343edo: Difference between revisions
m →Scales |
m →Scales |
||
| Line 101: | Line 101: | ||
== Scales == | == Scales == | ||
Scales are rotated to modes with lots of | Scales are rotated to modes with lots of consonances. | ||
* [[4 of 7-17-19-21-51 pentany]]: 96 50 55 96 46 (sounds like minor pentatonic) | * [[4 of 7-17-19-21-51 pentany]]: 96 50 55 96 46 (sounds like minor pentatonic) | ||
* [[4 of 7-17-19-21-51 tetrapentany]]: 3 9 46 4 34 21 29 8 47 3 9 46 38 46 | * [[4 of 7-17-19-21-51 tetrapentany]]: 3 9 46 4 34 21 29 8 47 3 9 46 38 46 | ||
Revision as of 05:38, 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 (gen. 97\343, per. 343\343). 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.
Use as a NEJI
Of all n-afdos where n is between 343 and 800, and where n is a multiple of a simple prime by any number of 2s, or a simple semiprime by any number of 2s, 476afdo (7x17x2x2) approximates 343edo with the least relative error. (See Approximating 343edo in afdos.)
343edo could be approximated into 476afdo as a neji scale. Doing so would make it an over-17-by-7 scale (when viewed through a primodal lens). (Scala file.)
It would make sense to use smaller over-17, over-7, or over-17-by-7 JI scales as subsets of this neji.
Regular temperament properties
343edo is on the optimal ET sequence of gammy temperament (343be, 10\343 generator, 2/1 period), protolangwidge temperament (343, 200\343 g, 2/1 p) and anthoine temperament (343dd, 110\343 g, 2/1 p).
343edo might potentially be useful for 49th-octave temperaments (see Fractional-octave temperaments), this is something which hasn't been explored yet.
| 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 either for a specific temperament, or because of its good primes 2, 7, 17 and 19, which will inform how one might want to octave stretch or compress it.
- Using for a temperament
TE octave stretch:
- For 13-limit gammy
- Octave size: 1200.437 ¢
- For 7-limit anthoine
- Octave size: 1199.630 ¢
- Using for primes 2, 7, 17, 19
If one is using 343 for its accurate 2.7.17.19 intervals, 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
- 343 (patent val)
- Octave size: 1199.950 ¢
- TE error: 0.395 ¢/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
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.96 | +1.59 | -0.25 | -1.58 | +0.81 | -1.57 | -0.95 | -0.76 | -0.93 | +0.71 | +0.62 |
| Relative (%) | +27.4 | +45.5 | -7.2 | -45.3 | +23.1 | -44.8 | -27.1 | -21.7 | -26.5 | +20.2 | +17.8 | |
| Step | 544 | 797 | 963 | 1087 | 1187 | 1269 | 1340 | 1402 | 1457 | 1507 | 1552 | |
Scales
Scales are rotated to modes with lots of consonances.
- 4 of 7-17-19-21-51 pentany: 96 50 55 96 46 (sounds like minor pentatonic)
- 4 of 7-17-19-21-51 tetrapentany: 3 9 46 4 34 21 29 8 47 3 9 46 38 46
- 7-17-19-21 hexany: 50 46 50 55 87 55 (sounds like minor hexatonic)
- 7-17-19-21 by 3/2 trihexany: 3 47 8 38 12 38 8 47 3 46 9 29 9 46
- 9afdo: 40 37 34 32 30 28 52 47 43
- 18afdo: 21 19 19 18 17 17 16 16 15 15 14 14 27 25 24 23 22 21
- 36afdo: 11 10 9 10 9 10 9 9 8 9 8 9 8 8 8 8 7 8 7 8 7 7 7 7 14 13 13 12 12 12 12 11 11 11 11 10
- 72afdo: 5 6 5 5 4 5 5 5 5 4 5 5 4 5 4 5 4 4 5 4 4 4 5 4 4 4 4 4 4 4 4 4 4 3 4 4 4 3 4 4 3 4 4 3 4 3 4 3 7 7 6 7 6 7 6 6 6 6 6 6 6 6 6 5 6 5 6 5 6 5 5 5