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:genewardsmith|genewardsmith]] and made on <tt>2010-05-11 06:44:07 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-11 06:53:13 UTC</tt>.<br>
: The original revision id was <tt>141051507</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 octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]] 
<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">
by putting a [[http://en.wikipedia.org/wiki/Normed_vector_space|norm]]  on the three dimensional real we can regard them as living in. If we define
A rank three temperament is a &lt;A HREF="regular.html"&gt;&lt;TT&gt;regular temperament&lt;/TT&gt;&lt;/A&gt;&lt;FONT
the norm by
COLOR="#C00000"&gt;&lt;TT&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 lattice, hence the name.&lt;/TT&gt;&lt;/FONT&gt;
|| (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc)
&lt;FONT COLOR="#C00000"&gt;&lt;TT&gt;The most elegant way to put a Euclidean metric, and hence a lattice structure, on
then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0),
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.
+-(0 1 0), +-(0 0 1), +-(1 -1 0), +-(1 0 -1) and +-(0 1 -1) at a distance of one from the unison, (0 0 0). These
 
lie on the verticies of a &lt;A HREF="http://en.wikipedia.org/wiki/Cuboctahedron"&gt;cubeoctahedron&lt;/A&gt;, a semiregular
For instance, 7-limit just intonation has a &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF="sevlat.htm"&gt;&lt;TT&gt;symmetrical lattice structure&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR="#C00000"&gt;&lt;TT&gt;and a 7-limit planar temperament is defined by a single comma. If u = |* a b c&gt; is
solid. The lattice has two types of holes--the shallow holes, which are &lt;A HREF="http://en.wikipedia.org/wiki/Tetrahedron"&gt;tetrahera&lt;/A&gt;  
the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two
and which correspond to the major and minor &lt;A HREF="http://tonalsoft.com/enc/tetrad.htm"&gt;tetrads&lt;/A&gt; 4:5:6:7 and
generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2  
1/4:1/5:1/6:1/7, and the deep holes which are &lt;A HREF="http://en.wikipedia.org/wiki/Octahedron"&gt;octaheda&lt;/A&gt; and
+ 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice.  
correspond to &lt;A HREF="http://tonalsoft.com/enc/hexany.htm"&gt;hexanies&lt;/A&gt;.
Here the dot product is defined by the &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF="http://mathworld.wolfram.com/SymmetricBilinearForm.html"&gt;&lt;TT&gt;bilinear
A similar lattice may be defined in any p-limit, by using a norm which is the square root of the quadratic form
form&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR="#C00000"&gt;&lt;TT&gt; giving the metric structure. One good, and canonical, choice for generators
x_i x_j, summed over all i &lt;= j; moreover as an alternative approach we can use the &lt;A HREF="hahn.htm"&gt;Hahn
are the generators found by using &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF="http://mathworld.wolfram.com/HermiteNormalForm.html"&gt;&lt;TT&gt;Hermite
norm&lt;/A&gt; in place of the Euclidean norm. In the two dimensional case of the 5-limit, this gives the plane lattice
reduction&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR="#C00000"&gt;&lt;TT&gt; with the proviso that if the generators so obtained are less than
of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer
1, we take their reciprocal.&lt;/TT&gt;&lt;/FONT&gt;
to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An,
&lt;FONT COLOR="#C00000"&gt;&lt;TT&gt;The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the  
in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another
projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where "a" is the exponent
family of lattices, called Dn, to which it also belongs as D3, the&lt;A HREF="http://en.wikipedia.org/wiki/Crystal_structure"&gt;face-centered
of 3 and "b" of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7},  
cubic lattice&lt;/A&gt;. If we take (b+c)^2+(a+c)^2+(a+b)^2 and expand it, we get 2 (a^2 + b^2 + c^2 + ab + ac + bc).  
and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given
If we therefore take our triples (a b c) and change basis by sending (1 0 0) to (0 1 1), (0 1 0) to (1 0 1), and
by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.&lt;/TT&gt;&lt;/FONT&gt;&lt;P ALIGN="CENTER"&gt;&lt;A HREF="index.html"&gt;&lt;FONT FACE="Courier New"&gt;&lt;TT&gt;home&lt;/TT&gt;&lt;/FONT&gt;&lt;/A&gt;
(0 0 1) to (1 1 0), we have the lattice in terms of perpendicular coordinates, in which we may use ordinary Euclidean
&lt;FONT COLOR="#C00000"&gt;&lt;TT&gt; &lt;/TT&gt;&lt;/FONT&gt;&lt;/BODY&gt;&lt;/HTML&gt;</pre></div>
length. In this form, all distances are scaled up by a factor of sqrt(2), so that the 7-limit consonances become
(+-1 +-1 0), (+-1 0 +-1), and (0 +-1 +-1), the verticies of a cuboctahedron in a more standard form. The lattice
now may be described as triples of integers (a b c), such that a+b+c is an even number, and using the ordinary
Euclidean norm of sqrt(a^2 + b^2 + c^2).
