1236edo: Difference between revisions
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Clarify, simplify comma basis, note it being an atomic system, and replace irrelevant number theory with factorization and subset edos |
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The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[ | The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. | ||
1236 = 2<sup>2</sup> × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. | 1236 = 2<sup>2</sup> × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. | ||
Revision as of 16:33, 19 April 2022
The 1236 divisions of the octave divides the octave into 1236 equal parts of 0.9709 cents each. It is a zeta peak edo, though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-odd-limit, with a 17-limit comma basis of {2601/2600, 4096/4095, 5832/5831, 6656/6655, 9801/9800, 105644/105625}.
1236 = 22 × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an atomic system.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.013 | +0.094 | +0.106 | +0.138 | +0.249 | -0.101 | -0.426 | -0.119 | -0.451 | -0.375 |
| Relative (%) | +0.0 | -1.4 | +9.7 | +10.9 | +14.3 | +25.7 | -10.4 | -43.8 | -12.3 | -46.5 | -38.7 | |
| Steps (reduced) |
1236 (0) |
1959 (723) |
2870 (398) |
3470 (998) |
4276 (568) |
4574 (866) |
5052 (108) |
5250 (306) |
5591 (647) |
6004 (1060) |
6123 (1179) | |