7-limit symmetrical lattices: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~guest~~|~~guest~~]] and made on <tt>2010-05-16 01:31:42 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-16 03:24:15 UTC</tt>.

: The original revision id was <tt>~~142266479~~</tt>.

: The original revision id was <tt>142272595</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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~~ 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

correspond to [[http://tonalsoft.com/enc/hexany.htm|hexanies]].

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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;.

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~~ 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

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

correspond to <a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow">hexanies</a>.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:guest|guest]] and made on <tt>2010-05-16 01:31:42 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-16 03:24:15 UTC</tt>.<br>
-: The original revision id was <tt>142266479</tt>.<br>
+: The original revision id was <tt>142272595</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 14: / Line 14: @@
 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;.
-The thirteen intervals of the 7-limit [[Tonality Diamond|tonality diamond]] are represented by the unison |* 0 0 0&gt; and twelve 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|octaheda]] and
+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|octaheda]] and
 correspond to [[http://tonalsoft.com/enc/hexany.htm|hexanies]].
@@ Line 42: / Line 42: @@
 If |-x-y-z z y z&amp;gt; is any element of symmetric interval class space, then by definition || |-x-y-z x y z&amp;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&amp;gt;.&lt;br /&gt;
 &lt;br /&gt;
-The thirteen intervals of the 7-limit &lt;a class="wiki_link" href="/Tonality%20Diamond"&gt;tonality diamond&lt;/a&gt; are represented by the unison |* 0 0 0&amp;gt; and twelve twelve lattice points at a distance of one from the unison, given by +-|* 1 0 0&amp;gt;, +-|* 0 1 0&amp;gt;, +-|* 0 0 1&amp;gt;, +-|* 1 -1 0&amp;gt;, +-|* 1 0 -1&amp;gt; and +-|* 0 1 -1&amp;gt;. These lie on the verticies of a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow"&gt;cubeoctahedron&lt;/a&gt;, a semiregular solid. The lattice has two types of holes--the shallow holes, which are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tetrahedron" rel="nofollow"&gt;tetrahera&lt;/a&gt; and which correspond to the major and minor &lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/tetrad.htm" rel="nofollow"&gt;tetrads&lt;/a&gt; 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Octahedron" rel="nofollow"&gt;octaheda&lt;/a&gt; and&lt;br /&gt;
+The thirteen intervals of the 7-limit &lt;a class="wiki_link" href="/Tonality%20Diamond"&gt;tonality diamond&lt;/a&gt; are represented by the unison |* 0 0 0&amp;gt; and twelve lattice points at a distance of one from the unison, given by +-|* 1 0 0&amp;gt;, +-|* 0 1 0&amp;gt;, +-|* 0 0 1&amp;gt;, +-|* 1 -1 0&amp;gt;, +-|* 1 0 -1&amp;gt; and +-|* 0 1 -1&amp;gt;. These lie on the verticies of a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cuboctahedron" rel="nofollow"&gt;cubeoctahedron&lt;/a&gt;, a semiregular solid. The lattice has two types of holes--the shallow holes, which are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tetrahedron" rel="nofollow"&gt;tetrahera&lt;/a&gt; and which correspond to the major and minor &lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/tetrad.htm" rel="nofollow"&gt;tetrads&lt;/a&gt; 4:5:6:7 and 1/4:1/5:1/6:1/7, and the deep holes which are &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Octahedron" rel="nofollow"&gt;octaheda&lt;/a&gt; and&lt;br /&gt;
 correspond to &lt;a class="wiki_link_ext" href="http://tonalsoft.com/enc/hexany.htm" rel="nofollow"&gt;hexanies&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;