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Keenanismic family: Difference between revisions

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{{Technical data page}}
The keenanismic family of rank four temperaments tempers out the [[385/384|keenanisma]], the 11-limit comma 385/384 = |-7 -1 1 1 1> of 4.503 cents.
The '''keenanismic family''' of [[rank-4 temperament|rank-4]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[keenanisma]], the 11-limit comma 385/384, {{monzo| -7 -1 1 1 1 }} of 4.503 [[cent]]s.


=Keenanismic=
== Keenanismic ==
Comma: 385/384
[[Subgroup]]: 2.3.5.7.11


Minimax tuning
[[Comma list]]: 385/384


[|1 0 0 0 0>, |0 1 0 0 0>, |7/3 1/3 2/3 -1/3 -1/3>,
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}
|7/3 1/3 -1/3 2/3 -1/3>, |7/3 1/3 -1/3 -1/3 2/3>]
: Mapping generators: ~2, ~3, ~5, ~7


Eigenmonzos: 2, 3, 7/5, 11/5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4104{{c}}, ~3/2 = 701.6916{{c}}, ~5/4 = 385.1761{{c}}, ~7/4 = 967.5421{{c}}
: [[error map]]: {{val| +0.410 +0.147 -0.317 -0.463 -0.703 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6933{{c}}, ~5/4 = 385.1491{{c}}, ~7/4 = 967.3004{{c}}
: error map: {{val| 0.000 -0.262 -1.165 -1.525 -2.074 }}


Map: [<1 0 0 0 7|, <0 1 0 0 1|, <0 0 1 0 -1|, <0 0 0 1 -1|]
[[Minimax tuning]]:  
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 0 1 0 0 0 }}, ~5 = {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, ~7 = {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5


Transpose: [2 3 5 7 385/35]
{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}


EDOs: [[19edo|19]], [[22edo|22]], [[31edo|31]], [[41edo|41]], [[53edo|53]], [[68edo|53]], [[72edo|72]], [[118edo|118]], [[159edo|159]], [[190edo|190]], [[212edo|212]] and [[284edo|284]]
[[Badness]] (Sintel): 0.277


10^9*Badness: 15.159
== Martwin ==
[[Subgroup]]: 2.3.5.7.11.13


=Martwin=
[[Comma list]]: 325/324, 385/384
Commas: 385/384, 325/324


Minimax tuning
{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 0 0 1 4 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 0 }}
||1 0 0 0 0 0>, |0 1 0 0 0 0>, |2/3 4/3 1/3 0 0 -1/3>,
|17/3 -5/3 1/3 0 -1 2/3>, |2/3 4/3 -2/3 0 1 -1/3>, |2/3 4/3 -2/3 0 0 2/3>]


Eigenmonzos: 2, 3, 11/5, 13/5
Lattice basis:  
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247


Map: [<1 0 0 0 7 2|, <0 1 0 0 1 4|, <0 0 1 0 -1 -2|, <0 0 0 1 -1 0|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3494{{c}}, ~3/2 = 702.2131{{c}}, ~5/4 = 384.8666{{c}}, ~7/4 = 967.8487{{c}}
: [[error map]]: {{val| +0.349 +0.608 -0.748 -0.278 -0.422 -0.710 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1659{{c}}, ~5/4 = 384.8721{{c}}, ~7/4 = 967.6109{{c}}
: error map: {{val| 0.000 +0.211 -1.442 -1.215 -1.635 -1.608 }}


Transpose: [2 3 5 7 385/35 324/25]
[[Minimax tuning]]:  
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~2 = {{val| 1 0 0 0 0 0 }}, ~3 = {{val| 0 1 0 0 0 0 }}, ~5 = {{val| 2/3 4/3 1/3 0 0 -1/3 }}, ~7 = {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5


EDOs: [[7edo|7]], [[15edo|15]], [[19edo|19]], [[26edo|26]], [[34edo|34]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[72edo|72]], [[87edo|87]], [[140edo|140]], [[159edo|159]], [[212edo|212]] and[[299edo|299]]
{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}


10^6*Badness: 2.206
[[Badness]] (Sintel): 0.579


=Ancient=
== Ancient ==
Commas: 385/384, 625/624
[[Subgroup]]: 2.3.5.7.11.13


Map: [<1 0 0 0 7 -4|, <0 1 0 0 1 -1|, <0 0 1 0 -1 4|, <0 0 0 1 -1 0|]
[[Comma list]]: 385/384, 625/624


Transpose: [2 3 5 7 385/35 625/48]
{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}


EDOs: [[6edo|6]], [[15edo|15]], [[19edo|19]], [[22edo|22]], [[31edo|31]], [[50edo|50]], [[53edo|53]], [[72edo|72]], [[87edo|87]], [[103edo|103]], [[140edo|140]], [[159edo|159]], [[190edo|190]]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4130{{c}}, ~3/2 = 701.6863{{c}}, ~5/4 = 385.2319{{c}}, ~7/4 = 967.5188{{c}}
: [[error map]]: {{val| +0.413 +0.144 -0.256 -0.481 -0.730 -0.047 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6838{{c}}, ~5/4 = 385.2497{{c}}, ~7/4 = 967.2556{{c}}
: error map: {{val| 0.000 -0.271 -1.064 -1.570 -2.139 -1.213 }}


10^6*Badness: 2.573
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
 
[[Badness]] (Sintel): 0.675
 
== Zaxa ==
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 352/351, 385/384
 
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 -2 | 0 0 1 0 -1 -1 | 0 0 0 1 -1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1117{{c}}, ~3/2 = 702.6588{{c}}, ~5/4 = 385.6821{{c}}, ~7/4 = 968.0061{{c}}
: [[error map]]: {{val| +0.112 +0.815 -0.408 -0.596 -1.901 +1.137 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.6029{{c}}, ~5/4 = 385.6441{{c}}, ~7/4 = 967.9030{{c}}
: error map: {{val| 0.000 +0.648 -0.670 -0.923 -2.262 +0.720 }}
 
{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
 
[[Badness]] (Sintel): 0.880
 
[[Category:Temperament families]]
[[Category:Keenanismic family| ]] <!-- main article -->
[[Category:Rank 4]]
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