357edo
| ← 356edo | 357edo | 358edo → |
357 equal divisions of the octave (abbreviated 357edo or 357ed2), also called 357-tone equal temperament (357tet) or 357 equal temperament (357et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 357 equal parts of about 3.36 ¢ each. Each step represents a frequency ratio of 21/357, or the 357th root of 2.
Theory
While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament tempers out 1600000/1594323 (amity comma), and [61 4 -29⟩ (squarschimidt comma) in the 5-limit; 10976/10935 (hemimage comma), 235298/234375 (triwellisma), 250047/250000 (landscape comma), 2100875/2097152 (rainy comma) in the 7-limit; 3025/3024, 5632/5625, 12005/11979 in the 11-limit; 676/675, 1001/1000, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647 in the 13-limit.
It supports 5-limit amity and 7-limit weak extensions calamity and chromat. It provides the optimal patent val for 11- and 13-limit hemichromat, the 159 & 198 temperament. It also supports avicenna, but 270edo is better suited for this purpose.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.57 | +0.24 | -0.76 | -0.06 | -0.19 | -0.75 | +1.65 | +0.30 | -1.01 | +1.18 |
| Relative (%) | +0.0 | +16.8 | +7.2 | -22.6 | -1.7 | -5.7 | -22.4 | +49.0 | +8.8 | -29.9 | +35.2 | |
| Steps (reduced) |
357 (0) |
566 (209) |
829 (115) |
1002 (288) |
1235 (164) |
1321 (250) |
1459 (31) |
1517 (89) |
1615 (187) |
1734 (306) |
1769 (341) | |
Subsets and supersets
Since 357 factors into 3 × 7 × 17, 357edo has subset edos 3, 7, 17, 21, 51, and 119.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [566 -357⟩ | [⟨357 566]] | −0.1786 | 0.1785 | 5.31 |
| 2.3.5 | 1600000/1594323, [61 4 -29⟩ | [⟨357 566 829]] | −0.1536 | 0.1500 | 4.46 |
| 2.3.5.7 | 10976/10935, 235298/234375, 2100875/2097152 | [⟨357 566 829 1002]] | −0.0477 | 0.2248 | 6.69 |
| 2.3.5.7.11 | 3025/3024, 5632/5625, 10976/10935, 102487/102400 | [⟨357 566 829 1002 1235]] | −0.0348 | 0.2027 | 6.03 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935 | [⟨357 566 829 1002 1235 1321]] | −0.0204 | 0.1879 | 5.59 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 101\357 | 339.50 | 243/200 | Amity (5-limit) |
| 1 | 118\357 | 396.64 | 44/35 | Squarschmidt |
| 1 | 163\357 | 547.90 | 48/35 | Calamity |
| 3 | 9\357 | 30.25 | 55/54 | Hemichromat |
| 3 | 18\357 | 60.50 | 28/27 | Chromat (7-limit) |
| 3 | 41\357 | 137.82 | 13/12 | Avicenna |
| 3 | 48\357 | 161.34 | 192/175 | Pnict |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct