316edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 501 -316 }}
| {{monzo| 501 -316 }}
| {{mapping| 316 501 }}
| {{mapping| 316 501 }}
| -0.182
| &minus;0.182
| 0.182
| 0.182
| 4.79
| 4.79
Line 34: Line 26:
| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
| {{mapping| 316 501 734 }}
| {{mapping| 316 501 734 }}
| -0.269
| &minus;0.269
| 0.193
| 0.193
| 5.08
| 5.08
Line 41: Line 33:
| 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 }}
| -0.160
| &minus;0.160
| 0.252
| 0.252
| 6.64
| 6.64
Line 48: Line 40:
| 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 }}
| -0.088
| &minus;0.088
| 0.267
| 0.267
| 7.04
| 7.04
Line 55: Line 47:
| 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 }}
| -0.016
| &minus;0.016
| 0.293
| 0.293
| 7.72
| 7.72
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 112: Line 98:
|-
|-
| 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]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


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

Revision as of 04:31, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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