Equal temperamentFigure 1: The syntonic tuning continuum, which includes manynotable equal temperamenttunings (Milne 2007).*[1]Chromatic scale on C, full octave ascending, notated only withsharps. Play ascending & descendingAn equal temperament is a musical temperament, or asystem of tuning, in which every pair of adjacent noteshas an identical frequency ratio. As pitch is perceivedroughly as the logarithm of frequency, this means that theperceiveddistancefromevery note to its nearest neigh-bor is the same for every note in the system.In equal temperament tunings, an interval usually theoctave is divided into a series of equal steps (equal fre-quency ratios between successive notes). For classicalmusic, the most common tuning system istwelve-toneequal temperament (also known as 12 equal temper-ament), inconsistently abbreviated as 12-TET, 12TET,12tET, 12tet, 12-ET, 12ET, or 12et, whichdividesthe octave into 12 parts, all of which are equal on alogarithmic scale. It is usually tuned relative to a stan-dard pitch of 440 Hz, called A440.Other equal temperaments exist (some music has beenwritten in 19-TET and 31-TET for example, and 24-TETis used in Arabic music), but in Western countries whenpeople use the term equal temperament without quali-cation, they usually mean 12-TET.Equal temperaments may also divide some interval otherthan the octave, a pseudo-octave, into a whole number ofequal steps. An example is an equal-tempered BohlenPierce scale. To avoid ambiguity, the term equal divi-sion of the octave, or EDO is sometimes preferred. Ac-cording 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 mechan-ical tuning limitations, often use a tuning much closer tojust intonation, as it is naturally more consonant. Otherinstruments, such as some wind, keyboard, and frettedinstruments, often only approximate equal temperament,where technical limitations prevent exact tunings.Somewind instruments that can easily and spontaneously bendtheir tone, most notably trombones, use tuning similar tostring ensembles and vocal groups.Thetuningcontinuumof thesyntonictemperament,shown in Figure 1, includes a number of notable equaltemperamenttunings, including those that divide the oc-tave equally into 5, 7, 12, 17, 19, 22, 26, 31, 43, 50,and 53 parts.On an isomorphic keyboard, the ngeringof music written in any of these syntonic tunings is pre-cisely the same as it is in any other syntonic tuning, solong as the notes are spelled properlythat is, with no as-sumption of enharmonicity. This consistency of ngeringmakes it possible to smoothly vary the tuning (and hencethe pitches of all notes, systematically) all along the syn-tonic tuning continuuma polyphonic tuning bend. Theuse of dynamic timbres lets consonance be maintained(or otherwise manipulated) across such tuning bends.1 HistoryThe two gures frequently credited with the achievementof exact calculation of equal temperament are Zhu Zaiyu(also romanized as Chu-Tsaiyu. Chinese: ) in1584 and Simon Stevin in 1585. According to Fritz A.12 1 HISTORYKuttner, a critic of the theory,*[2] it is known thatChu-Tsaiyu presented a highly precise, simple and ingeniousmethod for arithmetic calculation of equal temperamentmono-chords in 1584and that Simon Stevin oereda mathematical denition of equal temperament plus asomewhat less precise computation of the correspondingnumerical values in 1585 or later.The developments oc-curred independently.*[3]Kenneth Robinson attributes the invention of equal tem-perament to Zhu Zaiyu*[4] and provides textual quota-tions as evidence.*[5] Zhu Zaiyu is quoted as saying that,in a text dating from 1584, I have founded a new sys-tem. I establish one foot as the number from which theothers are to be extracted, and using proportions I extractthem. Altogether one has to nd the exact gures for thepitch-pipers in twelve operations.*[5] Kuttner disagreesand remarks that his claimcannot be considered correctwithout major qualications.*[2] Kuttner proposes thatneither Zhu Zaiyu or Simon Stevin achieved equal tem-perament, and that neither of the two should be treated asinventors.*[6]1.1 China1.1.1 Early historyThe origin of the Chinese pentatonic scale is traditionallyascribed to the mythical Ling Lun. Allegedly his writingsdiscussed the equal division of the scale in the 27th cen-tury BC.*[7] However, evidence of the origins of writingin this period (the early Longshan) in China is limited torudimentary inscriptions on oracle bones and pottery.*[8]A complete set of bronze chime bells, among many mu-sical instruments found in the tomb of the Marquis Yi ofZeng (early Warring States, c. 5th century BCE in theChinese Bronze Age), covers 5 full 7 note octaves in thekey of C Major, including 12 note semi-tones in the mid-dle of the range.