Rank-3 temperament: Difference between revisions
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Wikispaces>xenwolf **Imported revision 141117203 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 141240351 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-05-11 17:07:12 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>141240351</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> | ||
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<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// | <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 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. | 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 | 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> is | ||
the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two | 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 | ||
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. | ||
+ 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric | 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 | ||
Here the dot product is defined by the | 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 | ||
form | 1, we take their reciprocal. | ||
are the generators found by using | |||
reduction | 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}, | ||
1, we take their reciprocal. | |||
projected lattice | |||
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 | 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. | by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.</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><em>the following is extracted from <!-- ws:start:WikiTextUrlRule: | <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:20: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:20 --></em><br /> | ||
<br /> | <br /> | ||
A rank three temperament is a & | A rank three temperament is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a> , hence the name. 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 /> | 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 /> | <br /> | ||
For instance, 7-limit just intonation has a & | For instance, 7-limit just intonation has a <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">symmetrical lattice structure</a> on pitch classes and a 7-limit planar temperament is defined by a single <a class="wiki_link" href="/comma">comma</a> . 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 | 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 <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 structure of the planar temperament lattice. <br /> | ||
+ 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric | Here the dot product is defined by the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow">bilinear form</a> giving the metric structure. One good, and canonical, choice for generators <br /> | ||
Here the dot product is defined by the | are the generators found by using <a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow">Hermite reduction</a> with the proviso that if the generators so obtained are less than <br /> | ||
form& | 1, we take their reciprocal. <br /> | ||
are the generators found by using | <br /> | ||
reduction& | 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 &quot;a&quot; is the exponent of 3 and &quot;b&quot; of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, <br /> | ||
1, we take their reciprocal.& | |||
projected lattice | |||
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 /> | 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. | 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.</body></html></pre></div> | ||
Revision as of 17:07, 11 May 2010
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2010-05-11 17:07:12 UTC.
- The original revision id was 141240351.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
//the following is extracted from http://lumma.org/tuning/gws/planar.htm//
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.
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> 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.
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.Original HTML content:
<html><head><title>Planar Temperament</title></head><body><em>the following is extracted from <!-- ws:start:WikiTextUrlRule:20: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:20 --></em><br />
<br />
A rank three temperament is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a> , hence the name. 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 <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">symmetrical lattice structure</a> on pitch classes and a 7-limit planar temperament is defined by a single <a class="wiki_link" href="/comma">comma</a> . If u = |* a b c> 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 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 structure of the planar temperament lattice. <br />
Here the dot product is defined by the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow">bilinear form</a> giving the metric structure. One good, and canonical, choice for generators <br />
are the generators found by using <a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow">Hermite reduction</a> with the proviso that if the generators so obtained are less than <br />
1, we take their reciprocal. <br />
<br />
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}, <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 "a" is the exponent of 49/40, and "b" the exponent of 10/7.</body></html>