137edo: Difference between revisions

Cleanup and set straight the rank-2 temps it supports
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|137}}
{{ED intro}}


== 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]], [[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.  
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|Comma List]]
! rowspan="2" | [[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]] (%)
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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>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 271.53
| 271.53
| 75/64
| 75/64
| [[Orwell]] (137) / [[sabric]] (137d)
| [[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 it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Diagrams ==
== Diagrams ==