# 431edo

 ← 430edo 431edo 432edo →
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 2.78 ¢ 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

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 (%) +0.0 -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

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