7-limit symmetrical lattices: Difference between revisions

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If |-x-y-z x y z> is any element of symmetric interval class space, then by definition || |-x-y-z x y z> || = sqrt(2) sqrt(x^2+y^2+z^2+xy+yz+zx) where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by |* x y z>.

The thirteen intervals of the 7-limit [[Tonality_diamond|tonality diamond]] are represented by the unison |* 0 0 0> and twelve lattice points at a distance of one from the unison, given by ∓|* 1 0 0>, ∓|* 0 1 0>, ∓|* 0 0 1>, ∓|* 1 -1 0>, ∓|* 1 0 -1> and ∓|* 0 1 -1>. These lie on the verticies of a [http://en.wikipedia.org/wiki/Cuboctahedron cubeoctahedron], a semiregular solid. The lattice has two types of holes--the shallow holes, which are [http://en.wikipedia.org/wiki/Tetrahedron ~~tetrahera~~] and which correspond to the major and minor [http://tonalsoft.com/enc/tetrad.htm tetrads] 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are [http://en.wikipedia.org/wiki/Octahedron octahedra] and correspond to [http://tonalsoft.com/enc/hexany.htm hexanies].

The thirteen intervals of the 7-limit [[Tonality_diamond|tonality diamond]] are represented by the unison |* 0 0 0> and twelve lattice points at a distance of one from the unison, given by ∓|* 1 0 0>, ∓|* 0 1 0>, ∓|* 0 0 1>, ∓|* 1 -1 0>, ∓|* 1 0 -1> and ∓|* 0 1 -1>. These lie on the verticies of a [http://en.wikipedia.org/wiki/Cuboctahedron cubeoctahedron], a semiregular solid. The lattice has two types of holes--the shallow holes, which are [http://en.wikipedia.org/wiki/Tetrahedron tetrahedra] and which correspond to the major and minor [http://tonalsoft.com/enc/tetrad.htm tetrads] 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are [http://en.wikipedia.org/wiki/Octahedron octahedra] and correspond to [http://tonalsoft.com/enc/hexany.htm hexanies].

In the two dimensional case of the 5-limit, this gives the plane lattice 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 family of lattices, called Dn, to which it also belongs as D3, the [http://en.wikipedia.org/wiki/Crystal_structure face-centered cubic lattice].

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 If |-x-y-z x y z&gt; is any element of symmetric interval class space, then by definition || |-x-y-z x y z&gt; || = sqrt(2) sqrt(x^2+y^2+z^2+xy+yz+zx) where we may remove the sqrt(2) factor without changing anything substantial. We may also remove the two term, and write elements of symmetrical interval class space by |* x y z&gt;.
-The thirteen intervals of the 7-limit [[Tonality_diamond|tonality diamond]] are represented by the unison |* 0 0 0&gt; and twelve lattice points at a distance of one from the unison, given by ∓|* 1 0 0&gt;, ∓|* 0 1 0&gt;, ∓|* 0 0 1&gt;, ∓|* 1 -1 0&gt;, ∓|* 1 0 -1&gt; and ∓|* 0 1 -1&gt;. These lie on the verticies of a [http://en.wikipedia.org/wiki/Cuboctahedron cubeoctahedron], a semiregular solid. The lattice has two types of holes--the shallow holes, which are [http://en.wikipedia.org/wiki/Tetrahedron tetrahera] and which correspond to the major and minor [http://tonalsoft.com/enc/tetrad.htm tetrads] 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are [http://en.wikipedia.org/wiki/Octahedron octahedra] and correspond to [http://tonalsoft.com/enc/hexany.htm hexanies].
+The thirteen intervals of the 7-limit [[Tonality_diamond|tonality diamond]] are represented by the unison |* 0 0 0&gt; and twelve lattice points at a distance of one from the unison, given by ∓|* 1 0 0&gt;, ∓|* 0 1 0&gt;, ∓|* 0 0 1&gt;, ∓|* 1 -1 0&gt;, ∓|* 1 0 -1&gt; and ∓|* 0 1 -1&gt;. These lie on the verticies of a [http://en.wikipedia.org/wiki/Cuboctahedron cubeoctahedron], a semiregular solid. The lattice has two types of holes--the shallow holes, which are [http://en.wikipedia.org/wiki/Tetrahedron tetrahedra] and which correspond to the major and minor [http://tonalsoft.com/enc/tetrad.htm tetrads] 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are [http://en.wikipedia.org/wiki/Octahedron octahedra] and correspond to [http://tonalsoft.com/enc/hexany.htm hexanies].
 In the two dimensional case of the 5-limit, this gives the plane lattice 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 family of lattices, called Dn, to which it also belongs as D3, the [http://en.wikipedia.org/wiki/Crystal_structure face-centered cubic lattice].