Vals and tuning space
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The p-limit [[Monzos and Interval Space|monzos]] 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. Tne [[http://planetmath.org/encyclopedia/DualModule.html|dual Z-module]] M* is [[http://en.wikipedia.org/wiki/Group_isomorphism|isomorphic]] to M, but not in a canonical way. Hence it, the group (Z-module) of **vals**, is also a free abelian group of rank pi(p). Just as monzos are often written as [[http://mathworld.wolfram.com/Ket.html|kets]], vals are typically written as [[http://mathworld.wolfram.com/Bra.html|bras]]. If V is a val and M is a monzo of the same rank, then the [[http://mathworld.wolfram.com/AngleBracket.html|angle bracket]] <V|M>, which can also be written V(M), is the result of applying the [[http://en.wikipedia.org/wiki/Group_homomorphism|homomorphism]] V to M. For example, if V = <12 19 28 34| and M = |-5 2 2 -1> then <V|M> = 12*(-5) + 19*2 + 28*2 - 34 = 0. This tells us that in septimal 12 equal, represented by V, the interval 225/224, represented by M, is mapped to 0, which represents 1. Hence, 225/224 vanishes in septimal 12 equal; it is in the [[http://mathworld.wolfram.com/GroupKernel.html|kernel]]of V. One should note in particular that the coordinates of V represent where the successive primes 2, 3, 5 and 7 are mapped.
Original HTML content:
<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. Tne <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 /> <br /> If V is a val and M is a monzo of the same rank, then the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/AngleBracket.html" rel="nofollow">angle bracket</a> <V|M>, which can also be written V(M), is the result of applying the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_homomorphism" rel="nofollow">homomorphism</a> V to M. For example, if V = <12 19 28 34| and M = |-5 2 2 -1> then<br /> <V|M> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x12*(-5) + 19*2 + 28*2 - 34"></a><!-- ws:end:WikiTextHeadingRule:0 --> 12*(-5) + 19*2 + 28*2 - 34 </h1> 0. This tells us that in septimal 12 equal, represented by V, the interval 225/224, represented by M, is mapped to 0, which represents 1. Hence, 225/224 vanishes in septimal 12 equal; it is in the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/GroupKernel.html" rel="nofollow">kernel</a>of V. One should note in particular that the coordinates of V represent where the successive primes 2, 3, 5 and 7 are mapped.</body></html>