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Countercomp family: Difference between revisions

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The '''countercomp family''' tempers out the [[41-comma|Pythagorean countercomma]], {{monzo| 65 -41 }}, and hence the fifths form a closed 41-note circle of fifths, identical to [[41edo]].
{{Technical data page}}
The '''countercomp family''' tempers out the [[41-comma|Pythagorean countercomma]], {{monzo| 65 -41 }}, and hence the [[3/2|perfect fifths]] form a closed 41-note [[circle of fifths]], identical to [[41edo]].


== Countercomp ==
== Countercomp ==
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[[Comma list]]: {{monzo| 65 -41 }}
[[Comma list]]: {{monzo| 65 -41 }}


[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 41 65 0 | 0 0 1 }}
: mapping generators: ~531441/524288, ~5


Mapping generators: ~531441/524288, ~5/1
[[Optimal tuning]]s:
* [[WE]]: ~531441/524288 = 29.2646{{c}}, ~5/4 = 386.6191{{c}}
: [[error map]]: {{val| -0.153 +0.242 -0.000 }}
* [[CWE]]: ~531441/524288 = 29.2683{{c}}, ~5/4 = 386.5501{{c}}
: error map: {{val| 0.000 +0.484 +0.236 }}


[[Optimal tuning]] ([[POTE]]): ~531441/524288 = 1\41, ~5/4 = 386.668
{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc, 2911bbc }}


{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
[[Badness]] (Sintel): 21.9
 
[[Badness]]: 0.934310


== Gamelacomp ==
== Gamelacomp ==
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[[Comma list]]: 1029/1024, 537824/531441
[[Comma list]]: 1029/1024, 537824/531441


[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 41 65 0 115 | 0 0 1 0 }}
 
{{Multival|legend=1| 0 41 0 65 0 -115 }}


[[Optimal tuning]] ([[POTE]]): ~64/63 = 1\41, ~5/4 = 385.731
[[Optimal tuning]]s:
* [[WE]]: ~64/63 = 29.2683{{c}}, ~5/4 = 386.3137{{c}}
: [[error map]]: {{val| +0.250 +0.880 -0.002 -2.271 }}
* [[CWE]]: ~64/63 = 29.2683{{c}}, ~5/4 = 385.8764{{c}}
: error map: {{val| 0.000 +0.484 -0.437 -2.972 }}


{{Val list|legend=1| 41, 123, 164, 205d, 246d, 451dd }}
{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d }}


[[Badness]]: 0.161056
[[Badness]] (Sintel): 4.08


=== 11-limit ===
=== 11-limit ===
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Comma list: 385/384, 441/440, 537824/531441
Comma list: 385/384, 441/440, 537824/531441


Mapping: [{{val| 41 65 0 115 237 }}, {{val| 0 0 1 0 -1 }}]
{{Mapping|legend=0| 41 65 0 115 237 | 0 0 1 0 -1 }}


Optimal tuning (POTE): ~56/55 = 1\41, ~5/4 = 385.871
Optimal tunings:
* WE: ~56/55 = 29.2737{{c}}, ~5/4 = 385.9427{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~5/4 = 385.8909{{c}}


Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d, 451dd }}
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d }}


Badness: 0.076537
Badness (Sintel): 2.53


=== 13-limit ===
=== 13-limit ===
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Comma list: 196/195, 352/351, 385/384, 59150/59049
Comma list: 196/195, 352/351, 385/384, 59150/59049


Mapping: [{{val| 41 65 0 115 237 247 }}, {{val| 0 0 1 0 -1 -1 }}]
{{Mapping|legend=0| 41 65 0 115 237 247 | 0 0 1 0 -1 -1 }}


Optimal tuning (POTE): ~56/55 = 1\41, ~5/4 = 386.604
Optimal tunings:
* WE: ~56/55 = 29.2700{{c}}, ~5/4 = 386.6265{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~5/4 = 386.5997{{c}}


Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d }}
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d }}


Badness: 0.054921
Badness (Sintel): 2.27


== Mermacomp ==
== Mermacomp ==
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[[Comma list]]: 5120/5103, 2500000/2470629
[[Comma list]]: 5120/5103, 2500000/2470629


[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]
{{Mapping|legend=1| 41 65 0 20 | 0 0 1 1 }}
 
{{Multival|legend=1| 0 41 41 65 65 -20 }}


[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\41, ~5/4 = 385.667
[[Optimal tuning]]s:
* [[WE]]: ~50/49 = 29.2633{{c}}, ~5/4 = 385.6013{{c}}
: [[error map]]: {{val| -0.204 +0.161 -1.120 +1.634 }}
* [[CWE]]: ~50/49 = 29.2683{{c}}, ~5/4 = 385.4966{{c}}
: error map: {{val| 0.000 +0.484 -0.817 +2.037 }}


{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
{{Optimal ET sequence|legend=1| 41, 123d, 164d, 205, 246 }}


