576edo: Difference between revisions

{{EDO intro|576}}

576 is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is known as a [[Highly composite equal division #Highly factorable numbers|highly factorable edo]], which enables it to be played through JI-agnostic approaches that make use of its divisors (see [[#Subsets and supersets]] section below). This approach may be preferrable since the [[patent val]] will create sequences that fall aside by 1\576 of each other{{clarify}}, which may not "live up to the spirit" of a composite number like 576.  

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 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.

 

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.

 

576edo supports a messed-up variant of the [[rectified hebrew]] scale^[which?], but with step hardness of 5:3 instead of 3:2, and in which 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.

576edo's nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, 9, 12, 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 {{w|Byzantine music|Byzantine chanting}}, has been theoreticized by {{w|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 the compositeness, it may be preferrable to make references to smaller edos instead of using the best approximation.  

|+Table of rank-2 temperaments by generator

! Periods<br>per 8ve

! Associated<br>Ratio*

| 239\576<br>(1\576)

| 497.916<br>(2.083)

| 4/3<br>(32805/32768)

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct