Monzos and interval 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>2012-03-~~04 02~~:48:05 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-11 19:28:51 UTC</tt>.<br>

: The original revision id was <tt>~~307514102~~</tt>.<br>

: The original revision id was <tt>357345582</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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[[math]]

which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]; hence we may [[http://en.wikipedia.org/wiki/Embedding|embed]] the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos ~~->~~ I. The monzos under this embedding now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0> represents 2, so does |0 log3(2)>.

which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]; hence we may [[http://en.wikipedia.org/wiki/Embedding|embed]] the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0> represents 2, so does |0 log3(2)>.

Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Tenney-Euclidean norm, or TE norm, of it is

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--><script type="math/tex">\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p</script><br />

<br />

which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a>; hence we may <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Embedding" rel="nofollow">embed</a> the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos ~~-&gt;~~ I. The monzos under this embedding now define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a>, which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a>. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&gt; represents 2, so does |0 log3(2)&gt;.<br />

which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a>; hence we may <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Embedding" rel="nofollow">embed</a> the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a>, which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a>. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&gt; represents 2, so does |0 log3(2)&gt;.<br />

<br />

Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean interval space</a> instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep&gt; then the Tenney-Euclidean norm, or TE norm, of it is<br />

@@ 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>2012-03-04 02:48:05 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-11 19:28:51 UTC</tt>.<br>
-: The original revision id was <tt>307514102</tt>.<br>
+: The original revision id was <tt>357345582</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 21: / Line 21: @@
 [[math]]
-which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]; hence we may [[http://en.wikipedia.org/wiki/Embedding|embed]] the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos -&gt; I. The monzos under this embedding now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&gt; represents 2, so does |0 log3(2)&gt;.
+which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]; hence we may [[http://en.wikipedia.org/wiki/Embedding|embed]] the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&gt; represents 2, so does |0 log3(2)&gt;.
 Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep&gt; then the Tenney-Euclidean norm, or TE norm, of it is
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   --&gt;&lt;script type="math/tex"&gt;\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-which is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;vector space norm&lt;/a&gt;; hence we may &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Embedding" rel="nofollow"&gt;embed&lt;/a&gt; the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos -&amp;gt; I. The monzos under this embedding now define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&gt;, which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow"&gt;L1 norm&lt;/a&gt;. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&amp;gt; represents 2, so does |0 log3(2)&amp;gt;.&lt;br /&gt;
+which is a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;vector space norm&lt;/a&gt;; hence we may &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Embedding" rel="nofollow"&gt;embed&lt;/a&gt; the p-limit monzos into a normed vector I space of dimension n = pi(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow"&gt;lattice&lt;/a&gt;, which is a discrete subgroup spanning the finite dimensional real normed vector space I. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow"&gt;L1 norm&lt;/a&gt;. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&amp;gt; represents 2, so does |0 log3(2)&amp;gt;.&lt;br /&gt;
 &lt;br /&gt;
 Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean interval space&lt;/a&gt; instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep&amp;gt; then the Tenney-Euclidean norm, or TE norm, of it is&lt;br /&gt;