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]..." |
The assesssment of subgroups was too hasty and arbitrary. For edos like this an analysis on the tuning profile is required. |
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== 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 '''1089''' 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" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|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]] (%) | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo|1016 -641}} | | {{monzo| 1016 -641 }} | ||
| {{mapping|641 1016}} | | {{mapping| 641 1016 }} | ||
| -0.0231 | | -0.0231 | ||
| 0.0231 | | 0.0231 | ||
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|- | |- | ||
| 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 | ||
|} | |} | ||