11664edo: Difference between revisions
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11664edo is a very strong 7-limit system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[ | 11664edo is a very strong [[7-limit]] system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the [[27-odd-limit]] and with a lower [[23-limit]] relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, giving itself another edge over similar systems. | ||
Some of the simpler commas [[tempering out|tempered out]] include [[123201/123200]] and [[1990656/1990625]] in the [[13-limit]]; [[194481/194480]] and [[336141/336140]] in the [[17-limit]]; 23409/23408 and 89376/89375 in the [[19-limit]]; 43264/43263, 71875/71874, and 76545/76544 in the [[23-limit]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|11664| | {{Harmonics in equal|11664|intervals=prime|columns=9}} | ||
{{Harmonics in equal|11664|intervals=prime|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 11664edo (continued)}} | |||
=== Subsets and supersets === | |||
11664 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>6</sup> }}. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]]. | |||
Latest revision as of 13:20, 3 August 2025
| ← 11663edo | 11664edo | 11665edo → |
11664 equal divisions of the octave (abbreviated 11664edo or 11664ed2), also called 11664-tone equal temperament (11664tet) or 11664 equal temperament (11664et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 11664 equal parts of about 0.103 ¢ each. Each step represents a frequency ratio of 21/11664, or the 11664th root of 2.
11664edo is a very strong 7-limit system, with a lower 7-limit relative error than any division until 18355. It is a zeta peak edo unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27-odd-limit and with a lower 23-limit relative error than any division until 16808. Aside from this peculiar double threat property, it is also very composite, giving itself another edge over similar systems.
Some of the simpler commas tempered out include 123201/123200 and 1990656/1990625 in the 13-limit; 194481/194480 and 336141/336140 in the 17-limit; 23409/23408 and 89376/89375 in the 19-limit; 43264/43263, 71875/71874, and 76545/76544 in the 23-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0003 | +0.0032 | +0.0013 | +0.0195 | +0.0073 | -0.0171 | +0.0178 | +0.0178 |
| Relative (%) | +0.0 | -0.3 | +3.1 | +1.2 | +19.0 | +7.1 | -16.7 | +17.3 | +17.3 | |
| Steps (reduced) |
11664 (0) |
18487 (6823) |
27083 (3755) |
32745 (9417) |
40351 (5359) |
43162 (8170) |
47676 (1020) |
49548 (2892) |
52763 (6107) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.0504 | +0.0262 | -0.0066 | -0.0501 | +0.0049 | +0.0284 | -0.0478 | -0.0153 | -0.0124 |
| Relative (%) | -49.0 | +25.4 | -6.4 | -48.7 | +4.8 | +27.6 | -46.4 | -14.9 | -12.0 | |
| Steps (reduced) |
56663 (10007) |
57786 (11130) |
60763 (2443) |
62490 (4170) |
63292 (4972) |
64789 (6469) |
66810 (8490) |
68615 (10295) |
69176 (10856) | |
Subsets and supersets
11664 factors into primes as 23 × 36. Among its divisiors are 12, 16, 24, 27, 72, 81 and 243.