429edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|429}} == Theory == 429et is consistent to the 11-odd-limit. It tempers out 48828125/48771072, 65625/65536, 250047/250000, 359661568/358..."
 
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"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|429}}
{{ED intro}}


== Theory ==
== Theory ==
429et is consistent to the [[11-odd-limit]]. It tempers out 48828125/48771072, [[65625/65536]], [[250047/250000]], 359661568/358722675 and 5250987/5242880 in the 7-limit; [[117440512/117406179]], 806736/805255, 234375/234256, 26214400/26198073, 12005/11979, 1953125/1951488, 496125/495616, [[1953125/1948617]], [[200704/200475]], 42875/42768, 4302592/4296875, 352947/352000, 1879453125/1879048192, 184549376/184528125, [[19712/19683]], 1684375/1679616, 422576/421875, 184877/184320, 47265625/47258883, [[3025/3024]], 9453125/9437184, 160083/160000, 16808715/16777216, [[532400/531441]], 456533/455625, 102487/102400, [[1771561/1769472]] and 781258401/781250000 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 3 × 11 × 13, with subset edos {{EDOs|3, 11, 13, 33, 39, and 143}}.
429 factors into {{factorisation|429}}, with subset edos {{EDOs|3, 11, 13, 33, 39, and 143}}.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|680 -429}}
! rowspan="2" | [[Comma list]]
|{{mapping|429 680}}
! rowspan="2" | [[Mapping]]
| -0.0451
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 680 -429 }}
| {{mapping| 429 680 }}
| −0.0451
| 0.0451
| 0.0451
| 1.61
| 1.61
|-
|-
|2.3.5
| 2.3.5
|{{monzo|24 -21 4}}, {{monzo|-20 -24 25}}
| {{monzo| 24 -21 4 }}, {{monzo| -20 -24 25 }}
|{{mapping|429 680 996}}
| {{mapping| 429 680 996 }}
| +0.0130
| +0.0130
| 0.0900
| 0.0900
| 3.22
| 3.22
|-
|-
|2.3.5.7
| 2.3.5.7
|250047/250000, 823543/820125, 65625/65536
| 65625/65536, 250047/250000, 823543/820125
|{{mapping|429 680 996 1204}}
| {{mapping| 429 680 996 1204 }}
| +0.0982
| +0.0982
| 0.1670
| 0.1670
| 5.97
| 5.97
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 42875/42768, 160083/160000, 456533/455625
| 3025/3024, 12005/11979, 19712/19683, 102487/102400
|{{mapping|429 680 996 1204 1484}}
| {{mapping| 429 680 996 1204 1484 }}
| +0.0941
| +0.0941
| 0.1496
| 0.1496
| 5.35
| 5.35
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|1001/1000, 676/675, 3025/3024, 47432/47385, 12005/11979
| 676/675, 1001/1000, 3025/3024, 12005/11979, 19712/19683
|{{mapping|429 680 996 1204 1484 1587}}
| {{mapping| 429 680 996 1204 1484 1587 }}
| +0.1400
| +0.1400
| 0.1708
| 0.1708
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|40\429
| 40\429
|111.89
| 111.89
|16/15
| 16/15
|[[Vavoom]]
| [[Vavoom]]
|-
|-
|1
| 1
|113\429
| 113\429
|316.08
| 316.08
|6/5
| 6/5
|[[Counterhanson]]
| [[Counterhanson]]
|-
|-
|1
| 1
|170\429
| 170\429
|475.52
| 475.52
|320/243
| 320/243
|[[Vulture]]
| [[Vulture]]
|-
|-
|3
| 3
|138\429<br>(5\429)
| 138\429<br />(5\429)
|386.01<br>(13.99)
| 386.01<br />(13.99)
|5/4<br>(393216/390625)
| 5/4<br />(126/125)
|[[Mutt]]
| [[Mutt]]
|}
|}
 
<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
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Latest revision as of 13:32, 13 March 2026

← 428edo 429edo 430edo →
Prime factorization 3 × 11 × 13
Step size 2.7972 ¢ 
Fifth 251\429 (702.098 ¢)
Semitones (A1:m2) 41:32 (114.7 ¢ : 89.51 ¢)
Consistency limit 11
Distinct consistency limit 11

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

Approximation of prime harmonics in 429edo
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

Table of rank-2 temperaments by generator
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

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