429edo: 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]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 48: | Line 57: | ||
| 0.1708 | | 0.1708 | ||
| 6.11 | | 6.11 | ||
|} | |||
=== 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 76: | Line 92: | ||
| 5/4<br />(126/125) | | 5/4<br />(126/125) | ||
| [[Mutt]] | | [[Mutt]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 13:06, 16 November 2024
| ← 428edo | 429edo | 430edo → |
Theory
429et is consistent to the 11-odd-limit. The equal temperament tempers out 65625/65536, 250047/250000, and 823543/820125 in the 7-limit; 3025/3024, 12005/11979, 19712/19683, 42875/42768, and 102487/102400 in the 11-limit. It supports vavoom, turkey, counterhanson and aluminium.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.14 | -0.30 | -0.99 | -0.27 | -1.37 | +1.34 | -1.01 | +1.10 | -0.21 | -0.98 |
| Relative (%) | +0.0 | +5.1 | -10.7 | -35.5 | -9.6 | -48.9 | +47.8 | -36.1 | +39.2 | -7.4 | -35.0 | |
| Steps (reduced) |
429 (0) |
680 (251) |
996 (138) |
1204 (346) |
1484 (197) |
1587 (300) |
1754 (38) |
1822 (106) |
1941 (225) |
2084 (368) |
2125 (409) | |
Subsets and supersets
429 factors into 3 × 11 × 13, with subset edos 3, 11, 13, 33, 39, and 143.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [680 -429⟩ | [⟨429 680]] | −0.0451 | 0.0451 | 1.61 |
| 2.3.5 | [24 -21 4⟩, [-20 -24 25⟩ | [⟨429 680 996]] | +0.0130 | 0.0900 | 3.22 |
| 2.3.5.7 | 65625/65536, 250047/250000, 823543/820125 | [⟨429 680 996 1204]] | +0.0982 | 0.1670 | 5.97 |
| 2.3.5.7.11 | 3025/3024, 12005/11979, 19712/19683, 102487/102400 | [⟨429 680 996 1204 1484]] | +0.0941 | 0.1496 | 5.35 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 3025/3024, 12005/11979, 19712/19683 | [⟨429 680 996 1204 1484 1587]] | +0.1400 | 0.1708 | 6.11 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 40\429 | 111.89 | 16/15 | Vavoom |
| 1 | 113\429 | 316.08 | 6/5 | Counterhanson |
| 1 | 170\429 | 475.52 | 320/243 | Vulture |
| 3 | 138\429 (5\429) |
386.01 (13.99) |
5/4 (126/125) |
Mutt |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct