135edo: Difference between revisions
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{{Infobox ET | |||
| Prime factorization = 3<sup>3</sup> × 5 | |||
| Step size = 8.88889¢ | |||
| Fifth = 79\135 (702.22¢) | |||
| Semitones = 13:10 (115.56¢ : 88.89¢) | |||
| Consistency = 7 | |||
}} | |||
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 [[EDO|equal division of the octave]] into 135 parts of about 8.89 [[cent]]s each. | 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 [[EDO|equal division of the octave]] into 135 parts of about 8.89 [[cent]]s each. | ||
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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 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. | 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]]. | 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]]. | ||
Revision as of 01:29, 12 November 2021
| ← 134edo | 135edo | 136edo → |
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
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [214 -135⟩ | [⟨135 214]] | -0.0843 | 0.0843 | 0.95 |
| 2.3.7 | 33554432/33480783, 40353607/40310784 | [⟨135 214 379]] | -0.0637 | 0.0747 | 0.84 |
| 2.3.7.11 | 19712/19683, 41503/41472, 43923/43904 | [⟨135 214 379 467]] | -0.0328 | 0.0840 | 0.94 |