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22 equal temperament
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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 22√2, or 54.55 cents (ⓘ).
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 and 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 needs to slightly change the D note. These discrepancies arise because, unlike 12-ET, 22-ET does not temper out the syntonic comma of 81/80, but instead exaggerates its size by mapping it to one step.
In the 7-limit, the septimal minor seventh (7/4) can be distinguished from the sum of a fifth (3/2) and a minor third (6/5), and 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: meantone systems tune the fifth flat so that intervals of 5 are simple while intervals of 7 are complex, superpythagorean systems have the fifth tuned sharp 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 (and by extension 53 equal temperament), 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.
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 use of a 22-tone unequal division of the octave in the music theory of India, Bosanquet noted that a 22-tone 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 and electronic music composer Sevish.
22-EDO can be notated several ways. The first, Ups And Downs Notation, uses up and down arrows, written as a caret and a lower-case "v", usually in a sans-serif font. One arrow equals one edostep. In note names, the arrows come first, to facilitate chord naming. This yields the following chromatic scale:
C, ^C/D♭, vC♯/^D♭, C♯/vD,
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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 22√2, or 54.55 cents (ⓘ).
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 and 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 needs to slightly change the D note. These discrepancies arise because, unlike 12-ET, 22-ET does not temper out the syntonic comma of 81/80, but instead exaggerates its size by mapping it to one step.
In the 7-limit, the septimal minor seventh (7/4) can be distinguished from the sum of a fifth (3/2) and a minor third (6/5), and 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: meantone systems tune the fifth flat so that intervals of 5 are simple while intervals of 7 are complex, superpythagorean systems have the fifth tuned sharp 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 (and by extension 53 equal temperament), 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.
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 use of a 22-tone unequal division of the octave in the music theory of India, Bosanquet noted that a 22-tone 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 and electronic music composer Sevish.
22-EDO can be notated several ways. The first, Ups And Downs Notation, uses up and down arrows, written as a caret and a lower-case "v", usually in a sans-serif font. One arrow equals one edostep. In note names, the arrows come first, to facilitate chord naming. This yields the following chromatic scale:
C, ^C/D♭, vC♯/^D♭, C♯/vD,