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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|316}}
{{ED intro}}


== Theory ==
== Theory ==
While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is [[consistency|distinctly consistent]] in the [[11-odd-limit]]. It [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, the [[undim comma]], {{monzo| 41 -20 -4 }}, and the [[maquila comma]], {{monzo| 49 -6 -17 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] in the 7-limit; [[3025/3024]], [[3388/3375]], [[9801/9800]] and [[14641/14580]] in the 11-limit; and using the [[patent val]], [[1716/1715]], [[2080/2079]], [[2197/2187]], [[4096/4095]], [[4225/4224]], [[6656/6655]], and [[10648/10647]] in the 13-limit; notably supporting [[abigail]] and [[semiparakleismic]].  
While not highly accurate for its size, 316edo is the point where a few important temperaments meet, and is [[consistency|distinctly consistent]] in the [[11-odd-limit]]. It [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, the [[undim comma]], {{monzo| 41 -20 -4 }}, and the [[maquila comma]], {{monzo| 49 -6 -17 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] in the 7-limit; [[3025/3024]], [[3388/3375]], [[9801/9800]], and [[14641/14580]] in the 11-limit; and using the [[patent val]], [[1716/1715]], [[2080/2079]], [[2197/2187]], [[4096/4095]], [[4225/4224]], [[6656/6655]], and [[10648/10647]] in the 13-limit; notably supporting [[abigail]] and [[semiparakleismic]].  


It provides the [[optimal patent val]] for the rank-4 temperament tempering out 3388/3375, and [[triglav]], which also tempers out 3025/3024.
It provides the [[optimal patent val]] for the rank-4 temperament tempering out 3388/3375, and [[triglav]], which also tempers out 3025/3024.
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=== Subsets and supersets ===
=== Subsets and supersets ===
316 factors into 2<sup>2</sup> × 79, with subset edos {{EDOs| 2, 4, 79, and 158 }}.  
316 factors into {{nowrap| 2<sup>2</sup> × 79 }}, with subset edos {{EDOs| 2, 4, 79, and 158 }}.  


== Regular temperament properties ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 501 -316 }}
| {{Monzo| 501 -316 }}
| {{mapping| 316 501 }}
| {{Mapping| 316 501 }}
| &minus;0.182
| −0.182
| 0.182
| 0.182
| 4.79
| 4.79
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
| {{Monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
| {{mapping| 316 501 734 }}
| {{Mapping| 316 501 734 }}
| &minus;0.269
| −0.269
| 0.193
| 0.193
| 5.08
| 5.08
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| 2.3.5.7
| 2.3.5.7
| 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }}
| 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }}
| {{mapping| 316 501 734 887 }}
| {{Mapping| 316 501 734 887 }}
| &minus;0.160
| −0.160
| 0.252
| 0.252
| 6.64
| 6.64
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 3136/3125, 4375/4374, 131072/130977
| 3025/3024, 3136/3125, 4375/4374, 131072/130977
| {{mapping| 316 501 734 887 1093 }}
| {{Mapping| 316 501 734 887 1093 }}
| &minus;0.088
| −0.088
| 0.267
| 0.267
| 7.04
| 7.04
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125
| 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125
| {{mapping| 316 501 734 887 1093 1169 }}
| {{Mapping| 316 501 734 887 1093 1169 }}
| &minus;0.016
| −0.016
| 0.293
| 0.293
| 7.72
| 7.72
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 588.61
| 588.61
| 45927/32768
| 45927/32768
| [[Countritonic]] (7-limit)
| [[Ragitritonic]] (7-limit)
|-
|-
| 2
| 2
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|-
|-
| 2
| 2
| 83\316<br />(75\316)
| 83\316<br>(75\316)
| 315.19<br />(284.81)
| 315.19<br>(284.81)
| 6/5<br />(33/28)
| 6/5<br>(33/28)
| [[Semiparakleismic]]
| [[Semiparakleismic]]
|-
|-
| 4
| 4
| 131\316<br />(27\316)
| 131\316<br>(27\316)
| 497.47<br />(102.53)
| 497.47<br>(102.53)
| 4/3<br />(35/33)
| 4/3<br>(35/33)
| [[Unlit]]
| [[Unlit]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Semiparakleismic]]
[[Category:Triglav]]
[[Category:Triglav]]
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