641edo: Difference between revisions
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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 | ||
|} | |} | ||
Revision as of 07:10, 21 July 2024
| ← 640edo | 641edo | 642edo → |
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 1089 1800 2218 2372], tempers out 676/675, 1001/1000, 6144/6125, 10985/10976, and 85294/85184.
Odd harmonics
| 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
| 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 it is distinct