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).  Play  Each step represents a frequency ratio of ²²√2, or 54.55 cents ( Play ).

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 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 survey of tuning history, Tuning and Temperament. Contemporary advocates of 22 equal temperament include music theorist Paul Erlich.

Practical aspects

When composing with 22-ET, one needs to take into account different facts. Considering the 5-limit, there is a difference between 3 fifths and the sum of 1 fourth + 1 major third. It means that, starting from C, there are two A's - one 16 steps and one 17 steps away. There is also a difference between a major tone and a minor tone. In C major, the second note (D) will be 4 steps away. However, in A minor, where A is 6 steps below C, the fourth note (D) will be 9 steps above A, so 3 steps above C. So when switching from C major to A minor, one need to slightly change the note D. These discrepancies arise because, unlike 12-ET, 22-ET does not temper out the syntonic comma of 81/80, and in fact exaggerates its size by mapping it to one step.

Extending 22-ET to the 7-limit, we find the septimal minor seventh (7/4) can be distinguished from the sum of a fifth (3/2) and a minor third (6/5). Also the septimal subminor third (7/6) is different from the minor third (6/5). This mapping tempers out the septimal comma of 64/63, which allows 22-ET to function as a "Superpythagorean" system where four stacked fifths are equated with the septimal major third (9/7) rather than the usual pental third of 5/4. This system is a "mirror image" of septimal meantone in many ways. Instead of tempering the fifth narrow so that intervals of 5 are simple while intervals of 7 are complex, the fifth is tempered wide so that intervals of 7 are simple while intervals of 5 are complex. The enharmonic structure is also reversed: sharps are sharper than flats, similar to Pythagorean tuning, but to a greater degree.

Finally, 22-ET has a good approximation of the 11th harmonic, and is in fact the smallest equal temperament to be consistent in the 11-limit.

The net effect is that 22-ET allows (and to some extent even forces) the exploration of new musical territory, while still having excellent approximations of common practice consonances.

Interval size

Here are the sizes of some common intervals in this system:

interval name	size (steps)	size (cents)	midi	just ratio	just (cents)	midi	error
17:10 wide major sixth	17	927.27	Play	17:10	920.64		−08.63
major sixth	16	872.73	Play	5:3	884.36	Play	−11.63
perfect fifth	13	709.09	Play	3:2	701.95	Play	+07.14
septendecimal tritone	11	600.00	Play	17:12	603.00		−03.00
septimal tritone	11	600.00		7:5	582.51	Play	+17.49
11:8 wide fourth	10	545.45	Play	11:80	551.32	Play	−05.87
15:11 wide fourth	10	545.45		15:11	536.95	Play	+08.50
perfect fourth	09	490.91	Play	4:3	498.05	Play	−07.14
septendecimal supermajor third	08	436.36	Play	22:17	446.36		−10.00
septimal major third	08	436.36		9:7	435.08	Play	+01.28
undecimal major third	08	436.36		14:11	417.51	Play	+18.86
major third	07	381.82	Play	5:4	386.31	Play	−04.49
undecimal neutral third	06	327.27	Play	11:90	347.41	Play	−20.14
septendecimal supraminor third	06	327.27		17:14	336.13	Play	−08.86
minor third	06	327.27		6:5	315.64	Play	+11.63
septendecimal augmented second	05	272.73	Play	20:17	281.36		−08.63
septimal minor third	05	272.73		7:6	266.88	Play	+05.85
septimal whole tone	04	218.18	Play	8:7	231.17	Play	−12.99
septendecimal major second	04	218.18		17:15	216.69		+01.50
whole tone, major tone	04	218.18		9:8	203.91	Play	+14.27
whole tone, minor tone	03	163.64	Play	10:90	182.40	Play	−18.77
neutral second, greater undecimal	03	163.64		11:10	165.00	Play	−01.37
neutral second, lesser undecimal	03	163.64		12:11	150.64	Play	+13.00
septimal diatonic semitone	02	109.09	Play	15:14	119.44	Play	−10.35
diatonic semitone, just	02	109.09		16:15	111.73	Play	−02.64
17th harmonic	02	109.09		17:16	104.95	Play	+04.13
Arabic lute index finger	02	109.09		18:17	098.95	Play	+10.14
septimal chromatic semitone	02	109.09		21:20	084.47	Play	+24.62
chromatic semitone, just	01	054.55	Play	25:24	070.67	Play	−16.13
septimal third-tone	01	054.55		28:27	062.96	Play	−08.42
undecimal quarter tone	01	054.55		33:32	053.27	Play	+01.27
septimal quarter tone	01	054.55		36:35	048.77	Play	+05.78

External links

• Erlich, Paul, "Tuning, Tonality, and Twenty-Two Tone Temperament", William A. Sethares.
• Pachelbel's Canon in 22edo, Herman Miller
• Good devil, Johann Elsass

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)