*[9]An approximation for equal temperament was describedbyHeChengtian, amathematicianof SouthernandNorthern Dynasties around 400 AD.*[10]Historically, therewasaseven-equal temperament orhepta-equal temperament practice in Chinese tradi-tion.*[11]*[12]Zhu Zaiyu (), a prince of the Ming court, spentthirty years on research based on the equal temperamentidea originally postulated by his father. He describedhis new pitch theory in his Fusion of Music and Calen-dar published in 1580. This was followedby the publication of a detailed account of the new the-ory of the equal temperament with a precise numericalspecication for 12-TET in his 5,000-page work Com-plete Compendium of Music and Pitch (Yuel quan shu) in 1584.*[13] An extended account is alsogiven by Joseph Needham.*[14] Zhu obtained his resultPrince Zhu Zaiyu constructed 12 string equal temperament tuninginstrument, front and back viewmathematically by dividing the length of string and pipesuccessively by122=1.059463094359295264561825 , and for pipediameter by242 *[15](1.059463094359295264561825)121.99999999999988 ;(1.059463094359295264561825)84127.999999999946 (still in tune after 84/12 = 7octaves)1.1.2 Zhu ZaiyuAccording to Gene Cho,Zhu Zaiyu was the rst per-son to solve the equal temperament problem mathemat-ically.*[16] Matteo Ricci, a Jesuit in China, was at Chi-nese trade fair in Canton the year Zhu published his so-lution, and very likely brought it back to the West.*[17]Matteo Ricci recorded this work in his personal jour-nal.*[16] In 1620, Zhu's work was referenced by an Eu-ropean mathematician.*[17] Murray Barbour said,Therst known appearance in print of the correct gures forequal temperament was in China, where Prince Tsaiy'sbrilliant solution remains an enigma.*[18] The 19th-century German physicist Hermann von Helmholtz wrotein On the Sensations of Tone that a Chinese prince (seebelow) introduced a scale of seven notes, and that the di-1.2 Europe 3vision of the octave into twelve semitones was discoveredin China.*[19]Zhu Zaiyu's equal temperament pitch pipesZhu Zaiyu illustrated his equal temperament theory byconstruction of a set of 36 bamboo tuning pipes rang-ing in 3 octaves, with instructions of the type of bam-boo, color of paint, and detailed specication on theirlength and inner and outer diameters. He also constructeda 12-string tuning instrument, with a set of tuning pitchpipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon, curator of the Conservatoire museumin Brussels, duplicated a set of pitch pipes according toZhu Zaiyu's specication.He said that the Chinese the-ory of tones knew more about the diameter of pitch pipesthan its Western counterpart, and that the set of pipes du-plicated according to the Zaiyu data proved the accuracyof this theory.*[20]1.2 Europe1.2.1 Early historyOne of the earliest discussions of equal temperament oc-curs in the writing of Aristoxenus in the 4th century BC.Vincenzo Galilei (father of Galileo Galilei) was one ofthe rst practical advocates of twelve-tone equal temper-ament. He composed a set of dance suites on each of the12 notes of the chromatic scale in all the transpositionkeys, and published also, in his 1584 "Fronimo", 24+1 ricercars.*[21] He used the 18:17 ratio for fretting thelute (although some adjustment was necessary for pureSimon Stevin's Van de Spiegheling der singconst c. 1605.octaves).*[22]Galilei's countryman and fellow lutenist Giacomo Gorza-nis had written music based on equal temperament by1567.*[23]*[24] Gorzanis was not the only lutenist to ex-plore all modes or keys: Francesco Spinacino wrote aRecercare de tutti li Toni(Ricercar in all the Tones) asearly as 1507.*[25] In the 17th century lutenist-composerJohn Wilson wrote a set of 30 preludes including 24 in allthe major/minor keys.*[26]*[27]Henricus Grammateus drew aclose approximation toequal temperament in 1518. The rst tuning rules inequal temperament were given by Giovani Maria Lan-franco in his Scintille de musica.*[28] Zarlino in hispolemic with Galilei initially opposed equal temperamentbut eventually conceded to it in relation to the lute in hisSopplimenti musicali in 1588.1.2.2 Simon StevinThe rst mention of equal temperament related toTwelfthroot oftwointheWest appearedinSimonStevin'smanuscript VanDe Spiegheling der singconst(ca 1605) published posthumously nearly three centurieslater in 1884.*[29] However, due to insucient accuracyof his calculation, many of the chord length numbers heobtained were o by one or two units from the correctvalues.*[30] As a result, the frequency ratios of SimonStevin's chords has no unied ratio, but one ratio per tone,which is claimed by Gene Cho as incorrect.