390edo
| ← 389edo | 390edo | 391edo → |
390 equal divisions of the octave (abbreviated 390edo or 390ed2), also called 390-tone equal temperament (390tet) or 390 equal temperament (390et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 390 equal parts of about 3.08 ¢ each. Each step represents a frequency ratio of 21/390, or the 390th root of 2.
Theory
390et is enfactored in the 5-limit, with the same tuning as 65edo. But its approximation to higher harmonics are improved, so that it is suitable for use in the 2.3.7.11.13.17.23.31.41 subgroup.
Using the patent val nonetheless, it tempers out 2401/2400 and 3136/3125 in the 7-limit, supporting hemiwürschmidt.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.42 | +1.38 | +0.40 | -0.55 | -0.53 | -0.34 | +0.95 | -0.58 | +1.19 | -0.42 |
| Relative (%) | +0.0 | -13.5 | +44.8 | +13.2 | -17.8 | -17.1 | -11.1 | +30.8 | -18.9 | +38.7 | -13.7 | |
| Steps (reduced) |
390 (0) |
618 (228) |
906 (126) |
1095 (315) |
1349 (179) |
1443 (273) |
1594 (34) |
1657 (97) |
1764 (204) |
1895 (335) |
1932 (372) | |
Subsets and supersets
Since 390 factors into 2 × 3 × 5 × 13, 390edo has subset edos 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, and 195. 780edo, which doubles it, gives a good correction to the harmonic 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.7 | 118098/117649, 34451725707/34359738368 | [⟨390 618 1095]] | 0.0395 | 0.1685 | 5.48 |
| 2.3.7.11 | 118098/117649, 1362944/1361367, 235782657/234881024 | [⟨390 618 1095 1349]] | 0.0693 | 0.1548 | 5.03 |
| 2.3.7.11.13 | 729/728, 10648/10647, 16848/16807, 1574573/1572864 | [⟨390 618 1095 1349 1443]] | 0.0839 | 0.1415 | 4.60 |
| 2.3.7.11.13.17 | 729/728, 1089/1088, 16848/16807, 65637/65536, 95823/95744 | [⟨390 618 1095 1349 1443 1594]] | 0.0838 | 0.1292 | 4.20 |