Kleismic rank three family
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These are the rank three temperaments tempering out the kleisma, 15625/15552. 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. 7 and 9 limit minimax [|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>] Eigenmonzos: 2, 7/6, 4/3 Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|] Generators: 2, 6/5, 7 Edos: 15, 19, 23, 34, 53, 68, 72, 87, 121, 125, 140, 212, 333, 405 ===Enlil=== Commas: 15625/15552, 385/384 Minimax tuning [|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>, |3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>] Eigenmonzos: 2, 4/3, 14/11 Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|] Generators: 2, 6/5, 7 13-limit Commas: 15625/15552, 385/384, 325/324 13 and 15 limit minimax tuning [|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>, |19/9 0 0 -5/27 -5/27 10/27>, |28/9 0 0 13/27 -14/27 1/27>, |28/9 0 0 -14/27 13/27 1/27>, |28/9 0 0 -14/27 -14/27 28/27>] Eigenmonzos: 2, 13/11, 14/11 Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|] Generators: 2, 6/5, 7 Edos: 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299 ===Enki=== Commas: 15625/15552, 3025/3024 Minimax tuning [|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>, |-2/13 0 5/13 -5/13 10/13>, |-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>] Eigenmonzos: 2, 11/8, 7/5 Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|] Generators: 2, 6/5, 66/25 Edos: 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439 13-limit Commas: 15625/15552, 3025/3024, 325/324 13 and 15 limit [|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>, |14/9 0 -5/9 0 0 5/9>, |0 0 0 1 0 0>, |13/9 0 -4/9 1/2 0 4/9>, |14/9 0 -14/9 0 0 14/9>] Eigenmonzos: 2, 8/7 13/10 Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|] Generators: 2, 6/5, 66/25 Edos: 15, 72, 87, 159, 193, 280, 439
Original HTML content:
<html><head><title>Kleismic rank three family</title></head><body>These are the rank three temperaments tempering out the kleisma, 15625/15552. 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.<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |0 1 0 0>, |1 5/6 0 0>, |0 0 0 1>]<br /> Eigenmonzos: 2, 7/6, 4/3<br /> <br /> Map: [<1 0 1 0|, <0 6 5 0|, <0 0 0 1|]<br /> Generators: 2, 6/5, 7<br /> Edos: 15, 19, 23, 34, 53, 68, 72, 87, 121, 125, 140, 212, 333, 405<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h3> --><h3 id="toc0"><a name="x--Enlil"></a><!-- ws:end:WikiTextHeadingRule:0 -->Enlil</h3> Commas: 15625/15552, 385/384<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |0 1 0 0 0>, |1 5/6 0 0 0>, <br /> |3 1/12 0 1/2 -1/2>, |3 1/12 0 -1/2 1/2>]<br /> Eigenmonzos: 2, 4/3, 14/11<br /> <br /> Map: [<1 0 1 0 6|, <0 6 5 0 1|, <0 0 0 1 -1|]<br /> Generators: 2, 6/5, 7<br /> <br /> 13-limit<br /> Commas: 15625/15552, 385/384, 325/324<br /> <br /> 13 and 15 limit minimax tuning<br /> [|1 0 0 0 0 0>, |4/3 0 0 -2/9 -2/9 4/9>, <br /> |19/9 0 0 -5/27 -5/27 10/27>, <br /> |28/9 0 0 13/27 -14/27 1/27>, <br /> |28/9 0 0 -14/27 13/27 1/27>, <br /> |28/9 0 0 -14/27 -14/27 28/27>]<br /> Eigenmonzos: 2, 13/11, 14/11<br /> <br /> Map: [<1 0 1 0 6 0|, <0 6 5 0 1 14|, <0 0 0 1 -1 0|]<br /> Generators: 2, 6/5, 7<br /> Edos: 15, 19, 34, 53, 57, 68, 72, 87, 159, 212, 284, 299<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x--Enki"></a><!-- ws:end:WikiTextHeadingRule:2 -->Enki</h3> Commas: 15625/15552, 3025/3024<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |-18/13 0 6/13 -6/13 12/13>, <br /> |-2/13 0 5/13 -5/13 10/13>, <br /> |-2/13 0 -8/13 8/13 10/13>, |0 0 0 0 1>]<br /> Eigenmonzos: 2, 11/8, 7/5<br /> <br /> Map: [<1 0 1 0 1|, <0 6 5 0 4|, <0 0 0 2 1|]<br /> Generators: 2, 6/5, 66/25<br /> Edos: 15, 49, 57, 72, 87, 159, 193, 231, 265, 280, 337, 352, 439 <br /> <br /> 13-limit<br /> Commas: 15625/15552, 3025/3024, 325/324<br /> <br /> 13 and 15 limit<br /> [|1 0 0 0 0 0>, |2/3 0 -2/3 0 0 2/3>, <br /> |14/9 0 -5/9 0 0 5/9>, |0 0 0 1 0 0>, <br /> |13/9 0 -4/9 1/2 0 4/9>, <br /> |14/9 0 -14/9 0 0 14/9>]<br /> Eigenmonzos: 2, 8/7 13/10<br /> <br /> Map: [<1 0 1 0 1 0|, <0 6 5 0 4 14|, <0 0 0 2 1 0|]<br /> Generators: 2, 6/5, 66/25<br /> Edos: 15, 72, 87, 159, 193, 280, 439</body></html>