2129edo
| ← 2128edo | 2129edo | 2130edo → |
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. However, its representation of 5/3 and its octave complement 6/5 are extremely accurate, due to being a continued fraction convergent to their logarithms.
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 (%) | -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
| 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