137edo
| ← 136edo | 137edo | 138edo → |
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 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.
Prime harmonics
| 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 | -14 | -10 | +39 | +6 | +4 | +2 | +3 | +27 | +46 | +28 | |
| 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
Since 137 is the 33rd prime number, 137edo has no proper divisors aside from 1.
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
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 3\137 | 26.28 | 1594323/1562500 | Sfourth (5-limit) |
| 1 | 4\137 | 35.04 | 1990656/1953125 | Gammic |
| 1 | 31\137 | 271.53 | 75/64 | Orson |
| 1 | 36\137 | 315.33 | 6/5 | Parakleismic |
| 1 | 59\137 | 516.79 | 27/20 | Gravity |
| 1 | 63\137 | 551.82 | 9765625/7077888 | Emka (5-limit) |
Diagrams
A diagram of 7-limit orwell based on the 31\137edo generator:
