576edo
Theory
576 is equal to 24 squared, which in itself is double the world-predominant 12edo. It is known as a highly factorable edo, which enables it to be played through JI-agnostic approaches that make use of its divisors (see #Subsets and supersets section below). This approach may be preferrable since the patent val will create sequences that fall aside by 1\576 of each other[clarification needed], which may not "live up to the spirit" of a composite number like 576.
Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4, 576edo is consistent in the 7-odd-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⟩. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.
In the 5-limit, the patent val of 576edo supports the atomic temperament and the amity temperament. The 576c val supports maquila. The 576ccd val, ⟨576 913 1336 1618], is a tuning for the garibaldi temperament in the 7-limit. In addition, in this case 5/4 comes from 72edo, and 7/4 comes form 288edo.
576edo supports a messed-up variant of the rectified hebrew scale[which?], but with step hardness of 5:3 instead of 3:2, and in which 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.
Prime harmonics
| 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) | |
Subsets and supersets
576edo's nontrivial divisors are 2, 3, 4, 6, 8, 9, 12, 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 the compositeness, it may be preferrable to make references to smaller edos instead of using the best approximation.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 163\576 | 339.583 | 243/200 | Amity (576) |
| 12 | 239\576 (1\576) |
497.916 (2.083) |
4/3 (32805/32768) |
Atomic (576) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct