Countercomp family: Difference between revisions

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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]].
{{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]].


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


[[Comma list]]: {{monzo| 65 -41 }}
[[Comma list]]: {{monzo| 65 -41 }}
Line 8: Line 9:
[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]
[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]


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


[[POTE generator]]: ~5/4 = 386.668
[[Optimal tuning]]s:  
* [[CTE]]: ~531441/524288 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.3137¢
* [[CWE]]: ~531441/524288 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5501¢


{{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]] (Sintel): 21.917


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


[[Comma list]]: 1029/1024, 537824/531441
[[Comma list]]: 1029/1024, 537824/531441
Line 23: Line 26:
[[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]]s:
* [[CTE]]: ~64/63 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.3137¢
* [[CWE]]: ~64/63 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8764¢


[[POTE generator]]: ~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]] (Sintel): 4.076


[[Badness]]: 0.161056
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Mermapyth ==
Comma list: 385/384, 441/440, 537824/531441
Subgroup: 2.3.5.7
 
Mapping: [{{val| 41 65 0 115 237 }}, {{val| 0 0 1 0 -1 }}]
 
Optimal tunings:
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.9888¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8909¢
 
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d, 451dd }}
 
Badness (Sintel): 2.530
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 385/384, 59150/59049
 
Mapping: [{{val| 41 65 0 115 237 247 }}, {{val| 0 0 1 0 -1 -1 }}]
 
Optimal tunings:
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5763¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5997¢
 
{{Optimal ET sequence|legend=1| 41, 123e, 164, 205d }}
 
Badness (Sintel): 2.269
 
== Mermacomp ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 2500000/2470629
[[Comma list]]: 5120/5103, 2500000/2470629
Line 38: Line 71:
[[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]]s:
 
* [[CTE]]: ~50/49 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.1546¢
[[POTE generator]]: ~5/4 = 385.667
* [[CWE]]: ~50/49 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.4966¢


{{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]] (Sintel): 3.602


=== 11-limit ===
=== 11-limit ===
Line 53: Line 86:
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 tunings:  
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.1784¢
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.2847¢
 
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246 }}
 
Badness (Sintel): 2.532
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 75625/75264


Vals: {{Val list| 41, 164d, 205, 246 }}
Mapping: [{{val| 41 65 0 20 237 247 }}, {{val| 0 0 1 1 -1 -1 }}]


Badness: 0.076588
Optimal tunings:  
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.7331¢
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8418¢


== Hemicounterpyth ==
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246f }}
Subgroup: 2.3.5.7
 
Badness (Sintel): 2.293
 
== Hemicountercomp ==
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2401/2400, 52613349376/52301766015
[[Comma list]]: 2401/2400, 52613349376/52301766015


[[Mapping]]: [{{val| 41 65 95 115 }}, {{val| 0 0 2 1 }}]
[[Mapping]]: [{{val| 41 65 1 68 }}, {{val| 0 0 2 1 }}]


[[POTE generator]]: ~8/7 = 231.045
: mapping generators: ~100352/98415, ~567/256


{{Val list|legend=1| 41, 328, 369, 779 }}
[[Optimal tuning]]s:
* [[CTE]]: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.5314¢
* [[CWE]]: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6568¢


[[Badness]]: 0.134559
{{Optimal ET sequence|legend=1| 41, …, 328, 369, 779, 1927bc }}


=== 11-limit ===
[[Badness]] (Sintel): 3.405
 
=== Hemicocomp ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 41 65 95 115 142 }}, {{val| 0 0 2 1 -1 }}]
Mapping: [{{val| 41 65 1 68 189 }}, {{val| 0 0 2 1 -1 }}]


POTE generator: ~8/7 = 230.828
Optimal tunings:  
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6944¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.8825¢


Vals: {{Val list| 41, 328, 369 }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369, 1066cee }}


Badness: 0.064400
Badness (Sintel): 2.129


=== Hemermapyth ===
==== 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 tunings:
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.9389¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~72/65 = 179.0133¢
 
{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}
 
Badness (Sintel): 1.718
 
=== Hemermacomp ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 41 65 95 115 142 }}, {{val| 0 0 2 1 -2 }}]
Mapping: [{{val| 41 65 1 68 236 }}, {{val| 0 0 2 1 -2 }}]
 
Optimal tunings:
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~256/231 = 178.3836¢
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~256/231 = 178.3736¢
 
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}
 
Badness (Sintel): 3.311
 
==== 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 }}]


POTE generator: ~8/7 = 231.385
Optimal tunings:  
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.3755¢
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.3772¢


Vals: {{Val list| 41, 328e, 369e, 410, 451, 861e, 2173bcee }}
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}


Badness: 0.100152
Badness (Sintel): 2.500


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Counterpyth]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 11:20, 25 February 2026

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 tunings:

  • CTE: ~531441/524288 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.3137¢
  • CWE: ~531441/524288 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5501¢

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

Badness (Sintel): 21.917

Gamelacomp

Subgroup: 2.3.5.7

Comma list: 1029/1024, 537824/531441

Mapping: [⟨41 65 0 115], ⟨0 0 1 0]]

Optimal tunings:

  • CTE: ~64/63 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.3137¢
  • CWE: ~64/63 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8764¢

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

Badness (Sintel): 4.076

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 tunings:

  • CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.9888¢
  • CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8909¢

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

Badness (Sintel): 2.530

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 tunings:

  • CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5763¢
  • CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5997¢

Optimal ET sequence: 41, 123e, 164, 205d

Badness (Sintel): 2.269

Mermacomp

Subgroup: 2.3.5.7

Comma list: 5120/5103, 2500000/2470629

Mapping: [⟨41 65 0 20], ⟨0 0 1 1]]

Optimal tunings:

  • CTE: ~50/49 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.1546¢
  • CWE: ~50/49 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.4966¢

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

Badness (Sintel): 3.602

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 tunings:

  • CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.1784¢
  • CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.2847¢

Optimal ET sequence: 41, 164d, 205, 246

Badness (Sintel): 2.532

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 75625/75264

Mapping: [⟨41 65 0 20 237 247], ⟨0 0 1 1 -1 -1]]

Optimal tunings:

  • CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.7331¢
  • CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8418¢

Optimal ET sequence: 41, 164d, 205, 246f

Badness (Sintel): 2.293

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 tunings:

  • CTE: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.5314¢
  • CWE: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6568¢

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

Badness (Sintel): 3.405

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 tunings:

  • CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6944¢
  • CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.8825¢

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

Badness (Sintel): 2.129

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 tunings:

  • CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.9389¢
  • CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~72/65 = 179.0133¢

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

Badness (Sintel): 1.718

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 tunings:

  • CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~256/231 = 178.3836¢
  • CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~256/231 = 178.3736¢

Optimal ET sequence: 41, …, 410, 451, 861e

Badness (Sintel): 3.311

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 tunings:

  • CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.3755¢
  • CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.3772¢

Optimal ET sequence: 41, …, 410, 451, 861e

Badness (Sintel): 2.500

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