Didymus rank-3 family

Revision as of 17:38, 28 February 2011 by Wikispaces>genewardsmith (**Imported revision 205854232 - Original comment: **)

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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.

==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, 81
Badness: 0.000558

==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

==Thalia==
Commas: 81/80, 55/54

==Melpomene==
Commas: 81/80, 56/55

==Urania==
Commas: 81/80, 121/120

Map: [<1 1 0 0 2|, <0 2 8 0 5|, <0 0 0 1 0|]
EDOs: 7, 24, 31
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

==Terpsichore==
Commas: 81/80, 540/539

Original HTML content:

<html><head><title>Didymus rank three family</title></head><body>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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Euterpe"></a><!-- ws:end:WikiTextHeadingRule:0 -->Euterpe</h2>
Commas: 81/80, 99/98<br />
<br />
Minimax tuning<br />
[|1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;, <br />
|0 0 0 1 0&gt;, |-1 0 -1/2 2 0&gt;]<br />
Eigenmonzos: 2, 5, 7<br />
<br />
Map: [&lt;1 0 -4 0 1|, &lt;0 1 4 0 -2|, &lt;0 0 0 1 2|]<br />
EDOs: 5, 12, 26, 31, 88<br />
Badness: 0.000536<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Calliope"></a><!-- ws:end:WikiTextHeadingRule:2 -->Calliope</h2>
Commas: 81/80, 45/44<br />
<br />
Minimax tuning<br />
[|1 0 0 0 0&gt;, |1 0 0 0 1/6&gt;, |0 0 0 0 2/3&gt;, <br />
|1 -1 0 1 1/6&gt;, |0 0 0 0 1&gt;]<br />
Eigenmonzos: 2, 7/3, 11<br />
<br />
Map: [&lt;1 0 -4 0 -6|, &lt;0 1 4 0 6|, &lt;0 0 0 1 0|]<br />
EDOs: 7, 12, 19, 26, 45<br />
Badness: 0.000530<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Erato"></a><!-- ws:end:WikiTextHeadingRule:4 -->Erato</h2>
Commas: 81/80, 126/125<br />
<br />
Map: [&lt;1 0 -4 -13 0|, &lt;0 1 4 10 0], &lt;0 0 0 0 1|]<br />
EDOs: 12, 19, 31, 81<br />
Badness: 0.000558<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Clio"></a><!-- ws:end:WikiTextHeadingRule:6 -->Clio</h2>
Commas: 81/80, 176/175<br />
<br />
Minimax tuning<br />
||1 0 0 0 0&gt;, |1 0 1/4 0 0&gt;, |0 0 1 0 0&gt;, <br />
|0 0 0 1 0&gt;, |-4 0 2 1 0&gt;]<br />
Eigenmonzos: 2, 5, 7<br />
<br />
Map: [&lt;1 0 -4 0 -12|, &lt;0 1 4 0 8|, &lt;0 0 0 1 1|]<br />
EDOs: 7, 12, 24, 31, 105, 129<br />
Badness: 0.000738<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="x-Polyhymnia"></a><!-- ws:end:WikiTextHeadingRule:8 -->Polyhymnia</h2>
Commas: 81/80, 385/384<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x-Thalia"></a><!-- ws:end:WikiTextHeadingRule:10 -->Thalia</h2>
Commas: 81/80, 55/54<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Melpomene"></a><!-- ws:end:WikiTextHeadingRule:12 -->Melpomene</h2>
Commas: 81/80, 56/55<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="x-Urania"></a><!-- ws:end:WikiTextHeadingRule:14 -->Urania</h2>
Commas: 81/80, 121/120<br />
<br />
Map: [&lt;1 1 0 0 2|, &lt;0 2 8 0 5|, &lt;0 0 0 1 0|]<br />
EDOs: 7, 24, 31<br />
Badness: 0.000842<br />
<a class="wiki_link" href="/Spectrum%20of%20a%20temperament">Spectrum</a>: 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<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="x-Terpsichore"></a><!-- ws:end:WikiTextHeadingRule:16 -->Terpsichore</h2>
Commas: 81/80, 540/539</body></html>