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The '''cataharry family''' of rank-3 temperaments tempers out the [[cataharry comma]], 19683/19600.
{{Technical data page}}
 
The '''cataharry family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[cataharry comma]], 19683/19600.


== Cataharry ==
== Cataharry ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[19683/19600]]
[[Comma list]]: [[19683/19600]]


[[Mapping]]: [<1 0 0 -2|, <0 2 0 9|, <0 0 1 -1|]
{{Mapping|legend=1| 1 0 0 -2 | 0 2 0 9 | 0 0 1 -1 }}


Mapping generators: ~2, ~140/81, ~5
: mapping generators: ~2, ~140/81, ~5


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]:  
: [|1 0 0 0>, |4/9 0 2/9 2/9>, |0 0 1 0>, |0 0 0 1>]
: {{monzo list| 1 0 0 0 | 4/9 0 2/9 2/9 | 0 0 1 0 | 0 0 0 1 }}
: Eigenmonzos: 2, 8/7, 5/4
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7


{{Val list|legend=1| 19, 24, 48, 53, 58, 72, 130, 202, 457, 587, 659, 717 }}
{{Optimal ET sequence|legend=1| 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587 }}
 
[[Badness]]: 0.301 × 10<sup>-3</sup>


[[Projection pair]]s: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7
[[Projection pair]]s: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7


== Albus ==
== Albus ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 540/539, 8019/8000
[[Comma list]]: 540/539, 8019/8000


[[Mapping]]: [&lt;1 0 0 -2 6|, &lt;0 2 0 9 -12|, &lt;0 0 1 -1 3|]
{{Mapping|legend=1| 1 0 0 -2 6 | 0 2 0 9 -12 | 0 0 1 -1 3 }}


Mapping generators: ~2, ~140/81, ~5
: mapping generators: ~2, ~140/81, ~5


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [|1 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3&gt;, |5 0 7/2 -2 -3/2&gt;, |5 0 5/2 -1 -3/2&gt;, |5 0 5/2 -2 -1/2&gt;]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 }}, {{monzo| 5 0 7/2 -2 -3/2 }}, {{monzo| 5 0 5/2 -1 -3/2 }}, {{monzo| 5 0 5/2 -2 -1/2 }}]
: Eigenmonzos: 2, 11/10, 7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


{{Val list|legend=1| 19, 53, 58, 72, 130, 183, 255, 385, 640 }}
{{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}


[[Badness]]: 0.621 × 10<sup>-3</sup>
[[Badness]]: 0.621 × 10<sup>-3</sup>
Line 42: Line 46:
Comma list: 351/350, 540/539, 676/675
Comma list: 351/350, 540/539, 676/675


Mapping: [&lt;1 0 0 -2 6 -1|, &lt;0 2 0 9 -12 3|, &lt;0 0 1 -1 3 1|]
Mapping: {{mapping| 1 0 0 -2 6 -1 | 0 2 0 9 -12 3 | 0 0 1 -1 3 1 }}


Mapping generators: ~2, ~26/15, ~5
[[Mapping to lattice]]: [{{val| 0 -2 -3 -6 3 -6 }}, {{val| 0 0 -1 1 -3 -1 }}]
 
Map to lattice: [&lt;0 -2 -3 -6 3 -6|, &lt;0 0 -1 1 -3 -1|]


Lattice basis:  
Lattice basis:  
Line 52: Line 54:
: Angle (15/13, 26/25) = 81.6476
: Angle (15/13, 26/25) = 81.6476


[[Minimax tuning]]:  
Minimax tuning:  
* 13- and 15-odd-limit
* 13- and 15-odd-limit
: [|1 0 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3 0&gt;, |5 0 7/2 -2 -3/2 0&gt;, |5 0 5/2 -1 -3/2 0&gt;, |5 0 5/2 -2 -1/2 0&gt;, |8 0 11/2 -3 -5/2 0&gt;]
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 0 }}, {{monzo| 5 0 7/2 -2 -3/2 0 }}, {{monzo| 5 0 5/2 -1 -3/2 0 }}, {{monzo| 5 0 5/2 -2 -1/2 0 }}, {{monzo| 8 0 11/2 -3 -5/2 0 }}]
: Eigenmonzos: 2, 11/10, 7/5
: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5


Vals: {{Val list| 19, 53, 58, 72, 111, 130, 183, 313 }}
{{Optimal ET sequence|legend=1| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}


Badness: 0.560 × 10<sup>-3</sup>
Badness: 0.560 × 10<sup>-3</sup>
Line 68: Line 70:


== Snape ==
== Snape ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 441/440, 19683/19600
[[Comma list]]: 441/440, 19683/19600
{{Mapping|legend=1| 1 0 0 -2 -7 | 0 2 0 9 22 | 0 0 1 -1 -3 }}
{{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}
[[Badness]]: 0.988 × 10<sup>-3</sup>


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


Comma list: 351/350, 441/440, 729/728
Comma list: 351/350, 441/440, 676/675
 
Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}
 
{{Optimal ET sequence|legend=1| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}
 
Badness: 0.786 × 10<sup>-3</sup>


== Hagrid ==
== Hagrid ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 243/242, 9801/9800
[[Comma list]]: 243/242, 9801/9800
{{Mapping|legend=1| 2 0 0 -4 -1 | 0 2 0 9 5 | 0 0 1 -1 0 }}
: mapping generators: ~99/70, ~140/81, ~5
{{Optimal ET sequence|legend=1| 24, 34d, 48, 58, 72, 130, 202 }}
[[Badness]]: 1.116 × 10<sup>-3</sup>


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


Comma list: 243/242, 351/350, 729/728
Comma list: 243/242, 351/350, 364/363
 
Mapping: {{mapping| 2 0 0 -4 -1 -2 | 0 2 0 9 5 3 | 0 0 1 -1 0 1 }}
 
: mapping generators: ~99/70, ~26/15, ~5
 
{{Optimal ET sequence|legend=1| 24, 34d, 58, 72, 130 }}
 
Badness: 0.876 × 10<sup>-3</sup>


== Draco ==
== Draco ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 19683/19600
[[Comma list]]: 176/175, 19683/19600


[[Mapping]]: [&lt;1 0 0 -2 -6|, &lt;0 2 0 9 9|, &lt;0 0 1 -1 1|]
{{Mapping|legend=1| 1 0 0 -2 -6 | 0 2 0 9 9 | 0 0 1 -1 1 }}


{{Val list|legend=1| 24, 34d, 53, 58, 111, 357cd, 468cd }}
{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}


[[Badness]]: 1.934 × 10<sup>-3</sup>
[[Badness]]: 1.934 × 10<sup>-3</sup>
Line 103: Line 133:
Comma list: 176/175, 351/350, 676/675
Comma list: 176/175, 351/350, 676/675


Mapping: [&lt;1 0 0 -2 -6 -1|, &lt;0 2 0 9 9 3|, &lt;0 0 1 -1 1 1|]
Mapping: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}


Vals: {{Val list| 24, 34d, 53, 58, 77, 111, 357cdf, 468cdf }}
{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cdf }}


Badness: 0.999 × 10<sup>-3</sup>
Badness: 0.999 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Cataharry family| ]] <!-- main article -->
[[Category:Cataharry family| ]] <!-- main article -->
[[Category:Cataharry| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
{{todo|review}}
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