Vals and tuning space: 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>2012-03-03 22:15:29 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 22:18:17 UTC</tt>.

: The original revision id was <tt>~~307490906~~</tt>.

: The original revision id was <tt>307491174</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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**|<V|M>| ≤ ||V|| ||M||**

to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. In tuning space, ~~the~~ vals ~~now define a~~ lattice. ~~Similarly, the~~ dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by

to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual mebedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by

[[math]]

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|&lt;V|M&gt;| ≤ ||V|| ||M||

to be always true. The dual of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a> is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow">Linfty norm</a>, and the dual space of Tenney interval space is Tenney tuning space. In tuning space, ~~the~~ vals ~~now define a~~ lattice. ~~Similarly, the~~ dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is Tenney-Euclidean tuning space. The Euclidean norm on a val v is given by

to be always true. The dual of the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a> is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow">Linfty norm</a>, and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual mebedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is Tenney-Euclidean tuning space. The Euclidean norm on a val v is given by

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@@ 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>2012-03-03 22:15:29 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-03 22:18:17 UTC</tt>.<br>
-: The original revision id was <tt>307490906</tt>.<br>
+: The original revision id was <tt>307491174</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 28: / Line 28: @@
 **|&lt;V|M&gt;| ≤ ||V|| ||M||**
-to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by
+to be always true. The dual of the [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]] is the [[http://mathworld.wolfram.com/L-Infinity-Norm.html|Linfty norm]], and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual mebedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is //Tenney-Euclidean tuning space//. The Euclidean norm on a val v is given by
 [[math]]
@@ Line 73: / Line 73: @@
 &lt;strong&gt;|&amp;lt;V|M&amp;gt;| ≤ ||V|| ||M||&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
-to be always true. The dual of the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow"&gt;L1 norm&lt;/a&gt; is the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow"&gt;Linfty norm&lt;/a&gt;, and the dual space of Tenney interval space is Tenney tuning space. In tuning space, the vals now define a lattice. Similarly, the dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is &lt;em&gt;Tenney-Euclidean tuning space&lt;/em&gt;. The Euclidean norm on a val v is given by&lt;br /&gt;
+to be always true. The dual of the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow"&gt;L1 norm&lt;/a&gt; is the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L-Infinity-Norm.html" rel="nofollow"&gt;Linfty norm&lt;/a&gt;, and the dual space of Tenney interval space is Tenney tuning space. The embedding of monzos into a real normed vector space automatically induces a dual mebedding of vals into a corresponding normed vector space, tuning space, in which vals are lattice points. The dual norm to the L2 norm is the L2 norm, and the dual space to Tenney-Euclidean interval space is &lt;em&gt;Tenney-Euclidean tuning space&lt;/em&gt;. The Euclidean norm on a val v is given by&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0: