Equal temperament
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 twelvetone equal temperament (also known as 12 equal temperament, 12TET or 12ET), 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 (^{12}√2 ≈ 1.05946). That resulting smallest interval, ^{1}⁄_{12} the width of an octave, is called a semitone or half step. In modern times, 12TET 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 19TET and 31TET. Arabic music uses 24TET as a notational convention. In Western countries the term equal temperament, without qualification, generally means 12TET. 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, 12TET is called 12EDO, 31TET is called 31EDO, 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 equaltempered 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 "pseudooctave" 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. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups.
History
The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as ChuTsaiyu. 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 "ChuTsaiyu presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament monochords 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 pitchpipers 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 7note octaves in the key of C Major, including 12 note semitones 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]
Historically, there was a sevenequal temperament or heptaequal 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 12TET in his 5,000page 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 ^{12}√2 ≈ 1.059463, and for pipe length by ^{24}√2,[17] such that after twelve divisions (an octave) the length was divided by a factor of 2:
Similarly, after 84 divisions (7 octaves) the length was divided by a factor of 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.[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 19thcentury 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 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 12string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, VictorCharles 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
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 twelvetone 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 lutenistcomposer 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 
fourtone  6298  1.0597014  6299 
fourtone and a half  5944  1.0595558  5946 
fivetone  5611  1.0593477  5612.3 
fivetone 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 100cent 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 17thcentury 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 WellTempered Clavier to demonstrate the musical possibilities of what is now called well temperament, which allows for all keys to be played, but does not make half steps equidistant, giving each key signature its own distinct character. 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.)
Twelvetone equal temperament took hold for a variety of reasons. It was a convenient fit for the existing keyboard design, and permitted total harmonic freedom with the burden of moderate impurity in every interval, particularly imperfect consonances. 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.
The progress of equal temperament from the mid18th 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), and it became standard during the Early Romantic era (first decade of the 19th century), 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.)
A precise equal temperament is possible using the 17thcentury 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 2decimal 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 equaltempered scale is the ratio:
where the ratio r divides the ratio p (typically the octave, which is 2:1) into n equal parts. (See Twelvetone 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:
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.
General formulas for the equaltempered interval
Twelvetone equal temperament
In twelvetone 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:
This is equivalent to:
This interval is divided into 100 cents.
Calculating absolute frequencies
To find the frequency, P_{n}, of a note in 12TET, the following definition may be used:
In this formula P_{n} refers to the pitch, or frequency (usually in hertz), you are trying to find. P_{a} 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, A_{4} (the reference pitch) is the 49th key from the left end of a piano (tuned to 440Hz), and C_{4} (middle C),and F#_{4} are the 40th and 46th key respectively. These numbers can be used to find the frequency of C_{4} and F#_{4} :
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 p55p78; J. Murray Barbour Tuning and Temperament, Michigan State University Press 1951
Comparison with Just Intonation
The intervals of 12TET 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 equaltempered counterparts, given as a ratio as well as cents.
Name  Exact value in 12TET  Decimal value in 12TET  Cents  Just intonation interval  Cents in just intonation  Difference 

Unison (C)  2^{0⁄12} = 1  1  0  ^{1}⁄_{1} = 1  0  0 
Minor second (C♯/D♭)  2^{1⁄12} = ^{12}√2  1.059463  100  ^{16}⁄_{15} = 1.06666…  111.73  −11.73 
Major second (D)  2^{2⁄12} = ^{6}√2  1.122462  200  ^{9}⁄_{8} = 1.125  203.91  −3.91 
Minor third (D♯/E♭)  2^{3⁄12} = ^{4}√2  1.189207  300  ^{6}⁄_{5} = 1.2  315.64  −15.64 
Major third (E)  2^{4⁄12} = ^{3}√2  1.259921  400  ^{5}⁄_{4} = 1.25  386.31  +13.69 
Perfect fourth (F)  2^{5⁄12} = ^{12}√32  1.334840  500  ^{4}⁄_{3} = 1.33333…  498.04  +1.96 
Tritone (F♯/G♭)  2^{6⁄12} = √2  1.414214  600  ^{7}⁄_{5} = 1.4 ^{10}⁄_{7} = 1.42857... 
582.51 617.49 
+17.49 −17.49 
Perfect fifth (G)  2^{7⁄12} = ^{12}√128  1.498307  700  ^{3}⁄_{2} = 1.5  701.96  −1.96 
Minor sixth (G♯/A♭)  2^{8⁄12} = ^{3}√4  1.587401  800  ^{8}⁄_{5} = 1.6  813.69  −13.69 
Major sixth (A)  2^{9⁄12} = ^{4}√8  1.681793  900  ^{5}⁄_{3} = 1.66666…  884.36  +15.64 
Minor seventh (A♯/B♭)  2^{10⁄12} = ^{6}√32  1.781797  1000  ^{16}⁄_{9} = 1.77777…  996.09  +3.91 
Major seventh (B)  2^{11⁄12} = ^{12}√2048  1.887749  1100  ^{15}⁄_{8} = 1.875  1088.27  +11.73 
Octave (C)  2^{12⁄12} = 2  2  1200  ^{2}⁄_{1} = 2  1200.00  0 
Seventone 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 semitone ratio is slightly higher than in the conventional twelvetone 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 ^{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 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
Five and seven tone equal temperament (5TET
A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7TET. 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 5TET 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 wellaccepted that of the two primary tuning systems in gamelan music, slendro and pelog, only slendro somewhat resembles fivetone equal temperament while pelog is highly unequal; however, Surjodiningrat et al. (1972) has analyzed pelog as a sevennote subset of ninetone equal temperament (133cent steps
A South American Indian scale from a preinstrumental culture measured by Boiles (1969) featured 175cent seventone equal temperament, which stretches the octave slightly as with instrumental gamelan music.
5TET and 7TET mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure 1.
 In 5TET 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 7TET 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
24 EDO, the quarter tone scale (or 24TET), 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 12tone 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 quartertone apart, such as two pianos, which also allowed each performer (or one performer playing a different piano with each hand) to read familiar 12tone notation. Various composers including Charles Ives experimented with music for quartertone 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 12tet'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 12TET; 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 29TET.
31 EDO was advocated by Christiaan Huygens and Adriaan Fokker. 31 EDO has a slightly less accurate fifth than 12 EDO, but provides nearjust 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 5limit 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 600cent tritone, since it is an evennumbered 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 13limit 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 (CF♭), as it is an example of schismatic temperament, just like 41 EDO.
72 EDO approximates many just intonation intervals well, even into the 7limit and 11limit, 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. 5limit).
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 log_{2}(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: 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 nonoctave intervals
The equaltempered 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 (
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: ^{9}√^{3}⁄_{2} (78.0 cents)
Play  beta: ^{11}√^{3}⁄_{2} (63.8 cents)
Play  gamma: ^{20}√^{3}⁄_{2} (35.1 cents)
Play
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 12EDO 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, 12kEDO sets q = ^{1}⁄_{2}, and 19kEDO sets q = ^{1}⁄_{3}. 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 24EDO, the halfsharps and halfflats are not in the circle of fifths generated starting from C.) The extreme cases are 5kEDO, where q = 0 and the semitone becomes a unison, and 7kEDO, 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 4t − s 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 24EDO, six in 72EDO). (One must take the small semitone for this purpose: 19EDO has two semitones, one being ^{1}⁄_{3} tone and the other being ^{2}⁄_{3}.)
The smallest of these families is 12kEDO, and in particular, 12EDO 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 12EDO has become the most commonly used equal temperament. (Another reason is that 12EDO is the smallest equal temperament to closely approximate 5limit harmony, the nextsmallest being 19EDO.)
Each choice of fraction q for the relationship results in exactly one equal temperament family, but the converse is not true: 47EDO has two different semitones, where one is ^{1}⁄_{7} tone and the other is ^{8}⁄_{9}, which are not complements of each other like in 19EDO (^{1}⁄_{3} and ^{2}⁄_{3}). Taking each semitone results in a different choice of perfect fifth.
Related tuning systems
Regular diatonic tunings
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 fivetone 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 twelvetone 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 twelvetone system closely approximates to the historically important 1/3 comma meantone.
If the chromatic semitone is twothirds 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 twelvetone system closely approximates to the historically important 1/4 comma meantone.
See also
References
Citations
 Sethares compares several equal temperament scales in a graph with axes reversed from the axes in the first comparison of equal temperaments, and identical axes of the second. (fig. 4.6, p. 58)
 O'Donnell, Michael. "Perceptual Foundations of Sound". Retrieved 20170311.
 The History of Musical Pitch in Europe p493511 Herman Helmholtz, Alexander J. Ellis On The Sensations of Tone, Dover Publications, Inc., New York
 Fritz A. Kuttner. p. 163.
 Fritz A. Kuttner. "Prince Chu TsaiYü's Life and Work: A ReEvaluation of His Contribution to Equal Temperament Theory", p.200, Ethnomusicology, Vol. 19, No. 2 (May 1975), pp. 163–206.
 Kenneth Robinson: A critical study of Chu Tsaiyü'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 "ChuTsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world
 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.
 Fritz A. Kuttner. p. 200.
 Owen Henry Jorgensen, Tuning (East Lansing: Michigan State University Press, 1991)
 The Formation of Chinese Civilization: An Archaeological Perspective, Zhang Zhongpei, Shao Wangping and others, Yale (New Haven Conn)and New World Press (Beijing),2005
 The Formation of Chinese Civilization, 2005 (opt. cit.), the Eastern Zhou and the Growth of Regionalism, Lu Liancheng, pp 140 ff
 J. Murray Barbour Tuning and Temperament p5556, Michigan State University Press 1951
 有关"七平均律"新文献著作的发现 [Findings of new literatures concerning the hepta – equal temperament] (in Chinese). Archived from the original on 20071027.
'Heptaequal temperament' in our folk music has always been a controversial issue.
 七平均律"琐谈兼及旧式均孔曲笛制作与转调 [abstract of About "Seven equal tuning System"] (in Chinese). Archived from the original on 20070930. Retrieved 20070625.
From the flute for two thousand years of the production process, and the Japanese shakuhachi remaining in the production of Sui and Tang Dynasties and the actual temperament, identification of people using the socalled 'Seven Laws' at least two thousand years of history; and decided that this law system associated with the flute law.
 "Quantifying Ritual: Political Cosmology, Courtly Music, and Precision Mathematics in SeventeenthCentury China Roger Hart Departments of History and Asian Studies, University of Texas, Austin". Uts.cc.utexas.edu. Archived from the original on 20120305. Retrieved 20120320.
 Science and Civilisation in China, Vol IV:1 (Physics), Joseph Needham, Cambridge University Press, 1962–2004, pp 220 ff
 The Shorter Science & Civilisation in China, An abridgement by Colin Ronan of Joseph Needham's original text, p385
 Gene J. Cho "The Significance of the Discovery of the Musical Equal Temperament in the Cultural History," http://en.cnki.com.cn/Article_en/CJFDTOTALXHYY201002002.htm Archived 20120315 at the Wayback Machine
 "EQUAL TEMPERAMENT". University of Houston. Retrieved 20141005.
 Thomas S. Christensen, The Cambridge history of western music theory p205, Cambridge University Press
 J. Murray Barbour, Tuning and Temperament Michigan University Press, 1951 p7
 "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
 Lau Hanson, Abacus and Practical Mathematics p389 (in Chinese 劳汉生 《珠算与实用数学》 389页)
 True, Timothy (2018). "The Battle Between Impeccable Intonation and Maximized Modulation". Musical Offerings. 9 (2): 61–74. doi:10.15385/jmo.2018.9.2.2.
 Galilei, V. (1584). Il Fronimo... Dialogo sopra l'arte del bene intavolare. G. Scotto: Venice, ff. 80–89.
 J. Murray Barbour Tuning and Temperament p8 1951 Michigan State University Press
 "Resound – Corruption of Music". Philresound.co.uk. Retrieved 20120320.
 Giacomo Gorzanis, c. 1525 – c. 1575 Intabolatura di liuto. Geneva, 1982
 "Spinacino 1507a: Thematic Index". Appalachian State University. Archived from the original on 20110725. Retrieved 20120614.
