393edo
| ← 392edo | 393edo | 394edo → |
393 equal divisions of the octave (abbreviated 393edo or 393ed2), also called 393-tone equal temperament (393tet) or 393 equal temperament (393et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 393 equal parts of about 3.05 ¢ each. Each step represents a frequency ratio of 21/393, or the 393rd root of 2.
Theory
393et is only consistent to the 5-limit, with three mappings possible for the 7-limit:
- ⟨393 623 913 1103] (patent val),
- ⟨393 623 912 1103] (393c),
- ⟨393 623 913 1104] (393d).
Using the patent val, it tempers out 393216/390625 and [-46 51 -15⟩ in the 5-limit; 10976/10935, 393216/390625 and 5250987/5242880 in the 7-limit.
Using the 393c val, it tempers out 2109375/2097152 and [32 -48 19⟩ in the 5-limit; 2401/2400, 1071875/1062882 and 2109375/2097152 in the 7-limit.
Using the 393d val, it tempers out 393216/390625 and [-46 51 -15⟩ in the 5-limit; 250047/250000, 2460375/2458624 and 2097152/2083725 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.34 | +1.47 | -0.89 | +0.67 | +1.35 | -0.83 | -1.25 | -1.14 | -1.33 | -0.55 | +0.73 |
| relative (%) | +11 | +48 | -29 | +22 | +44 | -27 | -41 | -37 | -44 | -18 | +24 | |
| Steps (reduced) |
623 (230) |
913 (127) |
1103 (317) |
1246 (67) |
1360 (181) |
1454 (275) |
1535 (356) |
1606 (34) |
1669 (97) |
1726 (154) |
1778 (206) | |
Subsets and supersets
393 factors into 3 × 131, with 3edo and 131edo as its subset edos. 786edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [623 -393⟩ | [⟨393 623]] | -0.1057 | 0.1057 | 3.43 |
| 2.3.5 | 393216/390625, [-46 51 -15⟩ | -0.2819 | 0.2636 | 8.63 | |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 127\393 | 387.79 | 5/4 | Würschmidt |