIS)
52. Carlos, Wendy. "Three Asymmetric Divisions of the Octave". wendycarlos.com. Serendip LLC. Retrieved 2016-09-01.
53. Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum" Archived 2016-01-09 at the Wayback Machine, Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.