22 equal temperament
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In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the tempered scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of 21/22, 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.
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.
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[edit] Interval size
Here are the sizes of some common intervals in this system:
| interval name | size (steps) | size (cents) | just ratio | just (cents) | difference |
| perfect fifth | 13 | 709.09 | 3:2 | 701.95 | -7.14 |
| inverted 11th harmonic | 12 | 654.55 | 16:11 | 648.68 | -5.87 |
| tritone | 11 | 600 | 7:5 | 582.51 | -17.49 |
| eleventh harmonic | 10 | 545.45 | 11:8 | 551.32 | 5.87 |
| perfect fourth | 9 | 490.91 | 4:3 | 498.05 | 7.14 |
| septimal major third | 8 | 436.36 | 9:7 | 435.08 | -1.28 |
| major third | 7 | 381.82 | 5:4 | 386.31 | 4.49 |
| minor third | 6 | 327.27 | 6:5 | 315.64 | -11.63 |
| septimal minor third | 5 | 272.73 | 7:6 | 266.88 | -5.85 |
| septimal whole tone | 4 | 218.18 | 8:7 | 231.17 | 12.99 |
| whole tone, major tone | 4 | 218.18 | 9:8 | 203.91 | -14.27 |
| whole tone, minor tone | 3 | 163.63 | 10:9 | 182.40 | 18.77 |
| neutral second, greater undecimal | 3 | 163.63 | 11:10 | 165.00 | 1.37 |
| diatonic semitone, just | 2 | 109.09 | 16:15 | 111.73 | 2.64 |
| tone | 1 | 54.55 | 32:31 | 54.96 | 0.41 |
As can be seen by the table above, 22-et does not have a very good match for the smaller intervals such as the 8:7, 9:8 and 10:9; these intervals lie roughly near the middle of the intervals defined. It does offer a very close match to the greater undecimal neutral second and the diatonic semitone.
[edit] Comparison to other equal temperaments
The matches to the septimal major third and septimal minor third in 22-et are excellent, significantly better than in 19-et, and the major and minor thirds are still significantly better than in 12-et. Although the 31-et improves the matches of the perfect fifth, and major, minor, and septimal minor thirds, it does not fit the septimal major third at all, nor does it fit the greater undecimal neutral second.
[edit] Properties of 22 equal temperament
22-ET does not "temper out" the syntonic comma of 81/80, which is connected to the fact that it is not a 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 perfect fifth. 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.
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 from 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 an otonal 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.
[edit] External links
Erlich, Paul, Tuning, Tonality, and Twenty-Two Tone Temperament
[edit] 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. 26, 1879, pp. 272-284. Reproduced in Tagore, Sourindro Mohun, Hindu Music from Various Authors, Chowkhamba Sanskrit Series, Varanasi, India, 1965
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