Rank-3 temperament
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 06:53:13 UTC.
- The original revision id was 141052599.
- 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:
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><br />
A rank three temperament is a <A HREF="regular.html"><TT>regular temperament</TT></A><FONT <br />
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><br />
<FONT COLOR="#C00000"><TT>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 </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 <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 </TT></FONT><A HREF="<!-- ws:start:WikiTextUrlRule:20: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:20 -->"><TT>bilinear <br />
form</TT></A><FONT COLOR="#C00000"><TT> giving the metric structure. One good, and canonical, choice for generators <br />
are the generators found by using </TT></FONT><A HREF="<!-- ws:start:WikiTextUrlRule:21: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:21 -->"><TT>Hermite <br />
reduction</TT></A><FONT COLOR="#C00000"><TT> with the proviso that if the generators so obtained are less than <br />
1, we take their reciprocal.</TT></FONT><br />
<FONT COLOR="#C00000"><TT>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 "a" is the exponent <br />
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.</TT></FONT><P ALIGN="CENTER"><A HREF="index.html"><FONT FACE="Courier New"><TT>home</TT></FONT></A><br />
<FONT COLOR="#C00000"><TT> </TT></FONT></BODY></HTML></body></html>