137edo: Difference between revisions
Cleanup and set straight the rank-2 temps it supports |
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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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== 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]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| {{monzo| -217 137 }} | | {{monzo| -217 137 }} | ||
| {{mapping| 137 217 }} | | {{mapping| 137 217 }} | ||
| 0.3865 | | +0.3865 | ||
| 0.3866 | | 0.3866 | ||
| 4.41 | | 4.41 | ||
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| {{monzo| -21 3 7 }}, {{monzo| -13 17 -6 }} | | {{monzo| -21 3 7 }}, {{monzo| -13 17 -6 }} | ||
| {{mapping| 137 217 318 }} | | {{mapping| 137 217 318 }} | ||
| 0.3887 | | +0.3887 | ||
| 0.3157 | | 0.3157 | ||
| 3.60 | | 3.60 | ||
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=== 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| 271.53 | | 271.53 | ||
| 75/64 | | 75/64 | ||
| [[Orwell]] ( | | [[Orwell]] (137e) / [[sabric]] (137d) | ||
|- | |- | ||
| 1 | | 1 | ||
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| 6/5 | | 6/5 | ||
| [[Parakleismic]] | | [[Parakleismic]] | ||
|- | |||
| 1 | |||
| 53\137 | |||
| 464.23 | |||
| 72/55 | |||
| [[Borwell]] | |||
|- | |- | ||
| 1 | | 1 | ||
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| [[Emka]] (137d) / [[emkay]] (137) | | [[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 distinct | ||
== Diagrams == | == Diagrams == |