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
Line 38: Line 37:
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
Line 51: Line 50:
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]]

Latest revision as of 00:29, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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]]

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]]

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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