User:Francium/1931edo
| ← 1930edo | 1931edo | 1932edo → |
1931 equal divisions of the octave (abbreviated 1931edo or 1931ed2), also called 1931-tone equal temperament (1931tet) or 1931 equal temperament (1931et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1931 equal parts of about 0.621 ¢ each. Each step represents a frequency ratio of 21/1931, or the 1931st root of 2.
Theory
1931edo is consistent to the 7-limit and its harmonic 3 is about halfway its steps. As an equal temperament, it tempers out 2401/2400, 359661568/358722675 and [-42 -26 31 4⟩ in the 7-limit. It is strong in the 2.9.7.11.17.23 subgroup, tempering out 5832/5831, 1896129/1895936, 43614208/43610967, 40257567/40247296, 448204933/448084224. The harmonics 7 and 23 have a remarkably small relative error of 0.2 percent. 1931edo supports newt.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.272 | +0.222 | -0.001 | -0.078 | -0.101 | +0.280 | -0.128 | +0.068 | +0.157 | +0.270 | +0.001 |
| Relative (%) | +43.7 | +35.7 | -0.2 | -12.5 | -16.2 | +45.1 | -20.6 | +10.9 | +25.2 | +43.5 | +0.2 | |
| Steps (reduced) |
3061 (1130) |
4484 (622) |
5421 (1559) |
6121 (328) |
6680 (887) |
7146 (1353) |
7544 (1751) |
7893 (169) |
8203 (479) |
8482 (758) |
8735 (1011) | |
Subsets and supersets
1931edo is the 294th prime edo. 3862edo, which doubles it, gives a good correction to its harmonic 3.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-6121 1931⟩ | [⟨1931 6121]] | +0.0123 | 0.0123 | 1.98 |
| 2.9.5 | 6115295232/6103515625, [-360 148 -47⟩ | [⟨1931 6121 4484]] | −0.0237 | 0.0518 | 8.34 |
| 2.9.5.7 | 43046721/43025920, 6115295232/6103515625, [-33 9 -15 14⟩ | [⟨1931 6121 4484 5421]] | −0.0176 | 0.0461 | 7.42 |
| 2.9.5.7.11 | 2359296/2358125, 9779616/9765625, 43046721/43025920, 3294225/3294172 | [⟨1931 6121 4484 5421 6680]] | −0.0083 | 0.0452 | 7.27 |