388edo
| ← 387edo | 388edo | 389edo → |
388 equal divisions of the octave (abbreviated 388edo or 388ed2), also called 388-tone equal temperament (388tet) or 388 equal temperament (388et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 388 equal parts of about 3.09 ¢ each. Each step represents a frequency ratio of 21/388, or the 388th root of 2.
Theory
388edo is the first edo that is distinctly consistent through to the 27-odd-limit; it is also consistent through the 37-odd-limit.
The equal temperament tempers out the vishnuzma, [23 6 -14⟩, the tricot comma, [39 -29 3⟩, the minortone comma, [-16 35 -17⟩, and the raider comma, [71 -99 31⟩, in the 5-limit, giving a strong tuning. It tempers out 4375/4374 and 235298/234375 in the 7-limit, and 3025/3024, 5632/5625 and 9801/9800 in the 11-limit and 847/845, 1001/1000 and 4096/4095 in the 13-limit.
It provides the optimal patent val for the rank-5 cuthbert temperament, which tempers out 847/845, the cuthbert comma, and for a number of other temperaments tempering it out, e.g. neusec, the 190 & 198 temperament. By tempering out cuthbert it supports cuthbert chords, in addition to sinbadmic chords.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.11 | +0.28 | -0.78 | -0.80 | +0.71 | +0.20 | -0.61 | -0.44 | +0.32 | -0.71 | -0.83 |
| Relative (%) | +0.0 | +3.5 | +9.2 | -25.4 | -25.9 | +22.9 | +6.4 | -19.6 | -14.2 | +10.3 | -22.8 | -26.8 | |
| Steps (reduced) |
388 (0) |
615 (227) |
901 (125) |
1089 (313) |
1342 (178) |
1436 (272) |
1586 (34) |
1648 (96) |
1755 (203) |
1885 (333) |
1922 (370) |
2021 (81) | |
Subsets and supersets
Since 388 factors into 22 × 97, 388edo has subset edos 2, 4, 97, and 194.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [615 -388⟩ | [⟨388 615]] | +0.0337 | 0.0337 | 1.09 |
| 2.3.5 | [23 6 -14⟩, [39 -29 3⟩ | [⟨388 615 901]] | −0.0633 | 0.0501 | 1.62 |
| 2.3.5.7 | 4375/4374, 235298/234375, 2100875/2097152 | [⟨388 615 901 1089]] | +0.0224 | 0.1546 | 5.00 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 5632/5625, 235298/234375 | [⟨388 615 901 1089 1342]] | +0.0643 | 0.1617 | 5.23 |
| 2.3.5.7.11.13 | 847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374 | [⟨388 615 901 1089 1342 1436]] | +0.0216 | 0.1758 | 5.68 |
| 2.3.5.7.11.13.17 | 833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700 | [⟨388 615 901 1089 1342 1436 1586]] | +0.0116 | 0.1646 | 5.32 |
| 2.3.5.7.11.13.17.19 | 833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330 | [⟨388 615 901 1089 1342 1436 1586 1648]] | +0.0280 | 0.1600 | 5.17 |
- 388et has a lower absolute error in the 5-limit than any previous equal temperaments, past 323 and followed by 441.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 59\388 | 182.47 | 10/9 | Mitonic |
| 1 | 111\388 | 343.30 | 8000/6561 | Raider |
| 1 | 145\388 | 448.45 | 35/27 | Semidimfourth |
| 1 | 183\388 | 565.97 | 75/52 | Trillium / pseudotrillium |
| 2 | 23\388 | 71.13 | 25/24 | Vishnu / ananta |
| 2 | 49\388 | 151.54 | 12/11 | Neusec |
| 4 | 123\388 (26\388) |
380.41 (80.41) |
81/65 (22/21) |
Quasithird |
| 97 | 161\388 (1\388) |
497.938 (3.09) |
4/3 (?) |
Berkelium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct