Rank-3 temperament: Difference between revisions
Wikispaces>xenwolf **Imported revision 143079783 - Original comment: I think some hint will be useful (means: "Here you see the living version.")** |
Wikispaces>xenwolf **Imported revision 143079895 - Original comment: minor change: some garbage removed** |
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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: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-05-19 02:57:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>143079895</tt>.<br> | ||
: The revision comment was: <tt> | : The revision comment was: <tt>minor change: some garbage removed</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">//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">//the following is derived from the version on 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. | 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. | ||
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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> | 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> | ||
<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 derived from the version on <!-- ws:start:WikiTextUrlRule:12: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:12 --> | <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 derived from the version on <!-- ws:start:WikiTextUrlRule:12: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:12 --></em><br /> | ||
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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 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 /> | 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 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 /> | ||