2129edo
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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
| ← 2128edo | 2129edo | 2130edo → |
2129 equal divisions of the octave (2129edo), or 2129-tone equal temperament (2129tet), 2129 equal temperament (2129et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 2129 equal parts of about 0.564 ¢ each.
Theory
2129et tempers out 95703125/95664294, 5767168/5764801, 47265625/47258883, 67110351/67108864 and 43923/43904 in the 11-limit; 33792000/33787663, 200000/199927, 34034175/34027136, 2250423/2249390, 78125/78078, 1449459/1449175, 1990656/1990625, 67392/67375, 4225/4224, 8858304/8857805, 59319/59290 and 4084223/4084101 in the 13-limit.
Odd harmonics
| 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 (%) | -39 | -38 | +14 | +23 | -13 | -24 | +23 | -21 | +16 | -24 | +34 | |
| 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
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 442\2129 | 249.131 | 81/70 | Hemischis (5-limit) |
| 1 | 884\2129 | 498.262 | 4/3 | Helmholtz |