Countercomp family: Difference between revisions

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Line 12: Line 12:
[[Optimal tuning]] ([[POTE]]): ~531441/524288 = 1\41, ~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
Line 27: Line 27:
[[Optimal tuning]] ([[POTE]]): ~64/63 = 1\41, ~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 }}
{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d, 451dd }}


[[Badness]]: 0.161056
[[Badness]]: 0.161056
Line 40: Line 40:
Optimal tuning (POTE): ~56/55 = 1\41, ~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
Line 53: Line 53:
Optimal tuning (POTE): ~56/55 = 1\41, ~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
Line 68: Line 68:
[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\41, ~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 }}
{{Optimal ET sequence|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}


[[Badness]]: 0.142344
[[Badness]]: 0.142344
Line 81: Line 81:
Optimal tuning (POTE): ~55/54 = 1\41, ~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
Line 96: Line 96:
[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\41, ~567/512 = 178.5314 (~5120/5103 = 2.9216)
[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\41, ~567/512 = 178.5314 (~5120/5103 = 2.9216)


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


[[Badness]]: 0.134559
[[Badness]]: 0.134559
Line 109: Line 109:
Optimal tuning (CTE): ~56/55 = 1\41, ~567/512 = 178.6944 (~3025/3024 = 3.0846)
Optimal tuning (CTE): ~56/55 = 1\41, ~567/512 = 178.6944 (~3025/3024 = 3.0846)


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


Badness: 0.064400
Badness: 0.064400
Line 122: Line 122:
Optimal tuning (CTE): ~56/55 = 1\41, ~72/65 = 178.9389 (~352/351 = 3.3291)
Optimal tuning (CTE): ~56/55 = 1\41, ~72/65 = 178.9389 (~352/351 = 3.3291)


Optimal GPV sequence: {{Val list| 41, …, 328, 369f, 697cef }}
{{Optimal ET sequence|legend=1| 41, …, 328, 369f, 697cef }}


Badness: 0.0416
Badness: 0.0416
Line 135: Line 135:
Optimal tuning (CTE): ~55/54 = 1\41, ~256/231 = 178.3836 (~3024/3025 = 2.7738)
Optimal tuning (CTE): ~55/54 = 1\41, ~256/231 = 178.3836 (~3024/3025 = 2.7738)


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


Badness: 0.100152
Badness: 0.100152
Line 148: Line 148:
Optimal tuning (CTE): ~55/54 = 1\41, ~72/65 = 178.3755 (~352/351 = 2.7657)
Optimal tuning (CTE): ~55/54 = 1\41, ~72/65 = 178.3755 (~352/351 = 2.7657)


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


Badness: 0.0605
Badness: 0.0605

Revision as of 16:08, 7 May 2023

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

Optimal ET sequence: 41, 123, 164, 205, 369, 574, 779, 2132bc

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

Optimal ET sequence: 41, 123, 164, 205d, 246d, 451dd

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 ET sequence: 41, 123e, 164, 205d, 451dd

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 ET sequence: 41, 123e, 164, 205d

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

Optimal ET sequence: 41, 123d, 164d, 205, 246, 451d, 697dd

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 ET sequence: 41, 164d, 205, 246

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)

Optimal ET sequence: 41, …, 328, 369, 779, 1927bc

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 ET sequence: 41, …, 328, 369, 1066cee

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 ET sequence: 41, …, 328, 369f, 697cef

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 ET sequence: 41, …, 410, 451, 861e

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 ET sequence: 41, …, 410, 451, 861e

Badness: 0.0605

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