375edo
| ← 374edo | 375edo | 376edo → |
375 equal divisions of the octave (abbreviated 375edo or 375ed2), also called 375-tone equal temperament (375tet) or 375 equal temperament (375et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 375 equal parts of exactly 3.2 ¢ each. Each step represents a frequency ratio of 21/375, or the 375th root of 2.
Theory
375edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. It can be used as a temperament in the 2.9.5.7.13.17.19 subgroup, in which it tempers out 250047/250000, 589824/588245, and [8 7 -13⟩ (parakleisma).
For the full 7-limit, the 375cd val is the best, where it tempers out 1029/1024 and 15625/15552, supporting tritikleismic. In the 11-limit it tempers out 385/384 and 441/440, supporting undecimal tritikleismic.
Using the patent val, it tempers out 6144/6125 and 177147/175616, supporting aufo. In the 11-limit it tempers out 540/539 and 5632/5625, supporting aufic and persephone.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.16 | +0.89 | +0.77 | +0.89 | -0.92 | +1.07 | -0.27 | +0.64 | +0.09 | -0.38 | -1.07 |
| Relative (%) | -36.1 | +27.7 | +24.2 | +27.8 | -28.7 | +33.5 | -8.4 | +20.1 | +2.7 | -11.9 | -33.6 | |
| Steps (reduced) |
594 (219) |
871 (121) |
1053 (303) |
1189 (64) |
1297 (172) |
1388 (263) |
1465 (340) |
1533 (33) |
1593 (93) |
1647 (147) |
1696 (196) | |
Subsets and supersets
Since 375 factors into 3 × 53, 375edo has subset edos 3, 5, 15, 25, 75, and 125. 1175edo, which triples it, gives a good correction to the harmonic 3 and is consistent to the 15-odd-limit.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [1189 -375⟩ | [⟨375 1189]] | -0.1404 | 0.1404 | 4.39 |
| 2.9.5 | [8 7 -13⟩, [97 -24 -9⟩ | [⟨375 1189 871]] | -0.2208 | 0.1615 | 5.05 |
| 2.9.5.7 | 250047/250000, 589824/588245, [8 7 -13⟩ | [⟨375 1189 871 1053]] | -0.2345 | 0.1418 | 4.43 |