Countercomp family: Difference between revisions
Name change following pythagorean -> compton |
Update optimal tuning to include the period |
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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]]. | 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 }} | {{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }} | ||
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{{Multival|legend=1| 0 41 0 65 0 -115 }} | {{Multival|legend=1| 0 41 0 65 0 -115 }} | ||
[[Optimal tuning]] ([[POTE]]): ~5/4 = 385.731 | [[Optimal tuning]] ([[POTE]]): ~64/63 = 1\41, ~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 }}] | ||
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 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 }}] | ||
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 GPV sequence: {{Val list| 41, 123e, 164, 205d }} | ||
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{{Multival|legend=1| 0 41 41 65 65 -20 }} | {{Multival|legend=1| 0 41 41 65 65 -20 }} | ||
[[Optimal tuning]] ([[POTE]]): ~5/4 = 385.667 | [[Optimal tuning]] ([[POTE]]): ~50/49 = 1\41, ~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 }} | ||
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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 GPV sequence: {{Val list| 41, 164d, 205, 246 }} | ||
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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: ~ | Mapping generators: ~100352/98415, ~567/256 | ||
[[Optimal tuning]] ([[POTE]]): ~567/512 = 178.7115 | [[Optimal tuning]] ([[POTE]]): ~100352/98415 = 1\41, ~567/512 = 178.7115 | ||
{{Val list|legend=1| 41, 328, 369, 779 }} | {{Val list|legend=1| 41, 328, 369, 779 }} | ||
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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 | Optimal tuning (POTE): ~56/55 = 1\41, ~567/512 = 178.9285 | ||
Optimal GPV sequence: {{Val list| 41, 328, 369 }} | Optimal GPV sequence: {{Val list| 41, 328, 369 }} | ||
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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): ~ | Optimal tuning (POTE): ~55/54 = 1\41, ~256/231 = 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 }} | ||
Revision as of 06:26, 22 December 2022
The countercomp family tempers out the 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): ~531441/524288 = 1\41, ~5/4 = 386.668
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): ~64/63 = 1\41, ~5/4 = 385.731
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): ~56/55 = 1\41, ~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): ~56/55 = 1\41, ~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): ~50/49 = 1\41, ~5/4 = 385.667
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): ~55/54 = 1\41, ~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: ~100352/98415, ~567/256
Optimal tuning (POTE): ~100352/98415 = 1\41, ~567/512 = 178.7115
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]]
Optimal tuning (POTE): ~56/55 = 1\41, ~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): ~55/54 = 1\41, ~256/231 = 178.3714
Optimal GPV sequence: Template:Val list
Badness: 0.100152