# Equal temperament

Figure 1: The syntonic tuning continuum, which includes many notable "equal temperament" tunings (Milne 2007).[1]
Chromatic scale on C, full octave ascending, notated only with sharps.

An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for every note in the system.

In equal temperament tunings, an interval – usually the octave – is divided into a series of equal steps (equal frequency ratios between successive notes). For classical music, the most common tuning system is twelve-tone equal temperament (also known as 12 equal temperament), inconsistently abbreviated as 12-TET, 12TET, 12tET, 12tet, 12-ET, 12ET, or 12et, which divides the octave into 12 parts, all of which are equal on a logarithmic scale. It is usually tuned relative to a standard pitch of 440 Hz, called A440.

Other equal temperaments exist (some music has been written in 19-TET and 31-TET for example, and 24-TET is used in Arabic music), but in Western countries when people use the term equal temperament without qualification, they usually mean 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 equal-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.

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. Some wind instruments that can easily and spontaneously bend their tone, most notably double-reeds, use tuning similar to string ensembles and vocal groups.

The tuning continuum of the syntonic temperament, shown in Figure 1, includes a number of notable "equal temperament" tunings, including those that divide the octave equally into 5, 7, 12, 17, 19, 22, 26, 31, 43, 50, and 53 parts. On an isomorphic keyboard, the fingering of music written in any of these syntonic tunings is precisely the same as it is in any other syntonic tuning, so long as the notes are spelled properly—that is, with no assumption of enharmonicity. This consistency of fingering makes it possible to smoothly vary the tuning (and hence the pitches of all notes, systematically) all along the syntonic tuning continuum—a polyphonic tuning bend. The use of dynamic timbres lets consonance be maintained (or otherwise manipulated) across such tuning bends.

## 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,[2] 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.[3]

Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu[4] and provides textual quotations as evidence.[5] 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."[5] Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications."[2] Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors.[6]

### 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.[7] 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.[8]

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 5 full 7 note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.[9]

An approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD.[10]

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.[11][12]

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 乐律融通 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 乐律全书) in 1584.[13] An extended account is also given by Joseph Needham.[14] Zhu obtained his result mathematically by dividing the length of string and pipe successively by

${\displaystyle {\sqrt[{12}]{2}}=1.059463094359295264561825}$, and for pipe diameter by

${\displaystyle (1.059463094359295264561825)^{84}\approx 127.999999999946}$ (still in tune after 84/12 = 7 octaves)

#### Zhu Zaiyu

According to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically.[16] Matteo Ricci, a Jesuit in China, was at Chinese trade fair in Canton the year Zhu published his solution, and very likely brought it back to the West.[17] 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."[18] 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.[19]

Zhu Zaiyu's equal temperament pitch pipes

Zhu Zaiyu illustrated his equal temperament theory by 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 diameter 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.[20]

### 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.

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.[21] He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).[22]

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567.[23][24] 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.[25] In the 17th century lutenist-composer John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.[26][27]

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".[28] 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 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.[29] 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.[30] 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.[31]

The following were Simon Stevin's chord length from Vande Spiegheling der singconst:[32]

TONE CHORD 10000 from Simon Stevin RATIO CORRECTED CHORD
semitone 9438 1.0595465 9438.7
whole tone 8909 1.0593781
1.5 tone 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-half 5944 1.0595558 5946
five-tone 5611 1.0593477 5612.3
five-tone-and-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.[33]

In 1630 Johann Faulhaber published a 100 cent monochord table, with the exception of several errors due to his use of logarithmic tables . He did not explain how he obtained his results.[34]

#### Baroque era

From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament,[35] 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.[36] 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.[37]

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 believeTemplate:Weasel-inline 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.){{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Twelve tone equal temperament took hold for a variety of reasons. It conveniently fit the existing keyboard design, and permitted total harmonic freedom at the expense of just a little impurity in every interval. This allowed greater expression through enharmonic modulation, which became extremely important in the 18th century in music of such composers as Francesco Geminiani, Wilhelm Friedemann Bach, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

## Rational semitone

For any semitone that is a proper fraction of a whole tone, exactly one equal division of the octave lets the circle of fifths generate all the notes of the equal division while preserving the order of the notes. (That is, C is lower than D, D is lower than E, etc., and FTemplate:Music is indeed sharper than F.) The number of divisions needed for the octave is seven times the number of divisions of a whole tone minus twice the number of divisions of the semitone. The corresponding fifth spans a number of divisions equal to four whole tones minus one semitone. Hence, for a semitone of one-half of a whole tone, the corresponding equal temperament scheme is 12-EDO with a fifth of seven divisions. A semitone of one-third of a whole tone corresponds to 19-EDO with a fifth of eleven divisions.

