Countercomp family: Difference between revisions

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{{Technical data page}}
{{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]].
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]].


== Countercomp ==
== Countercomp ==
Line 7: Line 7:
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~531441/524288 = 1\41, ~5/4 = 386.668
{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc, 2911bbc }}


{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
[[Badness]] (Sintel): 21.9
 
[[Badness]]: 0.934310


== Gamelacomp ==
== Gamelacomp ==
Line 22: Line 25:
[[Comma list]]: 1029/1024, 537824/531441
[[Comma list]]: 1029/1024, 537824/531441


[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 41 65 0 115 | 0 0 1 0 }}


[[Optimal tuning]] ([[POTE]]): ~64/63 = 1\41, ~5/4 = 385.731
[[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, 451dd }}
{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d }}


[[Badness]]: 0.161056
[[Badness]] (Sintel): 4.08


=== 11-limit ===
=== 11-limit ===
Line 35: Line 42:
Comma list: 385/384, 441/440, 537824/531441
Comma list: 385/384, 441/440, 537824/531441


Mapping: [{{val| 41 65 0 115 237 }}, {{val| 0 0 1 0 -1 }}]
{{Mapping|legend=0| 41 65 0 115 237 | 0 0 1 0 -1 }}


Optimal tuning (POTE): ~56/55 = 1\41, ~5/4 = 385.871
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=1| 41, 123e, 164, 205d, 451dd }}
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d }}


Badness: 0.076537
Badness (Sintel): 2.53


=== 13-limit ===
=== 13-limit ===
Line 48: Line 57:
Comma list: 196/195, 352/351, 385/384, 59150/59049
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 }}]
{{Mapping|legend=0| 41 65 0 115 237 247 | 0 0 1 0 -1 -1 }}


Optimal tuning (POTE): ~56/55 = 1\41, ~5/4 = 386.604
Optimal tunings:
* WE: ~56/55 = 29.2700{{c}}, ~5/4 = 386.6265{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~5/4 = 386.5997{{c}}


{{Optimal ET sequence|legend=1| 41, 123e, 164, 205d }}
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d }}


Badness: 0.054921
Badness (Sintel): 2.27


== Mermacomp ==
== Mermacomp ==
Line 61: Line 72:
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\41, ~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 }}


{{Optimal ET sequence|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 74: 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
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 729/728, 75625/75264
 
{{Mapping|legend=0| 41 65 0 20 237 247 | 0 0 1 1 -1 -1 }}


Optimal tuning (POTE): ~55/54 = 1\41, ~5/4 = 385.309
Optimal tunings:
* 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=1| 41, 164d, 205, 246 }}
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246f }}


Badness: 0.076588
Badness (Sintel): 2.29


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


[[Mapping]]: [{{val| 41 65 1 68 }}, {{val| 0 0 2 1 }}]
{{Mapping|legend=1| 41 65 1 68 | 0 0 2 1 }}
 
: mapping generators: ~100352/98415, ~567/256
Mapping generators: ~100352/98415, ~567/256


[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\41, ~567/512 = 178.5314 (~5120/5103 = 2.9216)
[[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 }}


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


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


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


Mapping: [{{val| 41 65 1 68 189 }}, {{val| 0 0 2 1 -1 }}]
{{Mapping|legend=0| 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 tunings:
* WE: ~56/55 = 29.2624{{c}}, ~567/512 = 178.8927{{c}}
* CWE: ~56/55 = 29.2683{{c}}, ~567/512 = 178.8825{{c}}


{{Optimal ET sequence|legend=1| 41, …, 328, 369, 1066cee }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369, 1066cee }}


Badness: 0.064400
Badness (Sintel): 2.13


==== 13-limit ====
==== 13-limit ====
Line 115: Line 152:
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647
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 }}]
{{Mapping|legend=0| 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 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=1| 41, …, 328, 369f, 697cef }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}


Badness: 0.0416
Badness (Sintel): 1.72


=== Hemermacomp ===
=== Hemermacomp ===
Line 128: Line 167:
Comma list: 2401/2400, 59290/59049, 131072/130977
Comma list: 2401/2400, 59290/59049, 131072/130977


Mapping: [{{val| 41 65 1 68 236 }}, {{val| 0 0 2 1 -2 }}]
{{Mapping|legend=0| 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 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=1| 41, …, 410, 451, 861e }}
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}


Badness: 0.100152
Badness (Sintel): 3.31


==== 13-limit ====
==== 13-limit ====
Line 141: Line 182:
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049
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 }}]
{{Mapping|legend=0| 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 tunings:
* WE: ~55/54 = 29.2688{{c}}, ~72/65 = 178.3805{{c}}
* CWE: ~55/54 = 29.2683{{c}}, ~72/65 = 178.3772{{c}}


{{Optimal ET sequence|legend=1| 41, …, 410, 451, 861e }}
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}


Badness: 0.0605
Badness (Sintel): 2.50


[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Countercomp family| ]] <!-- main article -->
[[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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