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This is a collection of 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]].  


Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla|Meantone family]]
* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
* ''[[Satin]]'' (+2100875/2097152) → [[Septischismic clan #Satin|Septischismic clan]]
* ''[[Pentorwell]]'', {1728/1715, 179200/177147} → [[Orwellismic temperaments #Pentorwell|Orwellismic temperaments]]
* ''[[Kleischismic]]'' (+32805/32768) → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Amicable]]'', {2401/2400, 1600000/1594323} → [[Breedsmic temperaments #Amicable|Breedsmic temperaments]]
* ''[[Pentaorwell]]'' (+1728/1715) → [[Orwellismic temperaments #Pentaorwell|Orwellismic temperaments]]
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Septiquarter]]'' (+5120/5103) → [[Aberschismic temperaments #Septiquarter|Aberschismic temperaments]]
* ''[[Septiquarter]]'', {5120/5103, 420175/419904} → [[Hemifamity temperaments #Septiquarter|Hemifamity temperaments]]
* ''[[Betic]]'' (+225/224) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Marthirds]]'', {15625/15552, 2460375/2458624} → [[Kleismic family #Marthirds|Kleismic family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]
* ''[[Kleischismic]]'', {32805/32768, 1500625/1492992} → [[Schismatic family #Kleischismic|Schismatic family]]
* ''[[Turkey]]'' (+5250987/5242880) → [[Vulture family #Turkey|Vulture family]]
* ''[[Turkey (temperament)|Turkey]]'', {4802000/4782969, 5250987/5242880} → [[Vulture family #Turkey|Vulture family]]
* ''[[Kaboom]]'' (+65625/65536) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Semiluna]]'', {4802000/4782969, 95703125/95551488} → [[Luna family #Semiluna|Luna 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]]


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


== Satin ==
== Hemiquindromeda ==
The satin temperament (94&217) uses 11/10 as a generator, three of them gives 4/3, and tempers out both the [[rainy comma]] and the canousma.
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Quindromeda]]''.


=== 5-limit ===
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


[[Comma list]]: {{monzo| 104 -70 3 }}
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 2 12 }}, {{val| 0 -3 -70 }}]
[[Comma list]]: 4802000/4782969, 67108864/66976875


[[POTE generator]]: ~{{monzo|-34 23 -1}} = 165.907
{{Mapping|legend=1| 1 2 0 6| 0 -10 56 -77 }}
: mapping generators: ~2, ~36/35


{{Val list|legend=1| 94, 217, 528, 745, 1273 }}
[[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 }}


[[Badness]]: 2.853049
{{Optimal ET sequence|legend=1| 24, 169d, 193, 217, 410 }}


=== 7-limit ===
[[Badness]] (Sintel): 6.19
Subgroup: 2.3.5.7
 
[[Comma list]]: 2100875/2097152, 4802000/4782969
 
[[Mapping]]: [{{val| 1 2 12 -3 }}, {{val| 0 -3 -70 42 }}]
 
[[POTE generator]]: ~8575/7776 = 165.913
 
{{Val list|legend=1| 94, 217, 311, 839, 1150 }}
 
[[Badness]]: 0.197207


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 4000/3993, 19712/19683, 41503/41472
Comma list: 14700/14641, 16384/16335, 19712/19683


Mapping: [{{val| 1 2 12 -3 13 }}, {{val| 0 -3 -70 42 -69 }}]
{{Mapping|legend=0| 1 2 0 6 4 |0 -10 56 -77 -13 }}


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}}


Vals: {{Val list| 94, 217, 311 }}
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}


Badness: 0.057972
Badness (Sintel): 3.08


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 1575/1573, 2080/2079, 4096/4095, 13720/13689
Comma list: 364/363, 676/675, 4096/4095, 19712/19683


Mapping: [{{val| 1 2 12 -3 13 -1 }}, {{val| 0 -3 -70 42 -69 34 }}]
{{Mapping|legend=0| 1 2 0 6 4 2 | 0 -10 56 -77 -13 41 }}