[[Badness]]: 0.142344
[[Badness]] (Sintel): 3.60


=== 11-limit ===
=== 11-limit ===
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Comma list: 540/539, 5120/5103, 75625/75264
Comma list: 540/539, 5120/5103, 75625/75264


Mapping: [{{val| 41 65 0 20 237 }}, {{val| 0 0 1 1 -1 }}]
{{Mapping|legend=0| 41 65 0 20 237 | 0 0 1 1 -1 }}
 
Optimal tunings:
* WE: ~55/54 = 29.2657{{c}}, ~5/4 = 385.2755{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~5/4 = 385.2847{{c}}
 
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246 }}
 
Badness (Sintel): 2.53


Optimal tuning (POTE): ~55/54 = 1\41, ~5/4 = 385.309
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal GPV sequence: {{Val list| 41, 164d, 205, 246 }}
Comma list: 352/351, 540/539, 729/728, 75625/75264


Badness: 0.076588
{{Mapping|legend=0| 41 65 0 20 237 247 | 0 0 1 1 -1 -1 }}
 
Optimal tunings:
* CTE: ~55/54 = 29.2619{{c}}, ~5/4 = 385.7758{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~5/4 = 385.8418{{c}}
 
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246f }}
 
Badness (Sintel): 2.29


== Hemicountercomp ==
== Hemicountercomp ==
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[[Comma list]]: 2401/2400, 52613349376/52301766015
[[Comma list]]: 2401/2400, 52613349376/52301766015


[[Mapping]]: [{{val| 41 65 1 68 }}, {{val| 0 0 2 1 }}]
{{Mapping|legend=1| 41 65 1 68 | 0 0 2 1 }}
: mapping generators: ~100352/98415, ~567/256


Mapping generators: ~100352/98415, ~567/256
[[Optimal tuning]]s:
* [[WE]]: ~100352/98415 = 29.2652{{c}}, ~567/512 = 178.6924{{c}}
: [[error map]]: {{val| -0.129 +0.280 +0.079 -0.231 }}
* [[CWE]]: ~100352/98415 = 29.2683{{c}}, ~567/512 = 178.6568{{c}}
: error map: {{val| 0.000 +0.484 +0.268 +0.075 }}


[[Optimal tuning]] ([[POTE]]): ~100352/98415 = 1\41, ~567/512 = 178.7115
{{Optimal ET sequence|legend=1| 41, …, 328, 369, 779, 1927bc }}


{{Val list|legend=1| 41, 328, 369, 779 }}
[[Badness]] (Sintel): 3.41


[[Badness]]: 0.134559
=== Hemicocomp ===
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 2401/2400, 16384/16335, 19712/19683
Comma list: 2401/2400, 16384/16335, 19712/19683


Mapping: [{{val| 41 65 1 68 189 }}, {{val| 0 0 2 1 -1 }}]
{{Mapping|legend=0| 41 65 1 68 189 | 0 0 2 1 -1 }}


Optimal tuning (POTE): ~56/55 = 1\41, ~567/512 = 178.9285
Optimal tunings:
* WE: ~56/55 = 29.2624{{c}}, ~567/512 = 178.8927{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~567/512 = 178.8825{{c}}


Optimal GPV sequence: {{Val list| 41, 328, 369 }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369, 1066cee }}


Badness: 0.064400
Badness (Sintel): 2.13
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647
 
{{Mapping|legend=0| 41 65 1 68 189 246 | 0 0 2 1 -1 -2 }}
 
Optimal tunings:
* WE: ~56/55 = 29.2615{{c}}, ~72/65 = 178.9843{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~72/65 = 179.0133{{c}}
 
{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}
 
Badness (Sintel): 1.72


=== Hemermacomp ===
=== Hemermacomp ===
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Comma list: 2401/2400, 59290/59049, 131072/130977
Comma list: 2401/2400, 59290/59049, 131072/130977


Mapping: [{{val| 41 65 1 68 236 }}, {{val| 0 0 2 1 -2 }}]
{{Mapping|legend=0| 41 65 1 68 236 | 0 0 2 1 -2 }}
 
Optimal tunings:
* WE: ~55/54 = 29.2688{{c}}, ~256/231 = 178.3748{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~256/231 = 178.3736{{c}}
 
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}
 
Badness (Sintel): 3.31
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049


Optimal tuning (POTE): ~55/54 = 1\41, ~256/231 = 178.3714
{{Mapping|legend=0| 41 65 1 68 236 293 | 0 0 2 1 -2 -3 }}


Optimal GPV sequence: {{Val list| 41, 328e, 369e, 410, 451, 861e, 2173bcee }}
Optimal tunings:  
* WE: ~55/54 = 29.2688{{c}}, ~72/65 = 178.3805{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~72/65 = 178.3772{{c}}


Badness: 0.100152
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}


Badness (Sintel): 2.50
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Countercomp_family"