576edo: Difference between revisions

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| Prime factorization = 2<sup>6</sup> × 3<sup>2</sup>

| Step size = 2.08333¢

| Fifth = 337\576 (702.~~08333¢~~)

| Fifth = 337\576 (702.08¢)

| ~~Major 2nd~~ = ~~98\576~~ (~~204~~.~~16666¢~~)

| Semitones = 55:43 (114.58¢ : 89.58¢)

| Consistency = 7

}}

==Theory==

== Theory ==

~~{{Harmonics in equal|576}}~~

576 is a near-highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. Its xenharmonic divisors (that is, besides 12edo and its subsets) are {{EDOs| 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 [[wikipedia: Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia: Alois Hába|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. 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.

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 {{EDOs|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 [[~~Wikipedia~~:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|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⟩~~.

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 – {{monzo| 99 -66 2 }}, {{monzo| 110 -57 -7 }}, and {{monzo| 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]].

In the 5-limit, 576edo supports 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|garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.

~~The 576ccd val,~~ {{~~Val~~|576 ~~913 1336 1618~~}}~~, provides the [[optimal patent val]] for the [[Garibaldi temperament|garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.~~

=== Prime harmonics ===

[[Category:Equal divisions of the octave|###]]

[[Category:Amity]]

[[Category:Atomic]]

@@ Line 2: / Line 2: @@
 | Prime factorization = 2<sup>6</sup> × 3<sup>2</sup>
 | Step size = 2.08333¢
-| Fifth = 337\576 (702.08333¢)
+| Fifth = 337\576 (702.08¢)
-| Major 2nd = 98\576 (204.16666¢)
+| Semitones = 55:43 (114.58¢ : 89.58¢)
 | Consistency = 7
 }}
 {{EDO intro|576}}
-==Theory==
+== Theory ==
-{{Harmonics in equal|576}}
+is a near-highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. Its xenharmonic divisors (that is, besides 12edo and its subsets) are {{EDOs| 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 [[wikipedia: Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia: Alois Hába|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. 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.
-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 {{EDOs|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 [[Wikipedia:Byzantine music|Byzantine chanting]], has been theoreticized by [[wikipedia:Alois Hába|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⟩.
+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 – {{monzo| 99 -66 2 }}, {{monzo| 110 -57 -7 }}, and {{monzo| 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]].
+In the 5-limit, 576edo supports 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|garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.
-The 576ccd val, {{Val|576 913 1336 1618}}, provides the [[optimal patent val]] for the [[Garibaldi temperament|garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.
+=== Prime harmonics ===
+{{Harmonics in equal|576|columns=11}}
 [[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Amity]]
 [[Category:Atomic]]