Euler–Fokker genus: Difference between revisions
Wikispaces>genewardsmith **Imported revision 150743639 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 150760611 - 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>2010-06- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-28 04:14:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>150760611</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> | ||
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Because of the way it is constructed, an Euler genus has chords related to the prime divisors of n, with otonal and utonal chords appearing equally, and has scale size equal to d(n), the number of divisors of n. If |e2 e3 e5 ... ep> is the [[monzo]] for n, then d(n) = (e2+1)(e3+1)...(ep+1) and hence the size of the scale, d(n), is composite and tends to be highly composite. | Because of the way it is constructed, an Euler genus has chords related to the prime divisors of n, with otonal and utonal chords appearing equally, and has scale size equal to d(n), the number of divisors of n. If |e2 e3 e5 ... ep> is the [[monzo]] for n, then d(n) = (e2+1)(e3+1)...(ep+1) and hence the size of the scale, d(n), is composite and tends to be highly composite. | ||
The Euler genus can be generalized in a natural way which brings out its relation to [[Combination product sets|combination product multisets]]. If we start from any [[http://en.wikipedia.org/wiki/Multiset|multiset]] of positive | The Euler genus can be generalized in a natural way which brings out its relation to [[Combination product sets|combination product multisets]]. If we start from any [[http://en.wikipedia.org/wiki/Multiset|multiset]] of positive real numbers, we may define the corresponding genus to be the products of all the combinations of elements of the multiset, reduced to an octave. When we start from a set of rational numbers, this very often this will be an Euler genus as defined by Euler, but it needn't be. If we take the combination products 0 at a time, 1 at a time and so forth up to n at a time, we get the genus; combination product multisets are slices of a genus.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>Euler genera</title></head><body>As originally defined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Leonhard_Euler" rel="nofollow">Euler</a>, an Euler genus consists of all <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisor" rel="nofollow">divisors</a> of a given positive integer n, reduced to an octave. Since we reduce to an octave, without loss of generality we can restrict n to be odd, in which case there is a one-to-one relationship between the Euler genus Euler(n) and the odd integers. However the real interest attaches to composite numbers of low prime limit; Euler himself considered mostly the 5-limit. <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>Euler genera</title></head><body>As originally defined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Leonhard_Euler" rel="nofollow">Euler</a>, an Euler genus consists of all <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisor" rel="nofollow">divisors</a> of a given positive integer n, reduced to an octave. Since we reduce to an octave, without loss of generality we can restrict n to be odd, in which case there is a one-to-one relationship between the Euler genus Euler(n) and the odd integers. However the real interest attaches to composite numbers of low prime limit; Euler himself considered mostly the 5-limit. <br /> | ||
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Because of the way it is constructed, an Euler genus has chords related to the prime divisors of n, with otonal and utonal chords appearing equally, and has scale size equal to d(n), the number of divisors of n. If |e2 e3 e5 ... ep&gt; is the <a class="wiki_link" href="/monzo">monzo</a> for n, then d(n) = (e2+1)(e3+1)...(ep+1) and hence the size of the scale, d(n), is composite and tends to be highly composite.<br /> | Because of the way it is constructed, an Euler genus has chords related to the prime divisors of n, with otonal and utonal chords appearing equally, and has scale size equal to d(n), the number of divisors of n. If |e2 e3 e5 ... ep&gt; is the <a class="wiki_link" href="/monzo">monzo</a> for n, then d(n) = (e2+1)(e3+1)...(ep+1) and hence the size of the scale, d(n), is composite and tends to be highly composite.<br /> | ||
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The Euler genus can be generalized in a natural way which brings out its relation to <a class="wiki_link" href="/Combination%20product%20sets">combination product multisets</a>. If we start from any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a> of positive | The Euler genus can be generalized in a natural way which brings out its relation to <a class="wiki_link" href="/Combination%20product%20sets">combination product multisets</a>. If we start from any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a> of positive real numbers, we may define the corresponding genus to be the products of all the combinations of elements of the multiset, reduced to an octave. When we start from a set of rational numbers, this very often this will be an Euler genus as defined by Euler, but it needn't be. If we take the combination products 0 at a time, 1 at a time and so forth up to n at a time, we get the genus; combination product multisets are slices of a genus.</body></html></pre></div> | ||