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339edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|339}} == Theory == 339et is only consistent to the 5-odd-limit. It can be used for the 2.3.5.13.19.23.31.41 subgroup, tempering out 780/779..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|339}}
{{ED intro}}


== Theory ==
== Theory ==
339et is only consistent to the [[5-odd-limit]]. It can be used for the 2.3.5.13.19.23.31.41 [[subgroup]], tempering out 780/779, [[621/620]], 1426/1425, 1026/1025, 2945/2944, 14391/14375 and 73853/73800. Using the 339d val ({{val|339 537 787 '''951'''}}) in the 7-limit and the 339de val ({{val|339 537 787 '''951''' '''1172'''}}) in the 11-limit, it [[support]]s [[tritriple]].
339edo is [[enfactoring|enfactored]] in the [[3-limit]] with the same tuning as [[113edo]], and is only [[consistent]] to the [[5-odd-limit]]. Using the 339d val ({{val| 339 537 787 '''951''' }}) in the 7-limit and the 339de val ({{val| 339 537 787 '''951''' '''1172''' }}) in the 11-limit, it [[support]]s [[tritriple]].
 
It can be used for the 2.3.5.13.19.23.31.41 [[subgroup]], where it [[tempering out|tempers out]] [[621/620]], 780/779, 1026/1025, 1426/1425, 2945/2944, 14391/14375 and 73853/73800.  


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
339 factors into 3 × 113, with [[3edo]] and [[113edo]] as its subset edos. [[1017edo]], which triples it, gives a good correction to the harmonics 7 and 11.  
Since 339 factors into 3 × 113, 339edo has [[3edo]] and 113edo as its subsets. [[1017edo]], which triples it, gives a good correction to the harmonics 7 and 11.  


== 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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-179 113}}
! rowspan="2" | [[Comma list]]
|{{mapping|339 537}}
! rowspan="2" | [[Mapping]]
| 0.3376
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.3377
! colspan="2" | Tuning error
| 9.54
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-13 17 -6}}, {{monzo|-44 -3 21}}
| {{monzo| -13 17 -6 }}, {{monzo| -44 -3 21 }}
|{{mapping|339 537 787}}
| {{mapping| 339 537 787 }}
| 0.2930
| +0.2930
| 0.2828
| 0.2828
| 7.99
| 7.99
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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
|-
|-
|1
| 1
|32\339
| 32\339
|113.27
| 113.27
|16/15
| 16/15
|[[Misneb]]
| [[Misneb]]
|-
|-
|1
| 1
|146\339
| 146\339
|516.81
| 516.81
|27/20
| 27/20
|[[Gravity]]
| [[Gravity]]
|-
|-
|1
| 1
|158\339
| 158\339
|559.29
| 559.29
|864/625
| 864/625
|[[Tritriple]] (339d)
| [[Tritriple]] (339d)
|}
|}
 
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
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