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

Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

 .

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

Music of India

The music of India includes multiple varieties of folk, popular, pop, classical music and R&B. India's classical music tradition, including Carnatic and Hindustani music, has a history spanning millennia and developed over several eras. It remains fundamental to the lives of Indians today as...

, Bosanquet noted that such an equal division was capable of representing 5-limit

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

 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

19 equal temperament

In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...

, 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 Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...	13	709.09	3:2	701.95	+7.14
inverted 11th harmonic	12	654.55	16:11	648.68	+5.87
septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7...	11	600	7:5	582.51	+17.49
11:8 wide fourth	10	545.45	11:8	551.32	−5.87
perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...	9	490.91	4:3	498.05	−7.14
septimal major third Septimal major third In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...	8	436.36	9:7	435.08	+1.28
major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...	7	381.82	5:4	386.31	−4.49
minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...	6	327.27	6:5	315.64	+11.63
septimal minor third Septimal minor third In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...	5	272.73	7:6	266.88	+5.85
septimal whole tone Septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its...	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 Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... , 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 Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...	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
28:27 semitome	1	54.55	28:27	62.96	−8.42
Didymus Didymus the Musician Didymus the Musician was a music theorist in Rome of the end of the 1st century BC or beginning of the 1st century AD, who combined elements of earlier theoretical approaches with an appreciation of the aspect of performance... ' 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.