236edo: Difference between revisions
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m →Subsets and supersets: Fix math error |
RTT table should start with the 7-limit as the 5-limit is enfactored; misc. cleanup |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|236}} | {{EDO intro|236}} | ||
==Theory== | |||
236edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by tempering out the [[schisma]] and the [[parakleisma]]. The 7-limit mapping is worse over that of 118edo, as it leans on the very sharp side. It tempers out [[6144/6125]] and [[19683/19600]], supporting [[hemischis]]. Using the 236e [[val]] {{val| 236 374 548 663 '''817''' }}, it tempers out [[243/242]], 1375/1372, [[6250/6237]], 14700/14641 and [[16384/16335]]. | == Theory == | ||
236edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]]. The 7-limit mapping is worse over that of 118edo in terms of relative error, as it leans on the very sharp side. It tempers out [[6144/6125]] and [[19683/19600]], supporting [[hemischis]]. Using the 236e [[val]] {{val| 236 374 548 663 '''817''' }}, it tempers out [[243/242]], 1375/1372, [[6250/6237]], 14700/14641 and [[16384/16335]]. | |||
The 236bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning very close to [[quarter-comma meantone]], although [[205edo]] is even closer. Alternately, sharpening it to 236b gives a fifth that is in the golden [[diaschismic]] sequence. | The 236bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning very close to [[quarter-comma meantone]], although [[205edo]] is even closer. Alternately, sharpening it to 236b gives a fifth that is in the golden [[diaschismic]] sequence. | ||
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Since 236 factors into 2<sup>2</sup> × 53, 236edo has subset edos {{EDOs| 2, 4, 59 and 118 }}. [[472edo]], which doubles it, provides good correction to harmonics 7 and 11. | Since 236 factors into 2<sup>2</sup> × 53, 236edo has subset edos {{EDOs| 2, 4, 59 and 118 }}. [[472edo]], which doubles it, provides good correction to 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]] (%) | ||
|- | |- | ||
|2.3.5 | | 2.3.5.7 | ||
| | | 6144/6125, 19683/19600, 390625/388962 | ||
|{{ | | {{mapping| 236 374 548 663 }} | ||
| 0. | | -0.1830 | ||
| 0. | | 0.03883 | ||
| | | 7.64 | ||
|} | |} | ||
Revision as of 13:05, 25 March 2024
| ← 235edo | 236edo | 237edo → |
Theory
236edo is enfactored in the 5-limit, with the same tuning as 118edo, defined by tempering out the schisma and the parakleisma. The 7-limit mapping is worse over that of 118edo in terms of relative error, as it leans on the very sharp side. It tempers out 6144/6125 and 19683/19600, supporting hemischis. Using the 236e val ⟨236 374 548 663 817], it tempers out 243/242, 1375/1372, 6250/6237, 14700/14641 and 16384/16335.
The 236bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning very close to quarter-comma meantone, although 205edo is even closer. Alternately, sharpening it to 236b gives a fifth that is in the golden diaschismic sequence.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.26 | +0.13 | +2.36 | -2.17 | -1.54 | +1.82 | +2.49 | +2.23 | -2.46 | -0.97 |
| Relative (%) | +0.0 | -5.1 | +2.5 | +46.4 | -42.6 | -30.4 | +35.9 | +48.9 | +43.9 | -48.4 | -19.0 | |
| Steps (reduced) |
236 (0) |
374 (138) |
548 (76) |
663 (191) |
816 (108) |
873 (165) |
965 (21) |
1003 (59) |
1068 (124) |
1146 (202) |
1169 (225) | |
Subsets and supersets
Since 236 factors into 22 × 53, 236edo has subset edos 2, 4, 59 and 118. 472edo, which doubles it, provides good correction to harmonics 7 and 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 6144/6125, 19683/19600, 390625/388962 | [⟨236 374 548 663]] | -0.1830 | 0.03883 | 7.64 |