137edo: Difference between revisions

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The ''137 equal division'' divides the octave into 137 equal parts of 8.759 cents each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Semicomma_family|orwell temperament]] and for the planar temperament tempering out 2430/2401. It tempers out 2109375/2097152 (the semicomma) in the 5-limit; 225/224 and 1728/1715 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. Since it is the 33rd [[prime_numbers|prime number]], 137edo has no proper divisors aside from 1.
{{Infobox ET}}
{{ED intro}}


A diagram of 7-limit Orwell based on the 31\137edo generator:
== Theory ==
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 ===
{{Harmonics in equal|137}}
 
=== 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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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
| 4.41
|-
| 2.3.5
| {{monzo| -21 3 7 }}, {{monzo| -13 17 -6 }}
| {{mapping| 137 217 318 }}
| +0.3887
| 0.3157
| 3.60
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 3\137
| 26.28
| 1594323/1562500
| [[Sfourth]] (5-limit)
|-
| 1
| 4\137
| 35.04
| 1990656/1953125
| [[Gammic]] (137d) / [[gammy]] (137)
|-
| 1
| 31\137
| 271.53
| 75/64
| [[Orwell]] (137e) / [[sabric]] (137d)
|-
| 1
| 36\137
| 315.33
| 6/5
| [[Parakleismic]]
|-
| 1
| 53\137
| 464.23
| 72/55
| [[Borwell]]
|-
| 1
| 59\137
| 516.79
| 27/20
| [[Marvo]] (137)
|-
| 1
| 63\137
| 551.82
| 11/8
| [[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 ==
A diagram of 7-limit orwell based on the 31\137edo generator:


[[File:137edo_MOS_031_demo_correction.png|alt=137edo_MOS_031_demo_correction.png|137edo_MOS_031_demo_correction.png]]
[[File:137edo_MOS_031_demo_correction.png|alt=137edo_MOS_031_demo_correction.png|137edo_MOS_031_demo_correction.png]]
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[[:File:137edo_MOS_031.svg|137edo_MOS_031.svg]]
[[:File:137edo_MOS_031.svg|137edo_MOS_031.svg]]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Nuwell]]
[[Category:Nuwell]]
[[Category:Orwell]]
[[Category:Orwell]]
[[Category:Prime EDO]]
[[Category:Orson]]
[[Category:Semicomma]]