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 [[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 ===
=== 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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| {{monzo| 566 -357 }}
| {{monzo| 566 -357 }}
| {{mapping| 357 566 }}
| {{mapping| 357 566 }}
| -0.1786
| −0.1786
| 0.1785
| 0.1785
| 5.31
| 5.31
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| 1600000/1594323, {{monzo| 61 4 -29 }}
| 1600000/1594323, {{monzo| 61 4 -29 }}
| {{mapping| 357 566 829 }}
| {{mapping| 357 566 829 }}
| -0.1536
| −0.1536
| 0.1500
| 0.1500
| 4.46
| 4.46
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| 10976/10935, 235298/234375, 2100875/2097152
| 10976/10935, 235298/234375, 2100875/2097152
| {{mapping| 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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| 3025/3024, 5632/5625, 10976/10935, 102487/102400
| 3025/3024, 5632/5625, 10976/10935, 102487/102400
| {{mapping| 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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| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
| 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935
| {{mapping| 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
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Pnict]]
| [[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
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 06:30, 21 February 2025

← 356edo 357edo 358edo →
Prime factorization 3 × 7 × 17
Step size 3.36134 ¢ 
Fifth 209\357 (702.521 ¢)
Semitones (A1:m2) 35:26 (117.6 ¢ : 87.39 ¢)
Consistency limit 7
Distinct consistency limit 7

357 equal divisions of the octave (abbreviated 357edo or 357ed2), also called 357-tone equal temperament (357tet) or 357 equal temperament (357et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 357 equal parts of about 3.36 ¢ each. Each step represents a frequency ratio of 21/357, or the 357th root of 2.

Theory

While not highly accurate for its size, 357et is the point where a few important temperaments meet. It tempers out 1600000/1594323 (amity comma), and [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 supports 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, but 270edo is better suited for this purpose.

Prime harmonics

Approximation of prime harmonics in 357edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.57 +0.24 -0.76 -0.06 -0.19 -0.75 +1.65 +0.30 -1.01 +1.18
Relative (%) +0.0 +16.8 +7.2 -22.6 -1.7 -5.7 -22.4 +49.0 +8.8 -29.9 +35.2
Steps
(reduced) 		357
(0) 		566
(209) 		829
(115) 		1002
(288) 		1235
(164) 		1321
(250) 		1459
(31) 		1517
(89) 		1615
(187) 		1734
(306) 		1769
(341)

Subsets and supersets

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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [566 -357⟩ [⟨357 566]] −0.1786 0.1785 5.31
2.3.5 1600000/1594323, [61 4 -29⟩ [⟨357 566 829]] −0.1536 0.1500 4.46
2.3.5.7 10976/10935, 235298/234375, 2100875/2097152 [⟨357 566 829 1002]] −0.0477 0.2248 6.69
2.3.5.7.11 3025/3024, 5632/5625, 10976/10935, 102487/102400 [⟨357 566 829 1002 1235]] −0.0348 0.2027 6.03
2.3.5.7.11.13 676/675, 1001/1000, 3025/3024, 4096/4095, 10976/10935 [⟨357 566 829 1002 1235 1321]] −0.0204 0.1879 5.59

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 101\357 339.50 243/200 Amity (5-limit)
1 118\357 396.64 44/35 Squarschmidt
1 163\357 547.90 48/35 Calamity
3 9\357 30.25 55/54 Hemichromat
3 18\357 60.50 28/27 Chromat (7-limit)
3 41\357 137.82 13/12 Avicenna
3 48\357 161.34 192/175 Pnict

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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