Radical interval: Difference between revisions
Wikispaces>genewardsmith **Imported revision 163292829 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 163349667 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-09- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-17 00:00:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>163349667</tt>.<br> | ||
: The revision comment was: <tt></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> | 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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<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;white-space: pre-wrap ! important" class="old-revision-html">A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents. | <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;white-space: pre-wrap ! important" class="old-revision-html">A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents. | ||
Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. | Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it. | ||
===Fractional projection maps=== | ===Fractional projection maps=== | ||
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<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"><html><head><title>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26&gt; represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with &lt;cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.<br /> | <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"><html><head><title>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep&gt; is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26&gt; represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/26). By taking a dot product with &lt;cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.<br /> | ||
<br /> | <br /> | ||
Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. <br /> | Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it. <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> | ||