In this new coordinate system, the 4:5:6:7 tetrad consists of the notes (0 0 0), (1 0 0), (0 1 0), and (0 0
1); the centroid of this is (1/2 1/2 1/2); similarly the centroid of 1/4:1/5:1/6:1/7 is (-1/2 -1/2 -1/2). If we
shift the origin to (1/2 1/2 1/2), major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1
c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind
of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This,
once again, is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form
a lattice.
If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2)
7^((a+c-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2)
if a+b+c is odd. Each unit cube corresponds to a &lt;A HREF="http://tonalsoft.com/enc/stellat.htm"&gt;stellated hexany&lt;/A&gt;,
or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.
If we look at twice the generators, namely [2 0 0], [0 2 0] and [0 0 2] we find they correspond to transposition
up by 35/24 for [2 0 0], up 21/20 for [0 2 0], and up 15/14 for [0 0 2]. Temperaments where the generator can be
taken as one of these three, such as miracle, are particularly easy to work with in terms of the lattice of chord
relations because of this.
In any limit, we may consider the dual lattice of mappings to primes, or octave-equivalent vals. Dual to the
An norm defined from x_j x_j is a norm defined by the inverse to the symmetric matrix of the &lt;A HREF="http://en.wikipedia.org/wiki/Quadratic_form"&gt;quadratic
form&lt;/A&gt; for the An norm, which normalizes to the square root of the quantity n times the sum of squares of x_i
minus twice the product x_i x_j, for j &gt; i. This defines the dual lattice An* to An. In the two dimensions of
the 5-limit, A2 is isomorphic to A2* and the lattice of maps is a equilateral triangular ("hexagonal")
lattice also. In the three dimensions of the 7-limit, we again have an exceptional situation, where A3* is isomorphic
to the dual of D3, D3*. We have that the norm for A3* can be defined as the square root of (-x_1+x_2+x_3)^2 + (x_1-x_2+x_3)^2
+(x_1+x_2-x_3)^2, so if we change basis so that our basis maps are (-1 1 1), (1 -1 1) and (1 1 -1), then the norm
becomes the usual Euclidean norm. If we take linear combinations with integer coefficents of these, we obtain all
triples of integers which are either all even or all odd. The lattice with these points and the usual Euclidean
norm is the &lt;A HREF="http://en.wikipedia.org/wiki/Crystal_structure"&gt;body-centered cubic lattice&lt;/A&gt;.
It is easy to verify that the dot product of a triple of integers, either all even or all odd, times a triple
of integers whose sum is even, is always even; and we get the precise relationship between mappings and note-classes
by dividing by two, and taking the lattice of mappings to be triples of integers, plus triples of halves of odd
integers. So for example the meantone mapping, (1 4 10), transforms to 1*(-1/2 1/2 1/2) + 4*(1/2 -1/2 1/2) + 10*(1/2
1/2 -1/2) = (13/2 7/2 -5/2), and the fifth class (1 0 0) to (0 1 1); taking the dot product of (13/2 7/2 -5/2)
with (0 1 1) gives 1, as expected. However I think it is better to keep the coordinates as integers, and simply
keep in mind that to get the mapping we now need to divide the dot product by two.
For any lattice, the isometries, or distance-preserving maps, which take the lattice to itself form a group,
the group of affine automorphisms. It has a subgroup, called the automorphism group of the lattice, which consists
of those affine automorphisms which fix the origin. In the case of D3, D3* and the cubic grid of tetrads, the automorphism
group is the group of order 48 which consists of all permutations of the three coordinates and all changes of sign,  
and is called both the group of the cube and the group of the octahedron. It is easy to see that such a transformation
takes triples with an even sum to triples with an even sum, and triples either even or odd to triples either even
or odd. Hence it takes the cubic lattice of tetrads to itself, the face-centered cubic lattice of note-classes
to itself, and the body-centered cubic lattice of mappings of note-classes to itself. The first two types of transformation
includes the major/minor transformation, and can be regarded as a vast generalization of that. Robert Walker has
a piece, &lt;A HREF="http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase"&gt;Hexany Phrase&lt;/A&gt;, which takes
a theme through all 48 resulting variations.
