135edo: Difference between revisions

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== Theory ==

135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s.

135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. 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 [[tempering out|tempers out]] the [[garischisma]] and the [[symbiotic comma]]. If we consider the full 13-limit, the flat-tending {{val| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.

Using the 135f [[val~~]] {{val| 135 214 313 379 467 '''499''' }}~~, ~~which tends flat, 135et [[tempering out|~~tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (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, it tempers out 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (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, 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 ===

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=== Subsets and supersets ===

Since 135 factors into {{factorization|135}}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.

== Regular temperament properties ==

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 == Theory ==
-edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s.
+edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. 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 [[tempering out|tempers out]] the [[garischisma]] and the [[symbiotic comma]]. If we consider the full 13-limit, the flat-tending {{val| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.
-Using the 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (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, it tempers out 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (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, 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 ===
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 === Subsets and supersets ===
 Since 135 factors into {{factorization|135}}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.
 == Regular temperament properties ==