429edo
| ← 428edo | 429edo | 430edo → |
429 equal divisions of the octave (abbreviated 429edo or 429ed2), also called 429-tone equal temperament (429tet) or 429 equal temperament (429et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 429 equal parts of about 2.8 ¢ each. Each step represents a frequency ratio of 21/429, or the 429th root of 2.
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 | +5 | -11 | -36 | -10 | -49 | +48 | -36 | +39 | -7 | -35 | |
| 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