# 429edo

 ← 428edo 429edo 430edo →
Prime factorization 3 × 11 × 13
Step size 2.7972¢
Fifth 251\429 (702.098¢)
Semitones (A1:m2) 41:32 (114.7¢ : 89.51¢)
Consistency limit 11
Distinct consistency limit 11

429 equal divisions of the octave (abbreviated 429edo or 429ed2), also called 429-tone equal temperament (429tet) or 429 equal temperament (429et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 429 equal parts of about 2.8 ¢ each. Each step represents a frequency ratio of 21/429, or the 429th root of 2.

## Theory

429et is consistent to the 11-odd-limit. The equal temperament tempers out 65625/65536, 250047/250000, and 823543/820125 in the 7-limit; 3025/3024, 12005/11979, 19712/19683, 42875/42768, and 102487/102400 in the 11-limit. It supports vavoom, turkey, counterhanson and aluminium.

### Prime harmonics

Approximation of prime harmonics in 429edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.14 -0.30 -0.99 -0.27 -1.37 +1.34 -1.01 +1.10 -0.21 -0.98
Relative (%) +0.0 +5.1 -10.7 -35.5 -9.6 -48.9 +47.8 -36.1 +39.2 -7.4 -35.0
Steps
(reduced)
429
(0)
680
(251)
996
(138)
1204
(346)
1484
(197)
1587
(300)
1754
(38)
1822
(106)
1941
(225)
2084
(368)
2125
(409)

### Subsets and supersets

429 factors into 3 × 11 × 13, with subset edos 3, 11, 13, 33, 39, and 143.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [680 -429 [429 680]] -0.0451 0.0451 1.61
2.3.5 [24 -21 4, [-20 -24 25 [429 680 996]] +0.0130 0.0900 3.22
2.3.5.7 65625/65536, 250047/250000, 823543/820125 [429 680 996 1204]] +0.0982 0.1670 5.97
2.3.5.7.11 3025/3024, 12005/11979, 19712/19683, 102487/102400 [429 680 996 1204 1484]] +0.0941 0.1496 5.35
2.3.5.7.11.13 676/675, 1001/1000, 3025/3024, 12005/11979, 19712/19683 [429 680 996 1204 1484 1587]] +0.1400 0.1708 6.11

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 40\429 111.89 16/15 Vavoom
1 113\429 316.08 6/5 Counterhanson
1 170\429 475.52 320/243 Vulture
3 138\429
(5\429)
386.01
(13.99)
5/4
(126/125)
Mutt

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct