Diaschismic rank three family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-02 14:33:38 UTC</tt>.<br>
: The original revision id was <tt>190498930</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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==
==Julius==
Line 12: Line 5:


Minimax tuning
Minimax tuning
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,  
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,  
|-1/8 3/2 -5/4 0 1&gt;, |11/8 -1/2 -1/4 0 1&gt;]
|-1/8 3/2 -5/4 0 1&gt;, |11/8 -1/2 -1/4 0 1&gt;]
Eigenmonzos: 2, 12/11, 10/9
Eigenmonzos: 2, 12/11, 10/9


Lattice basis: 3/2, 9/7
Lattice basis: 3/2, 9/7
Map to lattice: [&lt;0 1 -2 2 -2|, &lt;0 0 0 -1 -1|]
Map to lattice: [&lt;0 1 -2 2 -2|, &lt;0 0 0 -1 -1|]
Angle(3/2, 9/7) = 82.289 degrees
Angle(3/2, 9/7) = 82.289 degrees


Map: [&lt;2 0 11 0 14|, &lt;0 1 -2 0 -4|, &lt;0 0 0 1 1 |]
Map: [&lt;2 0 11 0 14|, &lt;0 1 -2 0 -4|, &lt;0 0 0 1 1 |]
EDOs: 58, 68, 80, 126
EDOs: 58, 68, 80, 126
Badness: 0.000728
Badness: 0.000728


Julius[22] [[Hobbits|hobbit]] [[transversal]]:
Julius[22] [[Hobbits|hobbit]] [[Transversal|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
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
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
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


Line 33: Line 35:


Map: [&lt;2 0 11 0 14 -18|, &lt;0 1 -2 0 -4 8|, &lt;0 0 0 1 1 0|]
Map: [&lt;2 0 11 0 14 -18|, &lt;0 1 -2 0 -4 8|, &lt;0 0 0 1 1 0|]
EDOs: 12, 46, 68, 80
EDOs: 12, 46, 68, 80
Badness: 0.00178
Badness: 0.00178


Line 40: Line 44:


Minimax tuning
Minimax tuning
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,  
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,  
|45/16 3/4 -3/8 1/2 -1/2&gt;, |45/16 3/4 -3/8 -1/2 1/2&gt;]
|45/16 3/4 -3/8 1/2 -1/2&gt;, |45/16 3/4 -3/8 -1/2 1/2&gt;]
Eigenmonzos: 2, 10/9, 14/11
Eigenmonzos: 2, 10/9, 14/11


Map: [&lt;2 0 11 0 3|, &lt;0 1 -2 0 3|, &lt;0 0 0 1 -1|]
Map: [&lt;2 0 11 0 3|, &lt;0 1 -2 0 3|, &lt;0 0 0 1 -1|]
EDOs: 46, 68, 114, 124
EDOs: 46, 68, 114, 124
Badness: 0.00158
Badness: 0.00158


Line 52: Line 60:


Minimax tuning
Minimax tuning
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,  
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,  
|0 0 0 1 0&gt;, |-3 2 -1 2 0&gt;]
|0 0 0 1 0&gt;, |-3 2 -1 2 0&gt;]
Eigenmonzos: 2, 9/5, 7
Eigenmonzos: 2, 9/5, 7


Map: [&lt;2 0 11 0 -17|, &lt;0 1 -2 0 4|, &lt;0 0 0 1 2|]
Map: [&lt;2 0 11 0 -17|, &lt;0 1 -2 0 4|, &lt;0 0 0 1 2|]
EDOs: 58, 148, 160
EDOs: 58, 148, 160
Badness: 0.00171
Badness: 0.00171


Line 64: Line 76:


Minimax tuning
Minimax tuning
[|1 0 0 0 0&gt;, |9/7 0 -3/7 2/7 1/7&gt;, |41/14 0 6/7 -4/7 -2/7&gt;,
[|1 0 0 0 0&gt;, |9/7 0 -3/7 2/7 1/7&gt;, |41/14 0 6/7 -4/7 -2/7&gt;,
|41/14 0 -1/7 3/7 -2/7&gt;, |41/14 0 -1/7 -4/7 5/7&gt;]
|41/14 0 -1/7 3/7 -2/7&gt;, |41/14 0 -1/7 -4/7 5/7&gt;]
Eigenmonzos: 2, 7/5, 11/5
Eigenmonzos: 2, 7/5, 11/5


Map: [&lt;2 0 11 0 15|, &lt;0 1 -2 0 1|, &lt;0 0 0 1 -2|]
Map: [&lt;2 0 11 0 15|, &lt;0 1 -2 0 1|, &lt;0 0 0 1 -2|]
EDOs: 58, 80
EDOs: 58, 80
Badness: 0.00197</pre></div>
 
<h4>Original HTML content:</h4>
Badness: 0.00197
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Diaschismic rank three family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Julius"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Julius&lt;/h2&gt;
Commas: 176/175, 896/891&lt;br /&gt;
&lt;br /&gt;
Minimax tuning&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |11/8 1/2 -1/4 0 0&amp;gt;, |11/4 -1 1/2 0 0&amp;gt;, &lt;br /&gt;
|-1/8 3/2 -5/4 0 1&amp;gt;, |11/8 -1/2 -1/4 0 1&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 12/11, 10/9&lt;br /&gt;
&lt;br /&gt;
Lattice basis: 3/2, 9/7&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 -2 2 -2|, &amp;lt;0 0 0 -1 -1|]&lt;br /&gt;
Angle(3/2, 9/7) = 82.289 degrees&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 11 0 14|, &amp;lt;0 1 -2 0 -4|, &amp;lt;0 0 0 1 1 |]&lt;br /&gt;
EDOs: 58, 68, 80, 126&lt;br /&gt;
Badness: 0.000728&lt;br /&gt;
&lt;br /&gt;
Julius[22] &lt;a class="wiki_link" href="/Hobbits"&gt;hobbit&lt;/a&gt; &lt;a class="wiki_link" href="/transversal"&gt;transversal&lt;/a&gt;:&lt;br /&gt;
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&lt;br /&gt;
Julius[24] hobbit transversal&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Julius-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h3&gt;
Commas: 176/175, 325/324, 896/891&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 11 0 14 -18|, &amp;lt;0 1 -2 0 -4 8|, &amp;lt;0 0 0 1 1 0|]&lt;br /&gt;
EDOs: 12, 46, 68, 80&lt;br /&gt;
Badness: 0.00178&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Augustus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Augustus&lt;/h2&gt;
Commas: 385/384, 1232/1215&lt;br /&gt;
&lt;br /&gt;
Minimax tuning&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |11/8 1/2 -1/4 0 0&amp;gt;, |11/4 -1 1/2 0 0&amp;gt;, &lt;br /&gt;
|45/16 3/4 -3/8 1/2 -1/2&amp;gt;, |45/16 3/4 -3/8 -1/2 1/2&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 10/9, 14/11&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 11 0 3|, &amp;lt;0 1 -2 0 3|, &amp;lt;0 0 0 1 -1|]&lt;br /&gt;
EDOs: 46, 68, 114, 124&lt;br /&gt;
Badness: 0.00158&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Tiberius"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Tiberius&lt;/h2&gt;
Commas: 2048/2025, 441/440&lt;br /&gt;
&lt;br /&gt;
Minimax tuning&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |11/8 1/2 -1/4 0 0&amp;gt;, |11/4 -1 1/2 0 0&amp;gt;, &lt;br /&gt;
|0 0 0 1 0&amp;gt;, |-3 2 -1 2 0&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 9/5, 7&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 11 0 -17|, &amp;lt;0 1 -2 0 4|, &amp;lt;0 0 0 1 2|]&lt;br /&gt;
EDOs: 58, 148, 160&lt;br /&gt;
Badness: 0.00171&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Claudius"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Claudius&lt;/h2&gt;
Commas: 2048/2025, 540/539&lt;br /&gt;
&lt;br /&gt;
Minimax tuning&lt;br /&gt;
[|1 0 0 0 0&amp;gt;, |9/7 0 -3/7 2/7 1/7&amp;gt;, |41/14 0 6/7 -4/7 -2/7&amp;gt;,&lt;br /&gt;
|41/14 0 -1/7 3/7 -2/7&amp;gt;, |41/14 0 -1/7 -4/7 5/7&amp;gt;]&lt;br /&gt;
Eigenmonzos: 2, 7/5, 11/5&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;2 0 11 0 15|, &amp;lt;0 1 -2 0 1|, &amp;lt;0 0 0 1 -2|]&lt;br /&gt;
EDOs: 58, 80&lt;br /&gt;
Badness: 0.00197&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 00:00, 17 July 2018

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

Badness: 0.000728

Julius[22] 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

Badness: 0.00178

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

Badness: 0.00158

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

Badness: 0.00171

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

Badness: 0.00197

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