Xenharmonic Wiki

Hemimage family: Difference between revisions

← Older edit
VisualWikitext
Edo lists reviewed
m Units, misc. cleanup, - redundant category
 
(12 intermediate revisions by 6 users not shown)
Line 1: Line 1:
The '''hemimage family''' of temperaments are rank-3 temperaments tempering out [[10976/10935]].  
{{Technical data page}}
The '''hemimage family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] tempers out the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).  


= Hemimage =
== Hemimage ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935
[[Comma list]]: 10976/10935


[[Mapping]]: [{{val| 1 0 2 -1 }}, {{val| 0 1 2 3 }}, {{val| 0 0 -3 -1 }}]
{{Mapping|legend=1| 1 0 2 -1 | 0 1 2 3 | 0 0 -3 -1 }}
 
: mapping generators: ~2, ~3, ~27/14
Mapping generators: ~2, ~3, ~27/14


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/8 1/8 -1/8 3/8 }}, {{monzo| 5/8 -7/8 7/8 3/8 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 5/8 1/8 -1/8 3/8 | 5/8 -7/8 7/8 3/8 | 0 0 0 1 }}
: [[Eigenmonzo]]s: 2, 8/7, 6/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/9 2/9 -1/9 1/3 }}, {{monzo| 10/9 -14/9 7/9 2/3 }}, {{monzo| 0 0 0 1 }}]
: {{monzo list| 1 0 0 0 | 5/9 2/9 -1/9 1/3 | 10/9 -14/9 7/9 2/3 | 0 0 0 1 }}
: [[Eigenmonzo]]s: 2, 10/9, 8/7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.9/5


{{Val list|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}


[[Badness]]: 0.199 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.199 × 10<sup>-3</sup>


[[Projection pair]]: 5 10976/2187 to 2.3.7
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7


= Uni =
== Uni ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 896/891
[[Comma list]]: 540/539, 896/891


[[Mapping]]: [{{val| 1 0 2 -1 6 }}, {{val| 0 1 2 3 -1 }}, {{val| 0 0 -3 -1 -1 }}]
{{Mapping|legend=1| 1 0 2 -1 6 | 0 1 2 3 -1 | 0 0 -3 -1 -1 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 16/11 6/11 1/11 0 -3/11 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 19/11 14/11 6/11 0 -7/11 }}, {{monzo| 32/11 -10/11 2/11 0 5/11 }}]
: [[Eigenmonzo]]s: 2, 11/9, 5/4
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9


{{Val list|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}


[[Badness]]: 0.702 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.702 × 10<sup>-3</sup>


== 13-limit ==
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 364/363, 540/539
Comma list: 352/351, 364/363, 540/539


Mapping: [{{val| 1 0 2 -1 6 11 }}, {{val| 0 1 2 3 -1 -4 }}, {{val| 0 0 -3 -1 -1 -1 }}]
Mapping: {{mapping| 1 0 2 -1 6 11 | 0 1 2 3 -1 -4 | 0 0 -3 -1 -1 -1 }}


Vals: {{Val list| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}


Badness: 1.06 × 10<sup>-3</sup>
Badness (Smith): 1.06 × 10<sup>-3</sup>


== Tin ==
=== Tin ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 196/195, 729/728
Comma list: 144/143, 196/195, 729/728


Mapping: [{{val| 1 0 2 -1 6 -2 }}, {{val| 0 1 2 3 -1 3 }}, {{val| 0 0 -3 -1 -1 1 }}]
Mapping: {{mapping| 1 0 2 -1 6 -2 | 0 1 2 3 -1 3 | 0 0 -3 -1 -1 1 }}


Vals: {{Val list| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}
{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}


Badness: 1.07 × 10<sup>-3</sup>
Badness (Smith): 1.07 × 10<sup>-3</sup>


= Turan =
== Turan ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 10976/10935
[[Comma list]]: 441/440, 10976/10935


[[Mapping]]: [{{val| 1 0 2 -1 -7 }}, {{val| 0 1 2 3 6 }}, {{val| 0 0 -3 -1 1 }}]
{{Mapping|legend=1| 1 0 2 -1 -7 | 0 1 2 3 6 | 0 0 -3 -1 1 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/12 -2/3 1/12 0 1/4 }}, {{monzo| 19/6 -10/3 7/6 0 1/2 }}, {{monzo| 37/12 -8/3 7/12 0 3/4 }}, {{monzo| 19/6 -10/3 1/6 0 3/2 }}]
: [[Eigenmonzo]]s: 2, 10/9, 11/9
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


{{Val list|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}
{{Optimal ET sequence|legend=1| 17c, 19, 36, 39d, 41, 58, 99e, 118, 159, 217, 335 }}


[[Badness]]: 0.936 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.936 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Hemimage family| ]] <!-- main article -->
[[Category:Hemimage]]
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Hemimage_family"