431edo
Jump to navigation
Jump to search
Prime factorization | 431 (prime) |
Step size | 2.78422¢ |
Fifth | 252\431 (701.62¢) |
Semitones (A1:m2) | 40:33 (111.37¢ : 91.88¢) |
Consistency limit | 15 |
The 431 equal divisions of the octave (431edo), or the 431(-tone) equal temperament (431tet, 431et) when viewed from a regular temperament perspective, is the equal division of the octave into 431 parts of about 2.78 cents each.
Theory
431edo is consistent to the 15-odd-limit, tempering 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.
431edo is the 83rd prime edo.
Prime harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | +0.00 | -0.33 | +0.69 | +0.08 | -0.04 | +0.31 | +0.85 | +0.40 | +0.96 | +0.59 |
relative (%) | +0 | -12 | +25 | +3 | -2 | +11 | +30 | +14 | +34 | +21 | |
Steps (reduced) |
431 (0) |
683 (252) |
1001 (139) |
1210 (348) |
1491 (198) |
1595 (302) |
1762 (38) |
1831 (107) |
1950 (226) |
2094 (370) |
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 octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
---|---|---|---|---|
1 | 63\431 | 175.41 | 448/405 | Sesquiquartififths |
1 | 172\431 | 498.55 | 4/3 | Helmholtz |