135edo: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | == Theory == | ||
135edo is [[consistent]] to the [[7-odd-limit]], but | 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 makes most sense to use 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]], the [[chrysia]], and the [[olympia]]. 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. | |||
Using the | Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (pentadecal 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, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 ([[magus comma]]) 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. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|135|columns=11}} | ||
{{Harmonics in equal|135|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 135edo (continued)}} | |||
=== 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, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | |- | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
| Line 25: | Line 31: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 214 -135 }} | ||
| | | {{Mapping| 135 214 }} | ||
| | | −0.0843 | ||
| 0.0843 | | 0.0843 | ||
| 0.95 | | 0.95 | ||
| Line 33: | Line 39: | ||
| 2.3.7 | | 2.3.7 | ||
| 33554432/33480783, 40353607/40310784 | | 33554432/33480783, 40353607/40310784 | ||
| | | {{Mapping| 135 214 379 }} | ||
| | | −0.0637 | ||
| 0.0747 | | 0.0747 | ||
| 0.84 | | 0.84 | ||
| Line 40: | Line 46: | ||
| 2.3.7.11 | | 2.3.7.11 | ||
| 19712/19683, 41503/41472, 43923/43904 | | 19712/19683, 41503/41472, 43923/43904 | ||
| | | {{Mapping| 135 214 379 467 }} | ||
| | | −0.0328 | ||
| 0.0840 | | 0.0840 | ||
| 0.94 | | 0.94 | ||
|- | |||
| 2.3.7.11.17 | |||
| 1089/1088, 2058/2057, 5832/5831, 19712/19683 | |||
| {{Mapping| 135 214 379 467 552 }} | |||
| −0.1100 | |||
| 0.1716 | |||
| 1.93 | |||
|} | |} | ||
== Instruments == | |||
[[ | * [[Lumatone mapping for 135edo]] | ||