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Rank-3 temperament: Difference between revisions

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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:xenwolf|xenwolf]] and made on <tt>2010-05-19 02:57:21 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-02 17:33:01 UTC</tt>.<br>
: The original revision id was <tt>143079895</tt>.<br>
: The original revision id was <tt>160092625</tt>.<br>
: The revision comment was: <tt>minor change: some garbage removed</tt><br>
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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">//the following is derived from the version on http://lumma.org/tuning/gws/planar.htm//
<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">
A rank three temperament is a [[regular temperament]] 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 [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], hence the name. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Monzos and Interval Space|Euclidean interval space]].


A rank three temperament is a [[regular temperament]]  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 [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]] , hence the name. 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.  
===Example===
7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted monzo m = |-5 3.17 4.64 -2.81&gt; and call the weighted monzo |1 0 0 0&gt; for 2 t, then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2x4 matrix  U = [t, m]. If U` denotes the [[RMS tuning|Moore-Pensrose pseudoinverse]] of U, then letting Q = U`U take P = I - Q, where I is the identity matrix. P is the projection map from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector |0 0 0 0&gt; representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96 to the fifth, and the major third angled 129.84 to the fifth.


For instance, 7-limit just intonation has a [[The Seven Limit Symmetrical Lattices|symmetrical lattice structure]] on pitch classes and a 7-limit planar temperament is defined by a single [[comma]] . If u = |* a b c&gt; 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 structure of the planar temperament lattice. Here the dot product is defined by the [[http://mathworld.wolfram.com/SymmetricBilinearForm.html|bilinear form]]  giving the metric structure. One good, and canonical, choice for generators are the generators found by using [[http://mathworld.wolfram.com/HermiteNormalForm.html|Hermite reduction]]  with the proviso that if the generators so obtained are less than 1, we take their reciprocal.  
If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log2(q) for each of primes up to p, which allows us to directly project unweighted monzos.


The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the projected lattice structure 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.</pre></div>
===Planar temperaments===
 
 
//an earlier version of this article may be found on http://lumma.org/tuning/gws/planar.htm//</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">&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 derived from the version on &lt;!-- ws:start:WikiTextUrlRule:12: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:12 --&gt;&lt;/em&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;br /&gt;
A rank three temperament is a &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&gt;, hence the name. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;Euclidean interval space&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Example&lt;/h3&gt;
7-limit marvel temperament is defined by tempering out a single comma, 225/224. If we convert that to a weighted monzo m = |-5 3.17 4.64 -2.81&amp;gt; and call the weighted monzo |1 0 0 0&amp;gt; for 2 t, then the two-dimensional subspace perpendicular in the four-dimensional 7-limit Euclidean interval space is the space onto which we propose to orthogonally project all 7-limit intervals. One way to do this is by forming a 2x4 matrix  U = [t, m]. If U` denotes the &lt;a class="wiki_link" href="/RMS%20tuning"&gt;Moore-Pensrose pseudoinverse&lt;/a&gt; of U, then letting Q = U`U take P = I - Q, where I is the identity matrix. P is the projection map from weighted monzos onto the two-dimensional lattice of tempered pitch classes. We have that mP and tP are the zero vector |0 0 0 0&amp;gt; representing the unison pitch class, which is to say octaves, and other intervals are mapped elsewhere. We find in this way that the lattice point closest to the origin is the secor, 16/15 and 15/14, and the second closest independent point the fifth (or alternatively, fourth). The secor and the fifth give a Minkowski basis for the lattice, but we could also use the major third and fifth as a basis. The secor and fifth are at an angle of 106.96 to the fifth, and the major third angled 129.84 to the fifth.&lt;br /&gt;
&lt;br /&gt;
If we list 2 first in the list of commas, the matrix P for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for P, by monzo-weighting the columns of P, which is to say, scalar multiplying the successive rows by log2(q) for each of primes up to p, which allows us to directly project unweighted monzos.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A rank three temperament is a &lt;a class="wiki_link" href="/regular%20temperament"&gt;regular temperament&lt;/a&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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&gt; , hence the name. 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. &lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Planar temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Planar temperaments&lt;/h3&gt;
&lt;br /&gt;
&lt;br /&gt;
For instance, 7-limit just intonation has a &lt;a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices"&gt;symmetrical lattice structure&lt;/a&gt; on pitch classes and a 7-limit planar temperament is defined by a single &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt; . If u = |* a b c&amp;gt; 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 structure of the planar temperament lattice. Here the dot product is defined by the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow"&gt;bilinear form&lt;/a&gt;  giving the metric structure. One good, and canonical, choice for generators are the generators found by using &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow"&gt;Hermite reduction&lt;/a&gt;  with the proviso that if the generators so obtained are less than 1, we take their reciprocal. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the projected lattice structure is defined by the norm sqrt(11a^2-14ab+11b^2), where &amp;quot;a&amp;quot; is the exponent of 3 and &amp;quot;b&amp;quot; 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 &amp;quot;a&amp;quot; is the exponent of 49/40, and &amp;quot;b&amp;quot; the exponent of 10/7.&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;em&gt;an earlier version of this article may be found on &lt;!-- ws:start:WikiTextUrlRule:17: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:17 --&gt;&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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