# 1429edo

 ← 1428edo 1429edo 1430edo →
Prime factorization 1429 (prime)
Step size 0.839748¢
Fifth 836\1429 (702.029¢)
Semitones (A1:m2) 136:107 (114.2¢ : 89.85¢)
Consistency limit 9
Distinct consistency limit 9

1429 equal divisions of the octave (abbreviated 1429edo or 1429ed2), also called 1429-tone equal temperament (1429tet) or 1429 equal temperament (1429et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1429 equal parts of about 0.84 ¢ each. Each step represents a frequency ratio of 21/1429, or the 1429th root of 2.

## Theory

1429edo has a reasonable approximation of the full 17-limit. It is consistent to the 9-odd-limit with only 11/10 barely missing the line. The 11-limit optimal tuning of the equal temperament is consistent to the 18-integer-limit; however, the 13- and 17-limit optimal tunings, which have less of octave compression, are not, so one might want to keep the compression tight.

The equal temperament tempers out 4375/4374 in the 7-limit; 131072/130977, 759375/758912, 1953125/1951488, 2359296/2358125, 2657205/2656192, and 3294225/3294172 in the 11-limit; 2080/2079, 4096/4095, 4225/4224, 78125/78078, and 123201/123200 in the 13-limit; 2500/2499, 5832/5831, 11016/11011, and 12376/12375 in the 17-limit. It supports the gross temperament and provides the optimal patent val for the 11- and 13-limit trillium temperament.

### Prime harmonics

Approximation of prime harmonics in 1429edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.074 -0.030 +0.243 +0.397 +0.060 +0.013 -0.242 -0.143 -0.046 +0.381
Relative (%) +0.0 +8.9 -3.5 +29.0 +47.2 +7.2 +1.6 -28.8 -17.0 -5.5 +45.3
Steps
(reduced)
1429
(0)
2265
(836)
3318
(460)
4012
(1154)
4944
(657)
5288
(1001)
5841
(125)
6070
(354)
6464
(748)
6942
(1226)
7080
(1364)

### Subsets and supersets

1429edo is the 226th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [2265 -1429 [1429 2265]] -0.0235 0.0234 2.80
2.3.5 [39 -29 3, [-66 -36 53 [1429 2265 3318]] -0.0114 0.0257 3.06
2.3.5.7 4375/4374, [26 4 -3 -14, [40 -22 -1 -1 [1429 2265 3318 4012]] -0.0302 0.0395 4.70
2.3.5.7.11 4375/4374, 131072/130977, 759375/758912, 3294225/3294172 [1429 2265 3318 4012 4944]] -0.0471 0.0488 5.81
2.3.5.7.11.13 2080/2079, 4096/4095, 4375/4374, 78125/78078, 3294225/3294172 [1429 2265 3318 4012 4944 5288]] -0.0420 0.0460 5.48
2.3.5.7.11.13.17 2080/2079, 2500/2499, 4096/4095, 4375/4374, 11016/11011, 108086/108045 [1429 2265 3318 4012 4944 5288 5841]] -0.0364 0.0447 5.32

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio
Temperaments
1 109\1429 91.533 [144 -22 -47 Gross
1 674\1429 565.990 25/18 Trillium

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

Francium