# 137edo

 ← 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

137 equal divisions of the octave (abbreviated 137edo or 137ed2), also called 137-tone equal temperament (137tet) or 137 equal temperament (137et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 137 equal parts of about 8.76 ¢ each. Each step represents a frequency ratio of 21/137, or the 137th root of 2.

## Theory

137edo is a fairly accurate 5-limit temperament and also a strong no-7 19-limit temperament. The equal temperament tempers out 2109375/2097152 (semicomma), [-13 17 -6 (graviton), [8 14 -13 (parakleisma), and [-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

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 (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)

* 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: