Countercomp family: Difference between revisions
Update optimal tuning to include the period |
→Hemicountercomp: +minimal generator tunings; +13-limit; consolidate names |
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Mapping generators: ~100352/98415, ~567/256 | Mapping generators: ~100352/98415, ~567/256 | ||
[[Optimal tuning]] ([[ | [[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\41, ~567/512 = 178.5314 (~5120/5103 = 2.9216) | ||
{{Val list|legend=1| 41, 328, 369, 779 }} | {{Val list|legend=1| 41, …, 328, 369, 779, 1927bc }} | ||
[[Badness]]: 0.134559 | [[Badness]]: 0.134559 | ||
=== | === Hemicocomp === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 107: | Line 107: | ||
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 }}] | ||
Optimal tuning ( | Optimal tuning (CTE): ~56/55 = 1\41, ~567/512 = 178.6944 (~3025/3024 = 3.0846) | ||
Optimal GPV sequence: {{Val list| 41, 328, 369 }} | Optimal GPV sequence: {{Val list| 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 GPV sequence: {{Val list| 41, …, 328, 369f, 697cef }} | |||
Badness: 0.0416 | |||
=== Hemermacomp === | === Hemermacomp === | ||
| Line 120: | Line 133: | ||
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 ( | Optimal tuning (CTE): ~55/54 = 1\41, ~256/231 = 178.3836 (~3024/3025 = 2.7738) | ||
Optimal GPV sequence: {{Val list| 41, | Optimal GPV sequence: {{Val list| 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 GPV sequence: {{Val list| 41, …, 410, 451, 861e }} | |||
Badness: 0.0605 | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Countercomp family| ]] <!-- main article --> | [[Category:Countercomp family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 13:32, 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 (CTE): ~100352/98415 = 1\41, ~567/512 = 178.5314 (~5120/5103 = 2.9216)
Badness: 0.134559
Hemicocomp
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 (CTE): ~56/55 = 1\41, ~567/512 = 178.6944 (~3025/3024 = 3.0846)
Optimal GPV sequence: Template:Val list
Badness: 0.064400
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647
Mapping: [⟨41 65 1 68 189 246], ⟨0 0 2 1 -1 -2]]
Optimal tuning (CTE): ~56/55 = 1\41, ~72/65 = 178.9389 (~352/351 = 3.3291)
Optimal GPV sequence: Template:Val list
Badness: 0.0416
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 (CTE): ~55/54 = 1\41, ~256/231 = 178.3836 (~3024/3025 = 2.7738)
Optimal GPV sequence: Template:Val list
Badness: 0.100152
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049
Mapping: [⟨41 65 1 68 236 293], ⟨0 0 2 1 -2 -3]]
Optimal tuning (CTE): ~55/54 = 1\41, ~72/65 = 178.3755 (~352/351 = 2.7657)
Optimal GPV sequence: Template:Val list
Badness: 0.0605