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 [[3/2|perfect 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 }}


[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 41 65 0 | 0 0 1 }}
: mapping generators: ~531441/524288, ~5


Mapping generators: ~531441/524288, ~5/1
[[Optimal tuning]]s:
* [[WE]]: ~531441/524288 = 29.2646{{c}}, ~5/4 = 386.6191{{c}}
: [[error map]]: {{val| -0.153 +0.242 -0.000 }}
* [[CWE]]: ~531441/524288 = 29.2683{{c}}, ~5/4 = 386.5501{{c}}
: error map: {{val| 0.000 +0.484 +0.236 }}


[[POTE generator]]: ~5/4 = 386.668
{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc, 2911bbc }}


{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
[[Badness]] (Sintel): 21.9


[[Badness]]: 0.934310
== Gamelacomp ==
[[Subgroup]]: 2.3.5.7


== Gamelapyth ==
[[Comma list]]: 1029/1024, 537824/531441
Subgroup: 2.3.5.7


[[Comma list]]: 1029/1024, 537824/531441
{{Mapping|legend=1| 41 65 0 115 | 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~64/63 = 29.2683{{c}}, ~5/4 = 386.3137{{c}}
: [[error map]]: {{val| +0.250 +0.880 -0.002 -2.271 }}
* [[CWE]]: ~64/63 = 29.2683{{c}}, ~5/4 = 385.8764{{c}}
: error map: {{val| 0.000 +0.484 -0.437 -2.972 }}
 
{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d }}
 
[[Badness]] (Sintel): 4.08
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 537824/531441
 
{{Mapping|legend=0| 41 65 0 115 237 | 0 0 1 0 -1 }}
 
Optimal tunings:
* WE: ~56/55 = 29.2737{{c}}, ~5/4 = 385.9427{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~5/4 = 385.8909{{c}}
 
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d }}
 
Badness (Sintel): 2.53
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]
Comma list: 196/195, 352/351, 385/384, 59150/59049


{{Multival|legend=1| 0 41 0 65 0 -115 }}
{{Mapping|legend=0| 41 65 0 115 237 247 | 0 0 1 0 -1 -1 }}


[[POTE generator]]: ~5/4 = 385.731
Optimal tunings:  
* WE: ~56/55 = 29.2700{{c}}, ~5/4 = 386.6265{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~5/4 = 386.5997{{c}}


{{Val list|legend=1| 41, 123, 164, 205d, 246d, 451dd }}
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d }}


[[Badness]]: 0.161056
Badness (Sintel): 2.27


== Mermapyth ==
== Mermacomp ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 2500000/2470629
[[Comma list]]: 5120/5103, 2500000/2470629


[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]
{{Mapping|legend=1| 41 65 0 20 | 0 0 1 1 }}
 
{{Multival|legend=1| 0 41 41 65 65 -20 }}


[[POTE generator]]: ~5/4 = 385.667
[[Optimal tuning]]s:
* [[WE]]: ~50/49 = 29.2633{{c}}, ~5/4 = 385.6013{{c}}
: [[error map]]: {{val| -0.204 +0.161 -1.120 +1.634 }}
* [[CWE]]: ~50/49 = 29.2683{{c}}, ~5/4 = 385.4966{{c}}
: error map: {{val| 0.000 +0.484 -0.817 +2.037 }}


{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
{{Optimal ET sequence|legend=1| 41, 123d, 164d, 205, 246 }}


[[Badness]]: 0.142344
[[Badness]] (Sintel): 3.60


=== 11-limit ===
=== 11-limit ===
Line 51: Line 89:
Comma list: 540/539, 5120/5103, 75625/75264
Comma list: 540/539, 5120/5103, 75625/75264


Mapping: [{{val| 41 65 0 20 237 }}, {{val| 0 0 1 1 -1 }}]
{{Mapping|legend=0| 41 65 0 20 237 | 0 0 1 1 -1 }}
 
Optimal tunings:
* WE: ~55/54 = 29.2657{{c}}, ~5/4 = 385.2755{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~5/4 = 385.2847{{c}}
 
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246 }}
 
Badness (Sintel): 2.53


POTE generator: ~5/4 = 385.309
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Vals: {{Val list| 41, 164d, 205, 246 }}
Comma list: 352/351, 540/539, 729/728, 75625/75264


Badness: 0.076588
{{Mapping|legend=0| 41 65 0 20 237 247 | 0 0 1 1 -1 -1 }}


== Hemicounterpyth ==
Optimal tunings:
Subgroup: 2.3.5.7
* CTE: ~55/54 = 29.2619{{c}}, ~5/4 = 385.7758{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~5/4 = 385.8418{{c}}
 
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246f }}
 
Badness (Sintel): 2.29
 
== 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|legend=1| 41 65 1 68 | 0 0 2 1 }}
: mapping generators: ~100352/98415, ~567/256


[[POTE generator]]: ~8/7 = 231.045
[[Optimal tuning]]s:  
* [[WE]]: ~100352/98415 = 29.2652{{c}}, ~567/512 = 178.6924{{c}}
: [[error map]]: {{val| -0.129 +0.280 +0.079 -0.231 }}
* [[CWE]]: ~100352/98415 = 29.2683{{c}}, ~567/512 = 178.6568{{c}}
: error map: {{val| 0.000 +0.484 +0.268 +0.075 }}


