Countercomp family: Difference between revisions

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m FloraC moved page Counterpyth family to Countercomp family: Name change following pythagorean -> compton
Name change following pythagorean -> compton
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The '''counterpyth family''' tempers out [[41-comma|counterpyth comma]], {{monzo| 65 -41 }}, and hence the fifths form a closed 41-note circle of fifths, identical to [[41edo|41EDO]].
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]].


== Counterpyth ==
== Countercomp ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo| 65 -41 }}
[[Comma list]]: {{monzo| 65 -41 }}
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Mapping generators: ~531441/524288, ~5/1
Mapping generators: ~531441/524288, ~5/1


[[POTE generator]]: ~5/4 = 386.668
[[Optimal tuning]] ([[POTE]]): ~5/4 = 386.668


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


== Gamelapyth ==
== Gamelacomp ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 537824/531441
[[Comma list]]: 1029/1024, 537824/531441
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{{Multival|legend=1| 0 41 0 65 0 -115 }}
{{Multival|legend=1| 0 41 0 65 0 -115 }}


[[POTE generator]]: ~5/4 = 385.731
[[Optimal tuning]] ([[POTE]]): ~5/4 = 385.731


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


POTE generator: ~5/4 = 385.871
Optimal tuning (POTE): ~5/4 = 385.871


Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d, 451dd }}
Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d, 451dd }}
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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 }}]


POTE generator: ~5/4 = 386.604
Optimal tuning (POTE): ~5/4 = 386.604


Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d }}
Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d }}
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Badness: 0.054921
Badness: 0.054921


== Mermapyth ==
== Mermacomp ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 2500000/2470629
[[Comma list]]: 5120/5103, 2500000/2470629
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{{Multival|legend=1| 0 41 41 65 65 -20 }}
{{Multival|legend=1| 0 41 41 65 65 -20 }}


[[POTE generator]]: ~5/4 = 385.667
[[Optimal tuning]] ([[POTE]]): ~5/4 = 385.667


{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
Line 79: Line 79:
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 }}]


POTE generator: ~5/4 = 385.309
Optimal tuning (POTE): ~5/4 = 385.309


Optimal GPV sequence: {{Val list| 41, 164d, 205, 246 }}
Optimal GPV sequence: {{Val list| 41, 164d, 205, 246 }}
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Badness: 0.076588
Badness: 0.076588


== Hemicounterpyth ==
== Hemicountercomp ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 52613349376/52301766015
[[Comma list]]: 2401/2400, 52613349376/52301766015
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Mapping generators: ~531441/524288, ~567/256
Mapping generators: ~531441/524288, ~567/256


[[POTE generator]]: ~567/512 = 178.7115
[[Optimal tuning]] ([[POTE]]): ~567/512 = 178.7115


{{Val list|legend=1| 41, 328, 369, 779 }}
{{Val list|legend=1| 41, 328, 369, 779 }}
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Badness: 0.064400
Badness: 0.064400


=== Hemermapyth ===
=== Hemermacomp ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 120: Line 120:
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 }}]


POTE generator: ~567/512 = 178.3714
Optimal tuning (POTE): ~567/512 = 178.3714


Optimal GPV sequence: {{Val list| 41, 328e, 369e, 410, 451, 861e, 2173bcee }}
Optimal GPV sequence: {{Val list| 41, 328e, 369e, 410, 451, 861e, 2173bcee }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Counterpyth]]
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 04:55, 22 December 2022

The countercomp family tempers out Pythagorean countercomma, [65 -41⟩, and hence the fifths form a closed 41-note circle of fifths, identical to 41edo.

Countercomp

Subgroup: 2.3.5

Comma list: [65 -41⟩

Mapping: [⟨41 65 0], ⟨0 0 1]]

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

Optimal tuning (POTE): ~5/4 = 386.668

Template:Val list

Badness: 0.934310

Gamelacomp

Subgroup: 2.3.5.7

Comma list: 1029/1024, 537824/531441

Mapping: [⟨41 65 0 115], ⟨0 0 1 0]]

Wedgie: ⟨⟨ 0 41 0 65 0 -115 ]]

Optimal tuning (POTE): ~5/4 = 385.731

Template:Val list

Badness: 0.161056

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 537824/531441

Mapping: [⟨41 65 0 115 237], ⟨0 0 1 0 -1]]

Optimal tuning (POTE): ~5/4 = 385.871

Optimal GPV sequence: Template:Val list

Badness: 0.076537

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 59150/59049

Mapping: [⟨41 65 0 115 237 247], ⟨0 0 1 0 -1 -1]]

Optimal tuning (POTE): ~5/4 = 386.604

Optimal GPV sequence: Template:Val list

Badness: 0.054921

Mermacomp

Subgroup: 2.3.5.7

Comma list: 5120/5103, 2500000/2470629

Mapping: [⟨41 65 0 20], ⟨0 0 1 1]]

Wedgie: ⟨⟨ 0 41 41 65 65 -20 ]]

Optimal tuning (POTE): ~5/4 = 385.667

Template:Val list

Badness: 0.142344

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5120/5103, 75625/75264

Mapping: [⟨41 65 0 20 237], ⟨0 0 1 1 -1]]

Optimal tuning (POTE): ~5/4 = 385.309

Optimal GPV sequence: Template:Val list

Badness: 0.076588

Hemicountercomp

Subgroup: 2.3.5.7

Comma list: 2401/2400, 52613349376/52301766015

Mapping: [⟨41 65 1 68], ⟨0 0 2 1]]

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

Optimal tuning (POTE): ~567/512 = 178.7115

Template:Val list

Badness: 0.134559

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨41 65 1 68 189], ⟨0 0 2 1 -1]]

POTE generator: ~567/512 = 178.9285

Optimal GPV sequence: Template:Val list

Badness: 0.064400

Hemermacomp

Subgroup: 2.3.5.7.11

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

Mapping: [⟨41 65 1 68 236], ⟨0 0 2 1 -2]]

Optimal tuning (POTE): ~567/512 = 178.3714

Optimal GPV sequence: Template:Val list

Badness: 0.100152

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