POTE generator: ~11/10 = 165.914
Optimal tunings:
* WE: ~2 = 1199.8852{{c}}, ~36/35 = 49.7543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.7591{{c}}


Vals: {{Val list| 94, 217, 311, 839e, 1150e }}
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}


Badness: 0.030316
Badness (Sintel): 2.01


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 595/594, 833/832, 1156/1155, 1575/1573, 4096/4095
Comma list: 364/363, 595/594, 676/675, 1156/1155, 4096/4095


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


POTE generator: ~11/10 = 165.913
Optimal tunings:
* WE: ~2 = 1199.8894{{c}}, ~34/33 = 49.7544{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7591{{c}}


Vals: {{Val list| 94, 217, 311, 839e, 1150eg }}
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e, 627e }}


Badness: 0.020007
Badness (Sintel): 1.43


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573
Comma list: 364/363, 595/594, 676/675, 1156/1155, 1216/1215, 1729/1728


Mapping: [{{val| 1 2 12 -3 13 -1 11 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 }}]
{{Mapping|legend=0| 1 2 0 6 4 2 5 4 | 0 -10 56 -77 -13 41 -22 6 }}


POTE generator: ~11/10 = 165.913
Optimal tunings:
* WE: ~2 = 1199.8724{{c}}, ~34/33 = 49.7535{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.7589{{c}}


Vals: {{Val list| 94, 217, 311, 839e, 1150eg }}
{{Optimal ET sequence|legend=0| 24, 169d, 193, 217, 410e }}


Badness: 0.014479
Badness (Sintel): 1.18


=== 23-limit ===
== Superlimmal ==
Subgroup: 2.3.5.7.11.13.17.19.23
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]].  


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


Mapping: [{{val| 1 2 12 -3 13 -1 11 16 16 }}, {{val| 0 -3 -70 42 -69 34 -50 -85 -83 }}]
[[Subgroup]]: 2.3.5.7
 
POTE generator: ~11/10 = 165.914
 
Vals: {{Val list| 94, 217, 311, 839ei, 1150egi }}
 
Badness: 0.012158
 
== Superlimmal ==
The superlimmal temperament uses an ever slightly sharpened [[27/25|large limma]] as the generator, nine exceed the octave by [[126/125]]. It can be described as 80 &amp; 311. It gets all the primes up to 29 reasonably covered, but still acceptible just as a 13-limit microtemperament, judging from its [[comma-basis]]. The 80-tone [[MOS scale]] is presumably the place to start, and a 151-tone MOS is possible.
 
Subgroup: 2.3.5.7


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


[[Mapping]]: [{{val| 1 8 12 18 }}, {{val| 0 -57 -86 -135 }}]
{{Mapping|legend=1| 1 -49 -74 -117 | 0 57 86 135 }}
 
: mapping generators: ~2, ~50/27
{{Multival|legend=1| 57 86 135 3 53 72 }}


[[POTE generator]]: ~27/25 = 135.0464
[[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 }}


{{Val list|legend=1| 80, 231, 311, 1324b, 1635b }}
{{Optimal ET sequence|legend=1| 80, 231, 311, 1324b, 1635b }}


[[Badness]]: 0.252387
[[Badness]] (Sintel): 6.39


=== 11-limit ===
=== 11-limit ===
Line 131: Line 128:
Comma list: 3025/3024, 4000/3993, 1479016/1476225
Comma list: 3025/3024, 4000/3993, 1479016/1476225


Mapping: [{{val| 1 8 12 18 11 }}, {{val| 0 -57 -86 -135 -67 }}]
{{Mapping|legend=0| 1 -49 -74 -117 -56 | 0 57 86 135 67 }}


POTE generator: ~27/25 = 135.0455
Optimal tuning:  
* WE: ~2 = 1199.9235{{c}}, ~50/27 = 1064.8866{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}


