Rank-3 temperament
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author xenwolf and made on 2010-05-19 02:57:21 UTC.
- The original revision id was 143079895.
- The revision comment was: minor change: some garbage removed
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Original Wikitext content:
//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.
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 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 />
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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 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 <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 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 1, we take their reciprocal. <br />
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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.</body></html>