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 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) | |
Subsets and supersets
Since 388 factors into primes as 22 × 97, 388edo has subset edos 2, 4, 97, and 194.
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 388edo. 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 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 41 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, and 113. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, and 113. 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 or 494edo. Note that 311edo has generally higher absolute errors than 388edo due to its smaller size, but having a smaller size also means the system is easier to handle.
Another system notable in high limits around this size is 422edo, using the sharp-tending 422l val for prime 37.
37-odd-limit intervals
The following table shows how 37-odd-limit intervals are represented in 388edo. Prime harmonics are in bold.
As 388edo is consistent in the 37-odd-limit, the mappings by direct approximation and through the patent val are identical.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 29/27, 54/29 | 0.001 | 0.0 |
| 33/31, 62/33 | 0.010 | 0.3 |
| 17/9, 18/17 | 0.014 | 0.5 |
| 11/7, 14/11 | 0.018 | 0.6 |
| 37/22, 44/37 | 0.026 | 0.8 |
| 31/21, 42/31 | 0.028 | 0.9 |
| 29/20, 40/29 | 0.035 | 1.1 |
| 27/20, 40/27 | 0.036 | 1.2 |
| 37/28, 56/37 | 0.044 | 1.4 |
| 35/23, 46/35 | 0.061 | 2.0 |
| 9/5, 10/9 | 0.071 | 2.3 |
| 29/15, 30/29 | 0.071 | 2.3 |
| 21/19, 38/21 | 0.072 | 2.3 |
| 31/28, 56/31 | 0.079 | 2.6 |
| 17/10, 20/17 | 0.085 | 2.7 |
| 33/28, 56/33 | 0.089 | 2.9 |
| 33/19, 38/33 | 0.090 | 2.9 |
| 17/12, 24/17 | 0.092 | 3.0 |
| 31/22, 44/31 | 0.097 | 3.1 |
| 31/19, 38/31 | 0.100 | 3.2 |
| 35/19, 38/35 | 0.105 | 3.4 |
| 29/18, 36/29 | 0.106 | 3.4 |
| 3/2, 4/3 | 0.107 | 3.5 |
| 29/17, 34/29 | 0.120 | 3.9 |
| 27/17, 34/27 | 0.121 | 3.9 |
| 37/31, 62/37 | 0.123 | 4.0 |
| 21/11, 22/21 | 0.125 | 4.0 |
| 37/33, 66/37 | 0.133 | 4.3 |
| 25/13, 26/25 | 0.141 | 4.6 |
| 37/21, 42/37 | 0.151 | 4.9 |
| 23/19, 38/23 | 0.167 | 5.4 |
| 5/3, 6/5 | 0.177 | 5.7 |
| 19/14, 28/19 | 0.179 | 5.8 |
| 17/15, 30/17 | 0.192 | 6.2 |
| 35/33, 66/35 | 0.195 | 6.3 |
| 19/11, 22/19 | 0.197 | 6.4 |
| 17/16, 32/17 | 0.199 | 6.4 |
| 35/31, 62/35 | 0.205 | 6.6 |
| 29/24, 48/29 | 0.213 | 6.9 |
| 9/8, 16/9 | 0.214 | 6.9 |
| 37/19, 38/37 | 0.223 | 7.2 |
| 23/21, 42/23 | 0.239 | 7.7 |
| 27/25, 50/27 | 0.248 | 8.0 |
| 29/25, 50/29 | 0.249 | 8.0 |
| 33/23, 46/33 | 0.256 | 8.3 |
| 31/23, 46/31 | 0.266 | 8.6 |
| 5/4, 8/5 | 0.284 | 9.2 |
| 35/22, 44/35 | 0.302 | 9.8 |
| 15/13, 26/15 | 0.318 | 10.3 |
| 29/16, 32/29 | 0.320 | 10.3 |
| 27/16, 32/27 | 0.321 | 10.4 |
| 37/35, 70/37 | 0.328 | 10.6 |
| 23/14, 28/23 | 0.345 | 11.2 |
| 25/18, 36/25 | 0.355 | 11.5 |
| 23/22, 44/23 | 0.363 | 11.7 |
| 25/17, 34/25 | 0.369 | 11.9 |
