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:genewardsmith|genewardsmith]] and made on <tt>2012-11-24 11:10:23 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-11-24 11:12:03 UTC</tt>.<br>

: The original revision id was <tt>~~385525278~~</tt>.<br>

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: The revision comment was: <tt></tt><br>

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==Defintion for mathematicians==

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. The [[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]]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees".

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. The [[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]]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees".

=Vals and Monzos=

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<br />

<h2 id="toc1"><a name="Definition-Defintion for mathematicians"></a>Defintion for mathematicians</h2>

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>. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a> seems to have been the first to write about them, under the name &quot;degrees&quot;.<br />

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>. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a> seems to have been the first to write about them, under the name &quot;degrees&quot;.<br />

<br />

<h1 id="toc2"><a name="Vals and Monzos"></a>Vals and Monzos</h1>

@@ 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>2012-11-24 11:10:23 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-11-24 11:12:03 UTC</tt>.<br>
-: The original revision id was <tt>385525278</tt>.<br>
+: The original revision id was <tt>385525542</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 17: / Line 17: @@
 ==Defintion for mathematicians==
-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. The [[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]]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees".
+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. The [[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]]. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist [[Yves Hellegouarch]] seems to have been the first to write about them, under the name "degrees".
 =Vals and Monzos=
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Definition-Defintion for mathematicians"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Defintion for mathematicians&lt;/h2&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;. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist &lt;a class="wiki_link" href="/Yves%20Hellegouarch"&gt;Yves Hellegouarch&lt;/a&gt; seems to have been the first to write about them, under the name &amp;quot;degrees&amp;quot;.&lt;br /&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;. Vals are homomorphisms from a subgroup of finite rank of ℚ*, the abelian group of the positive rational numbers under multiplication, to the integers ℤ. The number theorist &lt;a class="wiki_link" href="/Yves%20Hellegouarch"&gt;Yves Hellegouarch&lt;/a&gt; seems to have been the first to write about them, under the name &amp;quot;degrees&amp;quot;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Vals and Monzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Vals and Monzos&lt;/h1&gt;