equal temperament - wikipedia, the free encyclopedia.pdf

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3/7/2016 Equal temperament - Wikipedia, the free encyclopedia

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A comparison of some "EDO" equal temperament scales.[1] The

graph spans one octave horizontally (please open the image to view

the f ull width), and each shaded rectangle is the width of one step in

a scale. The just interval ratios are separated in rows by their prime

limits.

12-tone equal temperament chromatic scale on C,

one full octave ascending, notated only with sharps.

Play ascending & descending

Equal temperamentFrom Wikipedia, the free encyclopedia

An equal temperament is a musicaltemperament, or a system of tuning, inwhich every pair of adjacent pitches isseparated by the same interval. In other

words, the pitches of an equaltemperament can be produced by repeatinga generating interval. Equal intervals alsomeans equal ratios between thefrequencies of any adjacent pair, and, sincepitch is perceived roughly as the logarithmof frequency, equal perceived "distance"from every note to its nearest neighbor.

In equal temperament tunings, the

generating interval is often found bydividing some larger desired interval, oftenthe octave (ratio 2:1), into a number of smaller equal steps (equal frequency ratiosbetween successive notes). For classicalmusic and Western music in general, themost common tuning system for the pastfew hundred years has been and remainstwelve-tone equal temperament (also known as 12equal temperament, 12-TET, or 12-ET), which divides

the octave into 12 parts, all of which are equal on alogarithmic scale. That resulting smallest interval, 1 ⁄ 12the width of an octave, is called a semitone or half step.In modern times, 12TET is usually tuned relative to astandard pitch of 440 Hz, called A440, meaning onepitch is tuned to A440, and all other pitches are some multiple of semitones away from that in eitherdirection, although the standard pitch has not always been 440 and has fluctuated and generally risen over

the past few hundred years.[2]

Other equal temperaments exist. They divide the octave differently. For example, some music has been

written in 19-TET and 31-TET. Arabic music uses 24-TET. In Western countries, when people use the termequal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equaltemperaments which divide the octave and ones which divide some other interval (or that use an arbitrarygenerator without first dividing a larger interval), the term equal division of the octave, or EDO is preferredfor the former. According to this naming system, 12-TET is called 12-EDO, 31-TET is called 31-EDO, andso on.

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

equal parts.

String ensembles and vocal groups, who have no mechanical tuning limitations, often use a tuning muchcloser 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

https://en.wikipedia.org/wiki/File:Chromatische_toonladder.pnghttps://en.wikipedia.org/wiki/File:Chromatische_toonladder.pnghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/Frethttps://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/19_equal_temperamenthttps://en.wikipedia.org/wiki/31_equal_temperamenthttps://en.wikipedia.org/wiki/A440_(pitch_standard)https://en.wikipedia.org/wiki/Classical_musichttps://en.wikipedia.org/wiki/Classical_musichttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/Pitch_(music)https://en.wikipedia.org/wiki/Logarithmhttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/Frequencyhttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/Musical_temperamenthttps://en.wikipedia.org/wiki/Musical_tuninghttps://en.wikipedia.org/wiki/Musical_temperamenthttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/Musical_temperamenthttps://en.wikipedia.org/wiki/File:Equal_Temper_w_limits.svghttps://en.wikipedia.org/wiki/Frethttps://en.wikipedia.org/wiki/Keyboard_instrumentshttps://en.wikipedia.org/wiki/Wind_instrumentshttps://en.wikipedia.org/wiki/Consonance_and_dissonancehttps://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/Pseudo-octavehttps://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scalehttps://en.wikipedia.org/wiki/Western_worldhttps://en.wikipedia.org/wiki/Arab_tone_systemhttps://en.wikipedia.org/wiki/31_equal_temperamenthttps://en.wikipedia.org/wiki/19_equal_temperamenthttps://en.wikipedia.org/wiki/A440_(pitch_standard)https://en.wikipedia.org/wiki/Logarithmic_scalehttps://en.wikipedia.org/wiki/Classical_musichttps://en.wikipedia.org/wiki/Octavehttps://en.wikipedia.org/wiki/Logarithmhttps://en.wikipedia.org/wiki/Pitch_(music)https://en.wikipedia.org/wiki/Frequencyhttps://en.wikipedia.org/wiki/Interval_(music)https://en.wikipedia.org/wiki/Musical_tuninghttps://en.wikipedia.org/wiki/Musical_temperamenthttps://upload.wikimedia.org/wikipedia/commons/f/f0/ChromaticScaleUpDown.ogghttps://en.wikipedia.org/wiki/File:ChromaticScaleUpDown.ogghttps://en.wikipedia.org/wiki/File:Chromatische_toonladder.pnghttps://en.wikipedia.org/wiki/Prime_limithttps://en.wikipedia.org/wiki/Just_intervalhttps://en.wikipedia.org/wiki/Octave


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exact tunings. Some wind instruments that can easily and spontaneously bend their tone, most notablytrombones, use tuning similar to string ensembles and vocal groups.

Contents

1 History

1.1 China

1.1.1 Early history

1.1.2 Zhu Zaiyu

1.2 Europe

1.2.1 Early history

1.2.2 Simon Stevin

1.2.3 Baroque era

2 General properties

3 Twelve-tone equal temperament

3.1 Calculating absolute frequencies

3.2 Historical comparison

3.3 Comparison to just intonation

3.4 Seven-tone equal division of the fifth

4 Other equal temperaments

4.1 5 and 7 tone temperaments in ethnomusicology

4.2 Various Western equal temperaments4.3 Equal temperaments of non-octave intervals

4.4 Proportions between semitone and whole tone

5 Related tuning systems

5.1 Syntonic temperament

6 See also

7 References

7.1 Citations

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7.2 Sources

8 External links

History

The two figures frequently credited with the achievement of exact calculation of equal temperament are ZhuZaiyu (also romanized as Chu-Tsaiyu. Chinese:朱載堉) in 1584 and Simon Stevin in 1585. According to

Fritz A. Kuttner, a critic of the theory,[3] it is known that "Chu-Tsaiyu presented a highly precise, simple andingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that "SimonStevin 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.[4]

Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu[5] and provides textual

quotations as evidence.[6] 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 usingproportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve

operations."[6] Kuttner disagrees and remarks that his claim "cannot be considered correct without major

qualifications."[3] Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and

that neither of the two should be treated as inventors.[7]

