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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 ==
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


[[Optimal tuning]] ([[POTE]]): ~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


== Gamelacomp ==
== Gamelacomp ==
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¢


[[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]] (Sintel): 4.076
 
[[Badness]]: 0.161056


=== 11-limit ===
=== 11-limit ===
Line 38: Line 41:
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 tunings:
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.9888¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 385.8909¢


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


Badness: 0.076537
Badness (Sintel): 2.530


=== 13-limit ===
=== 13-limit ===
Line 51: Line 56:
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 tunings:
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5763¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~5/4 = 386.5997¢


Optimal GPV sequence: {{Val list| 41, 123e, 164, 205d }}
{{Optimal ET sequence|legend=1| 41, 123e, 164, 205d }}


Badness: 0.054921
Badness (Sintel): 2.269


== Mermacomp ==
== Mermacomp ==
Line 64: 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¢
[[Optimal tuning]] ([[POTE]]): ~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 79: 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 }}]


Optimal tuning (POTE): ~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 GPV sequence: {{Val list| 41, 164d, 205, 246 }}
{{Optimal ET sequence|legend=0| 41, 164d, 205, 246 }}


Badness: 0.076588
Badness (Sintel): 2.532
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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 }}]
 
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|legend=0| 41, 164d, 205, 246f }}
 
Badness (Sintel): 2.293


== Hemicountercomp ==
== Hemicountercomp ==
Line 92: Line 116:
[[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]]s:
* [[CTE]]: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.5314¢
* [[CWE]]: ~100352/98415 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6568¢


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


[[Badness]]: 0.134559
[[Badness]] (Sintel): 3.405


=== 11-limit ===
=== Hemicocomp ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 107: Line 133:
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 tunings:  
* CTE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.6944¢
* CWE: ~56/55 = 29.2683¢ (1 ⧵ 41), ~567/512 = 178.8825¢


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


Badness: 0.064400
Badness (Sintel): 2.129
 
==== 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 ===
=== Hemermacomp ===
Line 120: Line 163:
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 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 }}]
 
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 GPV sequence: {{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:Temperament families]]
[[Category:Temperament families]]
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Countercomp family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Countercomp_family"