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


[[Badness]]: 0.134559
[[Badness]] (Sintel): 3.41


=== 11-limit ===
=== 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|legend=0| 41 65 1 68 189 | 0 0 2 1 -1 }}


POTE generator: ~8/7 = 230.828
Optimal tunings:  
* WE: ~56/55 = 29.2624{{c}}, ~567/512 = 178.8927{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~567/512 = 178.8825{{c}}


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


Badness: 0.064400
Badness (Sintel): 2.13


=== Hemermapyth ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647
 
{{Mapping|legend=0| 41 65 1 68 189 246 | 0 0 2 1 -1 -2 }}
 
Optimal tunings:
* WE: ~56/55 = 29.2615{{c}}, ~72/65 = 178.9843{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~72/65 = 179.0133{{c}}
 
{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}
 
Badness (Sintel): 1.72
 
=== 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|legend=0| 41 65 1 68 236 | 0 0 2 1 -2 }}
 
Optimal tunings:
* WE: ~55/54 = 29.2688{{c}}, ~256/231 = 178.3748{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~256/231 = 178.3736{{c}}
 
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}
 
Badness (Sintel): 3.31
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049
 
{{Mapping|legend=0| 41 65 1 68 236 293 | 0 0 2 1 -2 -3 }}


POTE generator: ~8/7 = 231.385
Optimal tunings:  
* WE: ~55/54 = 29.2688{{c}}, ~72/65 = 178.3805{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~72/65 = 178.3772{{c}}


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


Badness: 0.100152
Badness (Sintel): 2.50


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

Latest revision as of 12:32, 24 September 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 perfect 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

Optimal tunings:

  • WE: ~531441/524288 = 29.2646 ¢, ~5/4 = 386.6191 ¢
error map: ⟨-0.153 +0.242 -0.000]
  • CWE: ~531441/524288 = 29.2683 ¢, ~5/4 = 386.5501 ¢
error map: ⟨0.000 +0.484 +0.236]

Optimal ET sequence: 41, 123, 164, 205, 369, 574, 779, 2132bc, 2911bbc

Badness (Sintel): 21.9

Gamelacomp

Subgroup: 2.3.5.7

Comma list: 1029/1024, 537824/531441

Mapping: [⟨41 65 0 115], ⟨0 0 1 0]]

Optimal tunings:

  • WE: ~64/63 = 29.2683 ¢, ~5/4 = 386.3137 ¢
error map: ⟨+0.250 +0.880 -0.002 -2.271]
  • CWE: ~64/63 = 29.2683 ¢, ~5/4 = 385.8764 ¢
error map: ⟨0.000 +0.484 -0.437 -2.972]

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

Badness (Sintel): 4.08

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:

  • WE: ~56/55 = 29.2737 ¢, ~5/4 = 385.9427 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~5/4 = 385.8909 ¢

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

Badness (Sintel): 2.53

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:

  • WE: ~56/55 = 29.2700 ¢, ~5/4 = 386.6265 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~5/4 = 386.5997 ¢

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

Badness (Sintel): 2.27

Mermacomp

Subgroup: 2.3.5.7

Comma list: 5120/5103, 2500000/2470629

Mapping: [⟨41 65 0 20], ⟨0 0 1 1]]

Optimal tunings:

  • WE: ~50/49 = 29.2633 ¢, ~5/4 = 385.6013 ¢
error map: ⟨-0.204 +0.161 -1.120 +1.634]
  • CWE: ~50/49 = 29.2683 ¢, ~5/4 = 385.4966 ¢
error map: ⟨0.000 +0.484 -0.817 +2.037]

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

Badness (Sintel): 3.60

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:

  • WE: ~55/54 = 29.2657 ¢, ~5/4 = 385.2755 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~5/4 = 385.2847 ¢

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

Badness (Sintel): 2.53

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 = 29.2619 ¢, ~5/4 = 385.7758 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~5/4 = 385.8418 ¢

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

Badness (Sintel): 2.29

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:

  • WE: ~100352/98415 = 29.2652 ¢, ~567/512 = 178.6924 ¢
error map: ⟨-0.129 +0.280 +0.079 -0.231]
  • CWE: ~100352/98415 = 29.2683 ¢, ~567/512 = 178.6568 ¢
error map: ⟨0.000 +0.484 +0.268 +0.075]

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

Badness (Sintel): 3.41

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:

  • WE: ~56/55 = 29.2624 ¢, ~567/512 = 178.8927 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~567/512 = 178.8825 ¢

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

Badness (Sintel): 2.13

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:

  • WE: ~56/55 = 29.2615 ¢, ~72/65 = 178.9843 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~72/65 = 179.0133 ¢

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

Badness (Sintel): 1.72

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:

  • WE: ~55/54 = 29.2688 ¢, ~256/231 = 178.3748 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~256/231 = 178.3736 ¢

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

Badness (Sintel): 3.31

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:

  • WE: ~55/54 = 29.2688 ¢, ~72/65 = 178.3805 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~72/65 = 178.3772 ¢

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

Badness (Sintel): 2.50

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