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 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.

Interval size

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

interval name	size (steps)	size (cents)	just ratio	just (cents)	error
perfect fifth	13	709.09	3:2	701.95	+7.14
septendecimal tritone	11	600	17:12	603.00	−3.00
septimal tritone	11	600	7:5	582.51	+17.49
11:8 wide fourth	10	545.45	11:8	551.32	−5.87
15:11 wide fourth	10	545.45	15:11	536.95	+8.50
perfect fourth	9	490.91	4:3	498.05	−7.14
septendecimal supermajor third	8	436.36	22:17	446.36	−10.00
septimal major third	8	436.36	9:7	435.08	+1.28
undecimal major third	8	436.36	14:11	417.51	+18.86
major third	7	381.82	5:4	386.31	−4.49
undecimal neutral third	6	327.27	11:9	347.41	−20.14
septendecimal supraminor third	6	327.27	17:14	336.13	−8.86
minor third	6	327.27	6:5	315.64	+11.63
septendecimal augmented second	5	272.73	20:17	281.36	−8.63
septimal minor third	5	272.73	7:6	266.88	+5.85
septimal whole tone	4	218.18	8:7	231.17	−12.99
septendecimal major second	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	163.64	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
septimal diatonic semitone	2	109.09	15:14	119.44	−10.35
diatonic semitone, just	2	109.09	16:15	111.73	−2.64
17th harmonic	2	109.09	17:16	104.95	+4.13
Arabic lute index finger	2	109.09	18:17	98.95	+10.14
septimal chromatic semitone	2	109.09	21:20	84.47	+24.62
chromatic semitone, just	1	54.55	25:24	70.67	−16.13
septimal third-tone	1	54.55	28:27	62.96	−8.42
undecimal quarter tone	1	54.55	33:32	53.27	+1.27
septimal quarter tone	1	54.55	36:35	48.77	+5.78

External links

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

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)