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Countercomp family: Difference between revisions

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The '''countercomp family''' tempers out [[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 fifths form a closed 41-note circle of fifths, identical to [[41edo]].


== Countercomp ==
== Countercomp ==
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Mapping generators: ~531441/524288, ~5/1
Mapping generators: ~531441/524288, ~5/1


[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668
[[Optimal tuning]] ([[POTE]]): ~531441/524288 = 1\41, ~5/4 = 386.668


{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}


[[Badness]]: 0.934310
[[Badness]]: 0.934310
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[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]
[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 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]] ([[POTE]]): ~5/4 = 385.731
{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d, 451dd }}
 
{{Val list|legend=1| 41, 123, 164, 205d, 246d, 451dd }}


[[Badness]]: 0.161056
[[Badness]]: 0.161056
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Mapping: [{{val| 41 65 0 115 237 }}, {{val| 0 0 1 0 -1 }}]
Mapping: [{{val| 41 65 0 115 237 }}, {{val| 0 0 1 0 -1 }}]


Optimal tuning (POTE): ~5/4 = 385.871
Optimal tuning (POTE): ~56/55 = 1\41, ~5/4 = 385.871


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


Badness: 0.076537
Badness: 0.076537
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Mapping: [{{val| 41 65 0 115 237 247 }}, {{val| 0 0 1 0 -1 -1 }}]
Mapping: [{{val| 41 65 0 115 237 247 }}, {{val| 0 0 1 0 -1 -1 }}]


Optimal tuning (POTE): ~5/4 = 386.604
Optimal tuning (POTE): ~56/55 = 1\41, ~5/4 = 386.604


Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d }}
{{Optimal ET sequence|legend=1| 41, 123e, 164, 205d }}


Badness: 0.054921
Badness: 0.054921
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[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]
[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 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]] ([[POTE]]): ~5/4 = 385.667
{{Optimal ET sequence|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
 
{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}


[[Badness]]: 0.142344
[[Badness]]: 0.142344
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Mapping: [{{val| 41 65 0 20 237 }}, {{val| 0 0 1 1 -1 }}]
Mapping: [{{val| 41 65 0 20 237 }}, {{val| 0 0 1 1 -1 }}]


Optimal tuning (POTE): ~5/4 = 385.309
Optimal tuning (POTE): ~55/54 = 1\41, ~5/4 = 385.309


Optimal GPV sequence: {{Val list| 41, 164d, 205, 246 }}
{{Optimal ET sequence|legend=1| 41, 164d, 205, 246 }}


Badness: 0.076588
Badness: 0.076588
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[[Mapping]]: [{{val| 41 65 1 68 }}, {{val| 0 0 2 1 }}]
[[Mapping]]: [{{val| 41 65 1 68 }}, {{val| 0 0 2 1 }}]


Mapping generators: ~531441/524288, ~567/256
Mapping generators: ~100352/98415, ~567/256


[[Optimal tuning]] ([[POTE]]): ~567/512 = 178.7115
[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\41, ~567/512 = 178.5314 (~5120/5103 = 2.9216)


{{Val list|legend=1| 41, 328, 369, 779 }}
{{Optimal ET sequence|legend=1| 41, …, 328, 369, 779, 1927bc }}


[[Badness]]: 0.134559
[[Badness]]: 0.134559


=== 11-limit ===
=== Hemicocomp ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


POTE generator: ~567/512 = 178.9285
Optimal tuning (CTE): ~56/55 = 1\41, ~567/512 = 178.6944 (~3025/3024 = 3.0846)


Optimal GPV sequence: {{Val list| 41, 328, 369 }}
{{Optimal ET sequence|legend=1| 41, …, 328, 369, 1066cee }}


Badness: 0.064400
Badness: 0.064400
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647
Mapping: [{{val| 41 65 1 68 189 246 }}, {{val| 0 0 2 1 -1 -2 }}]
Optimal tuning (CTE): ~56/55 = 1\41, ~72/65 = 178.9389 (~352/351 = 3.3291)
{{Optimal ET sequence|legend=1| 41, …, 328, 369f, 697cef }}
Badness: 0.0416


=== Hemermacomp ===
=== Hemermacomp ===
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Mapping: [{{val| 41 65 1 68 236 }}, {{val| 0 0 2 1 -2 }}]
Mapping: [{{val| 41 65 1 68 236 }}, {{val| 0 0 2 1 -2 }}]


Optimal tuning (POTE): ~567/512 = 178.3714
Optimal tuning (CTE): ~55/54 = 1\41, ~256/231 = 178.3836 (~3024/3025 = 2.7738)


Optimal GPV sequence: {{Val list| 41, 328e, 369e, 410, 451, 861e, 2173bcee }}
{{Optimal ET sequence|legend=1| 41, …, 410, 451, 861e }}


Badness: 0.100152
Badness: 0.100152
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049
Mapping: [{{val| 41 65 1 68 236 293 }}, {{val| 0 0 2 1 -2 -3 }}]
Optimal tuning (CTE): ~55/54 = 1\41, ~72/65 = 178.3755 (~352/351 = 2.7657)
{{Optimal ET sequence|legend=1| 41, …, 410, 451, 861e }}
Badness: 0.0605


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Countercomp_family"