Monzos and interval space: Difference between revisions
Wikispaces>xenwolf **Imported revision 163068905 - Original comment: Fractional monzos linked** |
Wikispaces>genewardsmith **Imported revision 175438567 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-01 15:38:01 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>175438567</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]. The monzos with this norm now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning a finite dimensional real normed vector space. 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]]. The monzos with this norm now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning a finite dimensional real normed vector space. 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 Euclidean interval space instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Euclidean norm of it is | 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 norms|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 | ||
sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2) | sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2) | ||
and if the coordinates are the | and if the coordinates are the weighted interval space coordinates, then the TE norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]]. | ||
//see also [[Fractional monzos]]...//</pre></div> | //see also [[Fractional monzos]]...//</pre></div> | ||
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which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a>. The monzos with this norm 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 a finite dimensional real normed vector space. 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>. The monzos with this norm 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 a finite dimensional real normed vector space. 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 /> | <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 Euclidean interval space instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep&gt; then the Euclidean norm of it is<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%20norms">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 /> | ||
<br /> | <br /> | ||
sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2)<br /> | sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2)<br /> | ||
<br /> | <br /> | ||
and if the coordinates are the | and if the coordinates are the weighted interval space coordinates, then the TE norm is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L2-Norm.html" rel="nofollow">standard Euclidean, or L2, norm</a>.<br /> | ||
<br /> | <br /> | ||
<em>see also <a class="wiki_link" href="/Fractional%20monzos">Fractional monzos</a>...</em></body></html></pre></div> | <em>see also <a class="wiki_link" href="/Fractional%20monzos">Fractional monzos</a>...</em></body></html></pre></div> | ||