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 ~~2<sup>2</sup> × 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 ==

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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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| [[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 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 ==
@@ Line 28: / Line 28: @@
 | {{monzo| 501 -316 }}
 | {{mapping| 316 501 }}
-| &minus;0.182
+| −0.182
 | 0.182
 | 4.79
@@ Line 35: / Line 35: @@
 | {{monzo| 8 14 -13 }}, {{monzo| 41 -20 -4 }}
 | {{mapping| 316 501 734 }}
-| &minus;0.269
+| −0.269
 | 0.193
 | 5.08
@@ Line 42: / Line 42: @@
 | 3136/3125, 4375/4374, {{monzo| -26 -1 1 9 }}
 | {{mapping| 316 501 734 887 }}
-| &minus;0.160
+| −0.160
 | 0.252
 | 6.64
@@ Line 49: / Line 49: @@
 | 3025/3024, 3136/3125, 4375/4374, 131072/130977
 | {{mapping| 316 501 734 887 1093 }}
-| &minus;0.088
+| −0.088
 | 0.267
 | 7.04
@@ Line 56: / Line 56: @@
 | 1716/1715, 2080/2079, 2197/2187, 3025/3024, 3136/3125
 | {{mapping| 316 501 734 887 1093 1169 }}
-| &minus;0.016
+| −0.016
 | 0.293
 | 7.72
@@ Line 125: / Line 125: @@
 | [[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 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]]