Vals: {{val list| 80, 231, 311, 1013e, 1324be }}
{{Optimal ET sequence|legend=0| 80, 231, 311, 1013e, 1324be }}


Badness: 0.060667
Badness (Sintel): 2.01


=== 13-limit ===
=== 13-limit ===
Line 144: Line 143:
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455
Comma list: 3025/3024, 4000/3993, 4225/4224, 4459/4455


Mapping: [{{val| 1 8 12 18 11 1 }}, {{val| 0 -57 -86 -135 -67 24 }}]
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 | 0 57 86 135 67 -24 }}


POTE generator: ~27/25 = 135.0446
Optimal tuning:  
* WE: ~2 = 1199.8904{{c}}, ~50/27 = 1064.8582{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9547{{c}}


Vals: {{val list| 80, 231, 311, 702, 1013e }}
{{Optimal ET sequence|legend=0| 80, 231, 311, 702, 1013e }}


Badness: 0.039017
Badness (Sintel): 1.61


=== 17-limit ===
=== 17-limit ===
Line 157: Line 158:
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224
Comma list: 595/594, 1275/1274, 2500/2499, 3025/3024, 4225/4224


Mapping: [{{val| 1 8 12 18 11 1 6 }}, {{val| 0 -57 -86 -135 -67 24 -17 }}]
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 | 0 57 86 135 67 -24 17 }}


POTE generator: ~27/25 = 135.0462
Optimal tuning:  
* WE: ~2 = 1199.9634{{c}}, ~50/27 = 1064.9213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9536{{c}}


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.030077
Badness (Sintel): 1.53


=== 19-limit ===
=== 19-limit ===
Line 170: Line 173:
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499
Comma list: 595/594, 969/968, 1275/1274, 1445/1444, 1729/1728, 2500/2499


Mapping: [{{val| 1 8 12 18 11 1 6 11 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 }}]
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 -49 | 0 57 86 135 67 -24 17 60 }}


POTE generator: ~27/25 = 135.0464
Optimal tuning:  
* WE: ~2 = 1199.9800{{c}}, ~50/27 = 1064.9358{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9535{{c}}


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.020460
Badness (Sintel): 1.24


=== 23-limit ===
=== 23-limit ===
Line 183: Line 188:
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495
Comma list: 595/594, 760/759, 969/968, 1105/1104, 1275/1274, 1445/1444, 1496/1495


Mapping: [{{val| 1 8 12 18 11 1 6 11 7 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 }}]
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 -49 -15 | 0 57 86 135 67 -24 17 60 22 }}


POTE generator: ~27/25 = 135.0458
Optimal tuning:  
* WE: ~2 = 1199.9546{{c}}, ~50/27 = 1064.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9539{{c}}


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.016146
Badness (Sintel): 1.16


=== 29-limit ===
=== 29-limit ===
Line 196: Line 203:
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495
Comma list: 595/594, 760/759, 784/783, 969/968, 1045/1044, 1105/1104, 1275/1274, 1496/1495


Mapping: [{{val| 1 8 12 18 11 1 6 11 7 16 }}, {{val| 0 -57 -86 -135 -67 24 -17 -60 -22 -99 }}]
{{Mapping|legend=0| 1 -49 -74 -117 -56 25 -11 -49 -15 -83 | 0 57 86 135 67 -24 17 60 22 99 }}


POTE generator: ~27/25 = 135.0460
Optimal tuning:  
* WE: ~2 = 1199.9430{{c}}, ~50/27 = 1064.9035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~50/27 = 1064.9538{{c}}


Vals: {{val list| 80, 231, 311 }}
{{Optimal ET sequence|legend=0| 80, 231, 311 }}


Badness: 0.013054
Badness (Sintel): 1.09


[[Category:Regular temperament theory]]
[[Category:Cathartic temperaments| ]] <!-- main article -->
[[Category:Temperament collection]]
[[Category:Cathartic| ]] <!-- key article -->
[[Category:Canousmic temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Cathartic_temperaments"