431edo

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431edo
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

Approximation of prime harmonics in 431edo
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

Table of rank-2 temperaments by generator
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