*[31]The following were Simon Stevin's chord length fromVande Spiegheling der singconst:*[32]A generation later, French mathematician MarinMersenne presented several equal tempered chordlengths obtained by Jean Beaugrand,Ismael Bouillaudand Jean Galle.*[33]In 1630 Johann Faulhaber published a 100 cent mono-chord table, with the exception of several errors due tohis use of logarithmic tables . He did not explain how heobtained his results.*[34]4 2 GENERAL PROPERTIES1.2.3 Baroque eraFrom 1450 to about 1800, plucked instrument players(lutenistsandguitarists)generallyfavoredequal tem-perament,*[35] and the Brossard lute Manuscript com-piled in the last quarter of the 17th century contains aseries of 18 preludes attributed to Bocquet written inall keys, including the last prelude, entitled Prelude surtous les tons, which enharmonically modulates through allkeys.*[36] Angelo Michele Bartolotti published a seriesof passacaglias in all keys, with connecting enharmoni-cally modulating passages. Among the 17th-century key-board composers Girolamo Frescobaldi advocated equaltemperament. Some theorists,such as Giuseppe Tar-tini, were opposed to the adoption of equal temperament;they felt that degrading the purity of each chord degradedthe aesthetic appeal of music, although Andreas Werck-meister emphatically advocated equal temperament in his1707 treatise published posthumously.*[37]J. S. Bach wrote The Well-Tempered Clavier to demon-stratethemusical possibilities of well temperament,where in some keys the consonances are even more de-graded than in equal temperament. It is reasonable tobelieve that when composers and theoreticians of ear-lier times wrote of the moods and colorsof the keys,they each described the subtly dierent dissonances madeavailable within a particular tuning method. However,it is dicult to determine with any exactness the actualtunings used in dierent places at dierent times by anycomposer. (Correspondingly, there is a great deal of va-riety in the particular opinions of composers about themoods and colors of particular keys.)Twelve tone equal temperament took hold for a varietyof reasons. It conveniently t the existing keyboard de-sign, and permitted total harmonic freedom at the ex-pense of just a little impurity in every interval. Thisallowed greater expression through enharmonic modula-tion, which became extremely important in the 18th cen-tury in music of such composers as Francesco Gemini-ani, Wilhelm Friedemann Bach, Carl Philipp EmmanuelBach and Johann Gottfried Mthel.The progress of equal temperament from the mid-18thcentury on is described with detail in quite a few modernscholarly publications: it was already the temperamentof choice during the Classical era (second half of the18th century), and it became standard during the EarlyRomantic era (rst decade of the 19th century), exceptfor organs that switched to it more gradually, completingonly in the second decade of the 19th century. (In Eng-land, some cathedral organists and choirmasters held outagainst 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 17th-century Sabbatini method of splitting the octave rst intothree tempered major thirds.*[38] This was also pro-posed by several writers during the Classical era. Tuningwithout beat rates but employing several checks, achiev-ing virtually modern accuracy, was already done in therst decades of the 19th century.*[39] Using beat rates,rst proposed in 1749, became common after their dif-fusion by Helmholtz and Ellis in the second half of the19th century.*[40] The ultimate precision was availablewith 2-decimal tables published by White in 1917.*[41]It is intheenvironment of equal temperament thatthe new styles of symmetrical tonality and polytonality,atonal music such as that written with the twelve tonetechnique or serialism, and jazz (at least its piano com-ponent) developed and ourished.2 General propertiesIn an equal temperament, the distance between each stepof the scale is the same interval. Because the perceivedidentity of an interval depends on its ratio, this scale ineven steps is a geometric sequence of multiplications.(An arithmetic sequence of intervals would not soundevenly spaced, and would not permit transposition to dif-ferent keys.) Specically, thesmallestinterval inanequal-tempered scale is the ratio:rn= pr =npwhere the ratio r divides the ratio p (typically the octave,which is 2/1) into n equal parts. (See Twelve-tone equaltemperament below.)Scales are often measured in cents, which divide the oc-tave into 1200 equal intervals (each called a cent).Thislogarithmic scale makes comparison of dierent tuningsystems easier than comparing ratios, and has consider-able use in Ethnomusicology. The basic step in cents forany equal temperament can be found by taking the widthofp above in cents (usually the octave, which is 1200cents wide), called below w, and dividing it into n parts:c =wnIn musical analysis, material belonging to an equal tem-perament is often given an integer notation, meaning asingle integer is used to represent each pitch. This simpli-es and generalizes discussion of pitch material within thetemperament in the same way that taking the logarithmof a multiplication reduces it to addition. Furthermore,by applying the modular arithmetic where the modulus isthe number of divisions of the octave (usually 12), theseintegers can be reduced to pitch classes, which removesthe distinction (or acknowledges the similarity) betweenpitches of the same name, e.g. 