343edo
| ← 342edo | 343edo | 344edo → |
343 equal divisions of the octave (abbreviated 343edo or 343ed2), also called 343-tone equal temperament (343tet) or 343 equal temperament (343et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 343 equal parts of about 3.5 ¢ each. Each step represents a frequency ratio of 21/343, or the 343rd root of 2.
Theory
343edo is only consistent to the 3-odd-limit since its errors of harmonics 3 and 5 are quite large. To start with, consider the 2.9.15.7 subgroup, where it tempers out 5250987/5242880. In the 2.5.7 subgroup it tempers out 2100875/2097152 and in the 2.3.7 subgroup it tempers out 118098/117649.
For the full 7-limit, the 343c val tempers out 4375/4374 and 5120/5103, supporting amity. The 343cdd val tempers out 16875/16807 and 65536/64827. The patent val tempers out 10976/10935 and 390625/387072.
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
Since 343 factors into 73, 343edo has 7edo and 49edo as its subsets. 686edo, which doubles it, gives a good correction to the harmonics 3 and 5.
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 |