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. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et ~~probably makes more sense as a~~ [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]], ~~where~~ it is characterized by [[tempering out]] the [[garischisma]], the [[~~septiennealimma~~]], the [[~~symbiotic comma~~]], the [[~~argyria~~]], and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].

135edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et supports an easy but extremely accurate equal tuning of the [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]] [[gary]]. As an equal temperament, it is characterized by [[tempering out]] the [[garischisma]], the [[symbiotic comma]] and the [[argyria]], which indirectly also tempers out the [[septiennealimma]], the [[olympia]] and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].

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.

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=== Subsets and supersets ===

Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, provides good ~~correction~~ for the approximation to harmonics 5, 13, and 19.

Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, notably provides extremely good corrections 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. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et probably makes more sense as a [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]], where it is characterized by [[tempering out]] the [[garischisma]], the [[septiennealimma]], the [[symbiotic comma]], the [[argyria]], and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].
+edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et supports an easy but extremely accurate equal tuning of the [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]] [[gary]]. As an equal temperament, it is characterized by [[tempering out]] the [[garischisma]], the [[symbiotic comma]] and the [[argyria]], which indirectly also tempers out the [[septiennealimma]], the [[olympia]] and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].
 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.
@@ Line 16: / Line 16: @@
 === Subsets and supersets ===
-Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.
+Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19.
 == Regular temperament properties ==