12 equal temperament

[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.

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."

[3] Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu[4] and provides textual quotations as evidence.

"[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.

[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.

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.

][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.

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 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.

[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.

[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.

[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.

[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][clarification needed] Angelo Michele Bartolotti published a series of passacaglias in all keys, with connecting enharmonically modulating passages.

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.

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.

(In England, some cathedral organists and choirmasters held out against it even after that date; Samuel Sebastian Wesley, for instance, opposed it all along.

)[citation needed] A precise equal temperament is possible using the 17th century Sabbatini method of splitting the octave first into three tempered major thirds.

Tuning without beat rates but employing several checks, achieving virtually modern accuracy, was already done in the first decades of the 19th century.

[35] It is in the environment of equal temperament that the new styles of symmetrical tonality and polytonality, atonal music such as that written with the twelve tone technique or serialism, and jazz (at least its piano component) developed and flourished.

Because the major tone (⁠ 9 /8⁠) is simply two perfect fifths minus an octave, and its inversion, the Pythagorean minor seventh (⁠ 16 /9⁠), is simply two perfect fourths combined, they, for the most part, retain the accuracy of their predecessors; the error is doubled, but it remains small – so small, in fact, that humans cannot perceive it.

The major triad, therefore, sounds in tune as its frequency ratio is approximately 4:5:6, further, merged with its first inversion, and two sub-octave tonics, it is 1:2:3:4:5:6, all six lowest natural harmonics of the bass tone.

[citation needed] The eleventh harmonic (⁠ 11 /8⁠), at 551.32 cents, falls almost exactly halfway between the nearest two equally-tempered intervals in 12 ET and therefore is not approximated by either.

[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[39] could be made more consonant by tempering them to simpler ratios like 5⁄4, resulting in Europe developing a series of meantone temperaments that slightly modified the interval sizes but could still be viewed as an approximate of 12-TEDO.

Due to meantone temperaments' tendency to concentrate error onto one enharmonic perfect fifth, making it very dissonant, European music theorists, such as Andreas Werckmeister, Johann Philipp Kirnberger, Francesco Antonio Vallotti, and Thomas Young, created various well temperaments with the goal of dividing up the commas in order to reduce the dissonance of the worst-affected intervals.