2129edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.9 | | 2.9 | ||
| Line 55: | Line 64: | ||
| 0.0295 | | 0.0295 | ||
| 5.2 | | 5.2 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 65: | Line 81: | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Scales == | == Scales == | ||
Revision as of 13:06, 16 November 2024
| ← 2128edo | 2129edo | 2130edo → |
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