'C' is 0 regardless of oc-tave register. The MIDI encoding standard uses integernote designations.53 Twelve-tone equal temperamentIn twelve-tone equal temperament, which divides the oc-tave into 12 equal parts, the width of a semitone, i.e.the frequency ratio of the interval between two adjacentnotes, is the twelfth root of two:122 = 2112 1.059463This interval is divided into 100 cents.3.1 Calculating absolute frequenciesSee also: Piano key frequenciesTo nd the frequency, Pn, of a note in 12-TET, the fol-lowing denition may be used:Pn= Pa(122)(na)In this formula Pn refers to the pitch, or frequency (usu-ally in hertz), you are trying to nd. P refers to the fre-quency of a reference pitch (usually 440Hz). n and a referto numbers assigned to the desired pitch and the refer-ence pitch, respectively. These two numbers are from alist of consecutive integers assigned to consecutive semi-tones. For example, A4 (the reference pitch) is the 49thkey from the left end of a piano (tuned to 440 Hz), andC4 (middle C) is the 40th key. These numbers can beused to nd the frequency of C4:P40= 440(122)(4049) 261.626 Hz3.2 Historical comparison*[42]3.3 Comparison to just intonationThe intervals of 12-TET closely approximate some inter-vals in just intonation.The fths and fourths are almostindistinguishably close to just.In the following table the sizes of various just intervalsare compared against their equal-tempered counterparts,given as a ratio as well as cents.3.4 Seven-tone equal division of the fthViolins, violas and cellos are tuned in perfect fths (G D A E, for violins, and C G D A, for violasand cellos), which suggests that their semi-tone ratio isslightly higher than in the conventional twelve-tone equaltemperament. Because a perfect fth is in 3:2 relationwith its base tone, and this interval is covered in 7 steps,each tone is in the ratio of73/2 to the next (100.28cents), which provides for a perfect fth with ratio of 3:2but a slightly widened octave with ratio of 517:258 or 2.00388:1 rather than the usual 2:1 ratio, because twelveperfect fths do not equal seven octaves.*[43] During ac-tual play, however, the violinist chooses pitches by ear,and only the four unstopped pitches of the strings areguaranteed to exhibit this 3:2 ratio.4 Rational semitoneFor any semitone that is a proper fraction of a whole tone,exactly one equal division of the octave lets the circle offths generate all the notes of the equal division whilepreserving the order of the notes. (That is, C is lowerthan D, D is lower than E, etc., and F is indeed sharperthan F.) The number of divisions needed for the octave isseven times the number of divisions of a whole tone mi-nus twice the number of divisions of the semitone. Thecorresponding fth spans a number of divisions equal tofour whole tones minus one semitone. Hence, for a semi-tone of one-half of a whole tone, the corresponding equaltemperament scheme is 12-EDO with a fth of seven di-visions. A semitone of one-third of a whole tone corre-sponds to 19-EDO with a fth of eleven divisions.12-EDOis the equal temperament with the smallest num-ber of divisions that allows for a rational semitone to pre-serve the desired properties concerning note order and thecircle of fths. It also has the desirable property of mak-ing the semitone exactly one-half of a whole tone. Theseare additional reasons why 12-EDO became the predom-inant form of equal temperament.While each rational semitone corresponds to only oneequal temperament, the reverse is not the case. For ex-ample, both a semitone of one-seventh, and a semitone ofeight-ninths both use 47-EDO, which is the smallest num-ber of divisions that has two dierent semitones. How-ever, they have dierent values for the fth, as a semitoneof one-seventh uses a fth of twenty-seven divisions whilea semitone of eight ninths uses a fth of twenty-eight di-visions.5 Other equal temperamentsSee also: Sonido 136 5 OTHER EQUAL TEMPERAMENTS5.1 5and7tonetemperamentsinethno-musicologyFive and seven tone equal temperament (5-TET Playand 7-TET Play ), with 240 Play and 171 Playcent steps respectively, are fairly common. A Thai xy-lophone measured by Morton (1974) varied only plusor minus 5 cents,from 7-TET. A Ugandan Chopi xylo-phone measured by Haddon (1952) was also tuned to thissystem. According to Morton,Thai instruments of xedpitch are tuned to an equidistant system of seven pitchesper 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 ve ofthe seven are used in principal pitches in any mode, thusestablishing a pattern of nonequidistant intervals for themode.