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


{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
{{Optimal ET sequence|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
Line 24: Line 23:
[[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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~64/63 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.3137¢
* [[CTE]]: ~64/63 = 29.2683{{c}}, ~5/4 = 386.3137{{c}}
* [[CWE]]: ~64/63 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8764¢
* [[CWE]]: ~64/63 = 29.2683{{c}}, ~5/4 = 385.8764{{c}}


{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d, 451dd }}
{{Optimal ET sequence|legend=1| 41, 123, 164, 205d, 246d, 451dd }}
Line 39: Line 38:
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: {{mapping| 41 65 0 115 237 | 0 0 1 0 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.9888¢
* CTE: ~56/55 = 29.2683{{c}}, ~5/4 = 385.9888{{c}}
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8909¢
* CWE: ~56/55 = 29.2683{{c}}, ~5/4 = 385.8909{{c}}


{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d, 451dd }}
{{Optimal ET sequence|legend=0| 41, 123e, 164, 205d, 451dd }}
Line 54: Line 53:
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: {{mapping| 41 65 0 115 237 247 | 0 0 1 0 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5763¢
* CTE: ~56/55 = 29.2683{{c}}, ~5/4 = 386.5763{{c}}
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5997¢
* 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 (Sintel): 2.269
Badness (Sintel): 2.269
Line 69: Line 68:
[[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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~50/49 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.1546¢
* [[CTE]]: ~50/49 = 29.2683{{c}}, ~5/4 = 385.1546{{c}}
* [[CWE]]: ~50/49 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.4966¢
* [[CWE]]: ~50/49 = 29.2683{{c}}, ~5/4 = 385.4966{{c}}


{{Optimal ET sequence|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
{{Optimal ET sequence|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
Line 84: Line 83:
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: {{mapping| 41 65 0 20 237 | 0 0 1 1 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.1784¢
* CTE: ~55/54 = 29.2683{{c}}, ~5/4 = 385.1784{{c}}
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.2847¢
* CWE: ~55/54 = 29.2683{{c}}, ~5/4 = 385.2847{{c}}


{{Optimal ET sequence|legend=0| 41, 164d, 205, 246 }}
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246 }}
Line 99: Line 98:
Comma list: 352/351, 540/539, 729/728, 75625/75264
Comma list: 352/351, 540/539, 729/728, 75625/75264


Mapping: [{{val| 41 65 0 20 237 247 }}, {{val| 0 0 1 1 -1 -1 }}]
Mapping: {{mapping| 41 65 0 20 237 247 | 0 0 1 1 -1 -1 }}


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


{{Optimal ET sequence|legend=0| 41, 164d, 205, 246f }}
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246f }}
Line 114: Line 113:
[[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]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.5314¢
* [[CTE]]: ~100352/98415 = 29.2683{{c}}, ~567/512 = 178.5314{{c}}
* [[CWE]]: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6568¢
* [[CWE]]: ~100352/98415 = 29.2683{{c}}, ~567/512 = 178.6568{{c}}


{{Optimal ET sequence|legend=1| 41, …, 328, 369, 779, 1927bc }}
{{Optimal ET sequence|legend=1| 41, …, 328, 369, 779, 1927bc }}
Line 131: Line 129:
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: {{mapping| 41 65 1 68 189 | 0 0 2 1 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6944¢
* CTE: ~56/55 = 29.2683{{c}}, ~567/512 = 178.6944{{c}}
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.8825¢
* CWE: ~56/55 = 29.2683{{c}}, ~567/512 = 178.8825{{c}}


{{Optimal ET sequence|legend=0| 41, …, 328, 369, 1066cee }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369, 1066cee }}
Line 146: Line 144:
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: {{mapping| 41 65 1 68 189 246 | 0 0 2 1 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.9389¢
* CTE: ~56/55 = 29.2683{{c}}, ~72/65 = 178.9389{{c}}
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~72/65 = 179.0133¢
* CWE: ~56/55 = 29.2683{{c}}, ~72/65 = 179.0133{{c}}


{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}
Line 161: Line 159:
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: {{mapping| 41 65 1 68 236 | 0 0 2 1 -2 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~256/231 = 178.3836¢
* CTE: ~55/54 = 29.2683{{c}}, ~256/231 = 178.3836{{c}}
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~256/231 = 178.3736¢
* CWE: ~55/54 = 29.2683{{c}}, ~256/231 = 178.3736{{c}}


{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}
Line 176: Line 174:
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: {{mapping| 41 65 1 68 236 293 | 0 0 2 1 -2 -3 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.3755¢
* CTE: ~55/54 = 29.2683{{c}}, ~72/65 = 178.3755{{c}}
* CWE: ~55/54 = 29.2683¢ (1 ⧵ 41), ~72/65 = 178.3772¢
* CWE: ~55/54 = 29.2683{{c}}, ~72/65 = 178.3772{{c}}


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

Revision as of 06:56, 2 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:

  • CTE: ~531441/524288 = 29.2683 ¢, ~5/4 = 386.3137 ¢
  • CWE: ~531441/524288 = 29.2683 ¢, ~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 ¢, ~5/4 = 386.3137 ¢
  • CWE: ~64/63 = 29.2683 ¢, ~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 ¢, ~5/4 = 385.9888 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~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 ¢, ~5/4 = 386.5763 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~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 ¢, ~5/4 = 385.1546 ¢
  • CWE: ~50/49 = 29.2683 ¢, ~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 ¢, ~5/4 = 385.1784 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~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 ¢, ~5/4 = 385.7331 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~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 ¢, ~567/512 = 178.5314 ¢
  • CWE: ~100352/98415 = 29.2683 ¢, ~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 ¢, ~567/512 = 178.6944 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~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 ¢, ~72/65 = 178.9389 ¢
  • CWE: ~56/55 = 29.2683 ¢, ~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 ¢, ~256/231 = 178.3836 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~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 ¢, ~72/65 = 178.3755 ¢
  • CWE: ~55/54 = 29.2683 ¢, ~72/65 = 178.3772 ¢

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

Badness (Sintel): 2.500

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