Transforming maps to maps when they are generator maps for two temperaments with the same period is sometimes
interesting, since it sends one temperament to another while preserving 7-odd-limit (meaning, not including 9-odd-limit)
harmony to itself. For example, the dominant seventh temperament, the {27/25, 28/25} temperament, and the {28/27,
35/32} temperaments can each be transformed to the others, as can septimal kleismic (the {49/48, 126/125} temperament)
and the {225/224, 250/243} temperament, and hemifourths and the {49/48, 135/128} temperament. Temperaments with
a period a fraction of an octave can also sometimes be transformed; for instance injera and the {50/49, 135/128}
temperament.&lt;P ALIGN="CENTER"&gt;&lt;P ALIGN="CENTER"&gt;&lt;A HREF="home.htm"&gt;home&lt;/A&gt;&lt;/BODY&gt;&lt;/HTML&gt;</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;The octave-equivalent note classes of 7-limit harmony can be represented in vector (odd-only monzo) form as triples of integers (a b c). We can make this into a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&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;
by putting a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;norm&lt;/a&gt; on the three dimensional real we can regard them as living in. If we define &lt;br /&gt;
A rank three temperament is a &amp;lt;A HREF=&amp;quot;regular.html&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;regular temperament&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT &lt;br /&gt;
the norm by&lt;br /&gt;
COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;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 lattice, hence the name.&amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&lt;br /&gt;
|| (a b c) || = sqrt(a^2 + b^2 + c^2 + ab + ac + bc)&lt;br /&gt;
&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;The most elegant way to put a Euclidean metric, and hence a lattice structure, on &lt;br /&gt;
then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0), &lt;br /&gt;
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;
+-(0 1 0), +-(0 0 1), +-(1 -1 0), +-(1 0 -1) and +-(0 1 -1) at a distance of one from the unison, (0 0 0). These &lt;br /&gt;
&lt;br /&gt;
lie on the verticies of a &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:77:http://en.wikipedia.org/wiki/Cuboctahedron --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Cuboctahedron&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:77 --&gt;&amp;quot;&amp;gt;cubeoctahedron&amp;lt;/A&amp;gt;, a semiregular &lt;br /&gt;
For instance, 7-limit just intonation has a &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;sevlat.htm&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;symmetrical lattice structure&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;and a 7-limit planar temperament is defined by a single comma. If u = |* a b c&amp;gt; is &lt;br /&gt;
solid. The lattice has two types of holes--the shallow holes, which are &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:78:http://en.wikipedia.org/wiki/Tetrahedron --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tetrahedron" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Tetrahedron&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:78 --&gt;&amp;quot;&amp;gt;tetrahera&amp;lt;/A&amp;gt; &lt;br /&gt;
the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two &lt;br /&gt;
and which correspond to the major and minor &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:79:http://tonalsoft.com/enc/tetrad.htm --&gt;&lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/tetrad.htm" rel="nofollow"&gt;http://tonalsoft.com/enc/tetrad.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:79 --&gt;&amp;quot;&amp;gt;tetrads&amp;lt;/A&amp;gt; 4:5:6:7 and &lt;br /&gt;
generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 &lt;br /&gt;
1/4:1/5:1/6:1/7, and the deep holes which are &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:80:http://en.wikipedia.org/wiki/Octahedron --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Octahedron" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Octahedron&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:80 --&gt;&amp;quot;&amp;gt;octaheda&amp;lt;/A&amp;gt; and &lt;br /&gt;
+ 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. &lt;br /&gt;
correspond to &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:81:http://tonalsoft.com/enc/hexany.htm --&gt;&lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow"&gt;http://tonalsoft.com/enc/hexany.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:81 --&gt;&amp;quot;&amp;gt;hexanies&amp;lt;/A&amp;gt;.&lt;br /&gt;
