7-limit symmetrical lattices: Difference between revisions

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<h4>Original Wikitext content:</h4>

then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0),  

+-(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  

A similar lattice may be defined in any p-limit, by using a norm which is the square root of the quadratic form  

of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer  

to a given lattice point than any other, are hexagons.) The higher dimensional versions of this are called An,  

in n dimensions, so the 7-limit lattice is the A3 lattice. However, the 7-limit is unique in that there is another  

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  

(0 0 1) to (1 1 0), we have the lattice in terms of perpendicular coordinates, in which we may use ordinary Euclidean  

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)  

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  

relations because of this.

In any limit, we may consider the dual lattice of mappings to primes, or octave-equivalent vals. Dual to the  

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")  

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  

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  

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 theme through all 48 resulting variations.

Transforming maps to maps when they are generator maps for two temperaments with the same period is sometimes  

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}  

<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;The Seven Limit Symmetrical Lattices&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;

then the twelve consonant intervals of 7-limit harmony are represented by the twelve lattice points +-(1 0 0), &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;

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;

of equilateral triangles, called A2 or the hexagonal lattice (since the Voroni cells, regions of points closer &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;

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;

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;

(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;

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;

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;

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;

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;

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;

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;

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 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;

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;