316edo: Difference between revisions

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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]]

Latest revision as of 22:50, 20 February 2025

← 315edo 316edo 317edo →
Prime factorization 22 × 79
Step size 3.79747 ¢ 
Fifth 185\316 (702.532 ¢)
Semitones (A1:m2) 31:23 (117.7 ¢ : 87.34 ¢)
Consistency limit 11
Distinct consistency limit 11

316 equal divisions of the octave (abbreviated 316edo or 316ed2), also called 316-tone equal temperament (316tet) or 316 equal temperament (316et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 316 equal parts of about 3.8 ¢ each. Each step represents a frequency ratio of 21/316, or the 316th root of 2.

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, [8 14 -13⟩, the undim comma, [41 -20 -4⟩, and the maquila comma, [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.

Prime harmonics

Approximation of prime harmonics in 316edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.58 +1.03 -0.47 -0.69 -1.29 +1.37 -1.31 -1.69 -0.46 +1.80
Relative (%) +0.0 +15.2 +27.1 -12.4 -18.0 -33.9 +36.2 -34.5 -44.6 -12.2 +47.4
Steps
(reduced) 		316
(0) 		501
(185) 		734
(102) 		887
(255) 		1093
(145) 		1169
(221) 		1292
(28) 		1342
(78) 		1429
(165) 		1535
(271) 		1566
(302)

Subsets and supersets

316 factors into 22 × 79, with subset edos 2, 4, 79, and 158.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [501 -316⟩ [⟨316 501]] −0.182 0.182 4.79
2.3.5 [8 14 -13⟩, [41 -20 -4⟩ [⟨316 501 734]] −0.269 0.193 5.08
2.3.5.7 3136/3125, 4375/4374, [-26 -1 1 9⟩ [⟨316 501 734 887]] −0.160 0.252 6.64
2.3.5.7.11 3025/3024, 3136/3125, 4375/4374, 131072/130977 [⟨316 501 734 887 1093]] −0.088 0.267 7.04
2.3.5.7.11.13 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125 [⟨316 501 734 887 1093 1169]] −0.016 0.293 7.72

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 51\316 193.67 28/25 Didacus
1 83\316 315.19 6/5 Parakleismic (7-limit)
1 84\316 322.78 3087/2560 Seniority
1 141\316 535.44 512/375 Maquila (5-limit)
1 149\316 565.82 18/13 Threedic
1 155\316 588.61 45927/32768 Countritonic (7-limit)
2 55\316 208.86 44/39 Abigail
2 83\316
(75\316) 		315.19
(284.81) 		6/5
(33/28) 		Semiparakleismic
4 131\316
(27\316) 		497.47
(102.53) 		4/3
(35/33) 		Unlit

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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