In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of 2^1/22, or 54.55 cents.

The idea of dividing the octave into 22 steps of equal size seems to have originated with nineteenth century music theorist RHM Bosanquet. Inspired by the division of the octave into 22 unequal parts in the music theory of India, Bosenquet noted that such an equal division was capable of representing 5-limit music with tolerable accuracy. In this he was followed in the twentieth century by theorist José Würschmidt, who noted it as a possible next step after 19 equal temperament, and J. Murray Barbour in his classic survey of tuning history, Tuning and Temperament. Robert Holford Macdowall Bosanquet (1841-1913) was an English scientist and music theorist, brother of the philosopher Bernard Bosanquet. ... The music of India includes multiple varieties of folk, popular, pop, and classical music. ... Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ... In music, 19 equal temperament, called 19-tet, 19-edo, or 19-et, is the scale derived by dividing the octave into 19 equally large steps. ...

The 22-et system is in fact the third equal division, after 12 and 19, which is capable of tolerably dealing with 5-limit music. However, there is more to it than that; unlike 12 or 19 it is able to do rough justice to the 7- and 11-limits. While 31 equal temperament does much better, 22-et at least allows the use of these higher-limit harmonies. Moreover, 22-et, unlike 12 and 19, is not a meantone system. The net effect is that 22 allows, and to some extent even forces, the exploration of less familiar musical territory, yet is small enough that it can be used in live performances with suitably designed instruments, such as 22-tone guitars and the like. In music, 31 equal temperament, called 31-tet, 31-edo, or 31-et, is the scale derived by dividing the octave into 31 equally large steps. ... Meantone temperament is a system of musical tuning. ...

Properties of 22 equal temperament

Possibly the most striking characteristic of 22-et to those not used to it is that it does not "temper out" the syntonic comma of 81/80, and therefore is not a system of meantone temperament. It does, however, temper out the diaschisma, 2048/2025, the magic comma or small diesis, 3125/3072, and the porcupine comma, or maximal diesis, 250/243. In a diaschismic system, such as 12-et or 22-et, the diatonic tritone 45/32, which is a major third above a major whole tone representing 9/8, is equated to its inverted form, 64/45. That the magic comma is tempered out means that 22-et is a magic system, where five major thirds make up a 3. That the porcupine comma is tempered out means that 22-et is a porcupine system, where three minor whole tones (10/9 tones) give a fourth, and five give a minor sixth. The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... Meantone temperament is a system of musical tuning. ...

In the 7-limit 22-et tempers out certain commas also tempered out by 12-et; this relates 12 equal to 22 in a way different than the way in which meantone systems are akin to it. Both 50/49, the jubilee comma, and 64/63, the septimal comma, are tempered out in both systems. Hence because of 50/49 they both equate the two septimal tritons of 7/5 and 10/7, and because of 64/63 they both do not distinguish between a dominant seventh chord and a utonal tetrad. Hence both also temper out (50/49)/(64/63) = 225/224, the septimal kleisma, so that the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. A septimal comma not tempered out by 12-et which 22-et does temper out is 1728/1715, the orwell comma; and the orwell tetrad is also a chord of 22-et. Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones (multiples) or undertones (divisors) of a given fixed tone. ... In music, the ratio 225/224 is called the septimal kleisma. ...

Musical examples

Tibia by Paul Erlich

Glassic by Paul Erlich

Decatonic Swing by Paul Erlich and Ara Sarkissian

Revenge of the Inorganic Compounds by Igliashon Jones mp3 file metal rock

Night on Porcupine Mountain by Modest Moussorgsky as mutated by Gene Ward Smith ogg file

Exterior links

Erlich, Paul, Tuning, Tonality, and Twenty-Two Tone Temperament [1]

References

Barbour, James Murray, Tuning and temperament, a historical survey, East Lansing, Michigan State College Press, 1953 [c1951]

Bosanquet, R.H.M. On the Hindoo division of the octave, with additions to the theory of higher orders", Proceedings of the Royal Society of London vol. 25, 1877, pp. 540-541. Reproduced in Tagore, Sourindro Mohun, Hindu Music from Various Authors, Chowkhamba Sanskrit Series, Varanasi, India, 1965