31edo: Difference between revisions
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31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next | 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 72, 72, 41, and 46, respectively. | ||
31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly [[consistent]]ly: to [[consistency #Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and the quince comma [[823543/819200]], thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]]. In the 11-limit, 31edo can be defined as the unique temperament that tempers out [[81/80]], [[99/98]], [[121/120]] and [[126/125]], and it supports [[orwell]], [[mohajira]], and the relatively high-accuracy temperament [[miracle]]. In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five 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/meantone, squares, winston, lupercalia and nightengale in the 13-limit. In the 17-limit it tempers out [[120/119]], equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad. | 31edo excels in the 2.5.7 subgroup (the JI chord 4:5:7 is represented highly [[consistent]]ly: to [[consistency #Consistency to distance d|distance]] 10.36). In 2.5.7 it tempers out the didacus comma [[3136/3125]] and the quince comma [[823543/819200]], thus also tempering out the very small [[rainy comma]], the simplest 2.5.7 comma tempered out by the 7-limit microtemperament [[171edo]]. In the 11-limit, 31edo can be defined as the unique temperament that tempers out [[81/80]], [[99/98]], [[121/120]] and [[126/125]], and it supports [[orwell]], [[mohajira]], and the relatively high-accuracy temperament [[miracle]]. In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five 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/meantone, squares, winston, lupercalia and nightengale in the 13-limit. In the 17-limit it tempers out [[120/119]], equating the otonal tetrad of 4:5:6:7 and the inversion of the 10:12:15:17 minor tetrad. | ||
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! [[Cents]] | ! [[Cents]] | ||
! [[Color name]] | ! [[Color name]] | ||
! Name | ! Name | ||
|- | |- | ||
| 5 | | 5 | ||
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| 21.506 | | 21.506 | ||
| Gu | | Gu | ||
| [[Syntonic comma]] | | [[Syntonic comma]] | ||
|- | |- | ||
| 5 | | 5 | ||
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| 10.061 | | 10.061 | ||
| Lasepyo | | Lasepyo | ||
| [[Semicomma]] | | [[Semicomma]] | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 854: | Line 854: | ||
| 13.795 | | 13.795 | ||
| Zotrigu | | Zotrigu | ||
| | | Starling comma | ||
|- | |- | ||
| 7 | | 7 | ||
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| 13.074 | | 13.074 | ||
| Trizo-agu | | Trizo-agu | ||
| Orwellisma | | Orwellisma | ||
|- | |- | ||
| 7 | | 7 | ||
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| 7.7115 | | 7.7115 | ||
| Ruyoyo | | Ruyoyo | ||
| | | Marvel comma | ||
|- | |- | ||
| 7 | | 7 | ||
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| 6.0832 | | 6.0832 | ||
| Zozoquingu | | Zozoquingu | ||
| Hemimean | | Hemimean | ||
|- | |- | ||
| 7 | | 7 | ||
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| 348.39 | | 348.39 | ||
| [[3L 1s]], [[3L 4s]], [[7L 3s]], <br>[[7L 10s]], [[7L 17s]] | | [[3L 1s]], [[3L 4s]], [[7L 3s]], <br>[[7L 10s]], [[7L 17s]] | ||
| [[Mohaha]] / [[vicentino]] / | | [[Mohaha]] / [[vicentino]] /<br>[[mohajira]] / [[migration]] | ||
[[mohajira]] / [[migration]] | |||
| (P8, P5/2) | | (P8, P5/2) | ||
|- | |- | ||