137edo: Difference between revisions
+regular temperament properties |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
137edo | 137edo is a fairly accurate 5-limit temperament and also a strong no-7 19-limit temperament. The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]), {{monzo| -13 17 -6 }} ([[graviton]]), {{monzo| 8 14 -13 }} ([[parakleisma]]), and {{monzo| -29 -11 20 }} (gammic comma) in the 5-limit. Using the [[patent val]], it tempers out [[225/224]], [[1728/1715]], 2430/2401 in the 7-limit; [[243/242]] in the 11-limit; [[351/350]] in the 13-limit; [[375/374]] and [[442/441]] in the 17-limit; and [[324/323]] and [[495/494]] in the 19-limit. It provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament [[tempering out]] [[2430/2401]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
137edo is the 33rd [[prime edo]], following [[131edo]] and before [[139edo]]. [[274edo]], which doubles it, provides a correction for its approximation to harmonic 7. | |||
== Regular temperament properties == | |||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-217 137}} | ! rowspan="2" | [[Comma list]] | ||
|{{ | ! rowspan="2" | [[Mapping]] | ||
| 0.3865 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -217 137 }} | |||
| {{mapping| 137 217 }} | |||
| +0.3865 | |||
| 0.3866 | | 0.3866 | ||
| 4.41 | | 4.41 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|-21 3 7}}, {{monzo|-13 17 -6}} | | {{monzo| -21 3 7 }}, {{monzo| -13 17 -6 }} | ||
|{{ | | {{mapping| 137 217 318 }} | ||
| 0.3887 | | +0.3887 | ||
| 0.3157 | | 0.3157 | ||
| 3.60 | | 3.60 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|3\137 | | 3\137 | ||
|26.28 | | 26.28 | ||
|1594323/1562500 | | 1594323/1562500 | ||
|[[Sfourth]] (5-limit) | | [[Sfourth]] (5-limit) | ||
|- | |||
| 1 | |||
| 4\137 | |||
| 35.04 | |||
| 1990656/1953125 | |||
| [[Gammic]] (137d) / [[gammy]] (137) | |||
|- | |- | ||
|1 | | 1 | ||
| | | 31\137 | ||
| | | 271.53 | ||
| | | 75/64 | ||
|[[ | | [[Orwell]] (137e) / [[sabric]] (137d) | ||
|- | |- | ||
|1 | | 1 | ||
| | | 36\137 | ||
| | | 315.33 | ||
| | | 6/5 | ||
|[[ | | [[Parakleismic]] | ||
|- | |- | ||
|1 | | 1 | ||
| | | 53\137 | ||
| | | 464.23 | ||
| | | 72/55 | ||
|[[ | | [[Borwell]] | ||
|- | |- | ||
|1 | | 1 | ||
|59\137 | | 59\137 | ||
|516.79 | | 516.79 | ||
|27/20 | | 27/20 | ||
|[[ | | [[Marvo]] (137) | ||
|- | |- | ||
|1 | | 1 | ||
|63\137 | | 63\137 | ||
|551.82 | | 551.82 | ||
| | | 11/8 | ||
|[[Emka]] ( | | [[Emka]] (137d) / [[emkay]] (137) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Diagrams == | == Diagrams == | ||