343edo: Difference between revisions
| Line 57: | Line 57: | ||
* 343cf | * 343cf | ||
** Octave size: 1199.643 {{c}} | ** Octave size: 1199.643{{c}} | ||
** TE error: 0.363{{c}}/octave | ** TE error: 0.363{{c}}/octave | ||
* 343c | * 343c | ||
** Octave size: 1199.761 {{c}} | ** Octave size: 1199.761{{c}} | ||
** TE error: 0.382{{c}}/octave | ** TE error: 0.382{{c}}/octave | ||
* 343e | * 343e | ||
** Octave size: 1200.076 {{c}} | ** Octave size: 1200.076{{c}} | ||
** TE error: 0.418{{c}}/octave | ** TE error: 0.418{{c}}/octave | ||
* 343f | * 343f | ||
** Octave size: 1199.831 {{c}} | ** Octave size: 1199.831{{c}} | ||
** TE error: 0.431{{c}}/octave | ** TE error: 0.431{{c}}/octave | ||
* 343ce | * 343ce | ||
** Octave size: 1199.888 {{c}} | ** Octave size: 1199.888{{c}} | ||
** TE error: 0.461{{c}}/octave | ** TE error: 0.461{{c}}/octave | ||
Revision as of 00:26, 19 October 2025
| ← 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
343edo is on the optimal ET sequence of gammy temperament (10\343 generator, 2/1 period), protolangwidge temperament (200\343 g, 2/1 p) and anthoine temperament (110\343 g, 2/1 p).
| 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 |
Octave stretch or compression
If one is using 343edo, it's probably because of its good primes 2, 7, 17 and 19.
So, one will probably not want to use 343edo with warts a, d, g or h.
That leaves the following TE tunings for the 19-limit:
- 343cf
- Octave size: 1199.643 ¢
- TE error: 0.363 ¢/octave
- 343c
- Octave size: 1199.761 ¢
- TE error: 0.382 ¢/octave
- 343e
- Octave size: 1200.076 ¢
- TE error: 0.418 ¢/octave
- 343f
- Octave size: 1199.831 ¢
- TE error: 0.431 ¢/octave
- 343ce
- Octave size: 1199.888 ¢
- TE error: 0.461 ¢/octave