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357edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|357}}
{{ED intro}}


== Theory ==
== 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. 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.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 357 factors into 3 × 5 × 7, 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 566 -357 }}
| {{monzo| 566 -357 }}
| [{{val| 357 566 }}]
| {{mapping| 357 566 }}
| -0.1786
| −0.1786
| 0.1785
| 0.1785
| 5.31
| 5.31
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| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| 61 4 -29 }}
| 1600000/1594323, {{monzo| 61 4 -29 }}
| [{{val| 357 566 829 }}]
| {{mapping| 357 566 829 }}
| -0.1536
| −0.1536
| 0.1500
| 0.1500
| 4.46
| 4.46
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| 2.3.5.7
| 2.3.5.7
| 10976/10935, 235298/234375, 2100875/2097152
| 10976/10935, 235298/234375, 2100875/2097152
| [{{val| 357 566 829 1002 }}]
| {{mapping| 357 566 829 1002 }}
| -0.0477
| −0.0477
| 0.2248
| 0.2248
| 6.69
| 6.69
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 5632/5625, 10976/10935, 102487/102400
| 3025/3024, 5632/5625, 10976/10935, 102487/102400
| [{{val| 357 566 829 1002 1235 }}]
| {{mapping| 357 566 829 1002 1235 }}
| -0.0348
| −0.0348
| 0.2027
| 0.2027
| 6.03
| 6.03
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
| 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
| −0.0204
| 0.1879
| 0.1879
| 5.59
| 5.59
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 44/35
| 44/35
| [[Squarschmidt]]
| [[Squarschmidt]]
|-
| 1
| 163\357
| 547.90
| 48/35
| [[Calamity]]
|-
|-
| 3
| 3
| 48\357
| 9\357
| 161.34
| 30.25
| 192/175
| 55/54
| [[Pnict]]
| [[Hemichromat]]
|-
|-
| 3
| 3
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| 13/12
| 13/12
| [[Avicenna (temperament)|Avicenna]]
| [[Avicenna (temperament)|Avicenna]]
|-
| 3
| 48\357
| 161.34
| 192/175
| [[Pnict]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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