| 27/26, 52/27 | 0.389 | 12.6 |
| 37/23, 46/37 | 0.389 | 12.6 |
| 29/26, 52/29 | 0.390 | 12.6 |
| 15/8, 16/15 | 0.391 | 12.6 |
| 13/10, 20/13 | 0.425 | 13.7 |
| 23/16, 32/23 | 0.439 | 14.2 |
| 25/24, 48/25 | 0.462 | 14.9 |
| 13/9, 18/13 | 0.496 | 16.0 |
| 35/32, 64/35 | 0.500 | 16.2 |
| 17/13, 26/17 | 0.510 | 16.5 |
| 23/12, 24/23 | 0.546 | 17.7 |
| 25/16, 32/25 | 0.568 | 18.4 |
| 13/12, 24/13 | 0.603 | 19.5 |
| 19/16, 32/19 | 0.606 | 19.6 |
| 35/24, 48/35 | 0.607 | 19.6 |
| 23/17, 34/23 | 0.639 | 20.6 |
| 23/18, 36/23 | 0.653 | 21.1 |
| 21/16, 32/21 | 0.678 | 21.9 |
| 33/32, 64/33 | 0.696 | 22.5 |
| 35/34, 68/35 | 0.700 | 22.6 |
| 31/16, 32/31 | 0.706 | 22.8 |
| 13/8, 16/13 | 0.709 | 22.9 |
| 19/12, 24/19 | 0.713 | 23.0 |
| 35/18, 36/35 | 0.714 | 23.1 |
| 23/20, 40/23 | 0.724 | 23.4 |
| 29/23, 46/29 | 0.759 | 24.5 |
| 27/23, 46/27 | 0.760 | 24.6 |
| 7/4, 8/7 | 0.785 | 25.4 |
| 11/8, 16/11 | 0.802 | 25.9 |
| 19/17, 34/19 | 0.805 | 26.0 |
| 31/24, 48/31 | 0.813 | 26.3 |
| 19/18, 36/19 | 0.820 | 26.5 |
| 35/29, 58/35 | 0.820 | 26.5 |
| 35/27, 54/35 | 0.821 | 26.5 |
| 37/32, 64/37 | 0.829 | 26.8 |
| 23/15, 30/23 | 0.830 | 26.8 |
| 21/17, 34/21 | 0.877 | 28.4 |
| 19/10, 20/19 | 0.890 | 28.8 |
| 7/6, 12/7 | 0.892 | 28.8 |
| 33/17, 34/33 | 0.895 | 28.9 |
| 31/17, 34/31 | 0.905 | 29.3 |
| 11/6, 12/11 | 0.909 | 29.4 |
| 31/18, 36/31 | 0.919 | 29.7 |
| 29/19, 38/29 | 0.926 | 29.9 |
| 27/19, 38/27 | 0.926 | 30.0 |
| 37/24, 48/37 | 0.935 | 30.2 |
| 21/20, 40/21 | 0.962 | 31.1 |
| 33/20, 40/33 | 0.980 | 31.7 |
| 17/14, 28/17 | 0.984 | 31.8 |
| 31/20, 40/31 | 0.990 | 32.0 |
| 19/15, 30/19 | 0.997 | 32.2 |
| 29/21, 42/29 | 0.998 | 32.3 |
| 9/7, 14/9 | 0.998 | 32.3 |
| 17/11, 22/17 | 1.002 | 32.4 |
| 25/23, 46/25 | 1.008 | 32.6 |
| 33/29, 58/33 | 1.015 | 32.8 |
| 11/9, 18/11 | 1.016 | 32.9 |
| 31/29, 58/31 | 1.025 | 33.2 |
| 31/27, 54/31 | 1.026 | 33.2 |
| 37/34, 68/37 | 1.028 | 33.2 |
| 37/36, 72/37 | 1.042 | 33.7 |
| 7/5, 10/7 | 1.069 | 34.6 |
| 11/10, 20/11 | 1.087 | 35.1 |
| 31/30, 60/31 | 1.097 | 35.5 |
| 29/28, 56/29 | 1.104 | 35.7 |
| 27/14, 28/27 | 1.105 | 35.7 |
| 37/20, 40/37 | 1.113 | 36.0 |
| 29/22, 44/29 | 1.122 | 36.3 |
| 27/22, 44/27 | 1.123 | 36.3 |
| 37/29, 58/37 | 1.148 | 37.1 |
| 23/13, 26/23 | 1.149 | 37.1 |
| 37/27, 54/37 | 1.149 | 37.2 |
| 25/19, 38/25 | 1.174 | 38.0 |
| 15/14, 28/15 | 1.176 | 38.0 |
| 15/11, 22/15 | 1.194 | 38.6 |
| 35/26, 52/35 | 1.210 | 39.1 |
| 37/30, 60/37 | 1.220 | 39.4 |
| 25/21, 42/25 | 1.246 | 40.3 |
| 33/25, 50/33 | 1.264 | 40.9 |
| 31/25, 50/31 | 1.274 | 41.2 |
| 19/13, 26/19 | 1.315 | 42.5 |
| 25/14, 28/25 | 1.353 | 43.8 |
| 25/22, 44/25 | 1.371 | 44.3 |
| 21/13, 26/21 | 1.387 | 44.9 |
| 37/25, 50/37 | 1.397 | 45.2 |
| 33/26, 52/33 | 1.405 | 45.4 |
| 31/26, 52/31 | 1.415 | 45.8 |
| 13/7, 14/13 | 1.494 | 48.3 |
| 13/11, 22/13 | 1.512 | 48.9 |
| 37/26, 52/37 | 1.538 | 49.7 |
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