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:

• Godzilla (+49/48 or 81/80) → Slendro clan
• Betic (+225/224) → Sycamore family
• Pentaorwell (+1728/1715) → Orwellismic temperaments
• Amicable (+2401/2400) → Breedsmic temperaments
• Parakleismic (+3136/3125 or 4375/4374) → Ragismic microtemperaments
• Septiquarter (+5120/5103) → Hemifamity temperaments
• Marthirds (+15625/15552) → Kleismic family
• Kleischismic (+32805/32768) → Schismatic family
• Kaboom (+65625/65536) → Vavoom family
• Quartiquart (+390625/388962) → Quartonic family
• Turkey (+5250987/5242880) → Vulture family
• Hemiquindromeda (+67108864/66976875) → Quindromeda family
• Semiluna (+95703125/95551488) → Luna family

Considered below are satin and superlimmal.

Satin

For the 5-limit version of this temperament, see Miscellaneous 5-limit temperaments #Satin.

Satin tempers out the rainy comma, the canousma, and the garischisma, and may be described as the 94 & 217 temperament. It uses ~11/10 as a generator, three of which gives a perfect fourth, tempering out 4000/3993 in the 11-limit and onwards. Its ploidacot is omega-tricot.

Subgroup: 2.3.5.7

Comma list: 2100875/2097152, 4802000/4782969

Mapping: [⟨1 2 12 -3], ⟨0 -3 -70 42]]

mapping generators: ~2, ~8575/7776

Optimal tunings:

• WE: ~2 = 1200.0198 ¢, ~8575/7776 = 165.9161 ¢

error map: ⟨+0.020 +0.336 -0.200 -0.411]

• CWE: ~2 = 1200.0000 ¢, ~8575/7776 = 165.9133 ¢

error map: ⟨0.000 +0.305 -0.241 -0.469]

Optimal ET sequence: 94, 217, 311, 839, 1150

Badness (Sintel): 4.99

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 19712/19683, 41503/41472

Mapping: [⟨1 2 12 -3 13], ⟨0 -3 -70 42 -69]]

Optimal tunings:

• WE: ~2 = 1199.9931 ¢, ~11/10 = 165.9145 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9155 ¢

Optimal ET sequence: 94, 217, 311

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689

Mapping: [⟨1 2 12 -3 13 -1], ⟨0 -3 -70 42 -69 34]]

Optimal tunings:

• WE: ~2 = 1199.9607 ¢, ~11/10 = 165.9085 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9141 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 1.25

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095

Mapping: [⟨1 2 12 -3 13 -1 11], ⟨0 -3 -70 42 -69 34 -50]]

Optimal tunings:

• WE: ~2 = 1199.9843 ¢, ~11/10 = 165.9110 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9132 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 1.02

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573

Mapping: [⟨1 2 12 -3 13 -1 11 16], ⟨0 -3 -70 42 -69 34 -50 -85]]

Optimal tunings:

• WE: ~2 = 1199.9875 ¢, ~11/10 = 165.9111 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9129 ¢

Optimal ET sequence: 94, 217, 311, 839e

Badness (Sintel): 0.881

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155

Mapping: [⟨1 2 12 -3 13 -1 11 16 16], ⟨0 -3 -70 42 -69 34 -50 -85 -83]]

Optimal tunings:

• WE: ~2 = 1199.9745 ¢, ~11/10 = 165.9103 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9140 ¢

Optimal ET sequence: 94, 217, 311

Badness (Sintel): 0.871

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 tempering out (27/25)/(40/37) = 1000/999, where 40/37 is notably the mediant of 27/25 and 13/12, which could be interpreted as an explanation of 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

No-31's 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 595/594, 760/759, 784/783, 925/924, 969/968, 1000/999, 1045/1044, 1105/1104, 1275/1274

Mapping: [⟨1 -49 -74 -117 -56 25 -11 -49 -15 -83 -72], ⟨0 57 86 135 67 -24 17 60 22 99 87]]

Optimal tuning:

• WE: ~2 = 1199.9549 ¢, ~37/20 = 1064.9140 ¢
• CWE: ~2 = 1200.0000 ¢, ~37/20 = 1064.9538 ¢

Optimal ET sequence: 80, 231, 311

Badness (Sintel): 1.06