429edo: Difference between revisions
Jump to navigation
Jump to search
ArrowHead294 (talk | contribs) m Partial undo |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
429et is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|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 [[support]]s [[vavoom]], [[turkey]], [[counterhanson]] and [[aluminium]]. | 429et is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|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 [[support]]s [[vavoom]], [[turkey]], [[counterhanson]], and [[aluminium]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
429 factors into | 429 factors into {{factorisation|429}}, with subset edos {{EDOs|3, 11, 13, 33, 39, and 143}}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 26: | Line 26: | ||
| {{monzo| 680 -429 }} | | {{monzo| 680 -429 }} | ||
| {{mapping| 429 680 }} | | {{mapping| 429 680 }} | ||
| | | −0.0451 | ||
| 0.0451 | | 0.0451 | ||
| 1.61 | | 1.61 | ||
| Line 93: | Line 93: | ||
| [[Mutt]] | | [[Mutt]] | ||
|} | |} | ||
<nowiki />* [[Normal | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
Latest revision as of 13:32, 13 March 2026
| ← 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 (%) | +0.0 | +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 distinct