31edo: Difference between revisions
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== Just approximation == | == Just approximation == | ||
=== 15-odd-limit interval mappings === | |||
The following table shows how [[15-odd-limit intervals]] are represented in 31edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | The following table shows how [[15-odd-limit intervals]] are represented in 31edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. | ||
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=== Selected 19-limit intervals === | |||
[[File:31-edo.svg|alt=alt : Your browser has no SVG support.]] | [[File:31-edo.svg|alt=alt : Your browser has no SVG support.]] | ||
== Relationship to 12-edo == | == Relationship to 12-edo == | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
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 {{monzo|-15 0 -2 7}} ([[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. | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -49 31 }} | |||
| [{{val| 31 49 }}] | |||
| +1.63 | |||
| 1.64 | |||
| 4.22 | |||
|- | |||
| 2.3.5 | |||
| 81/80, 393216/390625 | |||
| [{{val| 31 49 72 }}] | |||
| +0.98 | |||
| 1.63 | |||
| 4.20 | |||
|- | |||
| 2.3.5.7 | |||
| 81/80, 126/125, 1029/1024 | |||
| [{{val| 31 49 72 87 }}] | |||
| +0.83 | |||
| 1.43 | |||
| 3.70 | |||
|- | |||
| 2.3.5.7.11 | |||
| 81/80, 99/98, 121/120, 126/125 | |||
| [{{val| 31 49 72 87 107 }}] | |||
| +1.21 | |||
| 1.49 | |||
| 3.84 | |||
|} | |||
31et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, and 17-limit. The next ETs 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 {{monzo|-15 0 -2 7}} ([[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. | |||
=== Commas === | === Commas === | ||