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Generalized Tenney norms and Tp interval space: Difference between revisions

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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:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 04:02:26 UTC</tt>.<br>
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-08-06 04:02:34 UTC</tt>.<br>
: The original revision id was <tt>356496102</tt>.<br>
: The original revision id was <tt>356496110</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;Basics&lt;/span&gt;=  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;Basics&lt;/span&gt;=  
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;It can be useful to define a notion of the "complexity" of an interval, so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. Alternatively, if one prefers to think of the monzos as forming a &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt;-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℝ&lt;/span&gt; instead of &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt; and giving it the added structure of being a vector space. In either case, the resulting space is called [[Monzos and Interval Space|interval space]], with the monzos forming the lattice of vectors with integer coordinates.&lt;/span&gt;
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;It can be useful to define a notion of the "complexity" of an interval, so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. Alternatively, if one prefers to think of the monzos as forming a &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt;-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℝ&lt;/span&gt; instead of &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt; and giving it the added structure of being a vector space. In either case, the resulting space is called [[Monzos and Interval Space|interval space]], with the monzos forming the lattice of vectors with integer coordinates.&lt;/span&gt;
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.&lt;/span&gt;
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.&lt;/span&gt;
=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;The Tenney Norm (T1 norm)&lt;/span&gt;=  
=&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;The Tenney Norm (T1 norm)&lt;/span&gt;=  
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\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1


where **V** is a [[xenharmonic/Subgroup Mapping Matrices (V-maps)|V-map]] in which the columns are monzos express&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;For interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval v can be represented by the expression&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;where **W** is a diagonal "weighting matrix" such that the nth entry in the diagonal is the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 unit sphere, but be dilated a different amount along each axis.&lt;/span&gt;</pre></div>
where **V** is a [[xenharmonic/Subgroup Mapping Matrices (V-maps)|V-map]] in which the columns are monzos express&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;For interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval v can be represented by the expression&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;where **W** is a diagonal "weighting matrix" such that the nth entry in the diagonal is the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 unit sphere, but be dilated a different amount along each axis.&lt;/span&gt;</pre></div>
<h4>Original HTML content:</h4>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Generalized Tenney Norms and Tp Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;Basics&lt;/span&gt;&lt;/h1&gt;
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Generalized Tenney Norms and Tp Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Basics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;Basics&lt;/span&gt;&lt;/h1&gt;
  &lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;It can be useful to define a notion of the &amp;quot;complexity&amp;quot; of an interval, so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. Alternatively, if one prefers to think of the monzos as forming a &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt;-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℝ&lt;/span&gt; instead of &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt; and giving it the added structure of being a vector space. In either case, the resulting space is called &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, with the monzos forming the lattice of vectors with integer coordinates.&lt;/span&gt;&lt;br /&gt;
  &lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;It can be useful to define a notion of the &amp;quot;complexity&amp;quot; of an interval, so that small-integer ratios such as 3/2 are less complex and intervals such as 32805/32768 are more complex. This can be accomplished for any free abelian group of monzos or smonzos by embedding the group in a normed vector space, so that the norm of any interval is taken to be its complexity. Alternatively, if one prefers to think of the monzos as forming a &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt;-module, the vector space can be constructed by simply allowing the existence of monzos with real coordinates, hence defining the module over &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℝ&lt;/span&gt; instead of &lt;span style="background-color: #f9f9f9; font-family: sans-serif; font-size: 15px;"&gt;ℤ&lt;/span&gt; and giving it the added structure of being a vector space. In either case, the resulting space is called &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, with the monzos forming the lattice of vectors with integer coordinates.&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;The most important and natural norm which arises in this scenario is the Tenney norm, which we will explore below.&lt;/span&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="The Tenney Norm (T1 norm)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;The Tenney Norm (T1 norm)&lt;/span&gt;&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="The Tenney Norm (T1 norm)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif;"&gt;The Tenney Norm (T1 norm)&lt;/span&gt;&lt;/h1&gt;
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\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1&lt;br /&gt;
\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \mathbf{V} \cdot \vec{v} \right \|_1&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
where &lt;strong&gt;V&lt;/strong&gt; is a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; in which the columns are monzos express&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;For interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval v can be represented by the expression&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt; &lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;where &lt;strong&gt;W&lt;/strong&gt; is a diagonal &amp;quot;weighting matrix&amp;quot; such that the nth entry in the diagonal is the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 unit sphere, but be dilated a different amount along each axis.&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
where &lt;strong&gt;V&lt;/strong&gt; is a &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Subgroup%20Mapping%20Matrices%20%28V-maps%29"&gt;V-map&lt;/a&gt; in which the columns are monzos express&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;For interval spaces corresponding to a group of monzos where the basis is set to consist of only primes or prime powers, the Tenney norm of any interval v can be represented by the expression&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;\left \| \vec{v} \right \|_{\textbf{T1}} = \left \| \mathbf{W} \cdot \vec{v} \right \|_1&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;&lt;br /&gt;
&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; display: block; font-family: arial,sans-serif;"&gt;where &lt;strong&gt;W&lt;/strong&gt; is a diagonal &amp;quot;weighting matrix&amp;quot; such that the nth entry in the diagonal is the log&lt;span style="font-size: 10px; vertical-align: sub;"&gt;2&lt;/span&gt; of the interval represented by the nth coordinate of the monzo. For Tenney norms on these spaces, the unit sphere will look like the ordinary L1 unit sphere, but be dilated a different amount along each axis.&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
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