31edo: Difference between revisions
Undue weight for a particular chord |
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|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| -49 31 }} | ||
| {{ | | {{Mapping| 31 49 }} | ||
| +1.637 | | +1.637 | ||
| 1.637 | | 1.637 | ||
| Line 943: | Line 943: | ||
| 2.3.5 | | 2.3.5 | ||
| 81/80, 393216/390625 | | 81/80, 393216/390625 | ||
| {{ | | {{Mapping| 31 49 72 }} | ||
| +0.976 | | +0.976 | ||
| 1.628 | | 1.628 | ||
| Line 950: | Line 950: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 81/80, 126/125, 1029/1024 | | 81/80, 126/125, 1029/1024 | ||
| {{ | | {{Mapping| 31 49 72 87 }} | ||
| +0.828 | | +0.828 | ||
| 1.432 | | 1.432 | ||
| Line 957: | Line 957: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 81/80, 99/98, 121/120, 126/125 | | 81/80, 99/98, 121/120, 126/125 | ||
| {{ | | {{Mapping| 31 49 72 87 107 }} | ||
| +1.205 | | +1.205 | ||
| 1.487 | | 1.487 | ||
| 3.841 | | 3.841 | ||
|- | |- | ||
| 2.3.5.7.11.23 | |||
| 81/80, 99/98, 126/125, 161/160, 231/230 | |||
| {{Mapping| 31 49 72 87 107 140 }} | |||
| +1.333 | |||
| 1.387 | |||
| 3.584 | |||
|- style="border-top: double;" | |||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 66/65, 81/80, 99/98, 105/104, 121/120 | | 66/65, 81/80, 99/98, 105/104, 121/120 | ||
| {{ | | {{Mapping| 31 49 72 87 107 115 }} | ||
| +0.502 | | +0.502 | ||
| 2.072 | | 2.072 | ||
| 5.353 | | 5.353 | ||
|} | |} | ||
* 31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are [[72edo|72]], 72, [[41edo|41]], and [[46edo|46]], respectively. | * 31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next equal temperaments doing better in those subgroups are [[72edo|72]], 72, [[41edo|41]], and [[46edo|46]], respectively. | ||
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* [[Syntonic–31 equivalence continuum]] | * [[Syntonic–31 equivalence continuum]] | ||
31edo provides the [[optimal patent val]] for the rank-5 temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, [[squares]], and [[casablanca]] in the 11-limit, and [[huygens|huygens/meantone]], squares, [[winston]], [[lupercalia]], and [[ | 31edo provides the [[optimal patent val]] for the rank-5 temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, [[squares]], and [[casablanca]] in the 11-limit, and [[huygens|huygens/meantone]], squares, [[winston]], [[lupercalia]], and [[nightingale]] in the 13-limit. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
| Line 1,289: | Line 1,289: | ||
| 38.71 | | 38.71 | ||
| | | | ||
| [[ | | [[Diesic]] / [[slender]] | ||
| (P8, P4/13) | | (P8, P4/13) | ||
|- | |- | ||
| Line 1,295: | Line 1,295: | ||
| 77.42 | | 77.42 | ||
| [[1L 14s]], [[15L 1s]] | | [[1L 14s]], [[15L 1s]] | ||
| [[Valentine]] / [[lupercalia]] | | [[Valentine]] / [[valentino]] / [[lupercalia]] | ||
| (P8, P5/9) | | (P8, P5/9) | ||
|- | |- | ||
| Line 1,301: | Line 1,301: | ||
| 116.13 | | 116.13 | ||
| [[1L 9s]], [[10L 1s]], [[10L 11s]] | | [[1L 9s]], [[10L 1s]], [[10L 11s]] | ||
| [[ | | [[Miracle]] | ||
| (P8, P5/6) | | (P8, P5/6) | ||
|- | |- | ||
| Line 1,307: | Line 1,307: | ||
| 154.84 | | 154.84 | ||
| [[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]] | | [[1L 6s]], [[7L 1s]], <br>[[8L 7s]], [[8L 15s]] | ||
| [[ | | [[Nusecond]] | ||
| (P8, P11/11) | | (P8, P11/11) | ||
|- | |- | ||
| Line 1,313: | Line 1,313: | ||
| 193.55 | | 193.55 | ||
| [[1L 5s]], [[6L 1s]], [[6L 7s]], <br>[[6L 13s]], [[6L 19s]] | | [[1L 5s]], [[6L 1s]], [[6L 7s]], <br>[[6L 13s]], [[6L 19s]] | ||
| [[Luna]] | | [[Luna]], [[didacus]] / [[hemithirds]] /<br>[[hemiwürschmidt]] / [[tutone]] | ||
| (P8, ccP4/15) | | (P8, ccP4/15) | ||
|- | |- | ||
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| 232.26 | | 232.26 | ||
| [[1L 4s]], [[5L 1s]], [[5L 6s]], <br>[[5L 11s]], [[5L 16s]], [[5L 21s]] | | [[1L 4s]], [[5L 1s]], [[5L 6s]], <br>[[5L 11s]], [[5L 16s]], [[5L 21s]] | ||
| [[Mothra]] / [[mosura]]<br>[[ | | [[Mothra]] / [[mosura]]<br>[[Oncle]] | ||
| (P8, P5/3) | | (P8, P5/3) | ||
|- | |- | ||
| Line 1,325: | Line 1,325: | ||
| 270.97 | | 270.97 | ||
| [[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]] | | [[1L 3s]], [[4L 1s]], [[4L 5s]], <br>[[9L 4s]], [[9L 13s]] | ||
| [[ | | [[Orwell]] / [[winston]] | ||
| (P8, P12/7) | | (P8, P12/7) | ||
|- | |- | ||
| Line 1,331: | Line 1,331: | ||
| 309.68 | | 309.68 | ||
| [[3L 1s]], [[4L 3s]], [[4L 7s]], <br>[[4L 11s]], [[4L 15s]], [[4L 19s]], <br>[[4L 23s]] | | [[3L 1s]], [[4L 3s]], [[4L 7s]], <br>[[4L 11s]], [[4L 15s]], [[4L 19s]], <br>[[4L 23s]] | ||
| [[Myna | | [[Myna]] | ||
| (P8, ccP5/10) | | (P8, ccP5/10) | ||
|- | |- | ||
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| 387.10 | | 387.10 | ||
| [[3L 1s]], [[3L 4s]], [[3L 7s]], <br>[[3L 10s]], [[3L 13s]], [[3L 16s]], <br>[[3L 19s]], [[3L 22s]], [[3L 25s]] | | [[3L 1s]], [[3L 4s]], [[3L 7s]], <br>[[3L 10s]], [[3L 13s]], [[3L 16s]], <br>[[3L 19s]], [[3L 22s]], [[3L 25s]] | ||
| [[Würschmidt]] / [[worschmidt]] | | [[Würschmidt]] / [[worschmidt]]<br>[[Grendel]] | ||
| (P8, ccP5/8) | | (P8, ccP5/8) | ||
|- | |- | ||
| Line 1,367: | Line 1,367: | ||
| 541.94 | | 541.94 | ||
| [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]] | | [[2L 3s]], [[2L 5s]], [[2L 7s]], <br>[[9L 2s]], [[11L 9s]] | ||
| [[Casablanca]]<br>[[Cypress]]<br>[[Oracle]] | | [[Casablanca]]<br>[[Cypress]] / [[necromanteion]]<br>[[Oracle]] | ||
| (P8, c<sup>5</sup>P4/12) | | (P8, c<sup>5</sup>P4/12) | ||
|- | |- | ||
| Line 1,376: | Line 1,376: | ||
| (P8, ccP4/5) | | (P8, ccP4/5) | ||
|} | |} | ||
<nowiki/>* [[ | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
31edo can additionally be detemepered to [[borwell]] (31 & 106), [[countermiracle]] (31 & 145), [[triwell]] (31 & 159), [[quadrawell]] (31 & 212), [[birds]] (31 & 217), [[rainwell]] (31 & 265), and [[quasiorwell]] (31 & 270). | |||
== Octave stretch or compression == | == Octave stretch or compression == | ||
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* 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent. | * 17 is sharp, like 13. In fact, the 17:13 ratio is matched within a tenth of a cent. | ||
* 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent (see [[ | * 19 is also sharp, like 13 and 17. The 19:17 ratio is about one cent sharp. 31edo could be considered a tuning of the 2.5.7.13.17.19 subgroup, on which it is consistent (see [[mercy]] temperament). | ||
* 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent. | * 23 is about as flat as 11. The chromatic semitone is about half a cent off from 23:22. 31edo could be considered a tuning of the 2.3.5.7.11.23 subgroup, on which it is consistent. | ||
* 27 is quite flat, as it's 3<sup>3</sup> and the error from the meantone fifths accumulates. | * 27 is quite flat, as it's 3<sup>3</sup> and the error from the meantone fifths accumulates. | ||