Cathartic 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 page collects rank-2 cathartic temperaments, which temper out the cathartic comma (monzo: [4 -14 3 4⟩, ratio: 4802000/4782969). For the family of rank-3 temperaments, see Cathartic family.

Temperaments discussed elsewhere are:

• Godzilla (+49/48 or 81/80) → Semaphoresmic clan
• Satin (+2100875/2097152) → Septischismic clan
• Kleischismic (+32805/32768) → Schismatic family
• Pentaorwell (+1728/1715) → Orwellismic temperaments
• Septiquarter (+5120/5103) → Aberschismic temperaments
• Betic (+225/224) → Sycamore family
• Parakleismic (+3136/3125 or 4375/4374) → Parakleismic family
• Turkey (+5250987/5242880) → Vulture family
• Kaboom (+65625/65536) → Vavoom family
• Amicable (+2401/2400) → Amity family
• Marthirds (+15625/15552) → Kleismic family
• Semiluna (+95703125/95551488) → Luna family
• Quartiquart (+390625/388962) → Quartonic family

Considered below are hemiquindromeda and superlimmal, in the order of increasing badness.

Hemiquindromeda

For the 5-limit version, see Schismic–commatic equivalence continuum #Quindromeda.

Named by Xenllium in 2021, hemiquindromeda tempers out the decovulture comma. It is essentially a 24-form, and may be described as the 193 & 217 temperament. It splits the perfect fourth in ten ~36/35 quartertone generators, thus having a ploidacot signature of omega-decacot. Three generator steps reach an accurate ~12/11, from which we can derive an 11-limit extension where 14700/14641, 16384/16335, and 19712/19683 are tempered out, and a 13-limit extension that follows from 67108864/66976875 = (676/675)⋅(4096/4095)².

Subgroup: 2.3.5.7

Comma list: 4802000/4782969, 67108864/66976875

Mapping: [⟨1 2 0 6], ⟨0 -10 56 -77]]

mapping generators: ~2, ~36/35

Optimal tunings:

• WE: ~2 = 1199.9255 ¢, ~36/35 = 49.7534 ¢

error map: ⟨-0.075 +0.362 -0.126 -0.281]

• CWE: ~2 = 1200.0000 ¢, ~36/35 = 49.7566 ¢

error map: ⟨0.000 +0.479 +0.053 -0.080]

Optimal ET sequence: 24, 169d, 193, 217, 410

Badness (Sintel): 6.19

11-limit

Subgroup: 2.3.5.7.11

Comma list: 14700/14641, 16384/16335, 19712/19683

Mapping: [⟨1 2 0 6 4], ⟨0 -10 56 -77 -13]]

Optimal tunings:

• WE: ~2 = 1199.8344 ¢, ~36/35 = 49.7571 ¢
• CWE: ~2 = 1200.0000 ¢, ~36/35 = 49.7499 ¢

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness (Sintel): 3.08

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 676/675, 4096/4095, 19712/19683

Mapping: [⟨1 2 0 6 4 2], ⟨0 -10 56 -77 -13 41]]

Optimal tunings:

• WE: ~2 = 1199.8852 ¢, ~36/35 = 49.7543 ¢
• CWE: ~2 = 1200.0000 ¢, ~36/35 = 49.7591 ¢

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness (Sintel): 2.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 595/594, 676/675, 1156/1155, 4096/4095

Mapping: [⟨1 2 0 6 4 2 5], ⟨0 -10 56 -77 -13 41 -22]]

Optimal tunings:

• WE: ~2 = 1199.8894 ¢, ~34/33 = 49.7544 ¢
• CWE: ~2 = 1200.0000 ¢, ~34/33 = 49.7591 ¢

Optimal ET sequence: 24, 169d, 193, 217, 410e, 627e

Badness (Sintel): 1.43

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 595/594, 676/675, 1156/1155, 1216/1215, 1729/1728

Mapping: [⟨1 2 0 6 4 2 5 4], ⟨0 -10 56 -77 -13 41 -22 6]]

Optimal tunings:

• WE: ~2 = 1199.8724 ¢, ~34/33 = 49.7535 ¢
• CWE: ~2 = 1200.0000 ¢, ~34/33 = 49.7589 ¢

Optimal ET sequence: 24, 169d, 193, 217, 410e

Badness (Sintel): 1.18

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