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 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~~; notably~~ [[support]]~~ing~~ [[amity]], [[chromat]], ~~and~~ [[Avicenna (temperament)|avicenna]].

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.

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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| 2.3

| {{monzo| 566 -357 }}

| [{{~~val~~| 357 566 }}]

| {{mapping| 357 566 }}

| -0.1786

| 0.1785

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| 2.3.5

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

| [{{~~val~~| 357 566 829 }}]

| {{mapping| 357 566 829 }}

| -0.1536

| 0.1500

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| 2.3.5.7

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

| [{{~~val~~| 357 566 829 1002 }}]

| {{mapping| 357 566 829 1002 }}

| -0.0477

| 0.2248

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| 2.3.5.7.11

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

| [{{~~val~~| 357 566 829 1002 1235 }}]

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

| -0.0348

| 0.2027

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| 2.3.5.7.11.13

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

| [{{~~val~~| 357 566 829 1002 1235 1321 }}]

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

| -0.0204

| 0.1879

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Pnict]]

|}

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

@@ Line 3: / Line 3: @@
 == Theory ==
-While not highly accurate for its size, 357et is the point where a few important temperaments meet. The equal temperament 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; notably [[support]]ing [[amity]], [[chromat]], and [[Avicenna (temperament)|avicenna]].
+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.
+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 24: / Line 26: @@
 | 2.3
 | {{monzo| 566 -357 }}
-| [{{val| 357 566 }}]
+| {{mapping| 357 566 }}
 | -0.1786
 | 0.1785
@@ Line 31: / Line 33: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| 61 4 -29 }}
-| [{{val| 357 566 829 }}]
+| {{mapping| 357 566 829 }}
 | -0.1536
 | 0.1500
@@ Line 38: / Line 40: @@
 | 2.3.5.7
 | 10976/10935, 235298/234375, 2100875/2097152
-| [{{val| 357 566 829 1002 }}]
+| {{mapping| 357 566 829 1002 }}
 | -0.0477
 | 0.2248
@@ Line 45: / Line 47: @@
 | 2.3.5.7.11
 | 3025/3024, 5632/5625, 10976/10935, 102487/102400
-| [{{val| 357 566 829 1002 1235 }}]
+| {{mapping| 357 566 829 1002 1235 }}
 | -0.0348
 | 0.2027
@@ Line 52: / Line 54: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
-| [{{val| 357 566 829 1002 1235 1321 }}]
+| {{mapping| 357 566 829 1002 1235 1321 }}
 | -0.0204
 | 0.1879
@@ Line 62: / Line 64: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 109: / Line 111: @@
 | [[Pnict]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct