431edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
Line 34: Line 26:
| 32805/32768, {{monzo| 7 63 -46 }}
| 32805/32768, {{monzo| 7 63 -46 }}
| {{mapping| 431 683 1001 }}
| {{mapping| 431 683 1001 }}
| -0.0230
| &minus;0.0230
| 0.2082
| 0.2082
| 7.48
| 7.48
Line 41: Line 33:
| 2401/2400, 32805/32768, {{monzo| 3 16 -11 -1 }}
| 2401/2400, 32805/32768, {{monzo| 3 16 -11 -1 }}
| {{mapping| 431 683 1001 1210 }}
| {{mapping| 431 683 1001 1210 }}
| -0.0299
| &minus;0.0299
| 0.1803
| 0.1803
| 6.48
| 6.48
Line 48: Line 40:
| 2401/2400, 5632/5625, 8019/8000, 43923/43904
| 2401/2400, 5632/5625, 8019/8000, 43923/43904
| {{mapping| 431 683 1001 1210 1491 }}
| {{mapping| 431 683 1001 1210 1491 }}
| -0.0215
| &minus;0.0215
| 0.1621
| 0.1621
| 5.82
| 5.82
Line 55: Line 47:
| 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655
| 729/728, 1001/1000, 1716/1715, 4096/4095, 6656/6655
| {{mapping| 431 683 1001 1210 1491 1595 }}
| {{mapping| 431 683 1001 1210 1491 1595 }}
| -0.0318
| &minus;0.0318
| 0.1498
| 0.1498
| 5.38
| 5.38
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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| 19/14
| 19/14
| [[Ostara]]
| [[Ostara]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Music ==
== Music ==
; [[Francium]]
; [[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]
* "Take Advantage" from ''Take Advantage'' (2024) &ndash; [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]
* "Help Me Understand" from ''Abbreviations Gone Wrong'' (2024) &ndash; [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]
 
[[Category:Listen]]

Revision as of 03:10, 16 November 2024

← 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

Template:EDO intro

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.

Regular temperament properties

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

Music

Francium
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