*[44] Play Indonesian gamelans are tuned to5-TET according to Kunst (1949), but according to Hood(1966) and McPhee (1966) their tuning varies widely,and according to Tenzer (2000) they contain stretchedoctaves. It is now well-accepted that of the two pri-mary tuning systems in gamelan music, slendro and pelog,only slendro somewhat resembles ve-tone equal temper-ament while pelog is highly unequal; however, Surjodin-ingrat et al. (1972) has analyzed pelog as a seven-notesubset of nine-tone equal temperament (133 cent stepsPlay ). A South American Indian scale from a preinstru-mental culture measured by Boiles (1969) featured 175cent seven tone equal temperament, which stretches theoctave slightly as with instrumental gamelan music.5-TET and 7-TET mark the endpoints of the syntonictemperament's valid tuning range, as shown in Figure 1.In 5-TET the tempered perfect fth is 720 centswide (at the top of the tuning continuum), and marksthe endpoint on the tuning continuum at which thewidth of the minor second shrinks to a width of 0cents.In 7-TET the tempered perfect fth is 686 centswide (at the bottom of the tuning continuum), andmarks the endpointon the tuning continuum, atwhich the minor second expands to be as wide asthe major second (at 171 cents each).5.2 Various Western equal temperamentsEasley Blackwood's notation system for 16 equal temperament:intervalsarenotatedsimilarlytothosetheyapproximateandthere are fewer enharmonic equivalents.*[45] PlayLIMIT 3LIMIT 7LIMIT 13LIMIT 5LIMIT 115-TET7-TET12-TET19-TET22-TET31-TETP1 M3 P5 P81:15:43:27:42:1INTERVALRATIO24-TETComparison of selected equal temperaments and just interval ratios, showing various prime limitsM29:8m37:66:5P44:3TT7:5m611:78:5m7 M65:313:8M741-TET72-TET53-TETm216:159:711:88:716:915:810:716:1114:9 12:714:1116:1334-TET5-TET7-TET12-TET19-TET22-TET31-TET24-TET41-TET72-TET53-TET34-TETCENTS 100 200 300 400 500 600 700 800 900 1000 1100 1200 0A comparison of some equal temperament scales.*[46]The graph spans one octave horizontally, and eachshaded rectangle is the width of one step in a scale. Thejust interval ratios are separated in rows by their primelimits.31 tone equal temperament was advocated by ChristiaanComparison of equal temperaments from 9 to 25 (after Sethares(2005), p.58).Huygens and Adriaan Fokker. 31-TET has a slightly lessaccurate fth than 12-TET, but provides near-just majorthirds, and provides decent matches for harmonics up toat least 13, of which the seventh harmonic is particularlyaccurate.In the 20th century, standardized Western pitch and no-tation practices having been placed on a 12-TET founda-tion made the quarter tone scale (or 24-TET) a popularmicrotonal tuning.29-TET is the lowest number of equal divisions of the oc-tave which produces a better perfect fth than 12-TET.5.3 Equal temperaments of non-octave intervals 7Its major third is roughly as inaccurate as 12-TET, how-ever it is tuned 14 cents at rather than 14 cents sharp.41-TET is the second lowest number of equal divisionsthat produces a better perfect fth than 12-TET. Its majorthird is more accurate than 12-ET and 29-ET, about 6cents at.53-TET is better at approximating the traditional justconsonances than 12, 19 or 31-TET, but has had onlyoccasional use. Its extremely good perfect fths makeit interchangeable with an extended Pythagorean tuning,but it also accommodates schismatic temperament, andis sometimes used in Turkish music theory.It does not,however, t the requirements of meantone temperaments,which put good thirds within easy reach via the cycle offths. In 53-TET the very consonant thirds would bereached instead by strange enharmonic relationships. Aconsequence 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 at (unless the motion by I-viwasn't by the 5-limit minor third).Another extension of 12-TET is 72-TET (dividing thesemitone into 6 equal parts), which though not ameantone tuning, approximates well most just intonationintervals, even less traditional ones such as 7/4, 9/7, 11/5,11/6 and 11/7. 