137edo: Difference between revisions
+regular temperament properties |
Cleanup and set straight the rank-2 temps it supports |
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== Theory == | == Theory == | ||
137edo provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament tempering out [[2430/2401]]. It tempers out 2109375/2097152 ([[semicomma]]) in the 5-limit; [[225/224]] | 137edo provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament [[tempering out]] [[2430/2401]]. It tempers out 2109375/2097152 ([[semicomma]]) in the 5-limit; [[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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== 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" | ||
! rowspan="2" |[[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-217 137}} | | {{monzo| -217 137 }} | ||
|{{ | | {{mapping| 137 217 }} | ||
| 0.3865 | | 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 | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! 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 | | 1 | ||
|4\137 | | 4\137 | ||
|35.04 | | 35.04 | ||
|1990656/1953125 | | 1990656/1953125 | ||
|[[Gammic]] | | [[Gammic]] (137d) / [[gammy]] (137) | ||
|- | |- | ||
|1 | | 1 | ||
|31\137 | | 31\137 | ||
|271.53 | | 271.53 | ||
|75/64 | | 75/64 | ||
|[[ | | [[Orwell]] (137) / [[sabric]] (137d) | ||
|- | |- | ||
|1 | | 1 | ||
|36\137 | | 36\137 | ||
|315.33 | | 315.33 | ||
|6/5 | | 6/5 | ||
|[[Parakleismic]] | | [[Parakleismic]] | ||
|- | |- | ||
|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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Diagrams == | == Diagrams == | ||