Didymus rank-3 family: Difference between revisions

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-These are the rank three temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
+The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth.
-__FORCETOC__
-=Euterpe=
-Commas: 81/80, 99/98
-Minimax tuning
+= Euterpe =
+[[Comma list]]: 81/80, 99/98
-[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
+[[Minimax tuning]]:
-|0 0 0 1 0&gt;, |-1 0 -1/2 2 0&gt;]
+* [[11-odd-limit]]:
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -1 0 -1/2 2 0 }}]
+: [[Eigenmonzo]]s: 2, 5, 7
-Eigenmonzos: 2, 5, 7
+[[Mapping]]: [{{val| 1 0 -4 0 1 }}, {{val| 0 1 4 0 -2 }}, {{val| 0 0 0 1 2 }}]
-Map: [&lt;1 0 -4 0 1|, &lt;0 1 4 0 -2|, &lt;0 0 0 1 2|]
+{{Val list|legend=1| 5, 12, 26, 31, 88 }}
-EDOs: 5, 12, 26, 31, 88
+[[Badness]]: 0.536 × 10<sup>-3</sup>
-Badness: 0.000536
+= Calliope =
+[[Comma list]]: 81/80, 45/44
-=Calliope=
+[[Minimax tuning]]:
-Commas: 81/80, 45/44
+* [[11-odd-limit]]:
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 0 1/6 }}, {{monzo| 0 0 0 0 2/3 }}, {{monzo| 1 -1 0 1 1/6 }}, {{monzo| 0 0 0 0 1 }}]
+: [[Eigenmonzo]]s: 2, 7/3, 11
-Minimax tuning
+[[Mapping]]: [{{val| 1 0 -4 0 -6 }}, {{val| 0 1 4 0 6 }}, {{val| 0 0 0 1 0 }}]
-[|1 0 0 0 0&gt;, |1 0 0 0 1/6&gt;, |0 0 0 0 2/3&gt;,
+{{Val list|legend=1| 7, 12, 19, 26, 45 }}
-|1 -1 0 1 1/6&gt;, |0 0 0 0 1&gt;]
-Eigenmonzos: 2, 7/3, 11
+[[Badness]]: 0.530 × 10<sup>-3</sup>
-Map: [&lt;1 0 -4 0 -6|, &lt;0 1 4 0 6|, &lt;0 0 0 1 0|]
+= Erato =
+[[Comma list]]: 81/80, 126/125
-EDOs: 7, 12, 19, 26, 45
+[[Mapping]]: [{{val| 1 0 -4 -13 0 }}, {{val| 0 1 4 10 0 }}, {{val| 0 0 0 0 1 }}]
-Badness: 0.000530
+{{Val list|legend=1| 12, 19, 31, 50, 81 }}
-=Erato=
+[[Badness]]: 0.558 × 10<sup>-3</sup>
-Commas: 81/80, 126/125
-Map: [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0|, &lt;0 0 0 0 1|]
+== 13-limit ==
+Comma list: 81/80, 105/104, 126/125
-EDOs: 12, 19, 31, 50, 81
+Mapping: [{{val| 1 0 -4 -13 0 -20 }}, {{val| 0 1 4 10 0 15 }}, {{val| 0 0 0 0 1 0 }}]
-Badness: 0.000558
+{{Val list|legend=1| 12f, 19, 31, 50, 81 }}
-==13-limit==
+= Clio =
-Commas: 81/80, 105/104, 126/125
+[[Comma list]]: 81/80, 176/175
-Map: [&lt;1 0 -4 -13 0 -20|, &lt;0 1 4 10 0 15|, &lt;0 0 0 0 1 0|]
+[[Minimax tuning]]
+* [[11-odd-limit]]:
+: {{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -4 0 2 1 0 }}]
+: [[Eigenmonzo]]s: 2, 5, 7
-EDOs: 12f, 19, 31, 50, 81
+Mapping: [{{val| 1 0 -4 0 -12 }}, {{val| 0 1 4 0 8 }}, {{val| 0 0 0 1 1 }}]
-=Clio=
+{{Val list|legend=1| 7, 12, 24, 31, 105, 129 }}
-Commas: 81/80, 176/175
-Minimax tuning
+Badness: 0.738 × 10<sup>-3</sup>
-||1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;,
-|0 0 0 1 0&gt;, |-4 0 2 1 0&gt;]
-Eigenmonzos: 2, 5, 7
+= Polyhymnia =
+[[Comma list]]: 81/80, 385/384
-Map: [&lt;1 0 -4 0 -12|, &lt;0 1 4 0 8|, &lt;0 0 0 1 1|]
+[[Mapping]]: [{{val| 1 0 -4 0 11 }}, {{val| 0 1 4 0 -3 }}, {{val| 0 0 0 1 -1 }}]
-EDOs: 7, 12, 24, 31, 105, 129
+{{Val list|legend=1| 7, 19, 24, 26, 31, 38, 45 }}
-Badness: 0.000738
+= Thalia =
+[[Comma list]]: 81/80, 55/54
-=Polyhymnia=
+[[Mapping]]: [{{val| 1 0 -4 0 5 }}, {{val| 0 1 4 0 -1 }}, {{val| 0 0 0 1 0 }}]
-Commas: 81/80, 385/384
-Map: [&lt;1 0 -4 0 11|, &lt;0 1 4 0 -3|, &lt;0 0 0 1 -1|]
+{{Val list|legend=1| 5, 7, 12e, 19e }}
-EDOs: 7, 19, 24, 26, 31, 38, 45
+= Melpomene =
+[[Comma list]]: 81/80, 56/55
-=Thalia=
+[[Mapping]]: [{{val| 1 0 -4 0 7 }}, {{val| 0 1 4 0 -4 }}, {{val| 0 0 0 1 1 }}]
-Commas: 81/80, 55/54
-Map: [&lt;1 0 -4 0 5|, &lt;0 1 4 0 -1|, &lt;0 0 0 1 0|]
+{{Val list|legend=1| 5, 12, 19, 24, 36 }}
-EDOs: 5, 7, 12e, 19e
+= Urania =
+[[Comma list]]: 81/80, 121/120
-=Melpomene=
+[[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 8 0 5 }}, {{val| 0 0 0 1 0 }}]
-Commas: 81/80, 56/55
-Map: [&lt;1 0 -4 0 7|, &lt;0 1 4 0 -4|, &lt;0 0 0 1 1|]
+Lattice basis:
+: 11/9 length = 0.2536, 8/7 length = 2.807
+: Angle (11/9, 8/7) = 90 degrees
-EDOs: 5, 12, 19, 24, 36
+Map to lattice: [{{val| 0 2 8 0 5 }}, {{val| 0 0 0 -1 0 }}]
-=Urania=
+{{Val list|legend=1| 7, 14c, 24, 31, 100de, 131bde, 162bde }}
-Commas: 81/80, 121/120
-Lattice basis: 11/9 length 0.2536 8/7 length 2.807
+[[Badness]]: 0.842 × 10<sup>-3</sup>
-Angle(11/9, 8/7) = 90 degrees
+[[Complexity spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
-Map to lattice: [&lt;0 2 8 0 5|, &lt;0 0 0 -1 0|]
-Map: [&lt;1 1 0 0 2|, &lt;0 2 8 0 5|, &lt;0 0 0 1 0|]
-EDOs: 7, 14c, 24, 31, 100de, 131bde, 162bde
-Badness: 0.000842
-[[Spectrum_of_a_temperament|Spectrum]]: 11/9, 4/3, 12/11, 11/10, 10/9, 9/8, 11/8, 6/5, 5/4, 8/7, 7/6, 9/7, 14/11, 7/5
 Scale: [[urania24]]
-=Terpsichore=
+= Terpsichore =
-Commas: 81/80, 540/539
+[[Comma list]]: 81/80, 540/539
-Map: [&lt;1 0 -4 0 -2|, &lt;0 1 4 0 7|, &lt;0 0 0 1 -2|]
+[[Mapping]]: [{{val| 1 0 -4 0 -2 }}, {{val| 0 1 4 0 7 }}, {{val| 0 0 0 1 -2 }}]
-EDOs: 19, 31, 81, 112b
+{{Val list|legend=1| 19, 31, 81, 112b }}
-Badness: 0.000850
+[[Badness]]: 0.850 × 10<sup>-3</sup>
-Spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
+[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
 [[Category:Theory]]