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Cataharry temperaments: Difference between revisions

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* ''[[Hemischis]]'' (+6144/6125) → [[Schismatic family #Hemischis|Schismatic family]]
* ''[[Hemischis]]'' (+6144/6125) → [[Schismatic family #Hemischis|Schismatic family]]
* [[Buzzard]] (+1728/1715) → [[Buzzardsmic clan #Septimal buzzard|Buzzardsmic clan]]
* [[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]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
* ''[[Hemiseven]]'' (+1029/1024) → [[Gamelismic clan #Hemiseven|Gamelismic clan]]
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== Swetneus ==
== Swetneus ==
The swetneus temperament takes its name from the fact that an approximate swetismic neutral second, ~49/45, is the generator.
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
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 19683/19600, 50421/50000
[[Comma list]]: 19683/19600, 50421/50000


{{Mapping|legend=1| 1 4 8 8 | 0 -20 -47 -43 }}
{{Mapping|legend=1| 1 -16 -39 -35 | 0 20 47 43 }}
: mapping generators: ~2, ~90/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/45 = 144.956{{c}}
[[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 }}


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


[[Badness]] (Smith): 0.133168
[[Badness]] (Sintel): 3.37


=== 11-limit ===
=== 11-limit ===
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Comma list: 441/440, 3388/3375, 19683/19600
Comma list: 441/440, 3388/3375, 19683/19600


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/45 = 144.941{{c}}
Optimal tunings:
* WE: ~2 = 1200.1548{{c}}, ~81/44 = 1055.0954{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~81/44 = 1055.0631{{c}}


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


Badness (Smith): 0.055163
Badness (Sintel): 1.82


=== 13-limit ===
=== 13-limit ===
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Comma list: 351/350, 441/440, 676/675, 847/845
Comma list: 351/350, 441/440, 676/675, 847/845


Mapping: {{mapping| 1 4 8 8 13 13 | 0 -20 -47 -43 -79 -77 }}
Mapping: {{mapping| 1 -16 -39 -35 -66 -64 | 0 20 47 43 79 77 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/45 = 144.939{{c}}
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, 563cde, 919bcdef }}
{{Optimal ET sequence|legend=0| 58, 149, 207, 356e }}


Badness (Smith): 0.028876
Badness (Sintel): 1.19


== Majvamoid ==
== Majvamoid ==
The ''majvamoid'' temperament (149 & 183) is an extension of the [[Subgroup temperaments #No-sevens subgroup|majvam]] temperament which tempers out the cataharry comma and the quasiorwellisma (29360128/29296875, sazoquinbigu) in the 7-limit.
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
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 19683/19600, 29360128/29296875
[[Comma list]]: 19683/19600, 29360128/29296875


{{Mapping|legend=1| 1 10 5 38 | 0 -22 -7 -92 }}
{{Mapping|legend=1| 1 -12 -2 -54 | 0 22 7 92 }}
: mapping generators: ~2, ~192/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/96 = 459.031{{c}}
[[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, 1362bc }}
{{Optimal ET sequence|legend=1| 34d, 115d, 149, 183, 332, 515, 847 }}


[[Badness]] (Smith): 0.207280
[[Badness]] (Sintel): 5.25


=== 11-limit ===
=== 11-limit ===
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Comma list: 3025/3024, 5632/5625, 19683/19600
Comma list: 3025/3024, 5632/5625, 19683/19600


Mapping: {{mapping| 1 10 5 38 31 | 0 -22 -7 -92 -72 }}
Mapping: {{mapping| 1 -12 -2 -54 -41 | 0 22 7 92 72 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~125/96 = 459.027{{c}}
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, 149, 183, 332e, 515e, 698e }}
{{Optimal ET sequence|legend=0| 34d, 115de, 149, 183, 332e, 515e }}


Badness (Smith): 0.065874
Badness (Sintel): 2.18


=== 13-limit ===
=== 13-limit ===
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Comma list: 351/350, 676/675, 1573/1568, 4225/4224
Comma list: 351/350, 676/675, 1573/1568, 4225/4224


Mapping: {{mapping| 1 10 5 38 31 19 | 0 -22 -7 -92 -72 -40 }}
Mapping: {{mapping| 1 -12 -2 -54 -41 -21 | 0 22 7 92 72 40 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~125/96 = 459.028{{c}}
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, 149, 183, 332ef, 515ef }}
{{Optimal ET sequence|legend=0| 34d, 115def, 149, 183, 332ef, 515ef }}


Badness (Smith): 0.034696
Badness (Sintel): 1.43


=== 17-limit ===
=== 17-limit ===
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Comma list: 351/350, 561/560, 676/675, 715/714, 4225/4224
Comma list: 351/350, 561/560, 676/675, 715/714, 4225/4224


Mapping: {{mapping| 1 10 5 38 31 19 6 | 0 -22 -7 -92 -72 -40 -5 }}
Mapping: {{mapping| 1 -12 -2 -54 -41 -21 1 | 0 22 7 92 72 40 5 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~125/96 = 459.027{{c}}
Optimal tunings:
* WE: ~2 = 1200.0995{{c}}, ~135/88 = 741.0345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~135/88 = 740.9741{{c}}


{{Optimal ET sequence|legend=0| 34d, 149, 183, 332ef, 515ef, 698ef }}
{{Optimal ET sequence|legend=0| 34d, 115def, 149, 183, 332ef, 515ef }}


Badness (Smith): 0.023302
Badness (Sintel): 1.19


[[Category:Cataharry temperaments| ]] <!-- main article -->
[[Category:Cataharry temperaments| ]] <!-- main article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
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