7-limit symmetrical lattices: Difference between revisions
Wikispaces>genewardsmith **Imported revision 216519602 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 230239550 - 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>2011-05-20 02:40:59 UTC</tt>.<br> | ||
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If T is the [[Vals and Tuning Space|val]] T = <1 1 1 1| (note that this is //not// the JI point) then we may define a subspace of 7-limit interval space, **symmetric interval class space**, as the subspace of all vectors M in interval space such that <T|M> = 0, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in s two-dimensional subspace. | If T is the [[Vals and Tuning Space|val]] T = <1 1 1 1| (note that this is //not// the JI point) then we may define a subspace of 7-limit interval space, **symmetric interval class space**, as the subspace of all vectors M in interval space such that <T|M> = 0, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in s two-dimensional subspace. | ||
If |-x-y-z z y z> is any element of symmetric interval class space, then by definition || |-x-y-z x y z> || = sqrt(2) sqrt(x^2 | If |-x-y-z z 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|octaheda]] and | 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|octaheda]] and | ||
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If T is the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> T = &lt;1 1 1 1| (note that this is <em>not</em> the JI point) then we may define a subspace of 7-limit interval space, <strong>symmetric interval class space</strong>, as the subspace of all vectors M in interval space such that &lt;T|M&gt; = 0, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in s two-dimensional subspace.<br /> | If T is the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> T = &lt;1 1 1 1| (note that this is <em>not</em> the JI point) then we may define a subspace of 7-limit interval space, <strong>symmetric interval class space</strong>, as the subspace of all vectors M in interval space such that &lt;T|M&gt; = 0, which has a norm induced on it by inclusion. There is one and only one element of each octave-equivalency interval class contained in symmetric interval class space, and interval classes thereby become a symmetric lattice in a three-dimensional space, with a sublattice of 5-limit interval classes in s two-dimensional subspace.<br /> | ||
<br /> | <br /> | ||
If |-x-y-z z 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 | If |-x-y-z z 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 +-|* 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">octaheda</a> and<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 +-|* 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">octaheda</a> and<br /> | ||