388edo
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 alphatricot 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) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.83 | -1.21 | -0.56 | -1.34 | -1.44 | -0.39 | +1.11 | -0.32 | +1.08 | +0.41 | +1.50 | +1.28 |
| Relative (%) | +27.0 | -39.1 | -18.0 | -43.3 | -46.6 | -12.6 | +35.7 | -10.2 | +34.8 | +13.3 | +48.5 | +41.5 | |
| Steps (reduced) |
2079 (139) |
2105 (165) |
2155 (215) |
2222 (282) |
2282 (342) |
2301 (361) |
2354 (26) |
2386 (58) |
2402 (74) |
2446 (118) |
2474 (146) |
2513 (185) | |
| Harmonic | 97 | 101 | 103 | 107 | 109 | 113 | 127 | 131 | 137 | 139 | 149 | 151 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.72 | -1.19 | -1.12 | +0.96 | -0.17 | -0.71 | +1.21 | +0.10 | -0.11 | -0.46 | -0.12 | +1.54 |
| Relative (%) | +23.4 | -38.6 | -36.2 | +31.1 | -5.6 | -22.9 | +39.0 | +3.2 | -3.6 | -14.9 | -3.7 | +49.9 | |
| Steps (reduced) |
2561 (233) |
2583 (255) |
2594 (266) |
2616 (288) |
2626 (298) |
2646 (318) |
2712 (384) |
2729 (13) |
2754 (38) |
2762 (46) |
2801 (85) |
2809 (93) | |
Approximation to JI
This EDO has a high consistency limit, although due to 311edo having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.
388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic 7. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.
311edo also deals better with composite harmonics than 388. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22, 41/28, 41/22, and their octave complements. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer 270edo. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.
Another system notable in high limits around this size is 422edo.
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 | Alphatrillium / 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 distinct