 John Wilson, 26 Preludes, Diapason press (DP49, Utrecht)
 in "English Song, 1600–1675, Facsimile of Twentysix 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
 "Scintille de musica", (Brescia, 1533), p. 132
 "Van de Spiegheling der singconst, ed by Rudolf Rasch, The Diapason Press". Diapason.xentonic.org. 20090630. Retrieved 20120320.
 Gene Cho, The discovery of musical equal temperament in China and Europe – Page 223, Mellen Press
 Gene Cho, The discovery of musical equal temperament in China and Europe – Page 222, Mellen Press
 Thomas S. Christensen, The Cambridge history of western music theory p207, Cambridge University Press
 Thomas S. Christensen, The Cambridge history of western music theory p.78, Cambridge University Press
 "Lutes, Viols, Temperaments" Mark Lindley ISBN 9780521288835
 Vm7 6214
 Andreas Werckmeister: Musicalische ParadoxalDiscourse, 1707
 Di Veroli, Claudio. Unequal Temperaments: Theory, History and Practice. 2nd edition, Bray Baroque, Bray, Ireland 2009, pp. 140, 142 and 256.
 Moody, Richard. "Early Equal Temperament, An Aural Perspective: Claude Montal 1836" in Piano Technicians Journal. Kansas City, USA, Feb. 2003.
 Helmholtz, Hermann L.F. Lehre von den Tonempfindungen… Heidelberg 1862. English edition: On the Sensations of Tone, Longmans, London, 1885, p.548.
 White, William Braid. Piano Tuning and Allied Arts. 1917, 5th enlarged edition, Tuners Supply Co., Boston 1946, p.68.
 Partch, Harry (1979). Genesis of a Music (2nd ed.). Da Capo Press. p. 134. ISBN 030680106X.
 Cordier, Serge. "Le tempérament égal à quintes justes" (in French). Association pour la Recherche et le Développement de la Musique. Retrieved 20100602.
 Morton, David (1980). "The Music of Thailand", Musics of Many Cultures, p.70. May, Elizabeth, ed. ISBN 0520047788.
 Myles Leigh Skinner (2007). Toward a Quartertone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.55. ISBN 9780542998478.
 Monzo, Joe (2005). "EqualTemperament". Tonalsoft Encyclopedia of Microtonal Music Theory. Joe Monzo. Retrieved 26 February 2019.
 "665edo". xenoharmonic (microtonal wiki). Retrieved 20140618.
 "convergents(log2(3), 10)". WolframAlpha. Retrieved 20140618.

 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)
 Carlos, Wendy. "Three Asymmetric Divisions of the Octave". wendycarlos.com. Serendip LLC. Retrieved 20160901.
 Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum" Archived 20160109 at the Wayback Machine, Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 1532.
Sources
 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 0870132903
 Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). London: SpringerVerlag. ISBN 1852337974.
 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 https://web.archive.org/web/20050127000731/http://web.telia.com/~u57011259/pelog_main.htm. Retrieved May 19, 2006.
 Stewart, P. J. (2006) "From Galaxy to Galaxy: Music of the Spheres"
 Khramov, Mykhaylo. "Approximation of 5limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave", Proceedings of the International Conference SIGMAP2008, 26–29 July 2008, Porto, pp. 181–184, ISBN 9789898111609
Further reading
 Sensations of Tone a foundational work on acoustics and the perception of sound by Hermann von Helmholtz. Especially Appendix XX: Additions by the Translator, pages 430556, (pdf pages 451577)]
External links
 HuygensFokker Foundation Centre for Microtonal Music
 A.Orlandini: Music Acoustics
 "Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)
 Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900. (2008) Latina, Il Levante Libreria Editrice
 Fractal Microtonal Music, Jim Kukula.
 All existing 18th century quotes on J.S. Bach and temperament
 Dominic Eckersley: "Rosetta Revisited: Bach's Very Ordinary Temperament"
 Well Temperaments, based on the Werckmeister Definition
 FAVORED CARDINALITIES OF SCALES by PETER BUCH