72-TET has been taught, written and per-formed in practice by Joe Maneri and his students (whoseatonal inclinations interestingly typically avoid any refer-ence to just intonation whatsoever).Other equal divisions of the octave that have found occa-sional 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 denominatorsof rst convergents of log2(3) , so 2, 5, 12, 41, 53, 306,665 and 15601 twelfths (and fths), being in correspon-dent equal temperaments equal to en integer number ofoctaves, are better approximation of 2, 5, 12, 41, 53, 306,665 and 15601 just twelfths/fths than for any equal tem-peraments with less tones.*[47]*[48]1, 2, 3, 5, 7, 12, 29, 41, 53, 200... (sequence A060528in OEIS) is the sequence of divisions of octave that pro-vide better and better approximations of the perfect fth.Related sequences contain divisions approximating otherjust intervals.*[49] It is noteworthy that many elementsof this sequences are sums of previous elements. Thisapplication:http://equal.meteor.com/ calculates the fre-quencies, amounts of cents and Pitch Bend values for anysystems of Equal Division of the Octave. Note, that both'rounded' and 'oored' notations are equivalent, producesthe same MIDI value.5.3 Equal temperaments of non-octave in-tervalsThe equal-tempered version of the BohlenPierce scaleconsists of the ratio 3:1,1902 cents,conventionally aperfect fth and an octave, called in this theory a tritave( play ), and split into a thirteen equal parts. This pro-vides a very close match to justly tuned ratios consistingonly of odd numbers. Each step is 146.3 cents ( play ),or133 .Wendy Carlos created three unusual equal temperamentsafter a thorough study of the properties of possible tem-peraments having a step size between 30 and 120 cents.These were called alpha, beta, and gamma. They can beconsidered as equal divisions of the perfect fth. Each ofthem provides a very good approximation of several justintervals.*[50] Their step sizes:alpha:93/2 (78.0 cents)beta:113/2 (63.8 cents)gamma:203/2 (35.1 cents)Alpha and Beta may be heard on the title track of her1986 album Beauty in the Beast.6 See alsoMusical acoustics (the physics of music)Music and mathematicsMicrotunerMicrotonal musicPiano tuningList of meantone intervalsDiatonic and chromaticElectronic tuner7 References7.1 Citations[1] Milne, A., Sethares, W.A. and Plamondon, J.,Isomor-phic Controllers and Dynamic Tuning: Invariant Finger-ings Across a Tuning Continuum, Computer Music Jour-nal, Winter 2007, Vol. 31, No. 4, Pages 15-32.[2] Fritz A. Kuttner. p. 163.[3] Fritz A. Kuttner. Prince Chu Tsai-Y's Life and Work:A Re-Evaluation of His Contribution to Equal Tempera-ment Theory, p.200, Ethnomusicology, Vol. 19, No. 2(May, 1975), pp. 163206.8 7 REFERENCES[4] Kenneth Robinson: A critical study of Chu Tsai-y's con-tribution to the theory of equal temperament in Chinese mu-sic. (Sinologica Coloniensia, Bd. 9.) x, 136 pp. Wies-baden: Franz Steiner Verlag GmbH, 1980. DM 36. p.viiChu-Tsaiyu the rst formulator of the mathematics ofequal temperamentanywhere in the world[5] Robinson, KennethG., andJosephNeedham. 1962.Physics and Physical Technology. In Science and Civil-isation in China, vol. 4: Physics and Physical Technol-ogy, Part 1: Physics, edited by Joseph Needham.Cambridge: University Press. p. 221.[6] Fritz A. Kuttner. p. 200.[7] Owen Henry Jorgensen, Tuning (East Lansing: MichiganState University Press, 1991)[8] The Formation of Chinese Civilization: An Archaeolog-ical Perspective, Zhang Zhongpei, Shao Wangping andothers, 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, LuLiancheng, pp 140 [10] J. MurrayBarbourTuningandTemperament p55-56,Michigan State University Press 1951[11] " " [Findings of newliteratures concerning the hepta equal temperament] (inChinese). Archived fromthe original on 27 October 2007.'Hepta-equal temperament' in our folk music has alwaysbeen a controversial issue.[12] " --[abstract of About Seven- equal- tuning System] (inChinese). From the ute for two thousand years of theproduction process, and the Japanese shakuhachi remain-ing in the production of Sui and Tang Dynasties and theactual temperament, identication of people using the so-called 'Seven Laws' at least two thousand years of history;and decided that this law system associated with the utelaw.[13] Quantifying Ritual: Political Cosmology, Courtly Mu-sic, and Precision Mathematics in Seventeenth-CenturyChinaRogerHart DepartmentsofHistoryandAsianStudies, University of Texas, Austin. Uts.cc.utexas.edu.Retrieved 2012-03-20.[14] Science and Civilisation in China, Vol IV:1 (Physics),JosephNeedham, CambridgeUniversityPress, 19622004, pp 220 [15] The Shorter Science & Civilisation in China, An abridge-ment by Colin Ronan of Joseph Needham's original text,p385[16] Gene J. Cho The Signicance of the Discoveryof the Musical Equal Temperament In the Cul-tural History,http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm[17] EQUAL TEMPERAMENT. University of Houston.Retrieved October 5, 2014.[18] J. Murray Barbour, Tuning and Temperament MichiganUniversity Press, 1951 p7[19] The division of Octave into twelve Semitones, and thetransposition of scales have also been discovered by thisintelligent and skillful nationHermann von HelmholtzOn the Sensations of Tone as a Physiological basis for thetheory of music, p 258, 3rd edition, Longmans, Green andCo, London, 1895[20] Lau Hanson, Abacus and Practical Mathematics p389 (inChinese 389 )[21] Galilei, V. (1584). Il Fronimo... Dialogo sopra l'arte delbene intavolare. G. Scotto: Venice, . 8089.[22] J. Murray BarbourTuningandTemperament p8 1951Michigan State University Press[23] Resound Corruption of Music. Philresound.co.uk.Retrieved 2012-03-20.[24] Giacomo Gorzanis, c.1525 - c.1575 Intabolatura di li-uto. Geneva, 1982[25] Spinacino 1507a: Thematic Index. Appalachian StateUniversity. Archived from the original on 2011-07-25.Retrieved 14 June 2012.[26] John Wilson, 26 Preludes, Diapason press (DP49,Utrecht)[27] inEnglish Song, 16001675, Facsimile of Twenty-six Manuscripts and an Edition of the Textsvol. 7,Manuscripts at Oxford, Part II. introduction by Elise Bick-foer Jorgens, Garland Publishing Inc., NewYork and Lon-don, 1987[28]Scintille de musica, (Brescia, 1533), p. 132[29] Van de Spiegheling der singconst, ed by Rudolf Rasch,The Diapason Press. Diapason.xentonic.org. 2009-06-30. Retrieved 2012-03-20.[30] Thomas S. Christensen, The Cambridge history of west-ern music theory p205, Cambridge University Press[31] Gene Cho, The discovery of musical equal temperamentin China and Europe Page 223, Mellen Press[32] Gene Cho, The discovery of musical equal temperamentin China and Europe Page 222, Mellen Press[33] Thomas S. Christensen, The Cambridge history of west-ern music theory p207, Cambridge University Press[34] Thomas S. Christensen, The Cambridge history of west-ern music theory p78, Cambridge University Press[35]Lutes, Viols, TemperamentsMark 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 256.9[39] Moody, Richard. Early Equal Temperament, An AuralPerspective: Claude Montal 1836in Piano TechniciansJournal. Kansas City, USA, Feb. 2003.[40] Helmholtz, Hermann L.F. Lehre von den Tonempndun-genHeidelberg 1862. English edition: On the Sensa-tions of Tone, Longmans, London, 1885, p.548.[41] White, William Braid. Piano Tuning and Allied Arts.1917, 5th enlarged edition, Tuners Supply Co., Boston1946, p.68.[42] Date,name,ratio,cents: from equal temperament mono-chord tables p55-p78; J.Murray Barbour TuningandTemperament, Michigan State University Press 1951[43] Cordier, Serge. Le temprament gal quintes justes(inFrench). Associationpour la Recherche et leDveloppement de la Musique. Retrieved 2 June 2010.[44] Morton, David (1980).The Music of Thailand, Musicsof Many Cultures, p.70. May, Elizabeth, ed. ISBN 0-520-04778-8.[45] Myles Leigh Skinner (2007). Toward a Quarter-tone Syn-tax: Analyses of Selected Works by Blackwood, Haba, Ives,and Wyschnegradsky, p.55. ISBN 9780542998478.[46] Sethares compares several equal temperament scales in agraph with axes reversed from the axes here. (g. 4.6, p.58)[47] 665edo. xenoharmonic (microtonal wiki). Retrieved2014-06-18.[48] convergents(log2(3), 10)". WolframAlpha. Retrieved2014-06-18.[49] 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, 16/11and 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, 10/9and 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, 10/9and 9/5, 16/15 and 15/8, 45/32 and 64/45, 27/20 and40/27, 32/27 and 27/16, 81/64 and 128/81, 256/243 and243/128 (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)[50] Three Asymmetric Divisions of the Octave by Wendy Car-los7.2 BibliographyCho, Gene Jinsiong. (2003). The Discovery of Mu-sical Equal Temperament inChinaandEuropeinthe Sixteenth Century. Lewiston, NY: Edwin MellenPress.Dun, Ross W. HowEqual Temperament Ru-ined Harmony (and Why You Should Care).W.W.Norton & Company, 2007.Jorgensen, Owen. Tuning. Michigan State Univer-sity Press, 1991. ISBN 0-87013-290-3Sethares, William A. (2005). Tuning, Timbre, Spec-trum, Scale (2nd ed. ed.). London: Springer-Verlag.ISBN 1-85233-797-4.Surjodiningrat, W., Sudarjana, P.J., and Su-santo, A. (1972) Tone measurements of out-standing Javanese gamelans in Jogjakarta andSurakarta, GadjahMada UniversityPress, Jog-jakarta 1972. Cited on http://web.telia.com/~{}u57011259/pelog_main.htm, accessed May 19,2006.Stewart, P. J. (2006)FromGalaxy to Galaxy: Mu-sic of the SpheresKhramov, Mykhaylo. Approximation of 5-limitjust intonation. Computer MIDI Modeling inNegative Systems of EqualDivisions of the Oc-tave, Proceedings of the International ConferenceSIGMAP-2008, 2629 July 2008, Porto, pp. 181184, ISBN 978-989-8111-60-98 External linksExplaining the Equal Temperament by YuvalNov, YuvalNov.org.An Introduction to Historical Tuningsby