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.[8] However, evidence of the originsof writing in this period (the early Longshan) in China is limited to rudimentary inscriptions on oracle bones

and pottery.[9]

A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yiof Zeng (early Warring States, c. 5th century BCE in the Chinese Bronze Age), covers 5 full 7 note octaves

in the key of C Major, including 12 note semi-tones in the middle of the range.[10]

An approximation for equal temperament was described by He Chengtian, a mathematician of Southern andNorthern Dynasties around 400 AD.[11]

Historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese

tradition.[12][13]

Zhu Zaiyu (朱載堉), a prince of the Ming court, spent thirty years on research based on the equaltemperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar乐律融通 published in 1580. This was followed by the publication of a detailed

account of the new theory of the equal temperament with a precise numerical specification for 12-TET in his5,000-page work Complete Compendium of Music and Pitch (Yuelü quan shu乐律全书) in 1584.[14] An

extended account is also given by Joseph Needham.[15] Zhu obtained his result mathematically by dividing

https://en.wikipedia.org/wiki/Ming_Dynastyhttps://en.wikipedia.org/wiki/Zhu_Zaiyuhttps://en.wikipedia.org/wiki/Music_of_Chinahttps://en.wikipedia.org/wiki/Southern_and_Northern_Dynastieshttps://en.wikipedia.org/wiki/Ling_Lunhttps://en.wikipedia.org/wiki/Simon_Stevinhttps://en.wikipedia.org/wiki/Zhu_Zaiyu


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Prince Zhu Zaiyu constructed 12 string equal

temperament tuning instrument, front and back

view

Zhu Zaiyu's equal temperament pitch

pipes

the length of string and pipe successively by12 √2 ≈

1.059463, and for pipe length by24 √2 ,[16] such that after

twelve divisions (an octave) the length was divided by afactor of 2:

(12 √2)12 = 2.

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

(12 √2)84 = 27 = 128.

Zhu Zaiyu

According to Gene Cho, Zhu Zaiyu was the first personto solve the equal temperament problem

mathematically.[17]

Matteo Ricci, a Jesuit in China, wasat Chinese trade fair in Canton the year Zhu publishedhis solution, and very likely brought it back to the

West.[18] Matteo Ricci recorded this work in his personal

ournal.[17] In 1620, Zhu's work was referenced by a

European mathematician.[18] Murray Barbour said, "Thefirst known appearance in print of the correct figures forequal temperament was in China, where Prince Tsaiyü's

brilliant solution remains an enigma."[19] The 19th-century German physicist Hermann von Helmholtz wrote

in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that thedivision of the octave into twelve semitones was discovered in China.[20]

Zhu Zaiyu illustrated his equal temperament theory by constructionof a set of 36 bamboo tuning pipes ranging in 3 octaves, withinstructions of the type of bamboo, color of paint, and detailedspecification on their length and inner and outer diameters. He alsoconstructed a 12-string tuning instrument, with a set of tuning pitchpipes hidden inside its bottom cavity. In 1890, Victor-CharlesMahillon, 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 lengthof pitch pipes than its Western counterpart, and that the set of pipesduplicated according to the Zaiyu data proved the accuracy of this

theory.[21]

Europe

Early history

One of the earliest discussions of equal temperament occurs in thewriting of Aristoxenus in the 4th century BC.

https://en.wikipedia.org/wiki/Aristoxenushttps://en.wikipedia.org/wiki/Victor-Charles_Mahillonhttps://en.wikipedia.org/wiki/Hermann_von_Helmholtzhttps://en.wikipedia.org/wiki/J._Murray_Barbourhttps://en.wikipedia.org/wiki/Guangzhouhttps://en.wikipedia.org/wiki/Jesuithttps://en.wikipedia.org/wiki/Matteo_Riccihttps://en.wikipedia.org/wiki/File:%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-1154.jpghttps://en.wikipedia.org/wiki/File:%E4%B9%90%E5%BE%8B%E5%85%A8%E4%B9%A6%E5%85%A8-2297.jpg


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Simon Stevin's Van de Spiegheling der singconst c.

1605.

Vincenzo Galilei (father of Galileo Galilei) was one of the first practical advocates of twelve-tone equaltemperament. He composed a set of dance suites on eachof the 12 notes of the chromatic scale in all the"transposition keys", and published also, in his 1584

"Fronimo", 24 + 1 ricercars.[22] He used the 18:17 ratiofor fretting the lute (although some adjustment was

necessary for pure octaves).[23]

Galilei's countryman and fellow lutenist GiacomoGorzanis had written music based on equal temperament

by 1567.[24][25] Gorzanis was not the only lutenist toexplore all modes or keys: Francesco Spinacino wrote a"Recercare de tutti li Toni" (Ricercar in all the Tones) as

early as 1507.[26] In the 17th century lutenist-composer

John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.[27][28]

Henricus Grammateus drew a close approximation to equal temperament in 1518. The first tuning rules inequal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".[29] Zarlino in hispolemic with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lutein his Sopplimenti musicali in 1588.

Simon Stevin

The first mention of equal temperament related to Twelfth root of two in the West appeared in SimonStevin's manuscript Van De Spiegheling der singconst (ca 1605) published posthumously nearly three

centuries later in 1884.[30] However, due to insufficient accuracy of his calculation, many of the chord lengthnumbers he obtained were off by one or two units from the correct values.[31] As a result, the frequencyratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is claimed by Gene Cho as

incorrect.[32]

The following were Simon Stevin's chord length from Vande Spiegheling der singconst :[33]

https://en.wikipedia.org/wiki/Simon_Stevinhttps://en.wikipedia.org/wiki/Twelfth_root_of_twohttps://en.wikipedia.org/wiki/Lutehttps://en.wikipedia.org/wiki/Polemichttps://en.wikipedia.org/wiki/Zarlinohttps://en.wikipedia.org/w/index.php?title=Giovani_Maria_Lanfranco&action=edit&redlink=1https://en.wikipedia.org/wiki/Henricus_Grammateushttps://en.wikipedia.org/wiki/John_Wilson_(composer)https://en.wikipedia.org/wiki/Ricercarhttps://en.wikipedia.org/wiki/Francesco_Spinacinohttps://en.wikipedia.org/w/index.php?title=Giacomo_Gorzanis&action=edit&redlink=1https://en.wikipedia.org/wiki/Lutenisthttps://en.wikipedia.org/wiki/Ricercarhttps://en.wikipedia.org/wiki/Fronimo_Dialogohttps://en.wikipedia.org/wiki/Galileo_Galileihttps://en.wikipedia.org/wiki/Vincenzo_Galileihttps://en.wikipedia.org/wiki/File:De_Spiegheling_der_signconst.jpg


