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 ( 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 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 survey of tuning history, Tuning and Temperament.

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
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
15:14 semitone	2	109.09	15:14	119.44	−10.35
diatonic semitone, just	2	109.09	16:15	111.73	−2.64
21:20 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
Didymus' quarter-tone	1	54.55	32:31	54.96	−0.41

shaded rows mark poor matches

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)