576edo

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← 575edo576edo577edo →
Prime factorization 26 × 32
Step size 2.08333¢ 
Fifth 337\576 (702.083¢)
Semitones (A1:m2) 55:43 (114.6¢ : 89.58¢)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of prime harmonics in 576edo
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

Table of rank-2 temperaments by generator
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