157edo
| ← 156edo | 157edo | 158edo → |
157 equal divisions of the octave (abbreviated 157edo or 157ed2), also called 157-tone equal temperament (157tet) or 157 equal temperament (157et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 157 equal parts of about 7.64 ¢ each. Each step represents a frequency ratio of 21/157, or the 157th root of 2.
Theory
157et tempers out 78732/78125 (sensipent comma) and [37 -16 -5⟩ (quinticosiennic comma) in the 5-limit; 2401/2400, 5120/5103, and 110592/109375 in the 7-limit (supporting the hemififths and the catafourth temperaments). Using the patent val, it tempers out 176/175, 1331/1323, 3773/3750 and 8019/8000 in the 11-limit; 351/350, 352/351, 847/845, 1573/1568, and 2197/2187 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.23 | +3.50 | +1.87 | +2.46 | -1.00 | +0.24 | -2.92 | +2.05 | +0.58 | +3.10 | -1.52 |
| Relative (%) | +16.1 | +45.7 | +24.5 | +32.2 | -13.1 | +3.1 | -38.2 | +26.8 | +7.5 | +40.6 | -19.9 | |
| Steps (reduced) |
249 (92) |
365 (51) |
441 (127) |
498 (27) |
543 (72) |
581 (110) |
613 (142) |
642 (14) |
667 (39) |
690 (62) |
710 (82) | |
Subsets and supersets
157edo is the 37th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [249 -157⟩ | [⟨157 249]] | −0.388 | 0.388 | 5.08 |
| 2.3.5 | 78732/78125, ⟨37 -16 -5] | [⟨157 249 365]] | −0.760 | 0.614 | 8.04 |
| 2.3.5.7 | 2401/2400, 5120/5103, 78732/78125 | [⟨157 249 365 441]] | −0.737 | 0.533 | 6.98 |
| 2.3.5.7.11 | 176/175, 1331/1323, 2401/2400, 5120/5103 | [⟨157 249 365 441 543]] | −0.532 | 0.629 | 8.24 |
| 2.3.5.7.11.13 | 176/175, 351/350, 847/845, 1331/1323, 2197/2187 | [⟨157 249 365 441 543 581]] | −0.454 | 0.600 | 7.86 |
| 2.3.5.7.11.13.17 | 176/175, 256/255, 351/350, 442/441, 715/714, 2197/2187 | [⟨157 249 365 441 543 581 642]] | −0.461 | 0.556 | 7.28 |
| 2.3.5.7.11.13.17.19 | 176/175, 256/255, 286/285, 351/350, 361/360, 442/441, 476/475 | [⟨157 249 365 441 543 581 642 667]] | −0.420 | 0.531 | 6.95 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 13\157 | 99.36 | 18/17 | Quinticosiennic |
| 1 | 23\157 | 175.80 | 72/65 | Quadrafifths |
| 1 | 46\157 | 351.59 | 49/40 | Hemififths |
| 1 | 56\157 | 428.03 | 2800/2187 | Geb / osiris |
| 1 | 58\157 | 443.31 | 162/125 | Warrior |
| 1 | 64\157 | 489.17 | 250/189 | Catafourth |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct