Countercomp family: Difference between revisions

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Line 42: Line 42:
Comma list: 385/384, 441/440, 537824/531441
Comma list: 385/384, 441/440, 537824/531441


Mapping: {{mapping| 41 65 0 115 237 | 0 0 1 0 -1 }}
{{Mapping|legend=0| 41 65 0 115 237 | 0 0 1 0 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 57: Line 57:
Comma list: 196/195, 352/351, 385/384, 59150/59049
Comma list: 196/195, 352/351, 385/384, 59150/59049


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


Optimal tunings:  
Optimal tunings:  
Line 89: Line 89:
Comma list: 540/539, 5120/5103, 75625/75264
Comma list: 540/539, 5120/5103, 75625/75264


Mapping: {{mapping| 41 65 0 20 237 | 0 0 1 1 -1 }}
{{Mapping|legend=0| 41 65 0 20 237 | 0 0 1 1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 104: Line 104:
Comma list: 352/351, 540/539, 729/728, 75625/75264
Comma list: 352/351, 540/539, 729/728, 75625/75264


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


Optimal tunings:  
Optimal tunings:  
Line 137: Line 137:
Comma list: 2401/2400, 16384/16335, 19712/19683
Comma list: 2401/2400, 16384/16335, 19712/19683


Mapping: {{mapping| 41 65 1 68 189 | 0 0 2 1 -1 }}
{{Mapping|legend=0| 41 65 1 68 189 | 0 0 2 1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 152: Line 152:
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647
Comma list: 2080/2079, 2401/2400, 3584/3575, 10648/10647


Mapping: {{mapping| 41 65 1 68 189 246 | 0 0 2 1 -1 -2 }}
{{Mapping|legend=0| 41 65 1 68 189 246 | 0 0 2 1 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 167: Line 167:
Comma list: 2401/2400, 59290/59049, 131072/130977
Comma list: 2401/2400, 59290/59049, 131072/130977


Mapping: {{mapping| 41 65 1 68 236 | 0 0 2 1 -2 }}
{{Mapping|legend=0| 41 65 1 68 236 | 0 0 2 1 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 182: Line 182:
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049
Comma list: 2080/2079, 2401/2400, 4096/4095, 59290/59049


Mapping: {{mapping| 41 65 1 68 236 293 | 0 0 2 1 -2 -3 }}
{{Mapping|legend=0| 41 65 1 68 236 293 | 0 0 2 1 -2 -3 }}


Optimal tunings:  
Optimal tunings:  

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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