Here the dot product is defined by the &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:20:http://mathworld.wolfram.com/SymmetricBilinearForm.html --&gt;&lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/SymmetricBilinearForm.html" rel="nofollow"&gt;http://mathworld.wolfram.com/SymmetricBilinearForm.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:20 --&gt;&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;bilinear &lt;br /&gt;
A similar lattice may be defined in any p-limit, by using a norm which is the square root of the quadratic form &lt;br /&gt;
form&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt; giving the metric structure. One good, and canonical, choice for generators &lt;br /&gt;
x_i x_j, summed over all i &amp;lt;= j; moreover as an alternative approach we can use the &amp;lt;A HREF=&amp;quot;hahn.htm&amp;quot;&amp;gt;Hahn &lt;br /&gt;
are the generators found by using &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:21:http://mathworld.wolfram.com/HermiteNormalForm.html --&gt;&lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/HermiteNormalForm.html" rel="nofollow"&gt;http://mathworld.wolfram.com/HermiteNormalForm.html&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:21 --&gt;&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;Hermite &lt;br /&gt;
norm&amp;lt;/A&amp;gt; in place of the Euclidean norm. In the two dimensional case of the 5-limit, this gives the plane lattice &lt;br /&gt;
reduction&amp;lt;/TT&amp;gt;&amp;lt;/A&amp;gt;&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt; with the proviso that if the generators so obtained are less than &lt;br /&gt;
of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer &lt;br /&gt;
1, we take their reciprocal.&amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&lt;br /&gt;
to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An, &lt;br /&gt;
&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the &lt;br /&gt;
in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another &lt;br /&gt;
projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where &amp;quot;a&amp;quot; is the exponent &lt;br /&gt;
family of lattices, called Dn, to which it also belongs as D3, the&amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:82:http://en.wikipedia.org/wiki/Crystal_structure --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Crystal_structure&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:82 --&gt;&amp;quot;&amp;gt;face-centered &lt;br /&gt;
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}, &lt;br /&gt;
cubic lattice&amp;lt;/A&amp;gt;. If we take (b+c)^2+(a+c)^2+(a+b)^2 and expand it, we get 2 (a^2 + b^2 + c^2 + ab + ac + bc). &lt;br /&gt;
and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given &lt;br /&gt;
If we therefore take our triples (a b c) and change basis by sending (1 0 0) to (0 1 1), (0 1 0) to (1 0 1), and &lt;br /&gt;
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.&amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;P ALIGN=&amp;quot;CENTER&amp;quot;&amp;gt;&amp;lt;A HREF=&amp;quot;index.html&amp;quot;&amp;gt;&amp;lt;FONT FACE=&amp;quot;Courier New&amp;quot;&amp;gt;&amp;lt;TT&amp;gt;home&amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;/A&amp;gt;&lt;br /&gt;
(0 0 1) to (1 1 0), we have the lattice in terms of perpendicular coordinates, in which we may use ordinary Euclidean &lt;br /&gt;
&amp;lt;FONT COLOR=&amp;quot;#C00000&amp;quot;&amp;gt;&amp;lt;TT&amp;gt; &amp;lt;/TT&amp;gt;&amp;lt;/FONT&amp;gt;&amp;lt;/BODY&amp;gt;&amp;lt;/HTML&amp;gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
length. In this form, all distances are scaled up by a factor of sqrt(2), so that the 7-limit consonances become &lt;br /&gt;
(+-1 +-1 0), (+-1 0 +-1), and (0 +-1 +-1), the verticies of a cuboctahedron in a more standard form. The lattice &lt;br /&gt;
now may be described as triples of integers (a b c), such that a+b+c is an even number, and using the ordinary &lt;br /&gt;
Euclidean norm of sqrt(a^2 + b^2 + c^2).&lt;br /&gt;
In this new coordinate system, the 4:5:6:7 tetrad consists of the notes (0 0 0), (1 0 0), (0 1 0), and (0 0 &lt;br /&gt;
1); the centroid of this is (1/2 1/2 1/2); similarly the centroid of 1/4:1/5:1/6:1/7 is (-1/2 -1/2 -1/2). If we &lt;br /&gt;
shift the origin to (1/2 1/2 1/2), major tetrads correspond to [a b c], a+b+c even, and minor tetrads to [a-1 b-1 &lt;br /&gt;
