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). audio speaker iconPlay  Each step represents a frequency ratio of 222, or 54.55 cents (audio speaker iconPlay ).

When composing with 22-ET, one needs to take into account a variety of considerations. 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.


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, Ehalf flat, G is C major triad). This yields the following chromatic scale:

C, Chalf sharp, C/D, Dhalf flat,

D, Dhalf sharp, D/E, Ehalf flat, E,

F, Fhalf sharp, F/G, Ghalf flat,

G, Ghalf sharp, G/A, Ahalf flat,

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

See also


  1. ^ 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).
  2. ^ Barbour, James Murray, Tuning and temperament, a historical survey, East Lansing, Michigan State College Press, 1953 [c1951].

External links

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