12 equal temperament

12 equal temperament (12-ET)^[a] is a musical temperament (tuning system) that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 (${\textstyle {\sqrt[{12}]{2}}}$ ≈ 1.05946). That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step.

Twelve-tone equal temperament 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, 12-ET is usually tuned relative to a standard pitch of 440 Hz, called A440, meaning one note, A4 (the A in the 4th octave of a typical 88-key piano), is tuned to 440 hertz and all other notes are defined as some multiple of semitones apart from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz. It has varied and generally risen over the past few hundred years.^[1]

The two figures frequently credited with the achievement of exact calculation of twelve-tone equal temperament are Zhu Zaiyu (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,^[2] 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.^[3]

Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu^[4] and provides textual quotations as evidence.^[5] Zhu Zaiyu is quoted as saying that, in a text dating from 1584, "I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations."^[5] Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications."^[2] Kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament and that neither of the two should be treated as inventors.^[3]

A complete set of bronze chime bells, among many musical instruments found in the tomb of the Marquis Yi of Zeng (early Warring States, c. 5th century BCE in the Chinese Bronze Age), covers five full 7-note octaves in the key of C Major, including 12 note semi-tones in the middle of the range.^[6]

An approximation for equal temperament was described by He Chengtian, a mathematician of the Southern and Northern Dynasties who lived from 370 to 447.^[7] 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.^[8]

Zhu Zaiyu (朱載堉), a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father. He described his new pitch theory in his Fusion of Music and Calendar 律暦融通 published in 1580. This was followed by the publication of a detailed account of the new theory of the equal temperament with a precise numerical specification for 12-ET in his 5,000-page work Complete Compendium of Music and Pitch (Yuelü quan shu 樂律全書) in 1584.^[9] An extended account is also given by Joseph Needham.^[5] Zhu obtained his result mathematically by dividing the length of string and pipe successively by ¹²√2 ≈ 1.059463, and for pipe length by ²⁴√2,^[10] such that after twelve divisions (an octave) the length was divided by a factor of 2:

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

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

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

Zhu Zaiyu has been credited as the first person to solve the equal temperament problem mathematically.^[11] At least one researcher has proposed that Matteo Ricci, a Jesuit in China recorded this work in his personal journal^[11][12] and may have transmitted the work back to Europe. (Standard resources on the topic make no mention of any such transfer.^[13]) In 1620, Zhu's work was referenced by a European mathematician.^[12] 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."^[14] 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.^[15]

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.

One of the earliest discussions of equal temperament occurs in the writing of Aristoxenus in the 4th century BC.^[16]

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.^[17] He used the 18:17 ratio for fretting the lute (although some adjustment was necessary for pure octaves).^[18]

Galilei's countryman and fellow lutenist Giacomo Gorzanis had written music based on equal temperament by 1567.^[19] Gorzanis was not the only lutenist to explore all modes or keys: Francesco Spinacino wrote a "Recercare de tutti li Toni" (Ricercar in all the Tones) as early as 1507.^[20] In the 17th century lutenist-composer John Wilson wrote a set of 30 preludes including 24 in all the major/minor keys.^[21][22] 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".^[23] Zarlino in his polemic with Galilei initially opposed equal temperament but eventually conceded to it in relation to the lute in his Sopplimenti musicali in 1588.

The first mention of equal temperament related to the twelfth root of two in the West appeared in Simon Stevin's manuscript Van De Spiegheling der singconst (c. 1605), published posthumously nearly three centuries later in 1884.^[24] 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.^[13] 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.^[25]

The following were Simon Stevin's chord length from Van de Spiegheling der singconst:^[26]

A generation later, French mathematician Marin Mersenne presented several equal tempered chord lengths obtained by Jean Beaugrand, Ismael Bouillaud, and Jean Galle.^[27]

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.^[28]

From 1450 to about 1800, plucked instrument players (lutenists and guitarists) generally favored equal temperament,^[29] 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 Prélude sur tous les tons, which enharmonically modulates through all keys.^[30] 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.^[31]

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. 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.^[b]

A precise equal temperament is possible using the 17th century Sabbatini method of splitting the octave first into three tempered major thirds.^[32] 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.^[33] Using beat rates, first proposed in 1749, became common after their diffusion by Helmholtz and Ellis in the second half of the 19th century.^[34] The ultimate precision was available with 2 decimal tables published by White in 1917.^[35] The intervals of 12-ET closely approximate some intervals in just intonation.^[36]

Chinese musicians developed 3-limit just intonation at least a century before He Chengtian.^[38] 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 81⁄64 could be made more consonant by tempering them to simpler ratios like 5⁄4, resulting in Europe developing a series of meantone temperaments.^[39]

• Equal temperament
• Just intonation
• Musical acoustics (the physics of music)
• Music and mathematics
• Microtonal music
• List of meantone intervals
• Diatonic and chromatic
• Electronic tuner
• Musical tuning

• 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 Archived 2009-02-15 at the Wayback Machine. (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