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TONECHORD 10000 from

Simon StevinRATIO

CORRECTEDCHORD

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 7491.5tritone 7071 1.0593975 7071.1

tritone and a half 6674 1.0594845 6674.2

four-tone 6298 1.0597014 6299

four-tone and a half 5944 1.0595558 5946

five-tone 5611 1.0593477 5612.3

five-tone and a half 5296 1.0594788 5297.2

full tone 1.0592000

A generation later, French mathematician Marin Mersenne presented several equal tempered chord lengths

obtained by Jean Beaugrand, Ismael Bouillaud and Jean Galle.[34]

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.[35]

Baroque era

From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equaltemperament,[36] and the Brossard lute Manuscript compiled in the last quarter of the 17th century contains aseries 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.[37] Angelo Michele Bartolotti published aseries of passacaglias in all keys, with connecting enharmonically modulating passages. Among the 17th-century keyboard composers Girolamo Frescobaldi advocated equal temperament. Some theorists, such asGiuseppe 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.[38]

J. S. Bach wrote The Well-Tempered Clavier to demonstrate the musical possibilities of well temperament,where in some keys the consonances are even more degraded than in equal temperament. It is possible thatwhen composers and theoreticians of earlier times wrote of the moods and "colors" of the keys, they eachdescribed the subtly different dissonances made available within a particular tuning method. However, it isdifficult to determine with any exactness the actual tunings used in different places at different times by anycomposer. (Correspondingly, there is a great deal of variety in the particular opinions of composers about themoods and colors of particular keys.)

Twelve tone equal temperament took hold for a variety of reasons. It was a convenient fit for the existingkeyboard 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 extremelyimportant in the 18th century in music of such composers as Francesco Geminiani, Wilhelm FriedemannBach, Carl Philipp Emmanuel Bach and Johann Gottfried Müthel.

https://en.wikipedia.org/wiki/Johann_Gottfried_M%C3%BCthelhttps://en.wikipedia.org/wiki/Carl_Philipp_Emmanuel_Bachhttps://en.wikipedia.org/wiki/Wilhelm_Friedemann_Bachhttps://en.wikipedia.org/wiki/Francesco_Geminianihttps://en.wikipedia.org/wiki/Modulation_(music)https://en.wikipedia.org/wiki/Well_temperamenthttps://en.wikipedia.org/wiki/The_Well-Tempered_Clavierhttps://en.wikipedia.org/wiki/J._S._Bachhttps://en.wikipedia.org/wiki/Andreas_Werckmeisterhttps://en.wikipedia.org/wiki/Giuseppe_Tartinihttps://en.wikipedia.org/wiki/Girolamo_Frescobaldihttps://en.wikipedia.org/wiki/Passacagliahttps://en.wikipedia.org/wiki/Angelo_Michele_Bartolottihttps://en.wikipedia.org/wiki/Johann_Faulhaberhttps://en.wikipedia.org/wiki/Marin_Mersenne


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The progress of equal temperament from the mid-18th century on is described with detail in quite a fewmodern scholarly publications: it was already the temperament of choice during the Classical era (secondhalf of the 18th century), and it became standard during the Early Romantic era (first decade of the 19thcentury), except for organs that switched to it more gradually, completing only in the second decade of the19th 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 17th-century Sabbatini method of splitting the octave first

into three tempered major thirds.[39] 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.[40] Using beat rates, first proposed in 1749, became common

after their diffusion by Helmholtz and Ellis in the second half of the 19th century.[41] The ultimate precision

was available with 2-decimal tables published by White in 1917.[42]

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 pianocomponent) developed and flourished.

General properties

In an equal temperament, the distance between two adjacent steps of the scale is the same interval. Becausethe 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 permittransposition to different keys.) Specifically, the smallest interval in an equal-tempered scale is the ratio:

rn = p

r =n

√ p

where the ratio r divides the ratio p (typically the octave, which is 2:1) into n equal parts. (See Twelve-toneequal 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 hasconsiderable use in Ethnomusicology. The basic step in cents for any equal temperament can be found bytaking the width of p above in cents (usually the octave, which is 1200 cents wide), called below w, anddividing it into n parts:

c = wn

In musical analysis, material belonging to an equal temperament is often given an integer notation, meaninga single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch materialwithin 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 acknowledgesthe similarity) between pitches of the same name, e.g. 'C' is 0 regardless of octave register. The MIDIencoding standard uses integer note designations.

General formula for calculate equal temperament:

--- equivalent ---

https://en.wikipedia.org/wiki/MIDIhttps://en.wikipedia.org/wiki/Pitch_classhttps://en.wikipedia.org/wiki/Modular_arithmetichttps://en.wikipedia.org/wiki/Logarithmhttps://en.wikipedia.org/wiki/Musical_notation#Integer_notationhttps://en.wikipedia.org/wiki/Ethnomusicologyhttps://en.wikipedia.org/wiki/Logarithmhttps://en.wikipedia.org/wiki/Cent_(music)https://en.wikipedia.org/wiki/Octavehttps://en.wikipedia.org/wiki/Interval_(music)https://en.wikipedia.org/wiki/Arithmetic_sequencehttps://en.wikipedia.org/wiki/Geometric_sequencehttps://en.wikipedia.org/wiki/Ratiohttps://en.wikipedia.org/wiki/Interval_(music)https://en.wikipedia.org/wiki/Jazzhttps://en.wikipedia.org/wiki/Serialismhttps://en.wikipedia.org/wiki/Twelve_tone_techniquehttps://en.wikipedia.org/wiki/Atonalityhttps://en.wikipedia.org/wiki/Polytonalityhttps://en.wikipedia.org/wiki/Samuel_Sebastian_Wesley


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where t is the tempered interval, r the ratio interval, b the base interval (2 for cents) and d divisor of baseinterval (1200 for cents), unit of tempered interval.

Twelve-tone equal temperament

In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e.the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:

12 √2 = 2

1 ⁄ 12 ≈ 1.059463

This interval is divided into 100 cents.