12-EDO is the equal temperament with the smallest number of divisions that allows for a rational semitone to preserve the desired properties concerning note order and the circle of fifths. It also has the desirable property of making the semitone exactly one-half of a whole tone. These are additional reasons why 12-EDO became the predominant form of equal temperament.

While each rational semitone corresponds to only one equal temperament, the reverse is not the case. For example, both a semitone of one-seventh, and a semitone of eight-ninths both use 47-EDO, which is the smallest number of divisions that has two different semitones. However, they have different values for the fifth, as a semitone of one-seventh uses a fifth of twenty-seven divisions while a semitone of eight ninths uses a fifth of twenty-eight divisions.

## 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. 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."[44]   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 preinstrumental 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.[45]
Comparison of equal temperaments from 9 to 25 (after Sethares (2005), p.58).

31 tone equal temperament was advocated by Christiaan Huygens and Adriaan Fokker. 31-TET has a slightly less accurate fifth than 12-TET, 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.

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.

29-TET is the lowest number of equal divisions of the octave which produces a better perfect fifth than 12-TET. Its major third is roughly as inaccurate as 12-TET, however it is tuned 14 cents flat rather than 14 cents sharp.

41-TET is the second lowest number of equal divisions that produces a better perfect fifth than 12-TET. Its major third is more accurate than 12-ET and 29-ET, about 6 cents flat.

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. A consequence of this is that chord progressions like I-vi-ii-V-I won't land you back where you started in 53-TET, but rather one 53-tone step flat (unless the motion by I-vi wasn't by the 5-limit minor third).

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 14-TET, 15-TET, 16-TET, 17-TET, 19-TET, 22-TET, 34-TET, 46-TET, 48-TET, 99-TET, and 171-TET.

2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of ${\displaystyle \log _{2}(3)}$, so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to en 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.[46][47]

1, 2, 3, 5, 7, 12, 29, 41, 53, 200... (sequence A060528 in 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.[48] It is noteworthy that many elements of this sequences are sums of previous elements.

 This application: http://equal.meteor.com/ calculates the frequencies, amounts of cents and Pitch Bend values


for any systems of Equal Division of the Octave. Note, that both 'rounded' and 'floored' notations are equivalent, produces the same MIDI 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 and 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 ${\displaystyle {\sqrt[{13}]{3}}}$.

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Each of them provides a very good approximation of several just intervals.[49] Their step sizes:

