Cataharry temperaments

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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.

This is a collection of rank-2 cataharry temperaments, which temper out the cataharry comma (monzo: [-4 9 -2 -2⟩, ratio: 19683/19600). For the family of rank-3 temperaments, see Cataharry family.

Discussed elsewhere are:

Considered below is swetneus and majvamoid, in the order of increasing badness.

Swetneus

Swetneus tempers out the trimyna comma and may be described as the 58 & 91 temperament. It takes its name from the fact that the generator is ~49/45, a neutral second that is a swetisma away from 12/11. Note that in the data that follow, the generator is its octave complement, ~90/49, so that twenty of them reach the interval class of 3. In the 11-limit, this generator is equated with not 11/6 but 81/44.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 50421/50000

Mapping: [⟨1 -16 -39 -35], ⟨0 20 47 43]]

mapping generators: ~2, ~90/49

Optimal tunings:

  • WE: ~2 = 1200.2604 ¢, ~90/49 = 1055.2729 ¢
error map: ⟨+0.260 -0.663 +1.357 -1.205]
  • CWE: ~2 = 1200.0000 ¢, ~90/49 = 1055.0513 ¢
error map: ⟨0.000 -0.928 +1.099 -1.618]

Optimal ET sequence: 33cd, 58, 91, 149

Badness (Sintel): 3.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 19683/19600

Mapping: [⟨1 -16 -39 -35 -66], ⟨0 20 47 43 79]]

Optimal tunings:

  • WE: ~2 = 1200.1548 ¢, ~81/44 = 1055.0954 ¢
  • CWE: ~2 = 1200.0000 ¢, ~81/44 = 1055.0631 ¢

Optimal ET sequence: 58, 149, 207, 356e

Badness (Sintel): 1.82

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 676/675, 847/845

Mapping: [⟨1 -16 -39 -35 -66 -64], ⟨0 20 47 43 79 77]]

Optimal tunings:

  • WE: ~2 = 1200.1473 ¢, ~81/44 = 1055.1904 ¢
  • CWE: ~2 = 1200.0000 ¢, ~81/44 = 1055.0644 ¢

Optimal ET sequence: 58, 149, 207, 356e

Badness (Sintel): 1.19

Majvamoid

Named by Xenllium in 2021, majvamoid tempers out the quasiorwellisma and may be described as the 149 & 183 temperament. It is an extension of the majvam temperament.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 29360128/29296875

Mapping: [⟨1 -12 -2 -54], ⟨0 22 7 92]]

mapping generators: ~2, ~192/125

Optimal tunings:

  • WE: ~2 = 1200.0181 ¢, ~192/125 = 740.9804 ¢
error map: ⟨+0.018 -0.603 +0.513 +0.395]
  • CWE: ~2 = 1200.0000 ¢, ~192/125 = 740.9695 ¢
error map: ⟨0.000 -0.627 +0.473 +0.365]

Optimal ET sequence: 34d, 115d, 149, 183, 332, 515, 847

Badness (Sintel): 5.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 19683/19600

Mapping: [⟨1 -12 -2 -54 -41], ⟨0 22 7 92 72]]

Optimal tunings:

  • WE: ~2 = 1200.0467 ¢, ~135/88 = 741.0021 ¢
  • CWE: ~2 = 1200.0000 ¢, ~135/88 = 740.9738 ¢

Optimal ET sequence: 34d, 115de, 149, 183, 332e, 515e

Badness (Sintel): 2.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 676/675, 1573/1568, 4225/4224

Mapping: [⟨1 -12 -2 -54 -41 -21], ⟨0 22 7 92 72 40]]

Optimal tunings:

  • WE: ~2 = 1200.1305 ¢, ~135/88 = 741.0523 ¢
  • CWE: ~2 = 1200.0000 ¢, ~135/88 = 740.9735 ¢

Optimal ET sequence: 34d, 115def, 149, 183, 332ef, 515ef

Badness (Sintel): 1.43

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 561/560, 676/675, 715/714, 4225/4224

Mapping: [⟨1 -12 -2 -54 -41 -21 1], ⟨0 22 7 92 72 40 5]]

Optimal tunings:

  • WE: ~2 = 1200.0995 ¢, ~135/88 = 741.0345 ¢
  • CWE: ~2 = 1200.0000 ¢, ~135/88 = 740.9741 ¢

Optimal ET sequence: 34d, 115def, 149, 183, 332ef, 515ef

Badness (Sintel): 1.19

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