Diaschismic rank three family
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These are the rank three temperaments tempering out the diaschisma, 2048/2025. 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. ==Julius== Commas: 176/175, 896/891 Minimax tuning [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>] Eigenmonzos: 2, 12/11, 10/9 Lattice basis: 3/2, 9/7 Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|] Angle(3/2, 9/7) = 82.289 degrees Map: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |] EDOs: 58, 68, 80, 126 Julius[22] [[Hobbits|hobbit]] [[transversal]]: 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2 Julius[24] hobbit transversal 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2 ===13-limit=== Commas: 176/175, 325/324, 896/891 Map: [<2 0 11 0 14 -18|, <0 1 -2 0 -4 8|, <0 0 0 1 1 0|] EDOs: 12, 46, 68, 80 ==Augustus== Commas: 385/384, 1232/1215 Minimax tuning [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |45/16 3/4 -3/8 1/2 -1/2>, |45/16 3/4 -3/8 -1/2 1/2>] Eigenmonzos: 2, 10/9, 14/11 Map: [<2 0 11 0 3|, <0 1 -2 0 3|, <0 0 0 1 -1|] EDOs: 46, 68, 114, 124 ==Tiberius== Commas: 2048/2025, 441/440 Minimax tuning [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, |0 0 0 1 0>, |-3 2 -1 2 0>] Eigenmonzos: 2, 9/5, 7 Map: [<2 0 11 0 -17|, <0 1 -2 0 4|, <0 0 0 1 2|] EDOs: 58, 148, 160 ==Claudius== Commas: 2048/2025, 540/539 Minimax tuning [|1 0 0 0 0>, |9/7 0 -3/7 2/7 1/7>, |41/14 0 6/7 -4/7 -2/7>, |41/14 0 -1/7 3/7 -2/7>, |41/14 0 -1/7 -4/7 5/7>] Eigenmonzos: 2, 7/5, 11/5 Map: [<2 0 11 0 15|, <0 1 -2 0 1|, <0 0 0 1 -2|] EDOs: 58, 80
Original HTML content:
<html><head><title>Diaschismic rank three family</title></head><body>These are the rank three temperaments tempering out the diaschisma, 2048/2025. 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 /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Julius"></a><!-- ws:end:WikiTextHeadingRule:0 -->Julius</h2> Commas: 176/175, 896/891<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, <br /> |-1/8 3/2 -5/4 0 1>, |11/8 -1/2 -1/4 0 1>]<br /> Eigenmonzos: 2, 12/11, 10/9<br /> <br /> Lattice basis: 3/2, 9/7<br /> Map to lattice: [<0 1 -2 2 -2|, <0 0 0 -1 -1|]<br /> Angle(3/2, 9/7) = 82.289 degrees<br /> <br /> Map: [<2 0 11 0 14|, <0 1 -2 0 -4|, <0 0 0 1 1 |]<br /> EDOs: 58, 68, 80, 126<br /> <br /> Julius[22] <a class="wiki_link" href="/Hobbits">hobbit</a> <a class="wiki_link" href="/transversal">transversal</a>:<br /> 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2<br /> Julius[24] hobbit transversal<br /> 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Julius-13-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->13-limit</h3> Commas: 176/175, 325/324, 896/891<br /> <br /> Map: [<2 0 11 0 14 -18|, <0 1 -2 0 -4 8|, <0 0 0 1 1 0|]<br /> EDOs: 12, 46, 68, 80<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Augustus"></a><!-- ws:end:WikiTextHeadingRule:4 -->Augustus</h2> Commas: 385/384, 1232/1215<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, <br /> |45/16 3/4 -3/8 1/2 -1/2>, |45/16 3/4 -3/8 -1/2 1/2>]<br /> Eigenmonzos: 2, 10/9, 14/11<br /> <br /> Map: [<2 0 11 0 3|, <0 1 -2 0 3|, <0 0 0 1 -1|]<br /> EDOs: 46, 68, 114, 124<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-Tiberius"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tiberius</h2> Commas: 2048/2025, 441/440<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |11/8 1/2 -1/4 0 0>, |11/4 -1 1/2 0 0>, <br /> |0 0 0 1 0>, |-3 2 -1 2 0>]<br /> Eigenmonzos: 2, 9/5, 7<br /> <br /> Map: [<2 0 11 0 -17|, <0 1 -2 0 4|, <0 0 0 1 2|]<br /> EDOs: 58, 148, 160<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-Claudius"></a><!-- ws:end:WikiTextHeadingRule:8 -->Claudius</h2> Commas: 2048/2025, 540/539<br /> <br /> Minimax tuning<br /> [|1 0 0 0 0>, |9/7 0 -3/7 2/7 1/7>, |41/14 0 6/7 -4/7 -2/7>,<br /> |41/14 0 -1/7 3/7 -2/7>, |41/14 0 -1/7 -4/7 5/7>]<br /> Eigenmonzos: 2, 7/5, 11/5<br /> <br /> Map: [<2 0 11 0 15|, <0 1 -2 0 1|, <0 0 0 1 -2|]<br /> EDOs: 58, 80</body></html>