576edo: Difference between revisions
Jump to navigation
Jump to search
Cleanup; clarify the title row of the rank-2 temp table; -redundant categories; and mark a few more things to clarify |
→Theory: cleanup |
||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4|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 {{monzo| 99 -66 0 2 }}, {{monzo| 110 -57 0 -7 }} , and {{monzo| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{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 [[19/16|19]] or [[29/16|29]], or fractional subgroups using [[13/10]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 15: | Line 11: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
576edo | Since 576 factors as {{Factorization|576}}, 576edo has subset edos {{EDOs| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is also a [[Highly composite equal division#Highly factorable numbers|highly factorable]] edo. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
Revision as of 12:46, 11 February 2024
| ← 575edo | 576edo | 577edo → |
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 also a highly factorable edo.
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