Calculating absolute frequencies

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

Pn = P

a(12 √2)(n − a)

In this formula Pn 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 andthe reference pitch, respectively. These two numbers are from a list of consecutive integers assigned toconsecutive 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:

P40 = 440(12 √2)(40 − 49) ≈ 261.626 Hz

Historical comparison

[43]

YEAR NAME RATIO CENTS

400 He Chengtian 1.060070671 101.01580 Vincenzo Galilei 18:17 [1.058823529] 99.01581 Zhu Zaiyu 1.059463094 100.0

1585 Simon Stevin 1.059546514 100.11630 Marin Mersenne 1.059322034 99.81630 Johann Faulhaber 1.059490385 100.0

Comparison to just intonation

The intervals of 12-TET closely approximate some intervals in just intonation. The fifths and fourths arealmost indistinguishably close to just.

In the following table the sizes of various just intervals are compared against their equal-temperedcounterparts, given as a ratio as well as cents.

NameExact value in 12- Decimal value

Cents

Justintonation Cents in just

Error

https://en.wikipedia.org/wiki/Cent_(music)https://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/Middle_Chttps://en.wikipedia.org/wiki/A440_(pitch_standard)https://en.wikipedia.org/wiki/Hertzhttps://en.wikipedia.org/wiki/Cent_(music)https://en.wikipedia.org/wiki/Twelfth_root_of_twohttps://en.wikipedia.org/wiki/Interval_ratiohttps://en.wikipedia.org/wiki/Semitone


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TET in 12-TET interval intonation

Unison (C) 1.000000 0 = 1.0000000 0.00 0

Minor second(C♯/D♭)

1.059463 100 = 1.06666… 111.73 −11.73

Major second

(D)

1.122462 200 = 1.1250000 203.91 −3.91

Minor third(D♯/E♭)

1.189207 300 = 1.2000000 315.64 −15.64

Major third (E) 1.259921 400 = 1.2500000 386.31 +13.69

Perfect fourth(F) 1.334840 500

= 1.33333… 498.04 +1.96

Tritone (F♯/G♭) 1.414214 600 = 1.4000000 582.51 +17.49

Perfect fifth

(G) 1.498307 700 = 1.5000000 701.96 −1.96Minor sixth

(G♯/A♭)1.587401 800 = 1.6000000 813.69 −13.69

Major sixth (A) 1.681793 900 = 1.66666… 884.36 +15.64

Minor seventh(A♯/B♭)

1.781797 1000 = 1.77777… 996.09 +3.91

Major seventh(B) 1.887749 1100

= 1.8750000 1088.27 +11.73

Octave (C) 2.000000 1200 = 2.0000000 1200.00 0

Seven-tone equal division of the fifth

Violins, violas and cellos are tuned in perfect fifths (G – D – A – E, for violins, and C – G – D – A, forviolas and cellos), which suggests that their semi-tone ratio is slightly higher than in the conventionaltwelve-tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval

is covered in 7 steps, each tone is in the ratio of7 √ 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 theusual 2:1 ratio, because twelve perfect fifths do not equal seven octaves.[44] During actual play, however, theviolinist chooses pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibitthis 3:2 ratio.

Other equal temperaments

5 and 7 tone temperaments in ethnomusicology

Five and seven tone equal temperament (5-TET Play and 7-TET Play ), with 240 Play and 171Play cent steps respectively, are fairly common. A Thai xylophone measured by Morton (1974) "variedonly plus or minus 5 cents," from 7-TET. A Ugandan Chopi xylophone measured by Haddon (1952) wasalso tuned to this system. According to Morton, "Thai instruments of fixed pitch are tuned to an equidistantsystem of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning

https://en.wikipedia.org/wiki/Thai_musichttps://upload.wikimedia.org/wikipedia/commons/6/66/1_step_in_7-et_on_C.midhttps://en.wikipedia.org/wiki/File:1_step_in_7-et_on_C.midhttps://upload.wikimedia.org/wikipedia/commons/f/f7/1_step_in_5-et_on_C.midhttps://en.wikipedia.org/wiki/File:1_step_in_5-et_on_C.midhttps://upload.wikimedia.org/wikipedia/commons/5/5e/7-tet_scale_on_C.midhttps://en.wikipedia.org/wiki/File:7-tet_scale_on_C.midhttps://upload.wikimedia.org/wikipedia/commons/8/8a/5-tet_scale_on_C.midhttps://en.wikipedia.org/wiki/File:5-tet_scale_on_C.midhttps://en.wikipedia.org/wiki/C_(musical_note)https://en.wikipedia.org/wiki/B_(musical_note)https://en.wikipedia.org/wiki/B%E2%99%AD_(musical_note)https://en.wikipedia.org/wiki/A%E2%99%AF_(musical_note)https://en.wikipedia.org/wiki/A_(musical_note)https://en.wikipedia.org/wiki/A%E2%99%AD_(musical_note)https://en.wikipedia.org/wiki/G%E2%99%AF_(musical_note)https://en.wikipedia.org/wiki/G_(musical_note)https://en.wikipedia.org/wiki/G%E2%99%AD_(musical_note)https://en.wikipedia.org/wiki/F%E2%99%AF_(musical_note)https://en.wikipedia.org/wiki/F_(musical_note)https://en.wikipedia.org/wiki/E_(musical_note)https://en.wikipedia.org/wiki/E%E2%99%AD_(musical_note)https://en.wikipedia.org/wiki/D%E2%99%AF_(musical_note)https://en.wikipedia.org/wiki/D_(musical_note)https://en.wikipedia.org/wiki/D%E2%99%AD_(musical_note)https://en.wikipedia.org/wiki/C%E2%99%AF_(musical_note)https://en.wikipedia.org/wiki/C_(musical_note)


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Easley Blackwood's notation system for 16 equal temperament:

intervals are notated similarly to those they approximate and there are

fewer enharmonic equivalents.[46] Play

Comparison of equal temperaments

from 9 to 25 (after Sethares (2005),

p.58).

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."[45] Play Indonesian gamelans are tuned to 5-TET according to Kunst (1949), but according to Hood (1966)

and McPhee (1966) their tuning varies widely, and according to Tenzer (2000) they contain stretchedoctaves. It is now well-accepted that of the two primary tuning systems in gamelan music, slendro and pelog,only slendro somewhat resembles five-tone equal temperament while pelog is highly unequal; however,Surjodiningrat et al. (1972) has analyzed pelog as a seven-note subset of nine-tone equal temperament (133cent steps Play ). 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 withinstrumental gamelan music.

5-TET and 7-TET mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure1.

In 5-TET the tempered perfect fifth is 720 cents wide (at the top of the tuning continuum), and marksthe endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0cents.In 7-TET the tempered perfect fifth is 686 cents wide (at the bottom of the tuning continuum), and

marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as themajor second (at 171 cents each).

Various Western equal temperaments

24-EDO, the quarter tone scale (or 24-TET), was a popular microtonal tuning Inthe 20th century probably because itrepresented a convenient access point forcomposers conditioned on standard

Western 12-EDO pitch and notationpractices who were also interested inmicrotonality. Because 24-EDO containsall of the pitches of 12-EDO plus new pitches halfway between eachadjacent pair of 12-EDO pitches, they could employ the additionalcolors without losing any tactics available in 12-tone harmony. Thefact that 24 is a multiple of 12 also made 24-EDO easy to achieveinstrumentally by employing two traditional 12-EDO instrumentspurposely tuned a quarter-tone apart, such as two pianos, which alsoallowed each performer (or one performer playing a different piano

with each hand) to read familiar 12-tone notation. Various composersincluding Charles Ives experimented with music for quarter-tonepianos.

29-TET is the lowest number of equal divisions of the octave whichproduces a better perfect fifth than 12-TET. Its major third is roughlyas inaccurate as 12-TET; however, it is tuned 14 cents flat rather than14 cents sharp.

31 tone equal temperament was advocated by Christiaan Huygensand Adriaan Fokker. 31-TET has a slightly less accurate fifth than 12-TET, but provides near-just major

thirds, and provides decent matches for harmonics up to at least 13, of which the seventh harmonic isparticularly accurate.

https://en.wikipedia.org/wiki/Adriaan_Fokkerhttps://en.wikipedia.org/wiki/Christiaan_Huygenshttps://en.wikipedia.org/wiki/Quarter_tone_scalehttps://en.wikipedia.org/wiki/Syntonic_temperamenthttps://upload.wikimedia.org/wikipedia/commons/4/4a/Semitone_Maximus_on_C.midhttps://en.wikipedia.org/wiki/File:Semitone_Maximus_on_C.midhttps://en.wikipedia.org/wiki/Peloghttps://en.wikipedia.org/wiki/Slendrohttps://en.wikipedia.org/wiki/Pseudo-octavehttps://en.wikipedia.org/wiki/Michael_Tenzerhttps://en.wikipedia.org/wiki/Colin_McPheehttps://en.wikipedia.org/wiki/Mantle_Hoodhttps://en.wikipedia.org/wiki/Jaap_Kunsthttps://en.wikipedia.org/wiki/Gamelanhttps://upload.wikimedia.org/wikipedia/commons/3/31/Thai_pentatonic_scale_mode_1.midhttps://en.wikipedia.org/wiki/File:Thai_pentatonic_scale_mode_1.midhttps://en.wikipedia.org/wiki/File:Comparison_of_equal_temperaments.pnghttps://upload.wikimedia.org/wikipedia/commons/4/47/16-tet_scale_on_C.midhttps://en.wikipedia.org/wiki/File:16-tet_scale_on_C.midhttps://en.wikipedia.org/wiki/Enharmonichttps://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.https://en.wikipedia.org/wiki/File:16-tet_scale_on_C.png


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

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

53-TET is better at approximating the traditional just consonances than 12, 19 or 31-TET, but has had only

occasional use. Its extremely good perfect fifths make it interchangeable with an extended Pythagoreantuning, but it also accommodates schismatic temperament, and is sometimes used in Turkish music theory. Itdoes not, however, fit the requirements of meantone temperaments, which put good thirds within easy reachvia the cycle of fifths. In 53-TET the very consonant thirds would be reached instead by strange enharmonicrelationships like C-F♭, as it is an example of schismatic temperament. A consequence of this is that chordprogressions like I-vi-ii-V-I won't land you back where you started in 53-TET, but rather one 53-tone stepflat (unless the motion by I-vi wasn't by the 5-limit minor third).

72-TET approximates many just intonation intervals well, even into the 7-limit and 11-limit, such as 7:4, 9:7,11:5, 11:6 and 11:7. 72-TET has been taught, written and performed in practice by Joe Maneri and his

students (whose atonal inclinations interestingly typically avoid any reference to just intonation whatsoever).It can be considered an extension of 12 EDO because 72 is a multiple of 12. 72 EDO has a smallest intervalthat is six times smaller than the smallest interval of 12 EDO and therefore contains six copies of 12 EDOstarting on different pitches. It also contains three copies of 24 EDO and two copies of 36 EDO, which arethemselves multiples of 12 EDO.

Other equal divisions of the octave that have found occasional use include, 15-TET, 17-TET, 19-TET and22-TET.

2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of log2(3), so 2, 5, 12, 41, 53, 306,665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer numberof 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.[47][48]

1, 2, 3, 5, 7, 12, 29, 41, 53, 200... (sequence A060528 in OEIS) is the sequence of divisions of octave thatprovide better and better approximations of the perfect fifth. Related sequences contain divisions

approximating other just intervals.[49] It is noteworthy that many elements of this sequences are sums of previous elements.

This application: [2] (http://equal.meteor.com/) calculates the frequencies, approximate cents, and MIDI

Pitch Bend values for any systems of Equal Division of the Octave. Note that 'rounded' and 'floored' producethe same MIDI Pitch Bend value.

Equal temperaments of non-octave intervals

The equal-tempered version of the Bohlen–Pierce scale consists of the ratio 3:1, 1902 cents, conventionally aperfect fifth and an octave, called in this theory a tritave ( play ), and split into a thirteen equal parts. Thisprovides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents (

play ), or13 √3 .

Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possibletemperaments having a step size between 30 and 120 cents. These were called alpha, beta, and gamma. Theycan be considered as equal divisions of the perfect fifth. Each of them provides a very good approximation o

several just intervals.[50] Their step sizes:

https://en.wikipedia.org/wiki/Gamma_scalehttps://en.wikipedia.org/wiki/Beta_scalehttps://en.wikipedia.org/wiki/Alpha_scalehttps://en.wikipedia.org/wiki/Wendy_Carloshttps://upload.wikimedia.org/wikipedia/commons/6/63/BP_scale_et.midhttps://en.wikipedia.org/wiki/File:BP_scale_et.midhttps://en.wikipedia.org/wiki/Just_intonationhttps://upload.wikimedia.org/wikipedia/commons/d/d3/Tritave_on_C.midhttps://en.wikipedia.org/wiki/File:Tritave_on_C.midhttps://en.wikipedia.org/wiki/Tritavehttps://en.wikipedia.org/wiki/Octavehttps://en.wikipedia.org/wiki/Perfect_fifthhttps://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scalehttp://equal.meteor.com/https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A060528https://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/Fifth_(disambiguation)#Musichttps://en.wikipedia.org/wiki/Twelfth_(disambiguation)#Musichttps://en.wikipedia.org/wiki/Convergent_(continued_fraction)https://en.wikipedia.org/wiki/Denominatorhttps://en.wikipedia.org/wiki/22_equal_temperamenthttps://en.wikipedia.org/wiki/19_equal_temperamenthttps://en.wikipedia.org/wiki/17_equal_temperamenthttps://en.wikipedia.org/wiki/15_equal_temperamenthttps://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/Joe_Manerihttps://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/72_equal_temperamenthttps://en.wikipedia.org/wiki/Comma_pumphttps://en.wikipedia.org/wiki/Schismatic_temperamenthttps://en.wikipedia.org/wiki/Turkish_musichttps://en.wikipedia.org/wiki/Schismatic_temperamenthttps://en.wikipedia.org/wiki/Pythagorean_tuninghttps://en.wikipedia.org/wiki/Perfect_fifthhttps://en.wikipedia.org/wiki/Just_intonationhttps://en.wikipedia.org/wiki/53_equal_temperamenthttps://en.wikipedia.org/wiki/41_equal_temperamenthttps://en.wikipedia.org/wiki/34_equal_temperament


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alpha:9 √ 3 ⁄ 2 (78.0 cents)

beta:11 √ 3 ⁄ 2 (63.8 cents)

gamma:20 √ 3 ⁄ 2 (35.1 cents)

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

Proportions between semitone and whole tone

In this section, semitone and whole tone may not have their usual 12-EDO meanings, as it discusses howthey may be tempered in different ways from their just versions to produce desired relationships. Let thenumber 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 wholetone, while keeping the notes in the right order (meaning that, for example, C, D, E, F, and F♯ are inascending order if they preserve their usual relationships to C). That is, fixing q to a proper fraction in therelationship qs = t also defines a unique family of one equal temperament and its multiples that fulfill thisrelationship.

For example, where k is an integer, 12k -EDO will set q = 1 ⁄ 2, and 19k -EDO will set q =1 ⁄ 3. The smallest

multiples in these families (e.g. 12 and 19 above) will have the additional property of having no notesoutside the circle of fifths. (This is not true in general; in 24-EDO, the half-sharps and half-flats are not inthe circle of fifths generated starting from C.) The extreme cases are 5k -EDO, where q = 0 and the semitonebecomes a unison, and 7k -EDO, where q = 1 and the semitone and tone are the same interval.

Once one knows how many steps a semitone and a tone are in this equal temperament, one can find thenumber 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 − ssteps. If there are notes outside the circle of fifths, one must then multiply these results by n, which is thenumber of nonoverlapping circles of fifths required to generate all the notes (e.g. two in 24-EDO, six in 72-

EDO). (One must take the small semitone for this purpose: 19-EDO has two semitones, one being 1 ⁄ 3 tone

and the other being 2 ⁄ 3.)

The smallest of these families is 12k -EDO, and in particular 12-EDO is the smallest equal temperament thathas the above properties. Additionally, it also makes the semitone exactly half a whole tone, the simplestpossible relationship. These are some of the reasons why 12-EDO has become the most commonly usedequal temperament. (Another reason is that 12-EDO is the smallest equal temperament to closelyapproximate 5-limit harmony, the next-smallest being 19-EDO.)

Each choice of fraction q for the relationship results in exactly one equal temperament family, but the

converse is not true: 47-EDO has two different semitones, where one is 1 ⁄ 7 tone and the other is8 ⁄ 9, which

are not complements of each other like in 19-EDO (1 ⁄ 3 and2 ⁄ 3). Taking each semitone results in a different

choice of perfect fifth.

Related tuning systems

Syntonic temperament

https://en.wikipedia.org/wiki/Circle_of_fifthshttps://en.wikipedia.org/wiki/Proper_fractionhttps://en.wikipedia.org/wiki/Beauty_in_the_Beast


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Figure 1: The syntonic tuning continuum,

which includes many notable "equal

temperament" tunings (Milne 2007).[51]

Syntonic temperament is a system for generating a spectrum of "regular" tunings or temperaments with two irreducible essentialinterval sizes, such as meantone temperaments, but underspecific limited conditions, the two intervals line up in such away as to produce regular tunings with one essential intervalsize, in other words equal temperaments. However, it can onlyproduce those equal temperaments that divide the octave into aninteger number of parts, the so-called EDO (equal division of the

octave) equal temperaments.

The tuning continuum of the syntonic temperament, shown inFigure 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 isomorphickeyboard, the fingering of music written in any of these syntonictunings is precisely the same as it is in any other syntonic tuning,so long as the notes are spelled properly—that is, with noassumption of enharmonicity. This consistency of fingering

makes it possible to smoothly vary the tuning (and hence thepitches of all notes, systematically) all along the syntonic tuningcontinuum—a polyphonic tuning bend. The use of dynamictimbres lets consonance be maintained (or otherwisemanipulated) across such tuning bends.

See also

Musical acoustics (the physics of music)Music and mathematics

MicrotunerMicrotonal musicPiano tuningList of meantone intervalsDiatonic and chromaticElectronic tuner

References

Citations

1. Sethares compares several equal temperament scales in a graph with axes reversed from the axes here. (fig. 4.6, p.

58) (http://books.google.com/books?

id=KChoKKhjOb0C&pg=PA58&vq=%22tuning+of+one+octave+of+notes%22&source=gbs_search_r&cad=1_1

&sig=ACfU3U2dq8ONjfazmWxdojVsQ3TFyz-kTg)

2. The History of Musical Pitch in Europe p493-511 Herman Helmholtz, Alexander J. Ellis On The Sensations of

Tone, Dover Publications, Inc., New York

3. Fritz A. Kuttner. p. 163.

4. Fritz A. Kuttner. "Prince Chu Tsai-Yü's Life and Work: A Re-Evaluation of His Contribution to Equal

Temperament Theory", p.200, Ethnomusicology, Vol. 19, No. 2 (May, 1975), pp. 163–206.

