Equal temperament
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 twelvetone equal temperament (also known as 12 equal temperament), inconsistently abbreviated as 12TET, 12TET, 12tET, 12tet, 12ET, 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 19TET and 31TET for example, and 24TET is used in Arabic music), but in Western countries when people use the term equal temperament without qualification, they usually mean 12TET.
Equal temperaments may also divide some interval other than the octave, a pseudooctave, into a whole number of equal steps. An example is an equaltempered Bohlen–Pierce scale. To avoid ambiguity, the term equal division of the octave, or EDO is sometimes preferred. According to this naming system, 12TET is called 12EDO, 31TET is called 31EDO, 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 doublereeds, 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.
Contents

History 1

China 1.1
 Early history 1.1.1
 Zhu Zaiyu 1.1.2

Europe 1.2
 Early history 1.2.1
 Simon Stevin 1.2.2
 Baroque era 1.2.3

China 1.1
 General properties 2

Twelvetone equal temperament 3
 Calculating absolute frequencies 3.1
 Historical comparison 3.2
 Comparison to just intonation 3.3
 Seventone equal division of the fifth 3.4
 Rational semitone 4

Other equal temperaments 5
 5 and 7 tone temperaments in ethnomusicology 5.1
 Various Western equal temperaments 5.2
 Equal temperaments of nonoctave intervals 5.3
 See also 6

References 7
 Citations 7.1
 Bibliography 7.2
 External links 8
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,^{[2]} 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.^{[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 pitchpipers 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 semitones 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]}
Historically, there was a sevenequal temperament or heptaequal 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 12TET in his 5,000page 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
\sqrt [12]{2}=1.059463094359295264561825, and for pipe diameter by
\sqrt [24]{2}^{[15]}
(1.059463094359295264561825)^{12} \approx 1.99999999999988 ;
(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 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.^{[19]}
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 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 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
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 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.^{[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 lutenistcomposer 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 
fourtone  6298  1.0597014  6299 
fourtoneandhalf  5944  1.0595558  5946 
fivetone  5611  1.0593477  5612.3 
fivetoneandhalf  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 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.^{[37]}
J. S. Bach wrote The WellTempered Clavier to demonstrate the musical possibilities of well temperament, where in some keys the consonances are even more degraded than in equal temperament. It is reasonable to believe that when composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they 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.)
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.
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.^{[38]} 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.^{[39]} Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.^{[40]} The ultimate precision was available with 2decimal tables published by White in 1917.^{[41]}
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 each step 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:
 r^n_{}=p
 r=\sqrt[n]{p}
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:
 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.
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:
 \sqrt[12]{2}=2^{\frac{1}{12}}\approx 1.059463
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:
 P_n=P_a(\sqrt[12]{2})^{(na)}
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 (usually 440Hz). 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 440 Hz), and C4 (middle C) is the 40th key. These numbers can be used to find the frequency of C4:
 P_{40}=440(\sqrt[12]{2})^{(4049)}\approx 261.626~\text{Hz}
Historical comparison
^{[42]}
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 
Comparison to just intonation
The intervals of 12TET closely approximate some intervals in just intonation. The fifths and fourths are almost indistinguishably close to just.
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  Error 

Unison (C)  2^{0/12} = 1  1.000000  0  \begin{matrix} \frac{1}{1} \end{matrix} = 1.0000000  0.00  0 
Minor second (C♯/D♭)  2^{1/12} = \sqrt[12]{2}  1.059463  100  \begin{matrix} \frac{16}{15} \end{matrix} ≈ 1.06666…  111.73  −11.73 
Major second (D)  2^{2/12} = \sqrt[6]{2}  1.122462  200  \begin{matrix} \frac{9}{8} \end{matrix} = 1.1250000  203.91  −3.91 
Minor third (D♯/E♭)  2^{3/12} = \sqrt[4]{2}  1.189207  300  \begin{matrix} \frac{6}{5} \end{matrix} = 1.2000000  315.64  −15.64 
Major third (E)  2^{4/12} = \sqrt[3]{2}  1.259921  400  \begin{matrix} \frac{5}{4} \end{matrix} = 1.2500000  386.31  +13.69 
Perfect fourth (F)  2^{5/12} = \sqrt[12]{32}  1.334840  500  \begin{matrix} \frac{4}{3} \end{matrix} ≈ 1.33333…  498.04  +1.96 
Augmented fourth (F♯/G♭)  2^{6/12} = \sqrt{2}  1.414214  600  \begin{matrix} \frac{7}{5} \end{matrix} = 1.4000000  582.51  +17.49 
Perfect fifth (G)  2^{7/12} = \sqrt[12]{128}  1.498307  700  \begin{matrix} \frac{3}{2} \end{matrix} = 1.5000000  701.96  −1.96 
Minor sixth (G♯/A♭)  2^{8/12} = \sqrt[3]{4}  1.587401  800  \begin{matrix} \frac{8}{5} \end{matrix} = 1.6000000  813.69  −13.69 
Major sixth (A)  2^{9/12} = \sqrt[4]{8}  1.681793  900  \begin{matrix} \frac{5}{3} \end{matrix} ≈ 1.66666…  884.36  +15.64 
Minor seventh (A♯/B♭)  2^{10/12} = \sqrt[6]{32}  1.781797  1000  \begin{matrix} \frac{16}{9} \end{matrix} ≈ 1.77777…  996.09  +3.91 
Major seventh (B)  2^{11/12} = \sqrt[12]{2048}  1.887749  1100  \begin{matrix} \frac{15}{8} \end{matrix} = 1.8750000  1088.27  +11.73 
Octave (C)  2^{12/12} = {2}  2.000000  1200  \begin{matrix} \frac{2}{1} \end{matrix} = 2.0000000  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 \sqrt[7]{3/2} to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2 but a slightly widened octave with ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1 ratio, because twelve perfect fifths do not equal seven octaves.^{[43]} 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.
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 F♯ 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 onehalf of a whole tone, the corresponding equal temperament scheme is 12EDO with a fifth of seven divisions. A semitone of onethird of a whole tone corresponds to 19EDO with a fifth of eleven divisions.
12EDO 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 onehalf of a whole tone. These are additional reasons why 12EDO 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 oneseventh, and a semitone of eightninths both use 47EDO, which is the smallest number of divisions that has two different semitones. However, they have different values for the fifth, as a semitone of oneseventh uses a fifth of twentyseven divisions while a semitone of eight ninths uses a fifth of twentyeight divisions.
Other equal temperaments
5 and 7 tone temperaments in ethnomusicology
Five and seven tone equal temperament (5TET and 7TET ), 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 7TET. 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 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 (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.
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
31 tone equal temperament was advocated by Christiaan Huygens and Adriaan Fokker. 31TET has a slightly less accurate fifth than 12TET, but provides nearjust 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 12TET foundation made the quarter tone scale (or 24TET) a popular microtonal tuning.
29TET is the lowest number of equal divisions of the octave which produces a better perfect fifth than 12TET. Its major third is roughly as inaccurate as 12TET, however it is tuned 14 cents flat rather than 14 cents sharp.
41TET is the second lowest number of equal divisions that produces a better perfect fifth than 12TET. Its major third is more accurate than 12ET and 29ET, about 6 cents flat.
53TET is better at approximating the traditional just consonances than 12, 19 or 31TET, 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 53TET the very consonant thirds would be reached instead by strange enharmonic relationships. A consequence of this is that chord progressions like IviiiVI won't land you back where you started in 53TET, but rather one 53tone step flat (unless the motion by Ivi wasn't by the 5limit minor third).
Another extension of 12TET is 72TET (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. 72TET 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 14TET, 15TET, 16TET, 17TET, 19TET, 22TET, 34TET, 46TET, 48TET, 99TET, and 171TET.
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 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.^{[48]}^{[49]}
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.^{[50]} 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 nonoctave intervals
The equaltempered 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 \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. Each of them provides a very good approximation of several just intervals.^{[51]} Their step sizes:
 alpha: \sqrt[9]{3/2} (78.0 cents)
 beta: \sqrt[11]{3/2} (63.8 cents)
 gamma: \sqrt[20]{3/2} (35.1 cents)
Alpha and Beta may be heard on the title track of her 1986 album Beauty in the Beast.
See also
 Musical acoustics (the physics of music)
 Music and mathematics
 Microtuner
 Microtonal music
 Piano tuning
 List of meantone intervals
 Diatonic and chromatic
 Electronic tuner
References
Citations
 ^ 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 1532.
 ^ ^{a} ^{b} 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
 ^ ^{a} ^{b} 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 27 October 2007.
'Heptaequal temperament' in our folk music has always been a controversial issue.

^ ]About "Seven equal tuning System"七平均律"琐谈兼及旧式均孔曲笛制作与转调 [abstract of (in Chinese).
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. 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
 ^ "EQUAL TEMPERAMENT". University of Houston. Retrieved October 5, 2014.
 ^ J. Murray Barbour, Tuning and Temperament Michigan University Press, 1951 p7
 ^ [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
 ^ Lau Hanson, Abacus and Practical Mathematics p389 (in Chinese 劳汉生 《珠算与实用数学》 389页)
 ^ 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 14 June 2012.
 ^ 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.
 ^ Thomas S. Christensen, The Cambridge history of western music theory p205, Cambridge University Press
 ^ 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 p78, 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.
 ^ Date,name,ratio,cents: from equal temperament monochord tables p55p78; J. Murray Barbour Tuning and Temperament, Michigan State University Press 1951
 ^ Cordier, Serge. "Le tempérament égal à quintes justes" (in French). Association pour la Recherche et le Développement de la Musique. Retrieved 2 June 2010.
 ^ 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.
 ^ Sethares compares several equal temperament scales in a graph with axes reversed from the axes here. (fig. 4.6, p. 58)
 ^ Sethares compares several equal temperament scales in a graph with axes reversed from the axes here. (fig. 4.6, p. 58)
 ^ "665edo". xenoharmonic (microtonal wiki). Retrieved 20140618.
 ^ "convergents(log2(3), 10)".

^ 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)  ^ 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 0870132903
 Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed. ed.). London: SpringerVerlag.
 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 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
External links
 "An Introduction to Historical Tunings" by Kyle Gann, KyleGann.com.
 HuygensFokker Foundation Centre for Microtonal Music
 A.Orlandini: Music Acoustics
 (1753)A supplement to Mr. Chambers's cyclopædia"Temperament" from
 "12TET Frequency Table Maker" – Creates a frequency table (Hz.) for all 12TET pitches
 "HANDBOOK OF ACOUSTICS – online book" Brief description of Scheibler's tonometer
 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: https://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament



 Articles with hAudio microformats
 All articles with unsourced statements
 Articles with unsourced statements from June 2011
 Articles with unsourced statements from October 2013
 Articles needing additional references from June 2011
 All articles needing additional references
 Articles with unsourced statements from September 2012
 Articles with unsourced statements from January 2012
 CS1 maint: Unrecognized language
 CS1 uses 中文language script (zh)
 CS1 Frenchlanguage sources (fr)
 Equal temperaments
 Chinese discoveries