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Cataharry temperaments: Difference between revisions

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This is a collection of rank-2 temperaments tempering out the [[cataharry comma]], {{monzo| -4 9 -2 -2 }} = 19683/19600. These include swetneus, considered below, as well as others discussed elsewhere:
{{Technical data page}}
* [[godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla]]
This is a collection of [[rank-2 temperament|rank-2]] '''cataharry temperaments''', which [[tempering out|temper out]] the [[cataharry comma]] ({{monzo|legend=1| -4 9 -2 -2 }}, [[ratio]]: 19683/19600). For the family of [[rank-3 temperament]]s, see [[Cataharry family]].
* ''[[immunized]]'', {64/63, 8748/8575} → [[Immunity family #Immunized]]
* ''[[unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn]]
* [[catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic]]
* ''[[hemiseven]]'', {1029/1024, 19683/19600} → [[Gamelismic clan #Hemiseven]]
* ''[[buzzard]]'', {1728/1715, 5120/5103} → [[Hemifamity temperaments #Buzzard]]
* [[harry]], {2401/2400, 19683/19600} → [[Breedsmic temperaments #Harry]]
* [[hamity]], {2430/2401, 4000/3969} → [[Amity family #Hamity]]
* ''[[subpental]]'', {3136/3125, 19683/19600} → [[Hemimean clan #Subpental]]
* ''[[hemischis]]'', {6144/6125, 19683/19600} → [[Porwell temperaments #Hemischis]]
* ''[[mirkat]]'', {16875/16807, 19683/19600} → [[Mirkwai clan #Mirkat]]
* ''[[hours]]'', {19683/19600, 33075/32768} → [[Compton family #Hours]]


= Swetneus =
Discussed elsewhere are:
The swetneus temperament takes its name from the fact that an approximate swetismic neutral second, ~49/45, is the generator.
* ''[[Mite]]'' (+27/25) → [[Bug family #Mite|Bug family]]
* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* ''[[Immunized]]'' (+64/63) → [[Immunity family #Immunized|Immunity family]]
* ''[[Hemischis]]'' (+6144/6125) → [[Schismatic family #Hemischis|Schismatic family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Septimal buzzard|Buzzardsmic clan]]
* ''[[Sruti]]'' (+2048/2025) → [[Diaschismic family #Sruti|Diaschismic family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* ''[[Weasel]]'' (+50/49) → [[Tetracot family #Weasel|Tetracot family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
* ''[[Tuskaloosa]]'' (+110592/109375) → [[Mabila family #Tuskaloosa|Mabila family]]
* ''[[Hemiquintile]]'' (+589824/588245) → [[Quintile family #Hemiquintile|Quintile family]]
* [[Harry]] (+2401/2400) → [[Gravity family #Harry|Gravity family]]
* ''[[Subpental]]'' (+3136/3125) → [[Hemimean clan #Subpental|Hemimean clan]]
* ''[[Mirkat]]'' (+16875/16807) → [[Canopic clan #Mirkat|Canopic clan]]
* ''[[Hours]]'' (+33075/32768) → [[Compton family #Hours|Compton family]]


Subgroup: 2.3.5.7
Considered below is swetneus and majvamoid, in the order of increasing [[badness]].
 
== Swetneus ==
Swetneus tempers out the [[trimyna comma]] and may be described as the {{nowrap| 58 & 91 }} temperament. It takes its name from the fact that the generator is [[~]][[49/45]], a [[neutral second]] that is a [[swetisma]] away from [[12/11]]. Note that in the data that follow, the generator is its [[octave complement]], ~[[90/49]], so that twenty of them reach the [[interval class]] of [[3/1|3]]. In the [[11-limit]], this generator is equated with not [[11/6]] but [[81/44]].
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 50421/50000
[[Comma list]]: 19683/19600, 50421/50000


[[Mapping]]: [{{val| 1 4 8 8 }}, {{val| 0 -20 -47 -43 }}]
{{Mapping|legend=1| 1 -16 -39 -35 | 0 20 47 43 }}
 
: mapping generators: ~2, ~90/49
{{Multival|legend=1| 20 47 43 28 12 -32 }}


[[POTE generator]]: ~49/45 = 144.956
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2604{{c}}, ~90/49 = 1055.2729{{c}}
: [[error map]]: {{val| +0.260 -0.663 +1.357 -1.205 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~90/49 = 1055.0513{{c}}
: error map: {{val| 0.000 -0.928 +1.099 -1.618 }}


{{Val list|legend=1| 58, 91, 149, 505d, 654bd, 803bd }}
{{Optimal ET sequence|legend=1| 33cd, 58, 91, 149 }}


[[Badness]]: 0.1332
[[Badness]] (Sintel): 3.37


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


Comma list: 441/440, 3388/3375, 19683/19600
Comma list: 441/440, 3388/3375, 19683/19600


Mapping: [{{val| 1 4 8 8 13 }}, {{val| 0 -20 -47 -43 -79 }}]
Mapping: {{mapping| 1 -16 -39 -35 -66 | 0 20 47 43 79 }}


POTE generator: ~49/45 = 144.941
Optimal tunings:  
* WE: ~2 = 1200.1548{{c}}, ~81/44 = 1055.0954{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~81/44 = 1055.0631{{c}}


Vals: {{Val list| 58, 149, 207, 356e, 563cde, 919bcde }}
{{Optimal ET sequence|legend=0| 58, 149, 207, 356e }}


Badness: 0.0552
Badness (Sintel): 1.82


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


Comma list: 351/350, 441/440, 676/675, 847/845
Comma list: 351/350, 441/440, 676/675, 847/845


Mapping: [{{val| 1 4 8 8 13 13 }}, {{val| 0 -20 -47 -43 -79 -77 }}]
Mapping: {{mapping| 1 -16 -39 -35 -66 -64 | 0 20 47 43 79 77 }}
 
Optimal tunings:
* WE: ~2 = 1200.1473{{c}}, ~81/44 = 1055.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~81/44 = 1055.0644{{c}}
 
{{Optimal ET sequence|legend=0| 58, 149, 207, 356e }}
 
Badness (Sintel): 1.19
 
== Majvamoid ==
Named by [[Xenllium]] in 2021, majvamoid tempers out the [[quasiorwellisma]] and may be described as the {{nowrap| 149 & 183 }} temperament. It is an [[extension]] of the [[majvam]] temperament.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 19683/19600, 29360128/29296875
 
{{Mapping|legend=1| 1 -12 -2 -54 | 0 22 7 92 }}
: mapping generators: ~2, ~192/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0181{{c}}, ~192/125 = 740.9804{{c}}
: [[error map]]: {{val| +0.018 -0.603 +0.513 +0.395 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/125 = 740.9695{{c}}
: error map: {{val| 0.000 -0.627 +0.473 +0.365 }}
 
{{Optimal ET sequence|legend=1| 34d, 115d, 149, 183, 332, 515, 847 }}
 
[[Badness]] (Sintel): 5.25
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 5632/5625, 19683/19600
 
Mapping: {{mapping| 1 -12 -2 -54 -41 | 0 22 7 92 72 }}
 
Optimal tunings:
* WE: ~2 = 1200.0467{{c}}, ~135/88 = 741.0021{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~135/88 = 740.9738{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 115de, 149, 183, 332e, 515e }}
 
Badness (Sintel): 2.18
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 676/675, 1573/1568, 4225/4224
 
Mapping: {{mapping| 1 -12 -2 -54 -41 -21 | 0 22 7 92 72 40 }}
 
Optimal tunings:
* WE: ~2 = 1200.1305{{c}}, ~135/88 = 741.0523{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~135/88 = 740.9735{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 115def, 149, 183, 332ef, 515ef }}
 
Badness (Sintel): 1.43
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 561/560, 676/675, 715/714, 4225/4224
 
Mapping: {{mapping| 1 -12 -2 -54 -41 -21 1 | 0 22 7 92 72 40 5 }}


POTE generator: ~49/45 = 144.939
Optimal tunings:  
* WE: ~2 = 1200.0995{{c}}, ~135/88 = 741.0345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~135/88 = 740.9741{{c}}


Vals: {{Val list| 58, 149, 207, 356e, 563cde, 919bcdef }}
{{Optimal ET sequence|legend=0| 34d, 115def, 149, 183, 332ef, 515ef }}


Badness: 0.0289
Badness (Sintel): 1.19


[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Cataharry temperaments| ]] <!-- main article -->
[[Category:Cataharry temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Rank 2]]
[[Category:Rank 2]]
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