Vals and tuning space: Difference between revisions

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\displaystyle

||v|| = \sqrt{\left({\frac{v_2}{log_2(2)}\right)^2 + \left({\frac{v_3}{log_2(3)}\right)^2 + \left(\frac{v_5}{log_2(5)}\right)^2 + ... + \left(\frac{v_p}{log_2(p)}\right)^2

[[math]]

==Example==  

<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;

[[math]]&amp;lt;br/&amp;gt;

\displaystyle&amp;lt;br /&amp;gt;

  --&gt;&lt;script type="math/tex"&gt;\displaystyle

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It useful to renormalize to the RMS (root mean square) instead, which requires dividing the above by sqrt(n), where n = pi(p) is the number of primes up to p. This is the TE, or Tenney-Euclidean, norm.&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 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;br /&gt;

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