Countercomp family: Difference between revisions

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Line 9: 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]]): ~531441/524288 = 1\41, ~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¢


{{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 }}


[[Badness]]: 0.934310
[[Badness]] (Sintel): 21.917


== Gamelacomp ==
== Gamelacomp ==
Line 24: Line 26:
[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]
[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]


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


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


=== 11-limit ===
=== 11-limit ===
Line 37: 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): ~56/55 = 1\41, ~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 ET sequence|legend=1| 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 50: 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): ~56/55 = 1\41, ~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 ET sequence|legend=1| 41, 123e, 164, 205d }}
{{Optimal ET sequence|legend=1| 41, 123e, 164, 205d }}


Badness: 0.054921
Badness (Sintel): 2.269


== Mermacomp ==
== Mermacomp ==
Line 63: Line 71:
[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]
[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]


[[Optimal tuning]] ([[POTE]]): ~50/49 = 1\41, ~5/4 = 385.667
[[Optimal tuning]]s:
* [[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|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 76: 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): ~55/54 = 1\41, ~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=1| 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 89: 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: ~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:
* [[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|legend=1| 41, …, 328, 369, 779, 1927bc }}
{{Optimal ET sequence|legend=1| 41, …, 328, 369, 779, 1927bc }}


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


=== Hemicocomp ===
=== Hemicocomp ===
Line 104: 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 }}]


Optimal tuning (CTE): ~56/55 = 1\41, ~567/512 = 178.6944 (~3025/3024 = 3.0846)
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|legend=1| 41, …, 328, 369, 1066cee }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369, 1066cee }}


Badness: 0.064400
Badness (Sintel): 2.129


==== 13-limit ====
==== 13-limit ====
Line 117: Line 148:
Mapping: [{{val| 41 65 1 68 189 246 }}, {{val| 0 0 2 1 -1 -2 }}]
Mapping: [{{val| 41 65 1 68 189 246 }}, {{val| 0 0 2 1 -1 -2 }}]


Optimal tuning (CTE): ~56/55 = 1\41, ~72/65 = 178.9389 (~352/351 = 3.3291)
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=1| 41, …, 328, 369f, 697cef }}
{{Optimal ET sequence|legend=0| 41, …, 328, 369f, 697cef }}


Badness: 0.0416
Badness (Sintel): 1.718


=== Hemermacomp ===
=== Hemermacomp ===
Line 130: 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 (CTE): ~55/54 = 1\41, ~256/231 = 178.3836 (~3024/3025 = 2.7738)
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=1| 41, …, 410, 451, 861e }}
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}


Badness: 0.100152
Badness (Sintel): 3.311


==== 13-limit ====
==== 13-limit ====
Line 143: Line 178:
Mapping: [{{val| 41 65 1 68 236 293 }}, {{val| 0 0 2 1 -2 -3 }}]
Mapping: [{{val| 41 65 1 68 236 293 }}, {{val| 0 0 2 1 -2 -3 }}]


Optimal tuning (CTE): ~55/54 = 1\41, ~72/65 = 178.3755 (~352/351 = 2.7657)
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|legend=1| 41, …, 410, 451, 861e }}
{{Optimal ET sequence|legend=0| 41, …, 410, 451, 861e }}


Badness: 0.0605
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]]

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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