307edo
| ← 306edo | 307edo | 308edo → |
307 equal divisions of the octave (abbreviated 307edo or 307ed2), also called 307-tone equal temperament (307tet) or 307 equal temperament (307et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 307 equal parts of about 3.91 ¢ each. Each step represents a frequency ratio of 21/307, or the 307th root of 2.
Theory
307edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. It can be considered in either the 2.9.5.7 subgroup or the 2.9.15.21 subgroup, but the former is more flexible as it lends itself to an extension to the 2.9.5.7.11.13.17.19.23.
Using the full 7-limit patent val nonetheless, the equal temperament tempers out 2401/2400 in the 7-limit, and in the 11-limit extension, 3388/3375, 6250/6237, 15488/15435, 16384/16335, and 43923/43904.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.63 | +0.66 | +0.56 | -0.65 | -0.18 | -0.14 | -1.62 | +0.58 | -0.44 | -1.73 | +1.04 |
| Relative (%) | +41.7 | +16.8 | +14.2 | -16.7 | -4.6 | -3.5 | -41.5 | +14.9 | -11.4 | -44.1 | +26.6 | |
| Steps (reduced) |
487 (180) |
713 (99) |
862 (248) |
973 (52) |
1062 (141) |
1136 (215) |
1199 (278) |
1255 (27) |
1304 (76) |
1348 (120) |
1389 (161) | |
Subsets and supersets
307edo is the 63rd prime edo. 614edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-973 307⟩ | [⟨307 973]] | +0.1029 | 0.1030 | 2.64 |
| 2.9.5 | 32805/32768, [2 47 -65⟩ | [⟨307 973 713]] | -0.0257 | 0.2004 | 5.13 |
| 2.9.5.7 | 32805/32768, 118098/117649, 589824/588245 | [⟨307 973 713 862]] | -0.0687 | 0.1889 | 4.87 |
| 2.9.5.7.11 | 5632/5625, 8019/8000, 32805/32768, 46656/46585 | [⟨307 973 713 862 1062]] | -0.0447 | 0.1756 | 4.49 |
| 2.9.5.7.11.13 | 729/728, 1001/1000, 4096/4095, 6656/6655, 10648/10647 | [⟨307 973 713 862 1062 1136]] | -0.0311 | 0.1632 | 4.18 |
| 2.9.5.7.11.13.17 | 729/728, 936/935, 1001/1000, 1377/1375, 2025/2023, 7744/7735 | [⟨307 973 713 862 1062 1136 1255]] | -0.0470 | 0.1560 | 3.99 |