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:<br>

: This revision was by author [[User:~~clumma~~|~~clumma~~]] and made on <tt>2010-05-~~16 19~~:57:28 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-01 15:54:16 UTC</tt>.<br>

: The original revision id was <tt>~~142372131~~</tt>.<br>

: The original revision id was <tt>175444627</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>

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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 Euclidean interval space is Euclidean tuning space.

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 Tenney-Euclidean norm, or TE norm, on a val v is given by

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in ~~normalized~~ coordinates is ~~JIP =~~ <1 1 1 ... 1|. It has the property that if M is a monzo in ~~normalized~~ coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name.</pre></div>

||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2)

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is <1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then <JIP|M>, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = <1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.</pre></div>

<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"><html><head><title>Vals and Tuning Space</title></head><body>The p-limit <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The <a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow">dual Z-module</a> M* is <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow">isomorphic</a> to M, but not in a canonical way. Hence it, the group (Z-module) of <strong>vals</strong>, is also a free abelian group of rank pi(p). Just as monzos are often written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">kets</a>, vals are typically written as <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow">bras</a>.<br />

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<br />

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 Euclidean interval space is Euclidean tuning space.<br />

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 <em>Tenney-Euclidean tuning space</em>. The Tenney-Euclidean norm, or TE norm, on a val v is given by<br />

<br />

||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2) <br />

<br />

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in ~~normalized~~ coordinates is ~~JIP =~~ &lt;1 1 1 ... 1|. It has the property that if M is a monzo in ~~normalized~~ coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name.</body></html></pre></div>

It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is &lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.</body></html></pre></div>

@@ 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:clumma|clumma]] and made on <tt>2010-05-16 19:57:28 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-01 15:54:16 UTC</tt>.<br>
-: The original revision id was <tt>142372131</tt>.<br>
+: The original revision id was <tt>175444627</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 16: / Line 16: @@
 **|&lt;V|M&gt;| &lt;= ||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 Euclidean interval space is Euclidean tuning space.
+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 Tenney-Euclidean norm, or TE norm, on a val v is given by
-It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in normalized coordinates is JIP = &lt;1 1 1 ... 1|. It has the property that if M is a monzo in normalized coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name.</pre></div>
+||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2)
+It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is &lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &lt;JIP|M&gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.</pre></div>
 <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;Vals and Tuning Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The p-limit &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt; M form a free abelian group, or Z-module, of finite rank pi(p), which is the number of primes up to and including p. The &lt;a class="wiki_link_ext" href="http://planetmath.org/encyclopedia/DualModule.html" rel="nofollow"&gt;dual Z-module&lt;/a&gt; M* is &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_isomorphism" rel="nofollow"&gt;isomorphic&lt;/a&gt; to M, but not in a canonical way. Hence it, the group (Z-module) of &lt;strong&gt;vals&lt;/strong&gt;, is also a free abelian group of rank pi(p). Just as monzos are often written as &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow"&gt;kets&lt;/a&gt;, vals are typically written as &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Bra.html" rel="nofollow"&gt;bras&lt;/a&gt;.&lt;br /&gt;
@@ Line 30: / Line 32: @@
 &lt;strong&gt;|&amp;lt;V|M&amp;gt;| &amp;lt;= ||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 Euclidean interval space is Euclidean tuning space.&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 Tenney-Euclidean norm, or TE norm, on a val v is given by&lt;br /&gt;
+&lt;br /&gt;
+||v|| = sqrt(v2^2 + (v3/log2(3))^2 + ... + (vp/log2(p))^2)  &lt;br /&gt;
 &lt;br /&gt;
-It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in normalized coordinates is JIP = &amp;lt;1 1 1 ... 1|. It has the property that if M is a monzo in normalized coordinates, then &amp;lt;JIP|M&amp;gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name.&lt;/body&gt;&lt;/html&gt;</pre></div>
+It should be noted that despite the name, only vectors in a small region of tuning space can reasonably be considered to be tunings. These are the points in tuning space close to the JI point, or JIP, which in weighted coordinates is &amp;lt;1 1 1 ... 1|. It has the property that if M is a monzo in weighted coordinates, then &amp;lt;JIP|M&amp;gt;, or JIP(M) if you prefer, is exactly the log base two of the interval M represents, hence the name. In unweighted coordinates, JIP = &amp;lt;1 log2(3) ... log2(p)|, and applied to a monzo this gives the log base two of the corresponding interval.&lt;/body&gt;&lt;/html&gt;</pre></div>