5. Kenneth Robinson: A critical study of Chu Tsai-yü's contribution to the theory of equal temperament in Chinese

music. (Sinologica Coloniensia, Bd. 9.) x, 136 pp. Wiesbaden: Franz Steiner Verlag GmbH, 1980. DM 36. p.vii"Chu-Tsaiyu the first formulator of the mathematics of "equal temperament" anywhere in the world

6. 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.

http://books.google.com/books?id=KChoKKhjOb0C&pg=PA58&vq=%22tuning+of+one+octave+of+notes%22&source=gbs_search_r&cad=1_1&sig=ACfU3U2dq8ONjfazmWxdojVsQ3TFyz-kTghttps://en.wikipedia.org/wiki/Electronic_tunerhttps://en.wikipedia.org/wiki/Diatonic_and_chromatichttps://en.wikipedia.org/wiki/List_of_meantone_intervalshttps://en.wikipedia.org/wiki/Piano_tuninghttps://en.wikipedia.org/wiki/Microtonal_musichttps://en.wikipedia.org/wiki/Microtunerhttps://en.wikipedia.org/wiki/Music_and_mathematicshttps://en.wikipedia.org/wiki/Musical_acousticshttps://en.wikipedia.org/wiki/Dynamic_tonality#Dynamic_sonancehttps://en.wikipedia.org/wiki/Dynamic_tonality#Dynamic_timbrehttps://en.wikipedia.org/wiki/Dynamic_tonality#New_musical_effectshttps://en.wikipedia.org/wiki/Enharmonichttps://en.wikipedia.org/wiki/Isomorphic_keyboardhttps://en.wikipedia.org/wiki/Syntonic_temperamenthttps://en.wikipedia.org/wiki/Integerhttps://en.wikipedia.org/wiki/Syntonic_temperamenthttps://en.wikipedia.org/wiki/Syntonic_temperamenthttps://en.wikipedia.org/wiki/File:Syntonic_Tuning_Continuum.jpg


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7. Fritz A. Kuttner. p. 200.

8. Owen Henry Jorgensen, Tuning (East Lansing: Michigan State University Press, 1991)

9. The Formation of Chinese Civilization: An Archaeological Perspective, Zhang Zhongpei, Shao Wangping and

others, Yale (New Haven Conn)and New World Press (Beijing),2005

10. The Formation of Chinese Civilization, 2005 (opt cit), the Eastern Zhou and the Growth of Regionalism, Lu

Liancheng, pp 140 ff

11. J. Murray Barbour Tuning and Temperament p55-56, Michigan State University Press 1951

12. 有关"七平均律"新文献著作的发现 [Findings of new literatures concerning the hepta – equal temperament] (inChinese). Archived from the original on 27 October 2007. "'Hepta-equal temperament' in our folk music has

always been a controversial issue."13. 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).

"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 so-called

'Seven Laws' at least two thousand years of history; and decided that this law system associated with the flute

law."

14. "Quantifying Ritual: Political Cosmology, Courtly Music, and Precision Mathematics in Seventeenth-Century

China Roger Hart Departments of History and Asian Studies, University of Texas, Austin". Uts.cc.utexas.edu.

Retrieved 2012-03-20.

15. Science and Civilisation in China, Vol IV:1 (Physics), Joseph Needham, Cambridge University Press, 1962–2004,

pp 220 ff

16. The Shorter Science & Civilisation in China, An abridgement by Colin Ronan of Joseph Needham's original text,p385

17. Gene J. Cho "The Significance of the Discovery of the Musical Equal Temperament In the Cultural History,"

http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm

18. "EQUAL TEMPERAMENT". University of Houston. Retrieved October 5, 2014.

19. J. Murray Barbour, Tuning and Temperament Michigan University Press, 1951 p7

20. [1] (http://books.google.com/books?id=GwE6AAAAIAAJ&pg=PA258#v=onepage&q&f=false) "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

21. Lau Hanson, Abacus and Practical Mathematics p389 (in Chinese 劳汉生《珠算与实用数学》 389页)

22. Galilei, V. (1584). Il Fronimo... Dialogo sopra l'arte del bene intavolare. G. Scotto: Venice, ff. 80–89.23. J. Murray Barbour Tuning and Temperament p8 1951 Michigan State University Press

24. "Resound – Corruption of Music". Philresound.co.uk. Retrieved 2012-03-20.

25. Giacomo Gorzanis, c. 1525 - c. 1575 Intabolatura di liuto. Geneva, 1982

26. "Spinacino 1507a: Thematic Index". Appalachian State University. Archived from the original on 2011-07-25.

Retrieved 14 June 2012.

27. John Wilson, 26 Preludes, Diapason press (DP49, Utrecht)

28. in "English Song, 1600–1675, Facsimile of Twenty-six Manuscripts and an Edition of the Texts" vol. 7,

Manuscripts at Oxford, Part II. introduction by Elise Bickfoer Jorgens, Garland Publishing Inc., New York and

London, 1987

29. "Scintille de musica", (Brescia, 1533), p. 132

30. "Van de Spiegheling der singconst, ed by Rudolf Rasch, The Diapason Press". Diapason.xentonic.org. 2009-06-

30. Retrieved 2012-03-20.31. Thomas S. Christensen, The Cambridge history of western music theory p205, Cambridge University Press

32. Gene Cho, The discovery of musical equal temperament in China and Europe – Page 223, Mellen Press

33. Gene Cho, The discovery of musical equal temperament in China and Europe – Page 222, Mellen Press

34. Thomas S. Christensen, The Cambridge history of western music theory p207, Cambridge University Press

35. Thomas S. Christensen, The Cambridge history of western music theory p78, Cambridge University Press

36. "Lutes, Viols, Temperaments" Mark Lindley ISBN 978-0-521-28883-5

37. Vm7 6214

38. Andreas Werckmeister: Musicalische Paradoxal-Discourse, 1707

39. Di Veroli, Claudio. Unequal Temperaments: Theory, History and Practice. 2nd edition, Bray Baroque, Bray,

Ireland 2009, pp. 140, 142 and 256.

40. Moody, Richard. “Early Equal Temperament, An Aural Perspective: Claude Montal 1836” in Piano TechniciansJournal. Kansas City, USA, Feb. 2003.

41. Helmholtz, Hermann L.F. Lehre von den Tonempfindungen… Heidelberg 1862. English edition: On the

Sensations of Tone, Longmans, London, 1885, p.548.

42. White, William Braid. Piano Tuning and Allied Arts. 1917, 5th enlarged edition, Tuners Supply Co., Boston

https://en.wikipedia.org/wiki/Special:BookSources/9780521288835http://diapason.xentonic.org/ttl/ttl21.htmlhttp://www.sunstar.com.ph/cebu/business/bickering-blocking-cebu-s-progresshttp://web.archive.org/web/20110725182053/http://www.library.appstate.edu/music/lute/16index/tspi07a.htmlhttp://www.philresound.co.uk/page4.htmhttps://en.wikipedia.org/wiki/Girolamo_Scottohttp://books.google.com/books?id=GwE6AAAAIAAJ&pg=PA258#v=onepage&q&f=falsehttp://www.uh.edu/engines/epi380.htmhttp://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htmhttp://uts.cc.utexas.edu/~rhart/papers/quantifying.htmlhttp://scholar.ilib.cn/Abstract.aspx?A=xhyyxyxb200102005http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htmhttp://web.archive.org/web/20071027064731/http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm


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1946, p.68.

43. Date,name,ratio,cents: from equal temperament monochord tables p55-p78; J. Murray Barbour Tuning and

Temperament , Michigan State University Press 1951

44. Cordier, Serge. "Le tempérament égal à quintes justes" (in French). Association pour la Recherche et le

Développement de la Musique. Retrieved 2 June 2010.

45. Morton, David (1980). "The Music of Thailand", Musics of Many Cultures, p.70. May, Elizabeth, ed. ISBN 0-

520-04778-8.

46. Myles Leigh Skinner (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba,

Ives, and Wyschnegradsky, p.55. ISBN 9780542998478.

47. "665edo". xenoharmonic (microtonal wiki). Retrieved 2014-06-18.48. "convergents(log2(3), 10)". WolframAlpha. Retrieved 2014-06-18.

49. 3/2 and 4/3, 5/4 and 8/5, 6/5 and 5/3 (A054540 (https://oeis.org/A054540));

3/2 and 4/3, 5/4 and 8/5 (A060525 (https://oeis.org/A060525));

3/2 and 4/3, 5/4 and 8/5, 7/4 and 8/7 (A060526 (https://oeis.org/A060526));

3/2 and 4/3, 5/4 and 8/5, 7/4 and 8/7, 16/11 and 11/8 (A060527 (https://oeis.org/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

(https://oeis.org/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

(https://oeis.org/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 (https://oeis.org/A061921));5/4 and 8/5 (A061918 (https://oeis.org/A061918));

6/5 and 5/3 (A061919 (https://oeis.org/A061919));

6/5 and 5/3, 7/5 and 10/7, 7/6 and 12/7 (A060529 (https://oeis.org/A060529));

11/8 and 16/11 (A061416 (https://oeis.org/A061416))

50. Three Asymmetric Divisions of the Octave (http://www.wendycarlos.com/resources/pitch.html) by Wendy Carlos

51. Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings

Across a Tuning Continuum" (http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15), Computer

Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.

Sources

Cho, Gene Jinsiong. (2003). The Discovery of Musical Equal Temperament in China and Europe inthe Sixteenth Century. Lewiston, NY: Edwin Mellen Press.Duffin, Ross W. How Equal Temperament Ruined Harmony (and Why You Should Care).W.W.Norton & Company, 2007.Jorgensen, Owen. Tuning. Michigan State University Press, 1991. ISBN 0-87013-290-3Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). London: Springer-Verlag.ISBN 1-85233-797-4.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" [3](https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxncmFucGhpNjE4fGd4OjRhN2VjNTNhOGY1ZmRkNDA)Khramov, Mykhaylo. "Approximation of 5-limit just intonation. Computer MIDI Modeling inNegative Systems of Equal Divisions of the Octave", Proceedings of the International ConferenceSIGMAP-2008 (http://www.sigmap.org/Abstracts/2008/abstracts.htm), 26–29 July 2008, Porto,pp. 181–184, ISBN 978-989-8111-60-9

External links

Huygens-Fokker Foundation Centre for Microtonal Music (http://www.huygens-fokker.org/index_en.html)A.Orlandini: Music Acoustics (http://www.fortunecity.com/tinpan/lennon/362/english/acoustics.htm)"Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)

http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?type=turn&entity=HistSciTech001000270617&isize=M&q1=temperamenthttp://www.fortunecity.com/tinpan/lennon/362/english/acoustics.htmhttp://www.huygens-fokker.org/index_en.htmlhttps://en.wikipedia.org/wiki/Special:BookSources/9789898111609https://en.wikipedia.org/wiki/Portohttp://www.sigmap.org/Abstracts/2008/abstracts.htmhttps://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxncmFucGhpNjE4fGd4OjRhN2VjNTNhOGY1ZmRkNDAhttp://web.telia.com/~u57011259/pelog_main.htmhttps://en.wikipedia.org/wiki/Special:BookSources/1-85233-797-4https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0870132903https://en.wikipedia.org/wiki/Edwin_Mellen_Presshttp://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15http://www.wendycarlos.com/resources/pitch.htmlhttps://oeis.org/A061416https://oeis.org/A060529https://oeis.org/A061919https://oeis.org/A061918https://oeis.org/A061921https://oeis.org/A061920https://oeis.org/A060233https://oeis.org/A060527https://oeis.org/A060526https://oeis.org/A060525https://oeis.org/A054540https://en.wikipedia.org/wiki/WolframAlphahttp://www.wolframalpha.com/input/?i=convergents%28log2%283%29%2C+10%29http://xenharmonic.wikispaces.com/665edohttps://en.wikipedia.org/wiki/Special:BookSources/9780542998478https://en.wikipedia.org/wiki/Special:BookSources/0520047788http://aredem.online.fr/aredem/page_cordier.html


16/16


(http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?type=turn&entity=HistSciTech001000270617&isize=M&q1=temperament)Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900(http://www.patriziobarbieri.it/1.htm). (2008) Latina, Il Levante Libreria EditriceFractal Microtonal Music (http://www.interdependentscience.com/music/calliopist.html), 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)Dominic Eckersley: "Rosetta Revisited: Bach's Very Ordinary Temperament

(https://www.academia.edu/3368760/Rosetta_Revisited_Bachs_Very_Ordinary_Temperament)"

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