22 equal temperament

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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). About this sound Play  Each step represents a frequency ratio of 222, or 54.55 cents (About this sound 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 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.

Practical aspects[edit]

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.

Interval size[edit]

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

interval name size (steps) size (cents) midi just ratio just (cents) midi error
17:10 wide major sixth 17 927.27 About this sound Play
 
17:10 920.64 08.63
major sixth 16 872.73 About this sound Play
 
5:3 884.36 About this sound Play
 
−11.63
perfect fifth 13 709.09 About this sound Play
 
3:2 701.95 About this sound Play
 
+07.14
septendecimal tritone 11 600.00 About this sound Play
 
17:12 603.00 03.00
septimal tritone 11 600.00 7:5 582.51 About this sound Play
 
+17.49
11:8 wide fourth 10 545.45 About this sound Play
 
11:80 551.32 About this sound Play
 
05.87
15:11 wide fourth 10 545.45 15:11 536.95 About this sound Play
 
+08.50
perfect fourth 09 490.91 About this sound Play
 
4:3 498.05 About this sound Play
 
07.14
septendecimal supermajor third 08 436.36 About this sound Play
 
22:17 446.36 −10.00
septimal major third 08 436.36 9:7 435.08 About this sound Play
 
+01.28
undecimal major third 08 436.36 14:11 417.51 About this sound Play
 
+18.86
major third 07 381.82 About this sound Play
 
5:4 386.31 About this sound Play
 
04.49
undecimal neutral third 06 327.27 About this sound Play
 
11:90 347.41 About this sound Play
 
−20.14
septendecimal supraminor third 06 327.27 17:14 336.13 About this sound Play
 
08.86
minor third 06 327.27 6:5 315.64 About this sound Play
 
+11.63
septendecimal augmented second 05 272.73 About this sound Play
 
20:17 281.36 08.63
septimal minor third 05 272.73 7:6 266.88 About this sound Play
 
+05.85
septimal whole tone 04 218.18 About this sound Play
 
8:7 231.17 About this sound Play
 
−12.99
septendecimal major second 04 218.18 17:15 216.69 +01.50
whole tone, major tone 04 218.18 9:8 203.91 About this sound Play
 
+14.27
whole tone, minor tone 03 163.64 About this sound Play
 
10:90 182.40 About this sound Play
 
−18.77
neutral second, greater undecimal 03 163.64 11:10 165.00 About this sound Play
 
01.37
neutral second, lesser undecimal 03 163.64 12:11 150.64 About this sound Play
 
+13.00
septimal diatonic semitone 02 109.09 About this sound Play
 
15:14 119.44 About this sound Play
 
−10.35
diatonic semitone, just 02 109.09 16:15 111.73 About this sound Play
 
02.64
17th harmonic 02 109.09 17:16 104.95 About this sound Play
 
+04.13
Arabic lute index finger 02 109.09 18:17 098.95 About this sound Play
 
+10.14
septimal chromatic semitone 02 109.09 21:20 084.47 About this sound Play
 
+24.62
chromatic semitone, just 01 054.55 About this sound Play
 
25:24 070.67 About this sound Play
 
−16.13
septimal third-tone 01 054.55 28:27 062.96 About this sound Play
 
08.42
undecimal quarter tone 01 054.55 33:32 053.27 About this sound Play
 
+01.27
septimal quarter tone 01 054.55 36:35 048.77 About this sound Play
 
+05.78

External links[edit]

References[edit]

  • 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)