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Hemimage family: Difference between revisions

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The '''hemimage family''' of temperaments are rank-3 temperaments tempering out [[10976/10935]].  
{{Technical data page}}
The '''hemimage family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).  


= Hemimage =
== Hemimage ==
Subgroup: 2.3.5.7
Hemimage is generated by a [[3/2|perfect fifth]] and a [[~]][[28/27]] subminor second. Two of the subminor seconds make ~[[15/14]], and three make ~[[10/9]]. The ubiquitous presence of this little interval gives rise to the highly intense [[hemimage bleeding chord]]s.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935
[[Comma list]]: 10976/10935
{{Mapping|legend=1| 1 0 -1 -2 | 0 1 2 3 | 0 0 3 1 }}
: mapping generators: ~2, ~3, ~28/27
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8555{{c}}, ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}
: [[error map]]: {{val| -0.145 +0.507 +0.155 -0.683 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.6137{{c}}, ~28/27 = 60.4761{{c}}
: error map: {{val| 0.000 +0.659 +0.342 -0.509 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]
: [|1 0 0 0>, |5/8 1/8 -1/8 3/8>, |5/8 -7/8 7/8 3/8>, |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]]
: [|1 0 0 0>, |5/9 2/9 -1/9 1/3>, |10/9 -14/9 7/9 2/3>, |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]]: 2.9/5.7


[[Mapping]]: [<1 0 2 -1|, <0 1 2 3|, <0 0 -3 -1|]
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}


Mapping generators: ~2, ~3, ~27/14
[[Badness]] (Sintel): 0.878


{{Val list|legend=1| 14, 19, 41, 58, 77, 80, 99, 118, 140, 217, 316, 338, 357, 415, 456, 654, 695 }}
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7


[[Badness]]: 0.199 × 10<sup>-3</sup>
== Uni ==
[[Subgroup]]: 2.3.5.7.11


[[Projection pair]]: 5 10976/2187 to 2.3.7
[[Comma list]]: 540/539, 896/891
 
= Uni =
Subgroup: 2.3.5.7.11


[[Comma list]]: 540/539, 896/891
{{Mapping|legend=1| 1 0 -1 -2 5 | 0 1 2 3 -1 | 0 0 3 1 1 }}


Mapping: [&lt;1 0 2 -1 6|, &lt;0 1 2 3 -1|, &lt;0 0 -3 -1 -1|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.1497{{c}}, ~3/2 = 703.3014{{c}}, ~28/27 = 60.0617{{c}}
: [[error map]]: {{val| -0.859 +0.496 -0.376 +0.290 +2.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6475{{c}}, ~28/27 = 59.9675{{c}}
: error map: {{val| 0.000 +1.693 +0.884 +2.084 +5.002 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |16/11 6/11 1/11 0 -3/11&gt;, |0 0 1 0 0&gt;, |19/11 14/11 6/11 0 -7/11&gt;, |32/11 -10/11 2/11 0 5/11&gt;]
: [{{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| 19, 22, 41, 58, 80, 121 }}
{{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]] (Sintel): 0.844


== 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: [&lt;1 0 2 -1 6 11|, &lt;0 1 2 3 -1 -4|, &lt;0 0 -3 -1 -1 -1|]
Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}


Vals: {{Val list| 22, 41, 58, 80, 121 }}
Optimal tunings:  
* WE: ~2 = 1199.0867{{c}}, ~3/2 = 703.4048{{c}}, ~28/27 = 60.0034{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.9457{{c}}, ~28/27 = 59.8167{{c}}


Badness: 1.06 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}


== Tin ==
Badness (Sintel): 0.993
 
=== 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: [&lt;1 0 2 -1 6 -2|, &lt;0 1 2 3 -1 3|, &lt;0 0 -3 -1 -1 1|]
Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}


Vals: {{Val list| 19, 36, 41, 58 }}
Optimal tunings:  
* WE: ~2 = 1198.8801{{c}}, ~3/2 = 702.5101{{c}}, ~28/27 = 60.9480{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.8952{{c}}, ~28/27 = 60.9129{{c}}


Badness: 1.07 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 17c, 19, 22f, 36, 39df, 41, 58, 99ef, 157eff }}


= Turan =
Badness (Sintel): 1.00
Subgroup: 2.3.5.7.11
 
== Turan ==
[[Subgroup]]: 2.3.5.7.11


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


[[Mapping]]: [&lt;1 0 2 -1 -7|, &lt;0 1 2 3 6|, &lt;0 0 -3 -1 1|]
{{Mapping|legend=1| 1 0 -1 -2 -6 | 0 1 2 3 6 | 0 0 3 1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0289{{c}}, ~3/2 = 702.1366{{c}}, ~28/27 = 60.8231{{c}}
: [[error map]]: {{val| +0.029 +0.211 +0.458 -1.564 +0.679 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1290{{c}}, ~28/27 = 60.8257{{c}}
: error map: {{val| 0.000 +0.174 +0.421 -1.613 +0.630 }}


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


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


[[Badness]]: 0.936 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.13


[[Category:Regular temperament theory]]
[[Category:Hemimage family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Hemimage]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
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