431edo
| ← 430edo | 431edo | 432edo → |
431 equal divisions of the octave (abbreviated 431edo or 431ed2), also called 431-tone equal temperament (431tet) or 431 equal temperament (431et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 431 equal parts of about 2.78 ¢ each. Each step represents a frequency ratio of 21/431, or the 431st root of 2.
Theory
431edo is consistent to the 15-odd-limit. The equal temperament tempers out the schisma in the 5-limit; 2401/2400 in the 7-limit; 5632/5625 and 8019/8000 in the 11-limit; 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 and 10648/10647 in the 13-limit. It supports the sesquiquartififths temperament.
It allows essentially tempered chords of squbemic chords and sinbadmic chords in the 13-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.33 | +0.69 | +0.08 | -0.04 | +0.31 | +0.85 | +0.40 | +0.96 | +0.59 | -0.72 |
| Relative (%) | +0.0 | -11.9 | +24.9 | +3.0 | -1.5 | +11.0 | +30.4 | +14.3 | +34.5 | +21.0 | -25.9 | |
| Steps (reduced) |
431 (0) |
683 (252) |
1001 (139) |
1210 (348) |
1491 (198) |
1595 (302) |
1762 (38) |
1831 (107) |
1950 (226) |
2094 (370) |
2135 (411) | |
Subsets and supersets
431edo is the 83rd prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-683 431⟩ | [⟨431 683]] | +0.1044 | 0.1044 | 3.75 |
| 2.3.5 | 32805/32768, [7 63 -46⟩ | [⟨431 683 1001]] | −0.0230 | 0.2082 | 7.48 |
| 2.3.5.7 | 2401/2400, 32805/32768, [3 16 -11 -1⟩ | [⟨431 683 1001 1210]] | −0.0299 | 0.1803 | 6.48 |
| 2.3.5.7.11 | 2401/2400, 5632/5625, 8019/8000, 43923/43904 | [⟨431 683 1001 1210 1491]] | −0.0215 | 0.1621 | 5.82 |
| 2.3.5.7.11.13 | 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 | [⟨431 683 1001 1210 1491 1595]] | −0.0318 | 0.1498 | 5.38 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 63\431 | 175.41 | 448/405 | Sesquiquartififths |
| 1 | 176\431 | 490.02 | 65/49 | Surmarvelpyth |
| 1 | 179\431 | 498.55 | 4/3 | Helmholtz |
| 1 | 190\431 | 529.00 | 19/14 | Ostara |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct