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.
History and use
[edit]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.[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
[edit]22-EDO can be notated several ways. The first, Ups And Downs Notation,[3] 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,
D, ^D/E♭, vD♯/^E♭, D♯/vE, E,
F, ^F/G♭, vF♯/^G♭, F♯/vG,
G, ^G/A♭, vG♯/^A♭, G♯/vA,
A, ^A/B♭, vA♯/^B♭, A♯/vB, B, C
The Pythagorean minor chord with 32/27 on C is still named Cm and still spelled C–E♭–G. But the 5-limit upminor chord uses the upminor 3rd 6/5 and is spelled C–^E♭–G. This chord is named C^m. Compare with ^Cm (^C–^E♭–^G).
The second, Quarter Tone Notation, uses half-sharps and half-flats instead of up and down arrows:
However, chords and some enharmonic equivalences are much different than they are in 12-EDO. For example, even though a 5-limit C minor triad is notated as C–E♭–G, C major triads are now C–E–G instead of C–E–G, and an A minor triad is now A–C–E even though an A major triad is still A–C♯–E. Additionally, while major seconds such as C–D are divided as expected into 4 quarter tones, minor seconds such as E–F and B–C are 1 quarter tone, not 2. Thus E♯ is now equivalent to F instead of F, F♭ is equivalent to E instead of E, F is equivalent to E, and E is equivalent to F. Furthermore, the note a fifth above B is not the expected F♯ but rather F or G, and the note that is a fifth below F is now B instead of B♭.
The third, Porcupine Notation, introduces no new accidentals, but significantly changes chord spellings (e.g. the 5-limit major triad is now C–E♯–G♯). In addition, enharmonic equivalences 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
[edit]The table below gives the sizes of some common intervals in 22 equal temperament. Intervals shown with a shaded background—such as the septimal tritones—are more than 1/4 of a step (approximately 13.6 cents) out of tune, when compared to the just ratios they approximate.
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 | + | 2.64||
septimal minor seventh | 18 | 981.818 | 7:4 | 968.82591 | + | 12.99||
17:10 wide major sixth | 17 | 927.27 | 17:10 | 918.64 | + | 8.63||
major sixth | 16 | 872.73 | 5:3 | 884.36 | −11.63 | ||
perfect fifth | 13 | 709.09 | 3:2 | 701.95 | + | 7.14||
septendecimal tritone | 11 | 600.00 | 17:12 | 603.00 | − | 3.00||
tritone | 11 | 600.00 | 45:32 | 590.22 | + | 9.78||
septimal tritone | 11 | 600.00 | 7:5 | 582.51 | +17.49 | ||
11:8 wide fourth | 10 | 545.45 | 11:8 | 551.32 | − | 5.87||
375th subharmonic | 10 | 545.45 | 512:375 | 539.10 | + | 6.35||
15:11 wide fourth | 10 | 545.45 | 15:11 | 536.95 | + | 8.50||
perfect fourth | 9 | 490.91 | 4:3 | 498.05 | − | 7.14||
septendecimal supermajor third | 8 | 436.36 | 22:17 | 446.36 | −10.00 | ||
septimal major third | 8 | 436.36 | 9:7 | 435.08 | + | 1.28||
diminished fourth | 8 | 436.36 | 32:25 | 427.37 | + | 8.99||
undecimal major third | 8 | 436.36 | 14:11 | 417.51 | +18.86 | ||
major third | 7 | 381.82 | 5:4 | 386.31 | − | 4.49||
undecimal neutral third | 6 | 327.27 | 11:9 | 347.41 | −20.14 | ||
septendecimal supraminor third | 6 | 327.27 | 17:14 | 336.13 | − | 8.86||
minor third | 6 | 327.27 | 6:5 | 315.64 | +11.63 | ||
septendecimal augmented second | 5 | 272.73 | 20:17 | 281.36 | − | 8.63||
augmented second | 5 | 272.73 | 75:64 | 274.58 | − | 1.86||
septimal minor third | 5 | 272.73 | 7:6 | 266.88 | + | 5.85||
septimal whole tone | 4 | 218.18 | 8:7 | 231.17 | −12.99 | ||
diminished third | 4 | 218.18 | 256:225 | 223.46 | − | 5.28||
septendecimal major second | 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 | 163.64 | 10:9 | 182.40 | −18.77 | ||
neutral second, greater undecimal | 3 | 163.64 | 11:10 | 165.00 | − | 1.37||
1125th harmonic | 3 | 163.64 | 1125:1024 | 162.85 | + | 0.79||
neutral second, lesser undecimal | 3 | 163.64 | 12:11 | 150.64 | +13.00 | ||
septimal diatonic semitone | 2 | 109.09 | 15:14 | 119.44 | −10.35 | ||
diatonic semitone, just | 2 | 109.09 | 16:15 | 111.73 | − | 2.64||
17th harmonic | 2 | 109.09 | 17:16 | 104.95 | + | 4.13||
Arabic lute index finger | 2 | 109.09 | 18:17 | 98.95 | +10.14 | ||
septimal chromatic semitone | 2 | 109.09 | 21:20 | 84.47 | +24.62 | ||
chromatic semitone, just | 1 | 54.55 | 25:24 | 70.67 | −16.13 | ||
septimal third-tone | 1 | 54.55 | 28:27 | 62.96 | − | 8.42||
undecimal quarter tone | 1 | 54.55 | 33:32 | 53.27 | + | 1.27||
septimal quarter tone | 1 | 54.55 | 36:35 | 48.77 | + | 5.78||
diminished second | 1 | 54.55 | 128:125 | 41.06 | +13.49 |
See also
[edit]References
[edit]- ^ 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).
- ^ Barbour, James Murray, Tuning and temperament, a historical survey, East Lansing, Michigan State College Press, 1953 [c1951].
- ^ "Ups_and_downs_notation", on Xenharmonic Wiki. Accessed 2023-8-12.
External links
[edit]- Erlich, Paul, "Tuning, Tonality, and Twenty-Two Tone Temperament", William A. Sethares.
- Pachelbel's Canon in 22edo (MIDI), Herman Miller