# 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 222, or 54.55 cents ( ).

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.

## History and use

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.[1] 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.[2] Contemporary advocates of 22 equal temperament include music theorist Paul Erlich.

## Notation

Circle of fifths in 22 tone equal temperament, "up/down" notation

Circle of fifths in the chromatic circle, "up/down" notation

22-EDO can be notated several ways. The first, Up/Down Notation, uses ups and downs in addition to sharps and flats, chord spellings may change(C, E, G is C major triad). This yields the following chromatic scale:

C, C, C, C,

D, E, E, E, E,

F, F, F, F,

G, G/A, G/A, G/A,

A, B, B, B, B, C

The second, Quarter Tone Notation, uses quarter tone notation to divide the notes of Up/Down Notation. However, some chord spellings may change(C, E , G is C major triad). This yields the following chromatic scale:

C, C , C/D, D ,

D, D , D/E, E , E,

F, F , F/G, G ,

G, G , G/A, A ,

A, A , A/B, B , B, C

The third, Porcupine Notation, introduces no new accidentals, but significantly changes chord spellings(C, E, G is C major triad). In addition, enharmonicities from 12-EDO are no longer valid. This yields the following chromatic scale:

C, C, D, D, D, E, E, E, F, F, F, G, G, G, G /A , A, A, A, B, B, B, C, C

## Interval size

The table below gives the sizes of some common intervals in 22 equal temperament. An interval shown with a shaded background — such as the septimal tritone — is one that is more than 1/4 of a step (approximately 13.6 cents) out of tune, when compared to the just ratio it approximates.

interval name size (steps) size (cents) midi just ratio just (cents) midi error (cents)
octave 22 1200 2:1 1200 0
major seventh 20 1090.91   15:8 1088.27   +02.64
17:10 wide major sixth 17 927.27   17:10 918.64 +08.63
major sixth 16 872.73   5:3 884.36   −11.63
perfect fifth 13 709.09   3:2 701.95   +07.14
septendecimal tritone 11 600.00   17:12 603.00 03.00
tritone 11 600.00 45:32 590.22   +09.78
septimal tritone 11 600.00 7:5 582.51   +17.49
11:8 wide fourth 10 545.45   11:80 551.32   05.87
375th subharmonic 10 545.45 512:375 539.10 +06.35
15:11 wide fourth 10 545.45 15:11 536.95   +08.50
perfect fourth 09 490.91   4:3 498.05   07.14
septendecimal supermajor third 08 436.36   22:17 446.36 −10.00
septimal major third 08 436.36 9:7 435.08   +01.28
diminished fourth 08 436.36 32:25 427.37   +08.99
undecimal major third 08 436.36 14:11 417.51   +18.86
major third 07 381.82   5:4 386.31   04.49
undecimal neutral third 06 327.27   11:90 347.41   −20.14
septendecimal supraminor third 06 327.27 17:14 336.13   08.86
minor third 06 327.27 6:5 315.64   +11.63
septendecimal augmented second 05 272.73   20:17 281.36 08.63
augmented second 05 272.73 75:64 274.58   01.86
septimal minor third 05 272.73 7:6 266.88   +05.85
septimal whole tone 04 218.18   8:7 231.17   −12.99
diminished third 04 218.18 256:225 223.46   05.28
septendecimal major second 04 218.18 17:15 216.69 +01.50
whole tone, major tone 04 218.18 9:8 203.91   +14.27
whole tone, minor tone 03 163.64   10:90 182.40   −18.77
neutral second, greater undecimal 03 163.64 11:10 165.00   01.37
1125th harmonic 03 163.64 1125:1024 162.85 +00.79
neutral second, lesser undecimal 03 163.64 12:11 150.64   +13.00
septimal diatonic semitone 02 109.09   15:14 119.44   −10.35
diatonic semitone, just 02 109.09 16:15 111.73   02.64
17th harmonic 02 109.09 17:16 104.95   +04.13
Arabic lute index finger 02 109.09 18:17 098.95   +10.14
septimal chromatic semitone 02 109.09 21:20 084.47   +24.62
chromatic semitone, just 01 054.55   25:24 070.67   −16.13
septimal third-tone 01 054.55 28:27 062.96   08.42
undecimal quarter tone 01 054.55 33:32 053.27   +01.27
septimal quarter tone 01 054.55 36:35 048.77   +05.78
diminished second 01 054.55 128:125 041.06   +13.49