Countercomp family

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