Rank-3 temperament: Difference between revisions

Revision as of 10:52, 11 May 2010

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This revision was by author xenwolf and made on 2010-05-11 10:52:33 UTC.

The original revision id was 141117203.

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//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 
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>
<FONT COLOR="#C00000"><TT>The most elegant way to put a Euclidean metric, and hence a lattice structure, on 
the pitch classes of a planar temperament is to start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. 

For instance, 7-limit just intonation has a </TT></FONT><A HREF="sevlat.htm"><TT>symmetrical lattice structure</TT></A><FONT COLOR="#C00000"><TT>and a 7-limit planar temperament is defined by a single comma. If u = |* a b c> is 
the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two 
generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 
+ 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. 
Here the dot product is defined by the </TT></FONT><A HREF="http://mathworld.wolfram.com/SymmetricBilinearForm.html"><TT>bilinear 
form</TT></A><FONT COLOR="#C00000"><TT> giving the metric structure. One good, and canonical, choice for generators 
are the generators found by using </TT></FONT><A HREF="http://mathworld.wolfram.com/HermiteNormalForm.html"><TT>Hermite 
reduction</TT></A><FONT COLOR="#C00000"><TT> with the proviso that if the generators so obtained are less than 
1, we take their reciprocal.</TT></FONT>
<FONT COLOR="#C00000"><TT>The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the 
projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where "a" is the exponent 
of 3 and "b" of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, 
and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given 
by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.</TT></FONT><P ALIGN="CENTER"><A HREF="index.html"><FONT FACE="Courier New"><TT>home</TT></FONT></A>
<FONT COLOR="#C00000"><TT> </TT></FONT></BODY></HTML>

Original HTML content:

<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 />
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 />
&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;The most elegant way to put a Euclidean metric, and hence a lattice structure, on <br />
the pitch classes of a planar temperament is to start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. <br />
<br />
For instance, 7-limit just intonation has a &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;sevlat.htm&quot;&gt;&lt;TT&gt;symmetrical lattice structure&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;and a 7-limit planar temperament is defined by a single comma. If u = |* a b c&gt; is <br />
the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two <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 />
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 />
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 />
1, we take their reciprocal.&lt;/TT&gt;&lt;/FONT&gt;<br />
&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the <br />
projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where &quot;a&quot; is the exponent <br />
of 3 and &quot;b&quot; of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, <br />
and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given <br />
by sqrt(11a^2+8b^2), where now &quot;a&quot; is the exponent of 49/40, and &quot;b&quot; the exponent of 10/7.&lt;/TT&gt;&lt;/FONT&gt;&lt;P ALIGN=&quot;CENTER&quot;&gt;&lt;A HREF=&quot;index.html&quot;&gt;&lt;FONT FACE=&quot;Courier New&quot;&gt;&lt;TT&gt;home&lt;/TT&gt;&lt;/FONT&gt;&lt;/A&gt;<br />
&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; &lt;/TT&gt;&lt;/FONT&gt;&lt;/BODY&gt;&lt;/HTML&gt;</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2010-05-11 06:53:13 UTC</tt>.<br>
+: 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>141052599</tt>.<br>
+: The original revision id was <tt>141117203</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">
+<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 &lt;A HREF="regular.html"&gt;&lt;TT&gt;regular temperament&lt;/TT&gt;&lt;/A&gt;&lt;FONT
 COLOR="#C00000"&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;
@@ Line 28: / Line 29: @@
 &lt;FONT COLOR="#C00000"&gt;&lt;TT&gt; &lt;/TT&gt;&lt;/FONT&gt;&lt;/BODY&gt;&lt;/HTML&gt;</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Planar Temperament&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;br /&gt;
+<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;Planar Temperament&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;the following is extracted from &lt;!-- ws:start:WikiTextUrlRule:21:http://lumma.org/tuning/gws/planar.htm --&gt;&lt;a class="wiki_link_ext" href="http://lumma.org/tuning/gws/planar.htm" rel="nofollow"&gt;http://lumma.org/tuning/gws/planar.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:21 --&gt;&lt;/em&gt;&lt;br /&gt;
+&lt;br /&gt;
 A rank three temperament is a &amp;lt;A HREF=&amp;quot;regular.html&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;regular temperament&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT &lt;br /&gt;
 COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;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.&amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&lt;br /&gt;
@@ Line 38: / Line 40: @@
 generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 &lt;br /&gt;
 + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. &lt;br /&gt;
-Here the dot product is defined by the &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:20:http://mathworld.wolfram.com/SymmetricBilinearForm.html --&gt;&lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow"&gt;http://mathworld.wolfram.com/SymmetricBilinearForm.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:20 --&gt;&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;bilinear &lt;br /&gt;
+Here the dot product is defined by the &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:22:http://mathworld.wolfram.com/SymmetricBilinearForm.html --&gt;&lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow"&gt;http://mathworld.wolfram.com/SymmetricBilinearForm.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:22 --&gt;&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;bilinear &lt;br /&gt;
 form&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt; giving the metric structure. One good, and canonical, choice for generators &lt;br /&gt;
-are the generators found by using &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:21:http://mathworld.wolfram.com/HermiteNormalForm.html --&gt;&lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow"&gt;http://mathworld.wolfram.com/HermiteNormalForm.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:21 --&gt;&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;Hermite &lt;br /&gt;
+are the generators found by using &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:23:http://mathworld.wolfram.com/HermiteNormalForm.html --&gt;&lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow"&gt;http://mathworld.wolfram.com/HermiteNormalForm.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:23 --&gt;&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;Hermite &lt;br /&gt;
 reduction&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt; with the proviso that if the generators so obtained are less than &lt;br /&gt;
 , we take their reciprocal.&amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&lt;br /&gt;

Rank-3 temperament: Difference between revisions

Revision as of 10:52, 11 May 2010

IMPORTED REVISION FROM WIKISPACES

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