576edo: Difference between revisions

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~~576edo~~ is ~~an excellent 2~~.3.~~7 subgroup tuning~~.

576 is a highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It's xenharmonic divisors are {{EDOs|8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288}}. Some of these like 72 and 96 have been put into historical use. Its approximation to the perfect fifth is just one step above the 12edo fifth.

~~576~~ is ~~a highly composite number which is equal to 24 squared~~, ~~which in itself is double~~ the ~~world-predominant~~ [[~~12edo~~]]~~. It's xenharmonic divisors are {{EDOs|8~~, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, ~~144~~, ~~192~~, and ~~288}}~~. ~~Some of these like 72~~ and ~~96 have been put into historical use~~. ~~Its approximation to~~ the ~~perfect fifth is just one step above~~ the ~~12edo fifth~~.

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⟩.

In the 11-limit, using the 576c and 576e vals, it tempers out [[9801/9800]]. Specifying on 576c removes [[3136/3125]], and on 576e removes [[1375/1372]]. In the 19-limit, 576e val eliminates [[715/714]] and [[1225/1224]]. In addition, using the 576e val preserves the [[24edo]] mapping for [[11/8]].

Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation, that is the patent val. For example, using the ⟨576 912 1616 1992] [[val]] for representing the 2.3.7.11 subgroup makes references to 24edo and 36edo: ⟨1\1 7\12 11\24 29\36] when octave reduced. 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 sound like an untempered comma.

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 {{primes in edo|576|columns=14}}
-edo is an excellent 2.3.7 subgroup tuning.
+is a highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It's xenharmonic divisors are {{EDOs|8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288}}. Some of these like 72 and 96 have been put into historical use. Its approximation to the perfect fifth is just one step above the 12edo fifth.
-is a highly composite number which is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It's xenharmonic divisors are {{EDOs|8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288}}. Some of these like 72 and 96 have been put into historical use. Its approximation to the perfect fifth is just one step above the 12edo fifth.
+edo 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⟩.
+In the 11-limit, using the 576c and 576e vals, it tempers out [[9801/9800]]. Specifying on 576c removes [[3136/3125]], and on 576e removes [[1375/1372]]. In the 19-limit, 576e val eliminates [[715/714]] and [[1225/1224]]. In addition, using the 576e val preserves the [[24edo]] mapping for [[11/8]].
 Because of composition, it may be preferrable to make references to smaller EDOs instead of using the best approximation, that is the patent val. For example, using the ⟨576 912 1616 1992] [[val]] for representing the 2.3.7.11 subgroup makes references to 24edo and 36edo: ⟨1\1 7\12 11\24 29\36] when octave reduced. 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 sound like an untempered comma.