576edo

From Xenharmonic Wiki
Jump to navigation Jump to search

576edo divides the octave into steps of 2.083 cents each.

Theory

Approximation of prime intervals in 576 EDO
Prime number 2 3 5 7 11 13 17 19 23 29 31 37 41 43
Error absolute (¢) +0.000 +0.128 -0.897 -0.076 +0.765 -0.944 -0.789 +0.404 +0.892 -0.411 +0.798 +0.739 +0.104 +0.982
relative (%) +0 +6 -43 -4 +37 -45 -38 +19 +43 -20 +38 +35 +5 +47
Steps (reduced) 576 (0) 913 (337) 1337 (185) 1617 (465) 1993 (265) 2131 (403) 2354 (50) 2447 (143) 2606 (302) 2798 (494) 2854 (550) 3001 (121) 3086 (206) 3126 (246)

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 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.