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Cataharry family: Difference between revisions

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The '''cataharry family''' of rank three temperaments tempers out the [[cataharry comma]], 19683/19600.
{{Technical data page}}


= Cataharry =
The '''cataharry family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[cataharry comma]], 19683/19600.
Comma: 19683/19600


7 and 9 limit minimax
== Cataharry ==
[[Subgroup]]: 2.3.5.7


[|1 0 0 0>, |4/9 0 2/9 2/9>,
[[Comma list]]: [[19683/19600]]
|0 0 1 0>, |0 0 0 1>]


Eigenmonzos: 2, 8/7, 5/4
{{Mapping|legend=1| 1 0 0 -2 | 0 2 0 9 | 0 0 1 -1 }}


Map: [<1 0 0 -2|, <0 2 0 9|, <0 0 1 -1|]
: mapping generators: ~2, ~140/81, ~5


Generators: 2, 140/81, 5
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]:
: {{monzo list| 1 0 0 0 | 4/9 0 2/9 2/9 | 0 0 1 0 | 0 0 0 1 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7


{{EDOs|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 ==
Commas: 540/539, 8019/8000
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 8019/8000
 
{{Mapping|legend=1| 1 0 0 -2 6 | 0 2 0 9 -12 | 0 0 1 -1 3 }}
 
: mapping generators: ~2, ~140/81, ~5
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{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 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
 
{{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}
 
[[Badness]]: 0.621 × 10<sup>-3</sup>
 
=== Madagascar ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 676/675
 
Mapping: {{mapping| 1 0 0 -2 6 -1 | 0 2 0 9 -12 3 | 0 0 1 -1 3 1 }}
 
[[Mapping to lattice]]: [{{val| 0 -2 -3 -6 3 -6 }}, {{val| 0 0 -1 1 -3 -1 }}]
 
Lattice basis:
: 15/13 length = 0.3019, 26/25 length = 0.9477
: Angle (15/13, 26/25) = 81.6476
 
Minimax tuning:
* 13- and 15-odd-limit
: [{{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 }}]
: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
 
{{Optimal ET sequence|legend=1| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}
 
Badness: 0.560 × 10<sup>-3</sup>
 
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


Minimax tuning
Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3


[|1 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3&gt;,
Scales: [[madagascar19]]
|5 0 7/2 -2 -3/2&gt;, |5 0 5/2 -1 -3/2&gt;,
|5 0 5/2 -2 -1/2&gt;]


Eigenmonzos: 2, 11/10, 7/5
== Snape ==
[[Subgroup]]: 2.3.5.7.11


Map: [&lt;1 0 0 -2 6|, &lt;0 2 0 9 -12|, &lt;0 0 1 -1 3|]
[[Comma list]]: 441/440, 19683/19600


Generators: 2, 140/81, 5
{{Mapping|legend=1| 1 0 0 -2 -7 | 0 2 0 9 22 | 0 0 1 -1 -3 }}


{{EDOs|legend=1| 19, 53, 58, 72, 130, 183, 255, 385, 640 }}
{{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}


Badness: 0.000621
[[Badness]]: 0.988 × 10<sup>-3</sup>


== Madagascar ==
=== 13-limit ===
Commas: 351/350, 540/539, 676/675
Subgroup: 2.3.5.7.11.13


13 and 15 limit minimax
Comma list: 351/350, 441/440, 676/675


[|1 0 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3 0&gt;,
Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}
|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
{{Optimal ET sequence|legend=1| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}


Lattice basis: 15/13 0.3019 26/25 0.9477
Badness: 0.786 × 10<sup>-3</sup>


Angle(15/13, 26/25) = 81.6476
== Hagrid ==
[[Subgroup]]: 2.3.5.7.11


Map to lattice: [&lt;0 -2 -3 -6 3 -6|, &lt;0 0 -1 1 -3 -1|]
[[Comma list]]: 243/242, 9801/9800


Map: [&lt;1 0 0 -2 6 -1|, &lt;0 2 0 9 -12 3|, &lt;0 0 1 -1 3 1|]
{{Mapping|legend=1| 2 0 0 -4 -1 | 0 2 0 9 5 | 0 0 1 -1 0 }}


Generators: 2, 26/15, 5
: mapping generators: ~99/70, ~140/81, ~5


{{EDOs|legend=1| 19, 53, 58, 72, 111, 130, 183, 313 }}
{{Optimal ET sequence|legend=1| 24, 34d, 48, 58, 72, 130, 202 }}


Badness: 0.000560
[[Badness]]: 1.116 × 10<sup>-3</sup>


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


Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3
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 }}


* [[madagascar19]]
: mapping generators: ~99/70, ~26/15, ~5


= Snape =
{{Optimal ET sequence|legend=1| 24, 34d, 58, 72, 130 }}
Commas: 441/440, 19683/19600


== 13-limit ==
Badness: 0.876 × 10<sup>-3</sup>
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|legend=1| 1 0 0 -2 -6 | 0 2 0 9 9 | 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|]
{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}


{{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: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}


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


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


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