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These are rank-2 temperaments that temper out the [[canousma]], 4802000/4782969 = {{monzo|4 -14 3 4}}. For the rank-3 temperament, see [[Canou family]].  
{{Technical data page}}
This page collects [[rank-2 temperament|rank-2]] '''cathartic temperaments''', which [[tempering out|temper out]] the [[cathartic comma]] ({{monzo|legend=1| 4 -14 3 4 }}, [[ratio]]: 4802000/4782969). For the family of [[rank-3 temperament]]s, see [[Cathartic family]].  


Note that 4802000/4782969 = 2×([[10/9]])<sup>3</sup>/([[9/7]])<sup>4</sup>, these intervals tend to have lower complexity.
Temperaments discussed elsewhere are:
* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Satin]]'' (+2100875/2097152) → [[Septischismic clan #Satin|Septischismic clan]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Aberschismic temperaments #Septiquarter|Aberschismic temperaments]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* ''[[Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Amicable]]'' (+2401/2400) → [[Amity family #Amicable|Amity family]]
* ''[[Marthirds]]'' (+15625/15552) → [[Kleismic family #Marthirds|Kleismic family]]
* ''[[Semiluna]]'' (+95703125/95551488) → [[Luna family #Semiluna|Luna family]]
* ''[[Quartiquart]]'' (+390625/388962) → [[Quartonic family #Quartiquart|Quartonic family]]


Temperaments not dicussed here include [[Meantone family #Godzilla|godzilla]] (a trivial case) and these:
Considered below are hemiquindromeda and superlimmal, in the order of increasing [[badness]].
* [[Sycamore family #Betic|Betic]]
* [[Breedsmic temperaments #Amicable|Amicable]]
* [[Ragismic microtemperaments #Parakleismic|Parakleismic]]
* [[Hemifamity temperaments #Septiquarter|Septiquarter]]
* [[Schismatic family #Kleischismic|Kleischismic]]


= Satin =
== Hemiquindromeda ==
Comma: |104 -70 3&gt;
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Quindromeda]]''.


[[POTE_tuning|POTE generator]]: ~|-34 23 -1&gt; = 165.907
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>.  


Map: [&lt;1 2 12|, &lt;0 -3 -70|]
[[Subgroup]]: 2.3.5.7


EDOs: 94, 123, 217, 311, 405, 528, 745
[[Comma list]]: 4802000/4782969, 67108864/66976875


Badness: 2.8530
{{Mapping|legend=1| 1 2 0 6| 0 -10 56 -77 }}
: mapping generators: ~2, ~36/35


==7-limit==
[[Optimal tuning]]s:
Commas: 2100875/2097152, 4802000/4782969
* [[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 }}


POTE generator: ~8575/7776 = 165.913
{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410 }}


Map: [&lt;1 2 12 -3|, &lt;0 -3 -70 42|]
[[Badness]] (Sintel): 6.19


EDOs: 94, 123, 217, 311, 405, 528
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.1972
Comma list: 14700/14641, 16384/16335, 19712/19683


==11-limit==
{{Mapping|legend=0| 1 2 0 6 4 |0 -10 56 -77 -13 }}
Commas: 4000/3993, 19712/19683, 41503/41472


POTE generator: ~11/10 = 165.915
Optimal tunings:
* WE: ~2 = 1199.8344{{c}}, ~36/35 = 49.7571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7499{{c}}


Map: [&lt;1 2 12 -3 13|, &lt;0 -3 -70 42 -69|]
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}


EDOs: 94, 123, 217, 311, 405, 528
Badness (Sintel): 3.08


Badness: 0.0580
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==13-limit==
Comma list: 364/363, 676/675, 4096/4095, 19712/19683
Commas: 1575/1573, 2080/2079, 4096/4095, 13720/13689


POTE generator: ~11/10 = 165.914
{{Mapping|legend=0| 1 2 0 6 4 2 | 0 -10 56 -77 -13 41 }}


Map: [&lt;1 2 12 -3 13 -1|, &lt;0 -3 -70 42 -69 34|]
Optimal tunings:
* WE: ~2 = 1199.8852{{c}}, ~36/35 = 49.7543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7591{{c}}


EDOs: 94, 123, 217, 311, 405, 528
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}


Badness: 0.0303
Badness (Sintel): 2.01


==17-limit==
=== 17-limit ===
Commas: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Subgroup: 2.3.5.7.11.13.17


POTE generator: ~11/10 = 165.913
Comma list: 364/363, 595/594, 676/675, 1156/1155, 4096/4095


Map: [&lt;1 2 12 -3 13 -1 11|, &lt;0 -3 -70 42 -69 34 -50|]
{{Mapping|legend=0| 1 2 0 6 4 2 5 | 0 -10 56 -77 -13 41 -22 }}


EDOs: 94, 123, 217, 311, 405, 528
Optimal tunings:
* WE: ~2 = 1199.8894{{c}}, ~34/33 = 49.7544{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7591{{c}}


Badness: 0.0200
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e, 627e }}


==19-limit==
Badness (Sintel): 1.43
Commas: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573


POTE generator: ~11/10 = 165.913
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Map: [&lt;1 2 12 -3 13 -1 11 16|, &lt;0 -3 -70 42 -69 34 -50 -85|]
Comma list: 364/363, 595/594, 676/675, 1156/1155, 1216/1215, 1729/1728


EDOs: 94, 123h, 217, 311, 405, 528
{{Mapping|legend=0| 1 2 0 6 4 2 5 4 | 0 -10 56 -77 -13 41 -22 6 }}


Badness: 0.0145
Optimal tunings:
* WE: ~2 = 1199.8724{{c}}, ~34/33 = 49.7535{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7589{{c}}


==23-limit==
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}
Commas: 595/594, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155


POTE generator: ~11/10 = 165.914
Badness (Sintel): 1.18


Map: [&lt;1 2 12 -3 13 -1 11 16 16|, &lt;0 -3 -70 42 -69 34 -50 -85 -83|]
== Superlimmal ==
Superlimmal is essentially an 80-form, and may be described as the {{nowrap| 80 & 311 }} temperament. It uses an ever slightly sharpened [[27/25|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 reduction|octave reduced]] make the [[3/2|perfect fifth]].


EDOs: 94, 217, 311, 405
Superlimmal gets all the primes up to [[29/1|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 [[37/1|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.


Badness: 0.0122
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4802000/4782969, 52734375/52706752


[[Category:Temperament]]
{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}
[[Category:Canou]]
: mapping generators: ~2, ~50/27
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9770{{c}}, ~50/27 = 1064.9332{{c}}
: [[error map]]: {{val| -0.023 +0.365 -0.356 -0.152 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9533{{c}}
: error map: {{val| 0.000 +0.386 -0.326 -0.124 }}
 
{{Optimal ET sequence|legend=1| 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|legend=0| 1 -49 -74 -117 -56 | 0 57 86 135 67 }}
 
Optimal tuning:
* WE: ~2 = 1199.9235{{c}}, ~50/27 = 1064.8866{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }}
 
Optimal tuning:
* WE: ~2 = 1199.8904{{c}}, ~50/27 = 1064.8582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9547{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }}
 
Optimal tuning:
* WE: ~2 = 1199.9634{{c}}, ~50/27 = 1064.9213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -49 -74 -117 -56 25 -11 -49 | 0 57 86 135 67 -24 17 60 }}
 
Optimal tuning:
* WE: ~2 = 1199.9800{{c}}, ~50/27 = 1064.9358{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9535{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }}
 
Optimal tuning:
* WE: ~2 = 1199.9546{{c}}, ~50/27 = 1064.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9539{{c}}
 
{{Optimal ET sequence|legend=0| 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|legend=0| 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{{c}}, ~50/27 = 1064.9035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9538{{c}}
 
{{Optimal ET sequence|legend=0| 80, 231, 311 }}
 
Badness (Sintel): 1.09
 
[[Category:Cathartic temperaments| ]] <!-- main article -->
[[Category:Cathartic| ]] <!-- key article -->
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
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