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

## References

### Citations

1. Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
2. Fritz A. Kuttner. p. 163.
3. Fritz A. Kuttner. "Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal Temperament Theory", p.200, Ethnomusicology, Vol. 19, No. 2 (May, 1975), pp. 163–206.
4. 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
5. Robinson, Kenneth G., and Joseph Needham. 1962. "Physics and Physical Technology". In Science and Civilisation in China, vol. 4: "Physics and Physical Technology", Part 1: "Physics", edited by Joseph Needham. Cambridge: University Press. p. 221. Cite error: Invalid <ref> tag; name "Robinson221" defined multiple times with different content
6. Fritz A. Kuttner. p. 200.
7. Owen Henry Jorgensen, Tuning (East Lansing: Michigan State University Press, 1991)
8. The Formation of Chinese Civilization: An Archaeological Perspective, Zhang Zhongpei, Shao Wangping and others, Yale (New Haven Conn)and New World Press (Beijing),2005
9. The Formation of Chinese Civilization, 2005 (opt cit), the Eastern Zhou and the Growth of Regionalism, Lu Liancheng, pp 140 ff
10. J. Murray Barbour Tuning and Temperament p55-56, Michigan State University Press 1951
11. Template:Cite web
12. Template:Cite web
13. Template:Cite web
14. Science and Civilisation in China, Vol IV:1 (Physics), Joseph Needham, Cambridge University Press, 1962–2004, pp 220 ff
15. The Shorter Science & Civilisation in China, An abridgement by Colin Ronan of Joseph Needham's original text, p385
16. Gene J. Cho "The Significance of the Discovery of the Musical Equal Temperament In the Cultural History," http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm
17. Template:Cite web
18. J. Murray Barbour, Tuning and Temperament Michigan University Press, 1951 p7
19. [1] "The division of Octave into twelve Semitones, and the transposition of scales have also been discovered by this intelligent and skillful nation" Hermann von Helmholtz On the Sensations of Tone as a Physiological basis for the theory of music, p 258, 3rd edition, Longmans, Green and Co, London, 1895
20. Lau Hanson, Abacus and Practical Mathematics p389 (in Chinese 劳汉生 《珠算与实用数学》 389页)
21. Galilei, V. (1584). Il Fronimo... Dialogo sopra l'arte del bene intavolare. G. Scotto: Venice, ff. 80–89.
22. J. Murray Barbour Tuning and Temperament p8 1951 Michigan State University Press
23. Template:Cite web
24. Giacomo Gorzanis, c. 1525 - c. 1575 Intabolatura di liuto. Geneva, 1982
25. Template:Cite web
26. John Wilson, 26 Preludes, Diapason press (DP49, Utrecht)
27. in "English Song, 1600–1675, Facsimile of Twenty-six Manuscripts and an Edition of the Texts" vol. 7, Manuscripts at Oxford, Part II. introduction by Elise Bickfoer Jorgens, Garland Publishing Inc., New York and London, 1987
28. "Scintille de musica", (Brescia, 1533), p. 132
29. Template:Cite web
30. Thomas S. Christensen, The Cambridge history of western music theory p205, Cambridge University Press
31. Gene Cho, The discovery of musical equal temperament in China and Europe – Page 223, Mellen Press
32. Gene Cho, The discovery of musical equal temperament in China and Europe – Page 222, Mellen Press
33. Thomas S. Christensen, The Cambridge history of western music theory p207, Cambridge University Press
34. Thomas S. Christensen, The Cambridge history of western music theory p78, Cambridge University Press
35. "Lutes, Viols, Temperaments" Mark Lindley ISBN 978-0-521-28883-5
36. Vm7 6214
37. Andreas Werckmeister: Musicalische Paradoxal-Discourse, 1707
38. Di Veroli, Claudio. Unequal Temperaments: Theory, History and Practice. 2nd edition, Bray Baroque, Bray, Ireland 2009, pp. 140, 142 and 256.
39. Moody, Richard. “Early Equal Temperament, An Aural Perspective: Claude Montal 1836” in Piano Technicians Journal. Kansas City, USA, Feb. 2003.
40. Helmholtz, Hermann L.F. Lehre von den Tonempfindungen… Heidelberg 1862. English edition: On the Sensations of Tone, Longmans, London, 1885, p.548.
41. White, William Braid. Piano Tuning and Allied Arts. 1917, 5th enlarged edition, Tuners Supply Co., Boston 1946, p.68.
42. Date,name,ratio,cents: from equal temperament monochord tables p55-p78; J. Murray Barbour Tuning and Temperament, Michigan State University Press 1951
43. Template:Cite web
44. Morton, David (1980). "The Music of Thailand", Musics of Many Cultures, p.70. May, Elizabeth, ed. ISBN 0-520-04778-8.
45. Myles Leigh Skinner (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.55. ISBN 9780542998478.
46. Template:Cite web
47. Template:Cite web
48. 3/2 and 4/3, 5/4 and 8/5, 6/5 and 5/3 (A054540);
3/2 and 4/3, 5/4 and 8/5 (A060525);
3/2 and 4/3, 5/4 and 8/5, 7/4 and 8/7 (A060526);
3/2 and 4/3, 5/4 and 8/5, 7/4 and 8/7, 16/11 and 11/8 (A060527);
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 (A060233);
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 (A061920);
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 (A061921);
5/4 and 8/5 (A061918);
6/5 and 5/3 (A061919);
6/5 and 5/3, 7/5 and 10/7, 7/6 and 12/7 (A060529);
11/8 and 16/11 (A061416)
49. Three Asymmetric Divisions of the Octave by Wendy Carlos

### Bibliography

• Cho, Gene Jinsiong. (2003). The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston, NY: Edwin Mellen Press.
• Duffin, Ross W. How Equal Temperament Ruined Harmony (and Why You Should Care). W.W.Norton & Company, 2007.
• Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3
• {{#invoke:citation/CS1|citation

|CitationClass=book }}

• 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" [2]
• Khramov, Mykhaylo. "Approximation of 5-limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave", Proceedings of the International Conference SIGMAP-2008, 26–29 July 2008, Porto, pp. 181–184, ISBN 978-989-8111-60-9