357edo: Difference between revisions

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== Theory ==

While not highly accurate for its size, 357et is the point where a few important temperaments meet. ~~The equal temperament~~ [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit.

While not highly accurate for its size, 357et is the point where a few important temperaments meet. It [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo| 61 4 -29 }} (squarschimidt comma) in the [[5-limit]]; 10976/10935 ([[hemimage comma]]), 235298/234375 ([[triwellisma]]), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the [[7-limit]]; [[3025/3024]], [[5632/5625]], [[12005/11979]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the [[13-limit]].

It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[~~Avicenna~~ (temperament)|avicenna]], but [[270edo]] is better suited for this purpose.

It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[avicenna (temperament)|avicenna]], but [[270edo]] is better suited for this purpose.

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 357 factors into ~~{{factorization|357}}~~, 357edo has subset edos {{EDOs| 3, 7, 17, 21, 51, and 119 }}.

Since 357 factors into 3 × 7 × 17, 357edo has subset edos {{EDOs| 3, 7, 17, 21, 51, and 119 }}.

== Regular temperament properties ==

{~~{comma basis begin}}~~

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

|-

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

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

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

! rowspan="2" | Optimal<br />8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{monzo| 566 -357 }}

| {{mapping| 357 566 }}

| ~~−0~~.1786

| −0.1786

| 0.1785

| 5.31

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| 1600000/1594323, {{monzo| 61 4 -29 }}

| {{mapping| 357 566 829 }}

| ~~−0~~.1536

| −0.1536

| 0.1500

| 4.46

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| 10976/10935, 235298/234375, 2100875/2097152

| {{mapping| 357 566 829 1002 }}

| ~~−0~~.0477

| −0.0477

| 0.2248

| 6.69

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| 3025/3024, 5632/5625, 10976/10935, 102487/102400

| {{mapping| 357 566 829 1002 1235 }}

| ~~−0~~.0348

| −0.0348

| 0.2027

| 6.03

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| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935

| {{mapping| 357 566 829 1002 1235 1321 }}

| ~~−0~~.0204

| −0.0204

| 0.1879

| 5.59

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

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| 192/175

| [[Pnict]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|357}}
+{{ED intro}}
 == Theory ==
-While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo| 61 4 -29 }} (squarschimidt comma) in the 5-limit; 10976/10935 ([[hemimage comma]]), 235298/234375 (triwellisma), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the 7-limit; [[3025/3024]], [[5632/5625]], 12005/11979 in the 11-limit; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit.
+While not highly accurate for its size, 357et is the point where a few important temperaments meet. It [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]), and {{monzo| 61 4 -29 }} (squarschimidt comma) in the [[5-limit]]; 10976/10935 ([[hemimage comma]]), 235298/234375 ([[triwellisma]]), 250047/250000 ([[landscape comma]]), 2100875/2097152 ([[rainy comma]]) in the [[7-limit]]; [[3025/3024]], [[5632/5625]], [[12005/11979]] in the [[11-limit]]; [[676/675]], [[1001/1000]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the [[13-limit]].
-It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[Avicenna (temperament)|avicenna]], but [[270edo]] is better suited for this purpose.
+It [[support]]s 5-limit [[amity]] and 7-limit weak extensions [[calamity]] and [[chromat]]. It provides the [[optimal patent val]] for 11- and 13-limit [[hemichromat]], the 159 & 198 temperament. It also supports [[avicenna (temperament)|avicenna]], but [[270edo]] is better suited for this purpose.
 === Prime harmonics ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 357 factors into {{factorization|357}}, 357edo has subset edos {{EDOs| 3, 7, 17, 21, 51, and 119 }}.
+Since 357 factors into 3 × 7 × 17, 357edo has subset edos {{EDOs| 3, 7, 17, 21, 51, and 119 }}.
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
 | {{monzo| 566 -357 }}
 | {{mapping| 357 566 }}
-| &minus;0.1786
+| −0.1786
 | 0.1785
 | 5.31
@@ Line 26: / Line 35: @@
 | 1600000/1594323, {{monzo| 61 4 -29 }}
 | {{mapping| 357 566 829 }}
-| &minus;0.1536
+| −0.1536
 | 0.1500
 | 4.46
@@ Line 33: / Line 42: @@
 | 10976/10935, 235298/234375, 2100875/2097152
 | {{mapping| 357 566 829 1002 }}
-| &minus;0.0477
+| −0.0477
 | 0.2248
 | 6.69
@@ Line 40: / Line 49: @@
 | 3025/3024, 5632/5625, 10976/10935, 102487/102400
 | {{mapping| 357 566 829 1002 1235 }}
-| &minus;0.0348
+| −0.0348
 | 0.2027
 | 6.03
@@ Line 47: / Line 56: @@
 | 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
 | {{mapping| 357 566 829 1002 1235 1321 }}
-| &minus;0.0204
+| −0.0204
 | 0.1879
 | 5.59
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 96: / Line 112: @@
 | 192/175
 | [[Pnict]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct