316edo: Difference between revisions

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== 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, 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.

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=== Subsets and supersets ===

316 factors into ~~22 × 79~~, with subset edos {{EDOs| 2, 4, 79, and 158 }}.

316 factors into {{factorisation|316}}, with subset edos {{EDOs| 2, 4, 79, and 158 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal ~~8ve~~ ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{monzo| 501 -316 }}

| {{mapping| 316 501 }}

| -0.182

| −0.182

| 0.182

| 4.79

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| {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}

| {{mapping| 316 501 734 }}

| -0.269

| −0.269

| 0.193

| 5.08

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| 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }}

| {{mapping| 316 501 734 887 }}

| -0.160

| −0.160

| 0.252

| 6.64

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| 3025/3024, 3136/3125, 4375/4374, 131072/130977

| {{mapping| 316 501 734 887 1093 }}

| -0.088

| −0.088

| 0.267

| 7.04

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| 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125

| {{mapping| 316 501 734 887 1093 1169 }}

| -0.016

| −0.016

| 0.293

| 7.72

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=== Rank-2 temperaments ===

{| 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 per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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|-

| 2

| 83\316 (75\316)

| 83\316 (75\316)

| 315.19 (284.81)

| 315.19 (284.81)

| 6/5 (33/28)

| 6/5 (33/28)

| [[Semiparakleismic]]

|-

| 4

| 131\316 (27\316)

| 131\316 (27\316)

| 497.47 (102.53)

| 497.47 (102.53)

| 4/3 (35/33)

| 4/3 (35/33)

| [[Unlit]]

|}

<nowiki>*~~</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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Parakleismic]]

[[Category:Semiparakleismic]]

[[Category:Triglav]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|316}}
+{{ED intro}}
 == 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, 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.
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-factors into 2<sup>2</sup> × 79, with subset edos {{EDOs| 2, 4, 79, and 158 }}.
+factors into {{factorisation|316}}, with subset edos {{EDOs| 2, 4, 79, and 158 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 27: / Line 28: @@
 | {{monzo| 501 -316 }}
 | {{mapping| 316 501 }}
-| -0.182
+| −0.182
 | 0.182
 | 4.79
@@ Line 34: / Line 35: @@
 | {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
 | {{mapping| 316 501 734 }}
-| -0.269
+| −0.269
 | 0.193
 | 5.08
@@ Line 41: / Line 42: @@
 | 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }}
 | {{mapping| 316 501 734 887 }}
-| -0.160
+| −0.160
 | 0.252
 | 6.64
@@ Line 48: / Line 49: @@
 | 3025/3024, 3136/3125, 4375/4374, 131072/130977
 | {{mapping| 316 501 734 887 1093 }}
-| -0.088
+| −0.088
 | 0.267
 | 7.04
@@ Line 55: / Line 56: @@
 | 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125
 | {{mapping| 316 501 734 887 1093 1169 }}
-| -0.016
+| −0.016
 | 0.293
 | 7.72
@@ Line 62: / Line 63: @@
 === Rank-2 temperaments ===
 {| 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 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 112: / Line 114: @@
 |-
 | 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]]
 |-
 | 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]]
 |}
-<nowiki>*</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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Parakleismic]]
 [[Category:Semiparakleismic]]
 [[Category:Triglav]]