c-1], a+b+c even, which is the same as saying [a b c], a+b+c odd. Hence the 7-limit tetrads form the simplest kind &lt;br /&gt;
of lattice, the cubic or grid lattice consisting of triples of integers with the ordinary Euclidean distance. This, &lt;br /&gt;
once again, is a unique feature of the 7-limit; in no other limit do the complete utonalities and otonalities form &lt;br /&gt;
a lattice.&lt;br /&gt;
If [a b c] is any triple of integers, then it represents the major tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) &lt;br /&gt;
7^((a+c-c)/2) if a+b+c is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-b+c)/2 7^((1+a+b-c)/2) &lt;br /&gt;
if a+b+c is odd. Each unit cube corresponds to a &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:83:http://tonalsoft.com/enc/stellat.htm --&gt;&lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/stellat.htm" rel="nofollow"&gt;http://tonalsoft.com/enc/stellat.htm&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:83 --&gt;&amp;quot;&amp;gt;stellated hexany&amp;lt;/A&amp;gt;, &lt;br /&gt;
or tetradekany, or dekatesserany, though chord cube would be less of a mouthful.&lt;br /&gt;
If we look at twice the generators, namely [2 0 0], [0 2 0] and [0 0 2] we find they correspond to transposition &lt;br /&gt;
up by 35/24 for [2 0 0], up 21/20 for [0 2 0], and up 15/14 for [0 0 2]. Temperaments where the generator can be &lt;br /&gt;
taken as one of these three, such as miracle, are particularly easy to work with in terms of the lattice of chord &lt;br /&gt;
relations because of this.&lt;br /&gt;
In any limit, we may consider the dual lattice of mappings to primes, or octave-equivalent vals. Dual to the &lt;br /&gt;
An norm defined from x_j x_j is a norm defined by the inverse to the symmetric matrix of the &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:84:http://en.wikipedia.org/wiki/Quadratic_form --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quadratic_form" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Quadratic_form&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:84 --&gt;&amp;quot;&amp;gt;quadratic &lt;br /&gt;
form&amp;lt;/A&amp;gt; for the An norm, which normalizes to the square root of the quantity n times the sum of squares of x_i &lt;br /&gt;
minus twice the product x_i x_j, for j &amp;gt; i. This defines the dual lattice An* to An. In the two dimensions of &lt;br /&gt;
the 5-limit, A2 is isomorphic to A2* and the lattice of maps is a equilateral triangular (&amp;quot;hexagonal&amp;quot;) &lt;br /&gt;
lattice also. In the three dimensions of the 7-limit, we again have an exceptional situation, where A3* is isomorphic &lt;br /&gt;
to the dual of D3, D3*. We have that the norm for A3* can be defined as the square root of (-x_1+x_2+x_3)^2 + (x_1-x_2+x_3)^2 &lt;br /&gt;
+(x_1+x_2-x_3)^2, so if we change basis so that our basis maps are (-1 1 1), (1 -1 1) and (1 1 -1), then the norm &lt;br /&gt;
becomes the usual Euclidean norm. If we take linear combinations with integer coefficents of these, we obtain all &lt;br /&gt;
triples of integers which are either all even or all odd. The lattice with these points and the usual Euclidean &lt;br /&gt;
norm is the &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:85:http://en.wikipedia.org/wiki/Crystal_structure --&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow"&gt;http://en.wikipedia.org/wiki/Crystal_structure&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:85 --&gt;&amp;quot;&amp;gt;body-centered cubic lattice&amp;lt;/A&amp;gt;.&lt;br /&gt;
It is easy to verify that the dot product of a triple of integers, either all even or all odd, times a triple &lt;br /&gt;
of integers whose sum is even, is always even; and we get the precise relationship between mappings and note-classes &lt;br /&gt;
by dividing by two, and taking the lattice of mappings to be triples of integers, plus triples of halves of odd &lt;br /&gt;
integers. So for example the meantone mapping, (1 4 10), transforms to 1*(-1/2 1/2 1/2) + 4*(1/2 -1/2 1/2) + 10*(1/2 &lt;br /&gt;
1/2 -1/2) = (13/2 7/2 -5/2), and the fifth class (1 0 0) to (0 1 1); taking the dot product of (13/2 7/2 -5/2) &lt;br /&gt;
with (0 1 1) gives 1, as expected. However I think it is better to keep the coordinates as integers, and simply &lt;br /&gt;
keep in mind that to get the mapping we now need to divide the dot product by two.&lt;br /&gt;
For any lattice, the isometries, or distance-preserving maps, which take the lattice to itself form a group, &lt;br /&gt;
the group of affine automorphisms. It has a subgroup, called the automorphism group of the lattice, which consists &lt;br /&gt;
