576edo: Difference between revisions
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Despite having bad 5/4, 576edo is [[consistent]] in the 7-limit. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the [[septimal ennealimma]], 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex - [99, -66, 2⟩, [110, -57, -7⟩, and [88, -75, 11⟩. | Despite having bad 5/4, 576edo is [[consistent]] in the 7-limit. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the [[septimal ennealimma]], 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex - [99, -66, 2⟩, [110, -57, -7⟩, and [88, -75, 11⟩. | ||
In the 5-limit, the patent val supports [[amity]] temperament | In the 5-limit, 576edo provides the [[optimal patent val]] for the [[atomic]] temperament and also supports [[amity]] temperament. The 576c val supports [[maquila]]. | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Amity]] | [[Category:Amity]] | ||
[[Category:Atomic]] | |||
Revision as of 12:50, 18 September 2022
| ← 575edo | 576edo | 577edo → |
Theory
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.128 | -0.897 | -0.076 | +0.765 | -0.944 | -0.789 | +0.404 | +0.892 | -0.411 | +0.798 |
| Relative (%) | +0.0 | +6.2 | -43.1 | -3.6 | +36.7 | -45.3 | -37.9 | +19.4 | +42.8 | -19.7 | +38.3 | |
| Steps (reduced) |
576 (0) |
913 (337) |
1337 (185) |
1617 (465) |
1993 (265) |
2131 (403) |
2354 (50) |
2447 (143) |
2606 (302) |
2798 (494) |
2854 (550) | |
576 is a near-highly composite number which is equal to 24 squared, which in itself is double the world-predominant 12edo. It's xenharmonic divisors (that is, besides 12edo and its subsets) are 8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288. Some of these have been put into practical use. 72edo has been used in Byzantine chanting, has been theoreticized by Alois Haba and Ivan Wyschnegradsky, and has been used by jazz musician Joe Maneri. 96edo has been used by Julian Carrillo. Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation, that is the patent val. In fact, this approach may be preferrable since the patent val will create sequences that fall aside by 1\576 of each other, which may not "live up to the spirit" of a composite number like 576.
Regular temperament-based approach
Nonetheless, 576edo does offer simple interpretations.
Despite having bad 5/4, 576edo is consistent in the 7-limit. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the septimal ennealimma, 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from 9edo. However, other commas being tempered out are far more complex - [99, -66, 2⟩, [110, -57, -7⟩, and [88, -75, 11⟩.
In the 5-limit, 576edo provides the optimal patent val for the atomic temperament and also supports amity temperament. The 576c val supports maquila.