Xenharmonic Wiki

Canousmic temperaments

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 temperaments that temper out the canousma (monzo: [4 -14 3 4⟩, ratio: 4802000/4782969). For the rank-3 temperament, see Canou family.

Temperaments discussed elsewhere are:

Considered below is superlimmal.

Superlimmal

Superlimmal is essentially an 80-form, and may be described as the 80 & 311 temperament. It uses an ever slightly sharpened large limma as the generator, nine exceed the octave by 126/125. Note that in the data that follow, the generator is its octave complement, ~50/27, so that 57 of them octave reduced make the perfect fifth.

Superlimmal gets all the primes up to 29 reasonably covered, but is acceptable just as a 13-limit microtemperament, given a relatively simple comma basis. It can also be extended to include prime 37 by mapping it to 87 generator steps, tempering out (27/25)/(40/37) = 1000/999. Since 40/37 is the mediant of 27/25 and 13/12, this extension further consolidates the sharpened limma.

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 52734375/52706752

Mapping: [⟨1 -49 -74 -117], ⟨0 57 86 135]]

mapping generators: ~2, ~50/27

Optimal tunings:

  • WE: ~2 = 1199.9770 ¢, ~50/27 = 1064.9332 ¢
error map: ⟨-0.023 +0.365 -0.356 -0.152]
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9533 ¢
error map: ⟨0.000 +0.386 -0.326 -0.124]

Optimal ET sequence: 80, 231, 311, 1324b, 1635b

Badness (Sintel): 6.39

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 1479016/1476225

Mapping: [⟨1 -49 -74 -117 -56], ⟨0 57 86 135 67]]

Optimal tuning:

  • WE: ~2 = 1199.9235 ¢, ~50/27 = 1064.8866 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9536 ¢

Optimal ET sequence: 80, 231, 311, 1013e, 1324be

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455

Mapping: [⟨1 -49 -74 -117 -56 25], ⟨0 57 86 135 67 -24]]

Optimal tuning:

  • WE: ~2 = 1199.8904 ¢, ~50/27 = 1064.8582 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9547 ¢

Optimal ET sequence: 80, 231, 311, 702, 1013e

Badness (Sintel): 1.61

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224

Mapping: [⟨1 -49 -74 -117 -56 25 -11], ⟨0 57 86 135 67 -24 17]]

Optimal tuning:

  • WE: ~2 = 1199.9634 ¢, ~50/27 = 1064.9213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9536 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.53

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499

Mapping: [⟨1 -49 -74 -117 -56 25 -11 -49], ⟨0 57 86 135 67 -24 17 60]]

Optimal tuning:

  • WE: ~2 = 1199.9800 ¢, ~50/27 = 1064.9358 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9535 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.24

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495

Mapping: [⟨1 -49 -74 -117 -56 25 -11 -49 -15], ⟨0 57 86 135 67 -24 17 60 22]]

Optimal tuning:

  • WE: ~2 = 1199.9546 ¢, ~50/27 = 1064.9138 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9539 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.16

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495

Mapping: [⟨1 -49 -74 -117 -56 25 -11 -49 -15 -83], ⟨0 57 86 135 67 -24 17 60 22 99]]

Optimal tuning:

  • WE: ~2 = 1199.9430 ¢, ~50/27 = 1064.9035 ¢
  • CWE: ~2 = 1200.0000 ¢, ~50/27 = 1064.9538 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.09

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