Cathartic temperaments: Difference between revisions

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* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Aberschismic temperaments #Septiquarter|Aberschismic temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Aberschismic temperaments #Septiquarter|Aberschismic temperaments]]
* ''[[Hemiquindromeda]]'' (+67108864/66976875) → [[Quindromeda family #Hemiquindromeda|Quindromeda family]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* ''[[Turkey|Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]
* ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]
Line 18: Line 17:
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]


Considered below is superlimmal.
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 [[4/3|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]])<sup>2</sup>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4802000/4782969, 67108864/66976875
 
{{Mapping|legend=1| 1 2 0 6| 0 -10 56 -77 }}
: mapping generators: ~2, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9255{{c}}, ~36/35 = 49.7534{{c}}
: [[error map]]: {{val| -0.075 +0.362 -0.126 -0.281 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 49.7566{{c}}
: error map: {{val| 0.000 +0.479 +0.053 -0.080 }}
 
{{Optimal ET sequence|legend=1| 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|legend=0| 1 2 0 6 4 |0 -10 56 -77 -13 }}
 
Optimal tunings:
* WE: ~2 = 1199.8344{{c}}, ~36/35 = 49.7571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7499{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 2 0 6 4 2 | 0 -10 56 -77 -13 41 }}
 
Optimal tunings:
* WE: ~2 = 1199.8852{{c}}, ~36/35 = 49.7543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7591{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 2 0 6 4 2 5 | 0 -10 56 -77 -13 41 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.8894{{c}}, ~34/33 = 49.7544{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7591{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 2 0 6 4 2 5 4 | 0 -10 56 -77 -13 41 -22 6 }}
 
Optimal tunings:
* WE: ~2 = 1199.8724{{c}}, ~34/33 = 49.7535{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7589{{c}}
 
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}
 
Badness (Sintel): 1.18


== Superlimmal ==
== Superlimmal ==

Latest revision as of 13:28, 10 November 2026

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:

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

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