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 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
| 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
| 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:
