576edo
| ← 575edo | 576edo | 577edo → |
576 equal divisions of the octave (abbreviated 576edo or 576ed2), also called 576-tone equal temperament (576tet) or 576 equal temperament (576et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 576 equal parts of about 2.08 ¢ each. Each step represents a frequency ratio of 21/576, or the 576th root of 2.
Theory
576edo is consistent in the 7-odd-limit, though the error on harmonic 5 is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the septimal ennealimma, assining 7/6 to 2\9, as well as [99 -66 0 2⟩, [110 -57 0 -7⟩ , and [88 -75 0 11⟩. 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.
In higher limits, the 2.3.7 subgroup can be used with optional additions of 19 or 29, or fractional subgroups using 13/10.
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
Since 576 factors as 26 × 32, 576edo has subset edos 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288, of which 12, 24, 72, and 96 are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a highly factorable edo.
1152edo, which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.
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