22edt
| ← 21edt | 22edt | 23edt → |
22 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 22edt or 22ed3), is a nonoctave tuning system that divides the interval of 3/1 into 22 equal parts of about 86.5 ¢ each. Each step represents a frequency ratio of 31/22, or the 22nd root of 3. It supports mintaka temperament.
Like 11edt, both the octave and small whole tone (10/9) are about 10c off (sharp and flat respectively) dissonant but recognizable. Akin to 16edt with Blackwood, admitting the octave induces an interpretation into a tritave-based version of Whitewood temperament, therefore allowing the system to function as an octave stretch of 14edo. However, it can just as well be treated as a pure no-twos system, which is the main interpretation used in the below article.
22edt has good approximations of the 7th, 11th, 19th and 20th harmonics, being better for its size in the 3.7.11 subgroup than even 13edt is in 3.5.7. In this subgroup, it tempers out the commas 1331/1323 and 387420489/386683451, with the former comma allowing a hard 5L 2s (macrodiatonic) scale generated by 11/7, two of which are equated to 27/11 and three of which are equated to 9/7 up a tritave. This 9/7 can also serve as the generator for a 4L 5s (BPS Lambda) scale, supporting Bohlen-Pierce-Stearns harmony by tempering out 245/243, although its representation of the 3.5.7 subgroup is less accurate than that of 13edt, and tempered in the wrong direction relative to 13edt for ideal BPS.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | +0.0 | -19.8 | +2.8 | -1.6 | -31.5 | +22.8 | +3.2 | +18.2 | -37.3 | +20.2 | -26.8 | -31.6 | -27.6 | -8.7 |
| Relative (%) | +12.0 | +0.0 | -22.9 | +3.3 | -1.8 | -36.4 | +26.4 | +3.7 | +21.1 | -43.1 | +23.4 | -31.0 | -36.5 | -31.9 | -10.0 | |
| Steps (reduced) |
14 (14) |
22 (0) |
32 (10) |
39 (17) |
48 (4) |
51 (7) |
57 (13) |
59 (15) |
63 (19) |
67 (1) |
69 (3) |
72 (6) |
74 (8) |
75 (9) |
77 (11) | |
Intervals
The notation schemes below are based on the BPS-Lambda enneatonic scale presented in the symmetric (sLsLsLsLs, Cassiopeian) mode in J, and the Mintaka macrodiatonic scale presented in the macro-Phrygian (sLLLsLL) mode in E.
| Degree | Note (BPS-Lambda notation) | Note (Macrodiatonic notation) | Approximate 3.7.11 subgroup interval | cents value | hekts |
|---|---|---|---|---|---|
| 0 | J | E | 1/1 | 0 | 0 |
| 1 | J# = Kb | F | 81/77, 363/343 | 86.453 | 59.091 |
| 2 | K | Gb = Dx | 2673/2401, 6561/5929 | 172.905 | 118.182 |
| 3 | K# | E# = Abb | 343/297, 847/729 | 259.358 | 177.273 |
| 4 | Lb | F# | 11/9, 147/121 | 345.810 | 236.364 |
| 5 | L | G | 9/7 | 432.263 | 295.455 |
| 6 | L# = Mb | Ab = Ex | 729/539 | 518.715 | 354.545 |
| 7 | M | Fx = Bbb | 343/243 | 605.168 | 413.636 |
| 8 | M# | G# | 49/33, 121/81 | 691.620 | 472.727 |
| 9 | Nb | A | 11/7 | 778.073 | 531.818 |
| 10 | N | Bb | 81/49 | 864.525 | 590.909 |
| 11 | N# = Ob | Cb = Gx | 3773/2187, 6561/3773 | 950.978 | 650. |
| 12 | O | A# = Dbb | 49/27 | 1037.430 | 709.091 |
| 13 | O# | B | 21/11 | 1123.883 | 768.182 |
| 14 | Pb | C | 99/49, 243/121 | 1210.335 | 827.273 |
| 15 | P | Db = Ax | 729/343 | 1296.788 | 886.364 |
| 16 | P# = Qb | B# = Ebb | 539/243 | 1383.240 | 945.455 |
| 17 | Q | C# | 7/3 | 1469.693 | 1004.545 |
| 18 | Q# | D | 27/11, 121/49 | 1556.145 | 1063.636 |
| 19 | Rb | Eb | 891/343, 2187/847 | 1642.598 | 1122.727 |
| 20 | R | Fb = Cx | 2401/891, 5929/2187 | 1729.050 | 1181.818 |
| 21 | R# = Jb | D# = Gbb | 77/27, 343/121 | 1815.503 | 1240.909 |
| 22 | J | E | 3/1 | 1901.955 | 1300. |
Audio examples
A short composition by Wensik, based on the 7:9:11 chord and its inversion, 63:77:99.