135edo: Difference between revisions
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Expand the theory with 2.3.7.11 interpretation. For full 13-limit, talk about 135c and 135f instead since they make more sense |
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== Theory == | == Theory == | ||
135edo is consistent to the 7-odd-limit, but there is a large relative delta for 5th | 135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the 5th and the 13th harmonics. | ||
Using the 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et tempers out 32805/32768 ([[schisma]]) and 30517578125/29386561536 (quintriyo comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit. | |||
Using the 135c val {{val| 135 214 '''314''' 379 467 500 }}, which tends sharp, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; [[126/125]], [[10976/10935]], and 589824/588245 in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and 16384/16335 in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and 6656/6655 in the 13-limit. | |||
As every other step of the full 13-limit monster – [[270edo|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it tempers out the [[garischisma]] and the [[symbiotic comma]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 12:49, 23 July 2021
The 135 equal divisions of the octave (135edo), or the 135(-tone) equal temperament (135tet, 135et) when viewed from a regular temperament perspective, is the equal division of the octave into 135 parts of about 8.89 cents each.
Theory
135edo is consistent to the 7-odd-limit, but there is a large relative delta for the 5th and the 13th harmonics.
Using the 135f val ⟨135 214 313 379 467 499], which tends flat, 135et tempers out 32805/32768 (schisma) and 30517578125/29386561536 (quintriyo comma) in the 5-limit; 225/224, 3125/3087, and 28824005/28697814 in the 7-limit, 385/384, 540/539, 2200/2187, 12005/11979 and the quartisma in the 11-limit; 169/168 and 364/363 in the 13-limit.
Using the 135c val ⟨135 214 314 379 467 500], which tends sharp, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; 126/125, 10976/10935, and 589824/588245 in the 7-limit; 176/175, 441/440, 14641/14580 and 16384/16335 in the 11-limit; 196/195, 351/350, 352/351, 676/675, and 6656/6655 in the 13-limit.
As every other step of the full 13-limit monster – 270et, 135et probably makes more sense as a 2.3.7.11 subgroup temperament, where it tempers out the garischisma and the symbiotic comma.
Prime harmonics
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