KyleGann, KyleGann.com.Huygens-Fokker Foundation Centre for MicrotonalMusicA.Orlandini: Music Acoustics Temperamentfrom A supplement to Mr. Cham-bers's cyclopdia (1753) 12TET Frequency Table Maker Creates a fre-quency table (Hz.) for all 12TET pitchesHANDBOOK OF ACOUSTICS online bookBrief description of Scheibler's tonometerBarbieri, Patrizio. Enharmonic instruments and mu-sic, 14701900. (2008) Latina, Il Levante LibreriaEditriceFractal Microtonal Music, Jim Kukula.All existing 18th century quotes on J.S. Bachand temperament: https://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament10 8 EXTERNAL LINKSDominic Eckersley: Rosetta Revisited: Bach's VeryOrdinary Temperament. https://www.academia.edu/3368760/Rosetta_Revisited_Bachs_Very_Ordinary_Temperament.119 Text and image sources, contributors, and licenses9.1 Text Equal temperament Source: http://en.wikipedia.org/wiki/Equal%20temperament?oldid=657405498 Contributors: Damian Yerrick, To-bias Hoevekamp, Derek Ross, Zundark, Timo Honkasalo, Tarquin, Jeronimo, Karl E. V. Palmen, Rgamble, Merphant, Nonenmac, Heron,Karl Palmen, Camembert, Patrick, Michael Hardy, Mic, Darkwind, Julesd, AugPi, Cema, Timwi, Stismail, Hyacinth, Saltine, Thue, Anon-Moos, Mordomo, PuzzletChung, Robbot, Altenmann, HaeB, SaltyPig, Gwalla, Gene Ward Smith, Beefman, Art Carlson, Ds13, MarkusKuhn, Jorend, Frencheigh, Eric22, Sigfpe, Abu badali, Antandrus, Timsabin, Xezbeth, Quinobi, Nickj, Alberto Orlandini, Army1987,Jeodesic, Jumbuck, Tablizer, Walter Grlitz, Anders Kaseorg, Arthena, Neonumbers, Keenan Pepper, ABCD, Caesura, Emvee~enwiki,Bookandcoee, Spettro9, StradivariusTV, Steven Luo, SDC, Waldir, Noetica, Gisling, Markkawika, Missmarple, Mr.Unknown, GreyCat,Bmicomp, Joonasl, Glenn L, Chobot, Bgwhite, Ksyrie, Kasajian, Yahya Abdal-Aziz, JDoorjam, Crasshopper, Zwobot, Bota47, Elkman,Wknight94, Light current, Ninly, MaratL, CharlieHuang, KJBracey, GrinBot~enwiki, Mhardcastle, SmackBot, Monz, Cazort, Chris thespeller, Stevage, Mireut, TheLeopard, Patriarch, Eudaemonic3, RAlafriz, Nomenclator, Ccero, Morqueozwald, A.R., Lamidave, Justplain Bill, Bdiscoe, Andeggs, Lambiam, Rigadoun, DaveHorne, Carneyfex~enwiki, Rainwarrior, Dicklyon, Meco, JoeBot, CmdrObot,Posaune1, Fixmanius, NTsilakis, Signo, Cydebot, Galassi, Xxanthippe, Rracecarr, Kotiwalo, Torc2, Friera, Thijs!bot, Kubanczyk, Head-bomb, Uruiamme, WinBot, Matias.Reccius, Dougher, Sabbetius, Brto 'd Sra, JAnDbot, MER-C, Nitrode, Prof.rick, Mtd2006, BlackStripe, Jtir, Gimcrackery, Joelthesecond, CommonsDelinker, Nono64, S.dedalus, Bongomatic, Bill.Flavell, Acalamari, Hellvig, Gypsy-doctor, Mbbradford, Plasticup, Cadwaladr, Frank Zamjatin, Coppelia~enwiki, Vanilor, Pjstewart, Houtlijm~enwiki, Commator~enwiki,Derekjc, SieBot, Droog Andrey, Toddst1, Flyer22, Paolo.dL, BartekChom, Akarkera, Donald Harkness, Binksternet, The Thing ThatShould Not Be, P0mbal, Niceguyedc, Jigsaw dog, Eboyjr, Watchduck, Keithbowden, SchreiberBike, Environnement2100, Sarindam7,Mark Pfannschmidt, JimPlamondon, Bazj, Redheylin, LinkFA-Bot, Barak Sh, Lightbot, CountryBot, Legobot, Yobot, AnomieBOT,Verec, Carolina wren, Another Stickler, Wrelwser43, Quebec99, 94jepd, Braybaroque, Joaquin008, SD5, Mu Mind, Rigaudon, FinnFroding, Diveroli, KosmischeSynth, Kiefer.Wolfowitz, PoopyFaceTomatoNose, MinorMaide, Double sharp, Obankston, Ornithikos, , ClueBot NG, Dkm125, Kick030, Chester Markel, Adrien1018, Rurik the Varangian, Helpful Pixie Bot, Tdimhcs, JeanneAy-monier, BG19bot, Andrew.john.phillipson, Kagefumimaru, Hmainsbot1, 80 Ursae Majoris, Dominiceckersley, Monkbot, Novanijntje,Edwardcao1984, Combinare and Anonymous: 1739.2 Images File:16-tet_scale_on_C.png Source: http://upload.wikimedia.org/wikipedia/commons/f/fe/16-tet_scale_on_C.png License: Public do-main Contributors: Own work Original artist: Hyacinth File:Chromatische_toonladder.pngSource: http://upload.wikimedia.org/wikipedia/commons/2/2f/Chromatische_toonladder.pngLi-cense: CCBY-SA3.0 Contributors: w:Image:Chromatic scale ascending .png by Hyacinth (talk) 05:04, 30 July 2008 Original artist: Quatrostein File:Comparison_of_equal_temperaments.png Source: http://upload.wikimedia.org/wikipedia/commons/4/4c/Comparison_of_equal_temperaments.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Hyacinth File:De_Spiegheling_der_signconst.jpg Source: http://upload.wikimedia.org/wikipedia/commons/8/8f/De_Spiegheling_der_signconst.jpg License: Public domain Contributors: mybook Original artist: Simon Steven File:Equal_Temper_w_limits.svg Source: http://upload.wikimedia.org/wikipedia/commons/a/a0/Equal_Temper_w_limits.svg License:CC BY-SA 3.0 Contributors: I created this work, with input from other editors, based on a previous GFDL image. 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