22 equal temperament

In music, 22 equal temperament, called 22-tet, 22-edo, or 22-et, is the tempered scale derived by dividing the octave into 22 equal steps (equal frequency ratios). 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, Bosanquet 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.

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
septimal tritone	11	600	7:5	582.51	−17.49
11:8 wide fourth	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
(17:15) ratio	4	218.18	17:15	216.69	−1.50
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	163.64	11:10	165.00	1.37
neutral second, lesser undecimal	3	163.64	12:11	150.64	−13.00
diatonic semitone, just	2	109.09	16:15	111.73	2.64
chromatic semitone, just	1	54.55	25:24	70.67	16.13
Didymus' quarter-tone	1	54.55	32:31	54.96	0.41

shaded rows mark poor matches

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.

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 as well, nor does it fit the greater undecimal neutral second. Furthermore, 22-ET also provides no real match to the harmonic seventh; A multiple of 22, 88-ET, has an excellent seventh harmonic, although 31-ET or even 36-ET (proposed by Hába and Busoni) are much more practical. However, in terms of the ancient Greek enharmonic genus, 22-ET serves excellently.

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 temperament, where five major thirds make up a perfect twelfth (a perfect fifth plus an octave). That the porcupine comma is tempered out means that 22-et is a porcupine temperament, 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 50:49 is equated to a unison, they both equate the two septimal tritones of 7:5 and 10:7; because 64:63 is equated to a unison, neither of them distinguish between a dominant seventh chord and an otonal tetrad. As 22-et equates both of these commas to a unison, they are also made equivalent to each other: the septimal kleisma 225:224, which is the difference between the two $\left({\tfrac {50}{49}}\div {\tfrac {64}{63}}={\tfrac {225}{224}}\right)$, is also tempered out, so the septimal kleisma augmented triad is a chord of 22-et, as it also is of any meantone tuning. But not all septimal commas tempered out by 12-et are also tempered out by 22-et—1728:1715, the orwell comma, is tempered out by 12-et but not 22-et; and therefore the orwell tetrad is also a chord of 22-et.

External links

• Erlich, Paul, "Tuning, Tonality, and Twenty-Two Tone Temperament", William A. Sethares.

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" (Archived 2009-10-22), Proceedings of the Royal Society of London vol. 26 (March 1, 1877 to December 20, 1877) Taylor & Francis, London 1878, pp. 372-384. (Reproduced in Tagore, Sourindro Mohun, Hindu Music from Various Authors, Chowkhamba Sanskrit Series, Varanasi, India, 1965)