# 2129edo

 ← 2128edo 2129edo 2130edo →
Prime factorization 2129 (prime)
Step size 0.563645¢
Fifth 1245\2129 (701.738¢)
Semitones (A1:m2) 199:162 (112.2¢ : 91.31¢)
Dual sharp fifth 1246\2129 (702.302¢)
Dual flat fifth 1245\2129 (701.738¢)
Dual major 2nd 362\2129 (204.039¢)
Consistency limit 5
Distinct consistency limit 5

2129 equal divisions of the octave (abbreviated 2129edo or 2129ed2), also called 2129-tone equal temperament (2129tet) or 2129 equal temperament (2129et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2129 equal parts of about 0.564 ¢ each. Each step represents a frequency ratio of 21/2129, or the 2129th root of 2.

## Theory

2129edo is only consistent to the 5-odd-limit, where it tempers out the schisma. Otherwise its poor approximation to both harmonics 3 and 5 commends itself to a 2.9.15.7.11.13.… subgroup interpretation.

### Odd harmonics

Approximation of odd harmonics in 2129edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.217 -0.217 +0.080 +0.129 -0.073 -0.133 +0.130 -0.117 +0.091 -0.137 +0.190
Relative (%) -38.5 -38.5 +14.1 +23.0 -13.0 -23.6 +23.0 -20.8 +16.2 -24.4 +33.7
Steps
(reduced)
3374
(1245)
4943
(685)
5977
(1719)
6749
(362)
7365
(978)
7878
(1491)
8318
(1931)
8702
(186)
9044
(528)
9351
(835)
9631
(1115)

### Subsets and supersets

2129edo is the 320th prime edo. 4258edo, which doubles it, gives a good correction to the harmonics 3 and 5.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-6749 2129 [2129 6749]] -0.0204 0.0204 3.62
2.9.15 [37 29 -33, [209 -61 -4 [2129 6749 8318]] -0.0247 0.0177 3.14
2.9.15.7 24414062500/24407490807, 13841287201/13839609375, 2199023255552/2197176384375 [2129 6749 8318 5977]] -0.0256 0.0154 2.73
2.9.15.7.11 9800/9801, 5767168/5764801, 104857600/104825259, 13841287201/13839609375 [2129 6749 8318 5977 7365]] -0.0162 0.0232 4.12
2.9.15.7.11.13 10648/10647, 9801/9800, 196625/196608, 36924979/36905625, 304117528/303807105 [2129 6749 8318 5977 7365 7878]] -0.0075 0.0288 5.11
2.9.15.7.11.13.17 2431/2430, 10648/10647, 9801/9800, 845325/845152, 297440/297381, 11275335/11275264, 15980544/15978655 [2129 6749 8318 5977 7365 7878 8702]] -0.0024 0.0295 5.2

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 884\2129 498.262 4/3 Helmholtz

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

Francium