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The '''cataharry family''' of rank three temperaments tempers out the [[cataharry comma]], 19683/19600.
The '''cataharry family''' of rank-3 temperaments tempers out the [[cataharry comma]], 19683/19600.


== Cataharry ==
== Cataharry ==
Comma: 19683/19600
Subgroup: 2.3.5.7


7 and 9 limit minimax
[[Comma list]]: [[19683/19600]]


[|1 0 0 0>, |4/9 0 2/9 2/9>,
[[Mapping]]: [<1 0 0 -2|, <0 2 0 9|, <0 0 1 -1|]
|0 0 1 0>, |0 0 0 1>]


Eigenmonzos: 2, 8/7, 5/4
Mapping generators: ~2, ~140/81, ~5


Map: [<1 0 0 -2|, <0 2 0 9|, <0 0 1 -1|]
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]:
: [|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


Generators: 2, 140/81, 5
{{Val list|legend=1| 19, 24, 48, 53, 58, 72, 130, 202, 457, 587, 659, 717 }}


{{EDOs|legend=1| 19, 24, 48, 53, 58, 72, 130, 202, 457, 587, 659, 717 }}
[[Projection pair]]s: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7
 
== Albus ==
Subgroup: 2.3.5.7.11
 
[[Comma list]]: 540/539, 8019/8000


[[Projection pair]]s: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7
[[Mapping]]: [<1 0 0 -2 6|, <0 2 0 9 -12|, <0 0 1 -1 3|]


== Albus  ==
Mapping generators: ~2, ~140/81, ~5
Commas: 540/539, 8019/8000


Minimax tuning
[[Minimax tuning]]:
* [[11-odd-limit]]
: [|1 0 0 0 0>, |8/3 0 4/3 -2/3 -2/3>, |5 0 7/2 -2 -3/2>, |5 0 5/2 -1 -3/2>, |5 0 5/2 -2 -1/2>]
: Eigenmonzos: 2, 11/10, 7/5


[|1 0 0 0 0>, |8/3 0 4/3 -2/3 -2/3>,  
{{Val list|legend=1| 19, 53, 58, 72, 130, 183, 255, 385, 640 }}
|5 0 7/2 -2 -3/2>, |5 0 5/2 -1 -3/2>,  
|5 0 5/2 -2 -1/2>]


Eigenmonzos: 2, 11/10, 7/5
[[Badness]]: 0.621 × 10<sup>-3</sup>


Map: [&lt;1 0 0 -2 6|, &lt;0 2 0 9 -12|, &lt;0 0 1 -1 3|]
=== Madagascar ===
Subgroup: 2.3.5.7.11.13


Generators: 2, 140/81, 5
Comma list: 351/350, 540/539, 676/675


{{EDOs|legend=1| 19, 53, 58, 72, 130, 183, 255, 385, 640 }}
Mapping: [&lt;1 0 0 -2 6 -1|, &lt;0 2 0 9 -12 3|, &lt;0 0 1 -1 3 1|]


Badness: 0.000621
Mapping generators: ~2, ~26/15, ~5


=== Madagascar  ===
Map to lattice: [&lt;0 -2 -3 -6 3 -6|, &lt;0 0 -1 1 -3 -1|]
Commas: 351/350, 540/539, 676/675


13 and 15 limit minimax
Lattice basis:
: 15/13 length = 0.3019, 26/25 length = 0.9477
: Angle (15/13, 26/25) = 81.6476


[|1 0 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3 0&gt;,  
[[Minimax tuning]]:
|5 0 7/2 -2 -3/2 0&gt;, |5 0 5/2 -1 -3/2 0&gt;,  
* 13- and 15-odd-limit
|5 0 5/2 -2 -1/2 0&gt;, |8 0 11/2 -3 -5/2 0&gt;]
: [|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;]
: Eigenmonzos: 2, 11/10, 7/5


Eigenmonzos: 2, 11/10, 7/5
Vals: {{Val list| 19, 53, 58, 72, 111, 130, 183, 313 }}


Lattice basis: 15/13 0.3019 26/25 0.9477
Badness: 0.560 × 10<sup>-3</sup>


Angle(15/13, 26/25) = 81.6476
Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11


Map to lattice: [&lt;0 -2 -3 -6 3 -6|, &lt;0 0 -1 1 -3 -1|]
Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3


Map: [&lt;1 0 0 -2 6 -1|, &lt;0 2 0 9 -12 3|, &lt;0 0 1 -1 3 1|]
Scales: [[madagascar19]]


Generators: 2, 26/15, 5
== Snape ==
Subgroup: 2.3.5.7.11


{{EDOs|legend=1| 19, 53, 58, 72, 111, 130, 183, 313 }}
[[Comma list]]: 441/440, 19683/19600


Badness: 0.000560
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Spectrum of a temperament|Spectrum]]: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11
Comma list: 351/350, 441/440, 729/728


Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3
== Hagrid ==
Subgroup: 2.3.5.7.11


* [[madagascar19]]
[[Comma list]]: 243/242, 9801/9800


== Snape  ==
=== 13-limit ===
Commas: 441/440, 19683/19600
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 243/242, 351/350, 729/728
Commas: 351/350, 441/440, 729/728


== Hagrid  ==
== Draco ==
Commas: 243/242, 9801/9800
Subgroup: 2.3.5.7.11


=== 13-limit  ===
[[Comma list]]: 176/175, 19683/19600
Commas: 243/242, 351/350, 729/728


== Draco  ==
[[Mapping]]: [&lt;1 0 0 -2 -6|, &lt;0 2 0 9 9|, &lt;0 0 1 -1 1|]
Commas: 176/175, 19683/19600


Map: [&lt;1 0 0 -2 -6|, &lt;0 2 0 9 9|, &lt;0 0 1 -1 1|]
{{Val list|legend=1| 24, 34d, 53, 58, 111, 357cd, 468cd }}


{{EDOs|legend=1| 24, 34d, 53, 58, 111, 357cd, 468cd }}
[[Badness]]: 1.934 × 10<sup>-3</sup>


Badness: 0.001934
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 176/175, 351/350, 676/675
Commas: 176/175, 351/350, 676/675


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


{{EDOs|legend=1| 24, 34d, 53, 58, 77, 111, 357cdf, 468cdf }}
Vals: {{Val list| 24, 34d, 53, 58, 77, 111, 357cdf, 468cdf }}


Badness: 0.000999
Badness: 0.999 × 10<sup>-3</sup>


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Cataharry]]
[[Category:Cataharry family| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]


{{todo|review}}
{{todo|review}}
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