431edo

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← 430edo431edo432edo →
Prime factorization 431 (prime)
Step size 2.78422¢
Fifth 252\431 (701.624¢)
Semitones (A1:m2) 40:33 (111.4¢ : 91.88¢)
Consistency limit 15
Distinct consistency limit 15

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 2.784 ¢ 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

Approximation of prime harmonics in 431edo
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 -12 +25 +3 -2 +11 +30 +14 +34 +21 -26
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

Table of rank-2 temperaments by generator
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