Rank-3 temperament: Difference between revisions
Wikispaces>genewardsmith **Imported revision 141052599 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 141117203 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-05-11 10:52:33 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>141117203</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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"> | <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">//the following is extracted from http://lumma.org/tuning/gws/planar.htm// | ||
A rank three temperament is a <A HREF="regular.html"><TT>regular temperament</TT></A><FONT | A rank three temperament is a <A HREF="regular.html"><TT>regular temperament</TT></A><FONT | ||
COLOR="#C00000"><TT> with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.</TT></FONT> | COLOR="#C00000"><TT> with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.</TT></FONT> | ||
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<FONT COLOR="#C00000"><TT> </TT></FONT></BODY></HTML></pre></div> | <FONT COLOR="#C00000"><TT> </TT></FONT></BODY></HTML></pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Planar Temperament</title></head><body><br /> | <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"><html><head><title>Planar Temperament</title></head><body><em>the following is extracted from <!-- ws:start:WikiTextUrlRule:21:http://lumma.org/tuning/gws/planar.htm --><a class="wiki_link_ext" href="http://lumma.org/tuning/gws/planar.htm" rel="nofollow">http://lumma.org/tuning/gws/planar.htm</a><!-- ws:end:WikiTextUrlRule:21 --></em><br /> | ||
<br /> | |||
A rank three temperament is a &lt;A HREF=&quot;regular.html&quot;&gt;&lt;TT&gt;regular temperament&lt;/TT&gt;&lt;/A&gt;&lt;FONT <br /> | A rank three temperament is a &lt;A HREF=&quot;regular.html&quot;&gt;&lt;TT&gt;regular temperament&lt;/TT&gt;&lt;/A&gt;&lt;FONT <br /> | ||
COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.&lt;/TT&gt;&lt;/FONT&gt;<br /> | COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.&lt;/TT&gt;&lt;/FONT&gt;<br /> | ||
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generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 <br /> | generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 <br /> | ||
+ 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. <br /> | + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. <br /> | ||
Here the dot product is defined by the &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | Here the dot product is defined by the &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:22:http://mathworld.wolfram.com/SymmetricBilinearForm.html --><a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow">http://mathworld.wolfram.com/SymmetricBilinearForm.html</a><!-- ws:end:WikiTextUrlRule:22 -->&quot;&gt;&lt;TT&gt;bilinear <br /> | ||
form&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; giving the metric structure. One good, and canonical, choice for generators <br /> | form&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; giving the metric structure. One good, and canonical, choice for generators <br /> | ||
are the generators found by using &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule: | are the generators found by using &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- ws:start:WikiTextUrlRule:23:http://mathworld.wolfram.com/HermiteNormalForm.html --><a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow">http://mathworld.wolfram.com/HermiteNormalForm.html</a><!-- ws:end:WikiTextUrlRule:23 -->&quot;&gt;&lt;TT&gt;Hermite <br /> | ||
reduction&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with the proviso that if the generators so obtained are less than <br /> | reduction&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with the proviso that if the generators so obtained are less than <br /> | ||
1, we take their reciprocal.&lt;/TT&gt;&lt;/FONT&gt;<br /> | 1, we take their reciprocal.&lt;/TT&gt;&lt;/FONT&gt;<br /> | ||