of those affine automorphisms which fix the origin. In the case of D3, D3* and the cubic grid of tetrads, the automorphism &lt;br /&gt;
group is the group of order 48 which consists of all permutations of the three coordinates and all changes of sign, &lt;br /&gt;
and is called both the group of the cube and the group of the octahedron. It is easy to see that such a transformation &lt;br /&gt;
takes triples with an even sum to triples with an even sum, and triples either even or odd to triples either even &lt;br /&gt;
or odd. Hence it takes the cubic lattice of tetrads to itself, the face-centered cubic lattice of note-classes &lt;br /&gt;
to itself, and the body-centered cubic lattice of mappings of note-classes to itself. The first two types of transformation &lt;br /&gt;
includes the major/minor transformation, and can be regarded as a vast generalization of that. Robert Walker has &lt;br /&gt;
a piece, &amp;lt;A HREF=&amp;quot;&lt;!-- ws:start:WikiTextUrlRule:86:http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase --&gt;&lt;a class="wiki_link_ext" href="http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase" rel="nofollow"&gt;http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:86 --&gt;&amp;quot;&amp;gt;Hexany Phrase&amp;lt;/A&amp;gt;, which takes &lt;br /&gt;
a theme through all 48 resulting variations.&lt;br /&gt;
Transforming maps to maps when they are generator maps for two temperaments with the same period is sometimes &lt;br /&gt;
interesting, since it sends one temperament to another while preserving 7-odd-limit (meaning, not including 9-odd-limit) &lt;br /&gt;
harmony to itself. For example, the dominant seventh temperament, the {27/25, 28/25} temperament, and the {28/27, &lt;br /&gt;
35/32} temperaments can each be transformed to the others, as can septimal kleismic (the {49/48, 126/125} temperament) &lt;br /&gt;
and the {225/224, 250/243} temperament, and hemifourths and the {49/48, 135/128} temperament. Temperaments with &lt;br /&gt;
a period a fraction of an octave can also sometimes be transformed; for instance injera and the {50/49, 135/128} &lt;br /&gt;
temperament.&amp;lt;P ALIGN=&amp;quot;CENTER&amp;quot;&amp;gt;&amp;lt;P ALIGN=&amp;quot;CENTER&amp;quot;&amp;gt;&amp;lt;A HREF=&amp;quot;home.htm&amp;quot;&amp;gt;home&amp;lt;/A&amp;gt;&amp;lt;/BODY&amp;gt;&amp;lt;/HTML&amp;gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

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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 &lt;A HREF=&quot;regular.html&quot;&gt;&lt;TT&gt;regular temperament&lt;/TT&gt;&lt;/A&gt;&lt;FONT <br />
COLOR=&quot;#C00000&quot;&gt;&lt;TT&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 lattice, hence the name.&lt;/TT&gt;&lt;/FONT&gt;<br />
&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;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 &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;sevlat.htm&quot;&gt;&lt;TT&gt;symmetrical lattice structure&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;and a 7-limit planar temperament is defined by a single comma. 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 <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 &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- 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 -->&quot;&gt;&lt;TT&gt;bilinear <br />
form&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; giving the metric structure. One good, and canonical, choice for generators <br />
are the generators found by using &lt;/TT&gt;&lt;/FONT&gt;&lt;A HREF=&quot;<!-- 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 -->&quot;&gt;&lt;TT&gt;Hermite <br />
reduction&lt;/TT&gt;&lt;/A&gt;&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; with the proviso that if the generators so obtained are less than <br />
1, we take their reciprocal.&lt;/TT&gt;&lt;/FONT&gt;<br />
&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt;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 &quot;a&quot; is the exponent <br />
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 />
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.&lt;/TT&gt;&lt;/FONT&gt;&lt;P ALIGN=&quot;CENTER&quot;&gt;&lt;A HREF=&quot;index.html&quot;&gt;&lt;FONT FACE=&quot;Courier New&quot;&gt;&lt;TT&gt;home&lt;/TT&gt;&lt;/FONT&gt;&lt;/A&gt;<br />
&lt;FONT COLOR=&quot;#C00000&quot;&gt;&lt;TT&gt; &lt;/TT&gt;&lt;/FONT&gt;&lt;/BODY&gt;&lt;/HTML&gt;</body></html>
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