641edo: Difference between revisions

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The assesssment of subgroups was too hasty and arbitrary. For edos like this an analysis on the tuning profile is required.
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|641}}
{{ED intro}}


== Theory ==
== Theory ==
641edo is only [[consistent]] to the [[5-odd-limit]]. Since both [[harmonic]]s [[7/1|7]] and [[11/1|11]] are about halfway between its steps, and since harmonic [[5/1|5]] is also off by more than a third step, it can be used as a dual-5 dual-7 dual-11 temperament. Alternatively, it can be used as a 2.3.5.13.17.19 [[subgroup]] temperament, as it is consistent in the no-7 no-11 19-odd-limit.  
641edo is only [[consistent]] to the [[5-odd-limit]]. Since both [[harmonic]]s [[7/1|7]] and [[11/1|11]] are about halfway between its steps, and since harmonic [[5/1|5]] is also off by more than a third step, it can be used as a dual-5 dual-7 dual-11 temperament. Alternatively, it can be used as a 2.3.5.13.17.19 [[subgroup]] temperament, as it is consistent in the no-7 no-11 19-odd-limit.  


To start with, consider the 641d val {{val| 641 1016 1488 '''1799''' 2217 2372 }} in the 13-limit, which [[tempering out|tempers out]] [[625/624]], [[2200/2197]], [[4459/4455]], 14641/14625, and [[19712/19683]]. The alternative 641df val, {{val| 641 1016 1488 '''1799''' 2217 '''2371''' }}, tempers out [[676/675]], [[1001/1000]], 19712/19683, [[31213/31104]], and 983125/979776. The 641ce val, {{val| 641 1016 '''1089''' 1800 '''2218''' 2372 }}, tempers out 676/675, 1001/1000, [[6144/6125]], [[10985/10976]], and 85294/85184.  
To start with, consider the 641d val {{val| 641 1016 1488 '''1799''' 2217 2372 }} in the 13-limit, which [[tempering out|tempers out]] [[625/624]], [[2200/2197]], [[4459/4455]], 14641/14625, and [[19712/19683]]. The alternative 641df val, {{val| 641 1016 1488 '''1799''' 2217 '''2371''' }}, tempers out [[676/675]], [[1001/1000]], 19712/19683, [[31213/31104]], and 983125/979776. The 641ce val, {{val| 641 1016 '''1489''' 1800 '''2218''' 2372 }}, tempers out 676/675, 1001/1000, [[6144/6125]], [[10985/10976]], and 85294/85184.  


=== Odd harmonics ===
=== Odd harmonics ===
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== 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
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| {{monzo| 1016 -641 }}
| {{monzo| 1016 -641 }}
| {{mapping| 641 1016 }}
| {{mapping| 641 1016 }}
| -0.0231
| −0.0231
| 0.0231
| 0.0231
| 1.23
| 1.23
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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
|-
|-
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| [[Vulture]]
| [[Vulture]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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
 
== Music ==
; [[Francium]]
* "Goofy Individual" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/3wNm7mtNBOuE4utwHFSoRW Spotify] | [https://francium223.bandcamp.com/track/goofy-individual Bandcamp] | [https://www.youtube.com/watch?v=c3ZGAeYHmrQ YouTube]
 
[[Category:Listen]]

Latest revision as of 13:32, 13 March 2026

← 640edo 641edo 642edo →
Prime factorization 641 (prime)
Step size 1.87207 ¢ 
Fifth 375\641 (702.028 ¢)
Semitones (A1:m2) 61:48 (114.2 ¢ : 89.86 ¢)
Consistency limit 5
Distinct consistency limit 5

641 equal divisions of the octave (abbreviated 641edo or 641ed2), also called 641-tone equal temperament (641tet) or 641 equal temperament (641et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 641 equal parts of about 1.87 ¢ each. Each step represents a frequency ratio of 21/641, or the 641st root of 2.

Theory

641edo is only consistent to the 5-odd-limit. Since both harmonics 7 and 11 are about halfway between its steps, and since harmonic 5 is also off by more than a third step, it can be used as a dual-5 dual-7 dual-11 temperament. Alternatively, it can be used as a 2.3.5.13.17.19 subgroup temperament, as it is consistent in the no-7 no-11 19-odd-limit.

To start with, consider the 641d val ⟨641 1016 1488 1799 2217 2372] in the 13-limit, which tempers out 625/624, 2200/2197, 4459/4455, 14641/14625, and 19712/19683. The alternative 641df val, ⟨641 1016 1488 1799 2217 2371], tempers out 676/675, 1001/1000, 19712/19683, 31213/31104, and 983125/979776. The 641ce val, ⟨641 1016 1489 1800 2218 2372], tempers out 676/675, 1001/1000, 6144/6125, 10985/10976, and 85294/85184.

Odd harmonics

Approximation of odd harmonics in 641edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.073 -0.666 +0.909 +0.146 -0.928 +0.034 -0.593 -0.119 +0.147 -0.890 +0.743
Relative (%) +3.9 -35.6 +48.5 +7.8 -49.6 +1.8 -31.7 -6.4 +7.8 -47.5 +39.7
Steps
(reduced) 		1016
(375) 		1488
(206) 		1800
(518) 		2032
(109) 		2217
(294) 		2372
(449) 		2504
(581) 		2620
(56) 		2723
(159) 		2815
(251) 		2900
(336)

Subsets and supersets

641edo is the 116th prime edo. 1282edo, which doubles it, gives a good correction to the harmonics 7 and 11.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1016 -641⟩ [⟨641 1016]] −0.0231 0.0231 1.23
2.3.5 [24 -21 4⟩, [-56 -13 33⟩ [⟨641 1016 1488]] +0.0803 0.1474 7.87

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 254\641 475.507 320/243 Vulture

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

Music

Francium
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