7-limit symmetrical lattices: Difference between revisions
Wikispaces>genewardsmith **Imported revision 254462484 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278340690 - Original comment: ** |
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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>2011- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 20:45:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278340690</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> | ||
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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>. | 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 | 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]]. | ||
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]]. | 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;.<br /> | 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;.<br /> | ||
<br /> | <br /> | ||
The thirteen intervals of the 7-limit <a class="wiki_link" href="/Tonality%20Diamond">tonality diamond</a> are represented by the unison |* 0 0 0&gt; and twelve lattice points at a distance of one from the unison, given by | The thirteen intervals of the 7-limit <a class="wiki_link" href="/Tonality%20Diamond">tonality diamond</a> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow">cubeoctahedron</a>, a semiregular solid. The lattice has two types of holes--the shallow holes, which are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tetrahedron" rel="nofollow">tetrahera</a> and which correspond to the major and minor <a class="wiki_link_ext" href="http://tonalsoft.com/enc/tetrad.htm" rel="nofollow">tetrads</a> 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Octahedron" rel="nofollow">octahedra</a> and correspond to <a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow">hexanies</a>.<br /> | ||
<br /> | <br /> | ||
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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">face-centered cubic lattice</a>.<br /> | 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Crystal_structure" rel="nofollow">face-centered cubic lattice</a>.<br /> | ||