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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 431 (prime)
{{ED intro}}
| Step size = 2.78422¢
| Fifth = 252\431 (701.62¢)
| Semitones = 40:33 (111.37¢ : 91.88¢)
| Consistency = 15
}}
The '''431 equal divisions of the octave''' ('''431edo'''), or the '''431(-tone) equal temperament''' ('''431tet''', '''431et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 431 parts of about 2.78 [[cent]]s each.


== Theory ==
== Theory ==
431edo is [[consistent]] to the [[15-odd-limit]], tempering out the [[schisma]] in the 5-limit; [[2401/2400]] in the 7-limit; [[5632/5625]] and [[8019/8000]] in the 11-limit; [[729/728]], [[1001/1000]], [[1716/1715]], [[4096/4095]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s the [[sesquiquartififths]] temperament.
431edo is [[consistent]] to the [[15-odd-limit]]. The equal temperament [[tempering out|tempers out]] the [[schisma]] in the 5-limit; [[2401/2400]] in the 7-limit; [[5632/5625]] and [[8019/8000]] in the 11-limit; [[729/728]], [[1001/1000]], [[1716/1715]], [[4096/4095]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s the [[sesquiquartififths]] temperament.


431edo is the 83rd [[prime edo]].  
It allows [[essentially tempered chord]]s of [[squbemic chords]] and [[sinbadmic chords]] in the [[13-odd-limit]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|431}}
{{Harmonics in equal|431}}
=== Subsets and supersets ===
431edo is the 83rd [[prime edo]].
== Intervals ==
{{Main|Table of 431edo intervals}}


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3
| 2.3
| {{monzo| -683 431 }}
| {{monzo| -683 431 }}
| [{{val| 431 683 }}]
| {{mapping| 431 683 }}
| +0.1044
| +0.1044
| 0.1044
| 0.1044
Line 36: Line 37:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 7 63 -46 }}
| 32805/32768, {{monzo| 7 63 -46 }}
| [{{val| 431 683 1001 }}]
| {{mapping| 431 683 1001 }}
| -0.0230
| −0.0230
| 0.2082
| 0.2082
| 7.48
| 7.48
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 32805/32768, {{monzo| 3 16 -11 -1 }}
| 2401/2400, 32805/32768, {{monzo| 3 16 -11 -1 }}
| [{{val| 431 683 1001 1210 }}]
| {{mapping| 431 683 1001 1210 }}
| -0.0299
| −0.0299
| 0.1803
| 0.1803
| 6.48
| 6.48
Line 50: Line 51:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 5632/5625, 8019/8000, 43923/43904
| 2401/2400, 5632/5625, 8019/8000, 43923/43904
| [{{val| 431 683 1001 1210 1491 }}]
| {{mapping| 431 683 1001 1210 1491 }}
| -0.0215
| −0.0215
| 0.1621
| 0.1621
| 5.82
| 5.82
Line 57: Line 58:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655
| 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655
| [{{val| 431 683 1001 1210 1491 1595 }}]
| {{mapping| 431 683 1001 1210 1491 1595 }}
| -0.0318
| −0.0318
| 0.1498
| 0.1498
| 5.38
| 5.38
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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 octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 1
| 1
| 172\431
| 176\431
| 490.02
| 65/49
| [[Surmarvelpyth]]
|-
| 1
| 179\431
| 498.55
| 498.55
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
|1
| 1
|190\431
| 190\431
|529.00
| 529.00
|19/14
| 19/14
|[[Ostara]]
| [[Ostara]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* "Take Advantage" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/1mIxaoZYVo7DHLhePsWqmu Spotify] | [https://francium223.bandcamp.com/track/take-advantage Bandcamp] | [https://www.youtube.com/watch?v=CiFES4z6bMM YouTube]
* "Help Me Understand" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/6AiaohFI5RNNTAVncif4Ob Spotify] | [https://francium223.bandcamp.com/track/help-me-understand Bandcamp] | [https://www.youtube.com/watch?v=wWgIV-MlrSw YouTube]
* "I Will Get You." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/1FkxDVzNv6SuSfB5vc8T5y Spotify] | [https://francium223.bandcamp.com/track/i-will-get-you Bandcamp] | [https://www.youtube.com/watch?v=_jioqFrcwz8 YouTube] – in Nordenmarkic, 431edo tuning


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Listen]]
[[Category:Prime EDOs]]

Latest revision as of 02:30, 17 April 2025

← 430edo 431edo 432edo →
Prime factorization 431 (prime)
Step size 2.78422 ¢ 
Fifth 252\431 (701.624 ¢)
Semitones (A1:m2) 40:33 (111.4 ¢ : 91.88 ¢)
Consistency limit 15
Distinct consistency limit 15

431 equal divisions of the octave (abbreviated 431edo or 431ed2), also called 431-tone equal temperament (431tet) or 431 equal temperament (431et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 431 equal parts of about 2.78 ¢ each. Each step represents a frequency ratio of 21/431, or the 431st root of 2.

Theory

431edo is consistent to the 15-odd-limit. The equal temperament tempers out the schisma in the 5-limit; 2401/2400 in the 7-limit; 5632/5625 and 8019/8000 in the 11-limit; 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 and 10648/10647 in the 13-limit. It supports the sesquiquartififths temperament.

It allows essentially tempered chords of squbemic chords and sinbadmic chords in the 13-odd-limit.

Prime harmonics

Approximation of prime harmonics in 431edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.33 +0.69 +0.08 -0.04 +0.31 +0.85 +0.40 +0.96 +0.59 -0.72
Relative (%) +0.0 -11.9 +24.9 +3.0 -1.5 +11.0 +30.4 +14.3 +34.5 +21.0 -25.9
Steps
(reduced) 		431
(0) 		683
(252) 		1001
(139) 		1210
(348) 		1491
(198) 		1595
(302) 		1762
(38) 		1831
(107) 		1950
(226) 		2094
(370) 		2135
(411)

Subsets and supersets

431edo is the 83rd prime edo.

Intervals

Main article: Table of 431edo intervals

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-683 431⟩ [⟨431 683]] +0.1044 0.1044 3.75
2.3.5 32805/32768, [7 63 -46⟩ [⟨431 683 1001]] −0.0230 0.2082 7.48
2.3.5.7 2401/2400, 32805/32768, [3 16 -11 -1⟩ [⟨431 683 1001 1210]] −0.0299 0.1803 6.48
2.3.5.7.11 2401/2400, 5632/5625, 8019/8000, 43923/43904 [⟨431 683 1001 1210 1491]] −0.0215 0.1621 5.82
2.3.5.7.11.13 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655 [⟨431 683 1001 1210 1491 1595]] −0.0318 0.1498 5.38

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 63\431 175.41 448/405 Sesquiquartififths
1 176\431 490.02 65/49 Surmarvelpyth
1 179\431 498.55 4/3 Helmholtz
1 190\431 529.00 19/14 Ostara

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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