Equal temperament

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12-tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. Template:Audio

Equal temperamentTemplate:Efn is a musical temperament (tuning system) that divides the octave into 12 identical parts. It is a logarithmic scale with a ratio equal to the 12th root of 2 ( $\sqrt[12]{2}$ ≈ 1.05946). The scale unit is known as a semitone or half step.

Twelve-tone equal temperament (12 TET) is the most widespread system in music today. It has been the predominant tuning system of Western music, starting with classical music, since the 18th century. It has also been used in other cultures.

In modern times, equal temperament is usually tuned relative to a standard pitch of 440 Hz. Over the past few centuries, standard pitch has risen significantly.Template:Sfn

Comparison with just intonation

The intervals of 12 Template:Sc closely approximate some intervals in just intonation.^[1] The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.

In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents. Template:Clear

Interval Name	Equal temperament interval	Cents in equal temperament	Just intonation interval	Cents in just intonation	Tuning error
Unison	2Template:Sup = 1	0	Template:Sfrac = 1	0.00	0.00
Minor second	2Template:Sup	100	Template:Sfrac	111.73	−11.73
Major second	2Template:Sup	200	Template:Sfrac	203.91	−3.91
Minor third	2Template:Sup	300	Template:Sfrac	315.64	−15.64
Major third	2Template:Sup	400	Template:Sfrac	386.31	+13.69
Perfect fourth	2Template:Sup	500	Template:Sfrac	498.04	+1.96
Tritone	2Template:Sup	600	Template:Sfrac	590.22	+9.78
Perfect fifth	2Template:Sup	700	Template:Sfrac	701.96	−1.96
Minor sixth	2Template:Sup	800	Template:Sfrac	813.69	−13.69
Major sixth	2Template:Sup	900	Template:Sfrac	884.36	+15.64
Minor seventh	2Template:Sup	1000	Template:Sfrac	1017.60	−17.60
Major seventh	2Template:Sup	1100	Template:Sfrac	1088.27	+11.73
Octave	2Template:Sup = 2	1200	Template:Sfrac = 2	1200.00	0.00

History

The two figures frequently credited with the achievement of exact calculation of twelve-tone equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu, Template:Lang-zh) in 1584 and Simon Stevin in 1585. According to Fritz A. Kuttner, a critic of the theory,Template:Sfn it is known that "Chu-Tsaiyu presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that "Simon Stevin offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently.Template:Sfn

Kenneth Robinson attributes the invention of equal temperament to Zhu ZaiyuTemplate:Sfn and provides textual quotations as evidence.Template:Sfn Zhu Zaiyu is quoted as saying that, in a text dating from 1584, "I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations."Template:Sfn Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications."Template:Sfn Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors.Template:Sfn

China

Early history

A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng (early Warring States, Template:Circa in the Chinese Bronze Age), covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.Template:Sfn

An approximation for equal temperament was described by Template:Ill, a mathematician of the Southern and Northern Dynasties who lived from 370 to 447.^[2] He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history: 900 849 802 758 715 677 638 601 570 536 509.5 479 450.Template:Clarify Template:Sfn

Zhu Zaiyu

Zhu Zaiyu (Template:Lang), a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar Template:Lang published in 1580. This was followed by the publication of a precise numerical specification for equal temperament in his 5,000-page work Complete Compendium of Music and Pitch (Yuelü quan shu Template:Lang) in 1584.Template:Sfn An extended account is also given by Joseph Needham.Template:Sfn Zhu obtained his result mathematically by dividing the length of string and pipe successively by Template:Radic ≈ 1.059463, and for pipe length by Template:Radic,Template:Sfn such that after twelve divisions (an octave) the length was divided by a factor of 2:

${(\sqrt[12]{2})}^{12} = 2$

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

${(\sqrt[12]{2})}^{84} = 2^{7} = 128$

Zhu Zaiyu has been credited as the first person to solve the equal temperament problem mathematically.Template:Sfn At least one researcher has proposed that Matteo Ricci, a Jesuit in China recorded this work in his personal journalTemplate:Sfn Template:Sfn and may have transmitted the work back to Europe. (Standard resources on the topic make no mention of any such transfer.Template:Sfn) In 1620, Zhu's work was referenced by a European mathematician.Template:Who Template:Sfn Murray Barbour said, "The first known appearance in print of the correct figures for equal temperament was in China, where Prince Tsaiyü's brilliant solution remains an enigma."Template:Sfn The 19th-century German physicist Hermann von Helmholtz wrote in On the Sensations of Tone that a Chinese prince (see below) introduced a scale of seven notes, and that the division of the octave into twelve semitones was discovered in China.Template:Sfn

Zhu Zaiyu illustrated his equal temperament theory by the construction of a set of 36 bamboo tuning pipes ranging in 3 octaves, with instructions of the type of bamboo, color of paint, and detailed specification on their length and inner and outer diameters. He also constructed a 12-string tuning instrument, with a set of tuning pitch pipes hidden inside its bottom cavity. In 1890, Victor-Charles Mahillon, curator of the Conservatoire museum in Brussels, duplicated a set of pitch pipes according to Zhu Zaiyu's specification. He said that the Chinese theory of tones knew more about the length of pitch pipes than its Western counterpart, and that the set of pipes duplicated according to the Zaiyu data proved the accuracy of this theory.

