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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 79
{{ED intro}}
| Step size = 3.79747¢
| Fifth = 189\316 (702.53¢)
| Semitones = 31:23 (117.72¢ : 87.34¢)
| Consistency = 11
}}
The '''316 equal divisions of the octave''' ('''316edo'''), or the '''316(-tone) equal temperament''' ('''316tet''', '''316et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 316 [[equal]] parts of about 3.80 [[cent]]s each.


== Theory ==
== Theory ==
While not highly accurate for its size, 316et is the point where a few important temperaments meet, and is distinctly [[consistent]] in the [[11-odd-limit]]. It 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]] and [[4096/4095]] in the 13-limit; notably supporting [[abigail]] and [[semiparakleismic]].  
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]].  


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.


316 factors into 2<sup>2</sup> × 79, with subset edos {{EDOs| 2, 4, 79, and 158 }}.
=== Prime harmonics ===
{{Harmonics in equal|316}}


=== Prime harmonics ===
=== Subsets and supersets ===
{{Primes in edo|316}}
316 factors into {{factorisation|316}}, with subset edos {{EDOs| 2, 4, 79, and 158 }}.


== 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]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 31: Line 27:
| 2.3
| 2.3
| {{monzo| 501 -316 }}
| {{monzo| 501 -316 }}
| [{{val| 316 501 }}]
| {{mapping| 316 501 }}
| -0.182
| −0.182
| 0.182
| 0.182
| 4.79
| 4.79
Line 38: Line 34:
| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
| [{{val| 316 501 734 }}]
| {{mapping| 316 501 734 }}
| -0.269
| −0.269
| 0.193
| 0.193
| 5.08
| 5.08
Line 45: Line 41:
| 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 }}
| [{{val| 316 501 734 887 }}]
| {{mapping| 316 501 734 887 }}
| -0.160
| −0.160
| 0.252
| 0.252
| 6.64
| 6.64
Line 52: Line 48:
| 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
| [{{val| 316 501 734 887 1093 }}]
| {{mapping| 316 501 734 887 1093 }}
| -0.088
| −0.088
| 0.267
| 0.267
| 7.04
| 7.04
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 3025/3024, 3136/3125, 4096/4095
| 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125
| [{{val| 316 501 734 887 1093 1169 }}]
| {{mapping| 316 501 734 887 1093 1169 }}
| -0.016
| −0.016
| 0.293
| 0.293
| 7.72
| 7.72
Line 67: Line 63:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 84: Line 81:
| 315.19
| 315.19
| 6/5
| 6/5
| [[Parakleismic]]
| [[Parakleismic]] (7-limit)
|-
|-
| 1
| 1
Line 96: Line 93:
| 535.44
| 535.44
| 512/375
| 512/375
| [[Maquila]]
| [[Maquila]] (5-limit)
|-
| 1
| 149\316
| 565.82
| 18/13
| [[Threedic]]
|-
| 1
| 155\316
| 588.61
| 45927/32768
| [[Countritonic]] (7-limit)
|-
|-
| 2
| 2
Line 105: Line 114:
|-
|-
| 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)
| [[Undim]] / [[unlit]]
| [[Unlit]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Semiparakleismic]]
[[Category:Semiparakleismic]]
[[Category:Triglav]]
[[Category:Triglav]]
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