Didymus rank-3 family: Difference between revisions
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Map: [<1 0 -4 -13 0|, <0 1 4 10 0], <0 0 0 0 1|] | Map: [<1 0 -4 -13 0|, <0 1 4 10 0], <0 0 0 0 1|] | ||
EDOs: 12, 19, 31, 81 | EDOs: 12, 19, 31, 50, 81 | ||
Badness: 0.000558 | Badness: 0.000558 | ||
==13-limit== | |||
Commas: 81/80, 105/104, 126/125 | |||
Map: [<1 0 -4 -13 0 -20|, <0 1 4 10 0 15], <0 0 0 0 1 0|] | |||
EDOs: 12f, 19, 31, 50, 81 | |||
=Clio= | =Clio= | ||
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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 | 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:didymus]] | [[Category:didymus]] | ||
[[Category:family]] | [[Category:family]] | ||
[[Category:rank_3]] | [[Category:rank_3]] | ||
[[Category:theory]] | [[Category:theory]] | ||
Revision as of 08:13, 25 March 2019
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.
Euterpe
Commas: 81/80, 99/98
Minimax tuning
[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |0 0 0 1 0>, |-1 0 -1/2 2 0>]
Eigenmonzos: 2, 5, 7
Map: [<1 0 -4 0 1|, <0 1 4 0 -2|, <0 0 0 1 2|]
EDOs: 5, 12, 26, 31, 88
Badness: 0.000536
Calliope
Commas: 81/80, 45/44
Minimax tuning
[|1 0 0 0 0>, |1 0 0 0 1/6>, |0 0 0 0 2/3>, |1 -1 0 1 1/6>, |0 0 0 0 1>]
Eigenmonzos: 2, 7/3, 11
Map: [<1 0 -4 0 -6|, <0 1 4 0 6|, <0 0 0 1 0|]
EDOs: 7, 12, 19, 26, 45
Badness: 0.000530
Erato
Commas: 81/80, 126/125
Map: [<1 0 -4 -13 0|, <0 1 4 10 0], <0 0 0 0 1|]
EDOs: 12, 19, 31, 50, 81
Badness: 0.000558
13-limit
Commas: 81/80, 105/104, 126/125
Map: [<1 0 -4 -13 0 -20|, <0 1 4 10 0 15], <0 0 0 0 1 0|]
EDOs: 12f, 19, 31, 50, 81
Clio
Commas: 81/80, 176/175
Minimax tuning ||1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |0 0 0 1 0>, |-4 0 2 1 0>]
Eigenmonzos: 2, 5, 7
Map: [<1 0 -4 0 -12|, <0 1 4 0 8|, <0 0 0 1 1|]
EDOs: 7, 12, 24, 31, 105, 129
Badness: 0.000738
Polyhymnia
Commas: 81/80, 385/384
Map: [<1 0 -4 0 11|, <0 1 4 0 -3|, <0 0 0 1 -1|]
EDOs: 7, 19, 24, 26, 31, 38, 45
Thalia
Commas: 81/80, 55/54
Map: [<1 0 -4 0 5|, <0 1 4 0 -1|, <0 0 0 1 0|]
EDOs: 5, 7, 12e, 19e
Melpomene
Commas: 81/80, 56/55
Map: [<1 0 -4 0 7|, <0 1 4 0 -4|, <0 0 0 1 1|]
EDOs: 5, 12, 19, 24, 36
Urania
Commas: 81/80, 121/120
Lattice basis: 11/9 length 0.2536 8/7 length 2.807
Angle(11/9, 8/7) = 90 degrees
Map to lattice: [<0 2 8 0 5|, <0 0 0 -1 0|]
Map: [<1 1 0 0 2|, <0 2 8 0 5|, <0 0 0 1 0|]
EDOs: 7, 14c, 24, 31, 100de, 131bde, 162bde
Badness: 0.000842
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
Commas: 81/80, 540/539
Map: [<1 0 -4 0 -2|, <0 1 4 0 7|, <0 0 0 1 -2|]
EDOs: 19, 31, 81, 112b
Badness: 0.000850
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