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:genewardsmith|genewardsmith]] and made on <tt>2011-08-13 17:02:22 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-13 17:04:10 UTC</tt>.

: The original revision id was <tt>~~245792679~~</tt>.

: The original revision id was <tt>245792815</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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|| |e2 e2 e5 e7> || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)

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 a two-dimensional subspace.

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>.

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|| |e2 e2 e5 e7&gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)

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 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 &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.

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 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 &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 a two-dimensional subspace.

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;.

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-08-13 17:02:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-08-13 17:04:10 UTC</tt>.<br>
-: The original revision id was <tt>245792679</tt>.<br>
+: The original revision id was <tt>245792815</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 10: / Line 10: @@
 || |e2 e2 e5 e7&gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)
-If T is the [[Vals and Tuning Space|val]] T = &lt;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 &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.
+If T is the [[Vals and Tuning Space|val]] T = &lt;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 &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 a two-dimensional subspace.
 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;.
@@ Line 41: / Line 41: @@
 || |e2 e2 e5 e7&amp;gt; || = sqrt(e2^2 + e3^3 + e5^2 + e7^2)&lt;br /&gt;
 &lt;br /&gt;
-If T is the &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; T = &amp;lt;1 1 1 1| (note that this is &lt;em&gt;not&lt;/em&gt; the JI point) then we may define a subspace of 7-limit interval space, &lt;strong&gt;symmetric interval class space&lt;/strong&gt;, as the subspace of all vectors M in interval space such that &amp;lt;T|M&amp;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.&lt;br /&gt;
+If T is the &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; T = &amp;lt;1 1 1 1| (note that this is &lt;em&gt;not&lt;/em&gt; the JI point) then we may define a subspace of 7-limit interval space, &lt;strong&gt;symmetric interval class space&lt;/strong&gt;, as the subspace of all vectors M in interval space such that &amp;lt;T|M&amp;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 a two-dimensional subspace.&lt;br /&gt;
 &lt;br /&gt;
 If |-x-y-z x 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;