Xenharmonic Wiki

Hemimage family: Difference between revisions

← Older edit
VisualWikitext
m Cleanup; recategory
m Recategorize
 
(19 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]]s [[tempering out|tempers out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).  


= Hemimage =
== Hemimage ==
[[Comma]]: hemimage 10976/10935
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.


7-limit minimax
[[Subgroup]]: 2.3.5.7


[|1 0 0 0>, |5/8 1/8 -1/8 3/8>,
[[Comma list]]: 10976/10935
|5/8 -7/8 7/8 3/8>, |0 0 0 1>]


[[Eigenmonzo]]s: 2, 8/7, 6/5
{{Mapping|legend=1| 1 0 -1 -2 | 0 1 2 3 | 0 0 3 1 }}
: mapping generators: ~2, ~3, ~28/27


9-limit minimax
[[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 }}


[|1 0 0 0>, |5/9 2/9 -1/9 1/3>,
[[Minimax tuning]]:
|10/9 -14/9 7/9 2/3>, |0 0 0 1>]
* [[7-odd-limit]]
: {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7
* [[9-odd-limit]]
: {{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7


[[Eigenmonzo]]s: 2, 10/9, 8/7
{{Optimal ET sequence|legend=1| 17c, 19, 39d, 41, 58, 77, 80, 99, 217, 239, 338, 357, 456, 695 }}


Map: [<1 0 2 -1|, <0 1 2 3|, <0 0 -3 -1|]
[[Badness]] (Sintel): 0.878


[[Generator]]s: 2, 3, 27/14
[[Projection pair]]: <code>5 10976/2187</code> to 2.3.7


[[EDO|Edos]]: 14, 19, 41, 58, 77, 80, [[99edo|99]], [[118edo|118]], [[140edo|140]], [[217edo|217]], [[316edo|316]], [[338edo|338]], [[357edo|357]], [[415edo|415]], [[456edo|456]], [[654edo|654]], [[695edo|695]]
== Uni ==
[[Subgroup]]: 2.3.5.7.11


Badness: 0.000199
[[Comma list]]: 540/539, 896/891


[[Projection pair]]: 5 10976/2187 to 2.3.7
{{Mapping|legend=1| 1 0 -1 -2 5 | 0 1 2 3 -1 | 0 0 3 1 1 }}


= Uni =
[[Optimal tuning]]s:
[[Comma]]s: 540/539, 896/891
* [[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]]
: [{{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 basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/9


[|1 0 0 0 0&gt;, |16/11 6/11 1/11 0 -3/11&gt;,  
{{Optimal ET sequence|legend=1| 17c, 19, 22, 39d, 41, 58, 80, 99e, 121, 179e, 220e, 300bdee, 399bcdeee }}
|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;]


[[Eigenmonzo]]s: 2, 11/9, 5/4
[[Badness]] (Sintel): 0.844


Map: [&lt;1 0 2 -1 6|, &lt;0 1 2 3 -1|, &lt;0 0 -3 -1 -1|]
=== 13-limit  ===
Subgroup: 2.3.5.7.11.13


[[Generator]]s: 2, 3, 27/14
Comma list: 352/351, 364/363, 540/539


[[EDO|Edos]]: [[19edo|19]], [[22edo|22]], [[41edo|41]], [[58edo|58]], [[80edo|80]], [[121edo|121]]
Mapping: {{mapping| 1 0 -1 -2 5 10 | 0 1 2 3 -1 -4 | 0 0 3 1 1 1 }}


Badness: 0.000702
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}}


== 13-limit ==
{{Optimal ET sequence|legend=0| 17c, 22, 36f, 39d, 41, 58, 80, 99ef, 121, 179ef, 300bdeef }}
Commas: 352/351, 364/363, 540/539


Map: [&lt;1 0 2 -1 6 11|, &lt;0 1 2 3 -1 -4|, &lt;0 0 -3 -1 -1 -1|]
Badness (Sintel): 0.993


EDOs: 22, 41, 58, 80, 121
=== Tin ===
Subgroup: 2.3.5.7.11.13


Badness: 0.00106
Comma list: 144/143, 196/195, 729/728


== Tin ==
Mapping: {{mapping| 1 0 -1 -2 5 -1 | 0 1 2 3 -1 3 | 0 0 3 1 1 -1 }}
Commas: 144/143, 196/195, 729/728


Map: [&lt;1 0 2 -1 6 -2|, &lt;0 1 2 3 -1 3|, &lt;0 0 -3 -1 -1 1|]
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}}


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


Badness: 0.00107
Badness (Sintel): 1.00


= Turan =
== Turan ==
[[Comma]]s: 441/440, 10976/10935
[[Subgroup]]: 2.3.5.7.11


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


[|1 0 0 0 0&gt;, |19/12 -2/3 1/12 0 1/4&gt;,
{{Mapping|legend=1| 1 0 -1 -2 -6 | 0 1 2 3 6 | 0 0 3 1 -1 }}
|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;]


[[Eigenmonzo]]s: 2, 10/9, 11/9
[[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 }}


Map: [&lt;1 0 2 -1 -7|, &lt;0 1 2 3 6|, &lt;0 0 -3 -1 1|]
[[Minimax tuning]]:
* [[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 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/9


[[Generator]]s: 2, 3, 27/14
{{Optimal ET sequence|legend=1| 17c, 19e, 36e, 39dee, 41, 58, 99e, 118, 159, 217, 335 }}


[[EDO|Edos]]: [[41edo|41]], [[58edo|58]], [[118edo|118]], [[159edo|159]], [[217edo|217]], [[335edo|335]]
[[Badness]] (Sintel): 1.13


Badness: 0.000936
[[Category:Hemimage family| ]] <!-- main article -->
 
[[Category:Temperament families]]
[[Category:Theory]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Temperament family]]
[[Category:Hemimage]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Hemimage_family"