22edt: Difference between revisions
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'''22edt''' is the '''equal division of the third harmonic''' ([[edt]]) into '''22 tones''', each 86.4525 [[cent]]s in size. | '''22edt''' is the '''equal division of the third harmonic''' ([[edt]]) into '''22 tones''', each 86.4525 [[cent]]s in size. | ||
22edt has good approximations of the 7th, 11th, 19th and 20th harmonics. | 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 (3/1-equivalent)|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 (3/1-equivalent)|4L 5s]] (BPS enneatonic) scale, supporting [[Bohlen-Pierce-Stearns]] harmony by tempering out [[245/243]], although its representation fo the 3.5.7 subgroup is less accurate than that of 13edt. | ||
Like [[11edt]], both the [[octave]] and [[small whole tone]] ([[10/9]]) are about 10c off (sharp and flat respectively) dissonant but recognizable. | 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. | ||
{{Harmonics in equal|22|3|1|intervals=prime|columns=15}} | {{Harmonics in equal|22|3|1|intervals=prime|columns=15}} | ||
Revision as of 06:36, 21 August 2024
| ← 21edt | 22edt | 23edt → |
22edt is the equal division of the third harmonic (edt) into 22 tones, each 86.4525 cents in size.
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 enneatonic) scale, supporting Bohlen-Pierce-Stearns harmony by tempering out 245/243, although its representation fo the 3.5.7 subgroup is less accurate than that of 13edt.
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.
| 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
| Steps | Cents | hekts |
|---|---|---|
| 1 | 86.453 | 59.091 |
| 2 | 172.905 | 118.182 |
| 3 | 259.358 | 177.273 |
| 4 | 345.81 | 236.364 |
| 5 | 432.263 | 295.4545 |
| 6 | 518.715 | 354.5455 |
| 7 | 605.168 | 413.636 |
| 8 | 691.620 | 472.727 |
| 9 | 778.073 | 531.818 |
| 10 | 864.525 | 590.909 |
| 11 | 950.978 | 650 |
| 12 | 1037.430 | 709.091 |
| 13 | 1123.883 | 768.182 |
| 14 | 1210.335 | 827.273 |
| 15 | 1296.788 | 886.364 |
| 16 | 1383.24 | 945.4545 |
| 17 | 1469.693 | 1004.5455 |
| 18 | 1556.145 | 1063.636 |
| 19 | 1642.598 | 1122.727 |
| 20 | 1729.05 | 1181.818 |
| 21 | 1815.503 | 1240.909 |
| 22 | 1901.955 | 1300 |