137edo: Difference between revisions

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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]] 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.  
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 ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 137 is the 33rd [[prime number]], 137edo has no proper divisors aside from 1.
137edo is the 33rd [[prime edo]], following [[131edo]] and before [[139edo]]. [[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.  


[[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 }}
|{{val|137 217}}
| {{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 }}
|{{val|137 217 318}}
| {{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<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! 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
|[[Orson]]
| [[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
|[[Gravity]]
| [[Marvo]] (137)
|-
|-
|1
| 1
|63\137
| 63\137
|551.82
| 551.82
|9765625/7077888
| 11/8
|[[Emka]] (5-limit)
| [[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 ==

Revision as of 10:05, 26 May 2024

← 136edo 137edo 138edo →
Prime factorization 137 (prime)
Step size 8.75912 ¢ 
Fifth 80\137 (700.73 ¢)
Semitones (A1:m2) 12:11 (105.1 ¢ : 96.35 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

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.

Prime harmonics

Approximation of prime harmonics in 137edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -1.23 -0.91 +3.44 +0.51 +0.35 +0.15 +0.30 +2.38 +4.00 +2.41
Relative (%) +0.0 -14.0 -10.4 +39.2 +5.8 +4.0 +1.8 +3.4 +27.2 +45.7 +27.5
Steps
(reduced) 		137
(0) 		217
(80) 		318
(44) 		385
(111) 		474
(63) 		507
(96) 		560
(12) 		582
(34) 		620
(72) 		666
(118) 		679
(131)

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

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-217 137⟩ [⟨137 217]] 0.3865 0.3866 4.41
2.3.5 [-21 3 7⟩, [-13 17 -6⟩ [⟨137 217 318]] 0.3887 0.3157 3.60

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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 (137) / sabric (137d)
1 36\137 315.33 6/5 Parakleismic
1 59\137 516.79 27/20 Marvo (137)
1 63\137 551.82 11/8 Emka (137d) / emkay (137)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Diagrams

A diagram of 7-limit orwell based on the 31\137edo generator:

137edo_MOS_031_demo_correction.png

137edo_MOS_031.svg

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