641edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|641}} == Theory == 641edo is consistent to the 5-odd-limit. It can be used in the 2.3.5.11.13.17 subgroup, tempering out [[625/624]..." |
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}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
641edo is [[consistent]] to the [[5-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 '''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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=== Subsets and supersets === | === Subsets and supersets === | ||
641edo is the 116th [[prime | 641edo is the 116th [[prime edo]]. [[1282edo]], which doubles it, gives a good correction to the harmonics 7 and 11. | ||
== 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 | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" |Optimal<br>8ve | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
|- | ! colspan="2" | Tuning error | ||
![[TE error|Absolute]] (¢) | |- | ||
![[TE simple badness|Relative]] (%) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo|1016 -641}} | | {{monzo| 1016 -641 }} | ||
| {{mapping|641 1016}} | | {{mapping| 641 1016 }} | ||
| | | −0.0231 | ||
| 0.0231 | | 0.0231 | ||
| 1.23 | | 1.23 | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo|24 -21 4}}, {{monzo|-56 -13 33}} | | {{monzo| 24 -21 4 }}, {{monzo| -56 -13 33 }} | ||
| {{mapping|641 1016 1488}} | | {{mapping| 641 1016 1488 }} | ||
| +0.0803 | | +0.0803 | ||
| 0.1474 | | 0.1474 | ||
| 7.87 | | 7.87 | ||
|} | |} | ||
=== 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> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Vulture]] | | [[Vulture]] | ||
|} | |} | ||
<nowiki>* | <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]] | |||