Europe

Early history

One of the earliest discussions of equal temperament occurs in the writing of Aristoxenus in the 4th century BC.Template:Sfn

Vincenzo Galilei (father of Galileo Galilei) was one of the first practical advocates of twelve-tone equal temperament. He composed a set of dance suites on each of the 12 notes of the chromatic scale in all the "transposition keys", and published also, in his 1584 "Fronimo", 24 + 1 ricercars.Template:Sfn He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).Template:Sfn

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567.Template:Sfn Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino wrote a Template:Lang (Ricercar in all the Tones) as early as 1507.^[3] In the 17th century lutenist-composer John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.Template:Sfn Template:Sfn Henricus Grammateus drew a close approximation to equal temperament in 1518. The first tuning rules in equal temperament were given by Giovani Maria Lanfranco in his "Scintille de musica".^[4] Zarlino in his polemic with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute in his Template:Lang in 1588.

Simon Stevin

The first mention of equal temperament related to the twelfth root of two in the West appeared in Simon Stevin's manuscript Template:Lang (c. 1605), published posthumously nearly three centuries later in 1884.Template:Sfn However, due to insufficient accuracy of his calculation, many of the chord length numbers he obtained were off by one or two units from the correct values.Template:Sfn As a result, the frequency ratios of Simon Stevin's chords has no unified ratio, but one ratio per tone, which is claimed by Gene Cho as incorrect.Template:Sfn

The following were Simon Stevin's chord length from Template:Lang:Template:Sfn

Tone	Chord 10000 from Simon Stevin	Ratio	Corrected chord
semitone	9438	1.0595465	9438.7
whole tone	8909	1.0593781
tone and a half	8404	1.0600904	8409
ditone	7936	1.0594758	7937
ditone and a half	7491	1.0594046	7491.5
tritone	7071	1.0593975	7071.1
tritone and a half	6674	1.0594845	6674.2
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.Template:Sfn

In 1630 Johann Faulhaber published a 100-cent monochord table, which contained several errors due to his use of logarithmic tables. He did not explain how he obtained his results.Template:Sfn

Baroque era

From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament,^[5] and the Brossard lute manuscript compiled in the last quarter of the 17th century contains a series of 18 preludes attributed to Bocquet written in all keys, including the last prelude, entitled Template:Lang, which enharmonically modulates through all keys.^[6]Template:Huh? Angelo Michele Bartolotti published a series of passacaglias in all keys, with connecting enharmonically modulating passages. Among the 17th-century keyboard composers Girolamo Frescobaldi advocated equal temperament. Some theorists, such as Giuseppe Tartini, were opposed to the adoption of equal temperament; they felt that degrading the purity of each chord degraded the aesthetic appeal of music, although Andreas Werckmeister emphatically advocated equal temperament in his 1707 treatise published posthumously.^[7]

Twelve-tone equal temperament took hold for a variety of reasons. It was a convenient fit for the existing keyboard design, and permitted total harmonic freedom with the burden of moderate impurity in every interval, particularly imperfect consonances. This allowed greater expression through enharmonic modulation, which became extremely important in the 18th century in music of such composers as Francesco Geminiani, Wilhelm Friedemann Bach, Carl Philipp Emmanuel Bach, and Johann Gottfried Müthel.Template:Citation needed Twelve-tone equal temperament did have some disadvantages, such as imperfect thirds, but as Europe switched to equal temperament, it changed the music that it wrote in order to accommodate the system and minimize dissonance.Template:Fact

A precise equal temperament is possible using the 17th century Sabbatini method of splitting the octave first into three tempered major thirds.Template:Sfn 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.Template:Sfn Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.Template:Sfn The ultimate precision was available with 2 decimal tables published by White in 1917.^[8] The intervals of equal temperament closely approximate some intervals in just intonation.^[1]

Comparison of historical approximations of the semitone

Year	Name	RatioTemplate:Sfn	Cents
400	He Chengtian	1.060070671	100.99
1580	Vincenzo Galilei	18:17 [1.058823529]	98.95
1581	Zhu Zaiyu	1.059463094	100.00
1585	Simon Stevin	1.059546514	100.14
1630	Marin Mersenne	1.059322034	99.76
1630	Johann Faulhaber	1.059490385	100.04

Varieties of equal temperament

Template:Main

Other equal temperaments divide the octave differently. For example, some music has been written in [[19 equal temperament|Template:Nobr]] and [[31 equal temperament|Template:Nobr]], while the Arab tone system uses [[24-TET|Template:Nobr]].

Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts.

Similar tuning systems

Chinese musicians developed 3-limit just intonation at least a century before He Chengtian.Template:Sfn Likewise, Pythagorean tuning, which was developed by ancient Greeks, was the predominant system in Europe until during the Renaissance, when Europeans realized that dissonant intervals such as Template:Frac could be made more consonant by tempering them to simpler ratios like Template:Frac, resulting in Europe developing a series of meantone temperaments.Template:Sfn

References

External links

Xenharmonic wiki on EDOs vs. Equal Temperaments
Huygens-Fokker Foundation Centre for Microtonal Music
A.Orlandini: Music Acoustics
"Temperament" from A supplement to Mr. Chambers's cyclopædia (1753)
Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900 Template:Webarchive. (2008) Latina, Il Levante Libreria Editrice
Fractal Microtonal Music, Jim Kukula.
All existing 18th century quotes on J.S. Bach and temperament
Dominic Eckersley: "Rosetta Revisited: Bach's Very Ordinary Temperament"
Well Temperaments, based on the Werckmeister Definition
"Favored Cardinalities of Scales" by Peter Buch

Template:Musical tuning

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Equal temperament

Contents

Comparison with just intonation