2129edo

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← 2128edo2129edo2130edo →
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

Scales

Music

Francium