343edo
Theory
343et is only consistent to the 3-odd-limit and the harmonic 3 is about inbetween its steps. Using the patent val, it tempers out 40500000/40353607, 67108864/66976875, 10976/10935 and 2100875/2097152 in the 7-limit; 100663296/100656875, 806736/805255, 25165824/25109315, 820125/819896, 14700/14641, 16384/16335, 496125/495616, 1296000/1294139, 1265625/1261568, 5767168/5764801, 1375/1372, 1879453125/1879048192, 41503/41472, 1362944/1361367, 166375/165888 and 322102/321489 in the 11-limit. It supports protolangwidge.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.25 | -1.47 | +0.27 | -0.99 | +1.45 | -0.88 | -0.22 | +0.00 | -0.14 | +1.52 | +1.46 |
| Relative (%) | +35.8 | -42.1 | +7.7 | -28.4 | +41.5 | -25.1 | -6.3 | +0.0 | -3.9 | +43.5 | +41.8 | |
| Steps (reduced) |
544 (201) |
796 (110) |
963 (277) |
1087 (58) |
1187 (158) |
1269 (240) |
1340 (311) |
1402 (30) |
1457 (85) |
1507 (135) |
1552 (180) | |
Subsets and supersets
343 factors into 73, with 7edo and 49edo as its subset edos. 686edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1087 343⟩ | [⟨343 1087]] | 0.1569 | 0.1569 | 4.48 |
| 2.9.5 | [-27 -1 13⟩, [40 -28 21⟩ | [⟨343 1087 796]] | 0.3162 | 0.2592 | 7.41 |
| 2.9.5.7 | 118098/117649, 7381125/7340032, 9765625/9680832 | [⟨343 1087 796 963]] | 0.2130 | 0.2869 | 8.20 |