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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | As originally defined by [http://en.wikipedia.org/wiki/Leonhard_Euler Euler], an Euler genus consists of all [http://en.wikipedia.org/wiki/Divisor divisors] 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, and [http://en.wikipedia.org/wiki/Adriaan_Fokker Adriaan Fokker] the 7-limit. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-24 16:48:04 UTC</tt>.<br>
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| : The original revision id was <tt>479290308</tt>.<br>
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| : The revision comment was: <tt></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>
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| <h4>Original Wikitext content:</h4>
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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">As originally defined by [[http://en.wikipedia.org/wiki/Leonhard_Euler|Euler]], an Euler genus consists of all [[http://en.wikipedia.org/wiki/Divisor|divisors]] 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, and [[http://en.wikipedia.org/wiki/Adriaan_Fokker|Adriaan Fokker]] the 7-limit.
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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|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. |
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| 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]] S of positive real numbers, we may define the corresponding genus Euler(S) to be the set of products of all the combinations of elements of the multiset, reduced to an octave. When we start from a multiset 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> | | 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] S of positive real numbers, we may define the corresponding genus Euler(S) to be the set of products of all the combinations of elements of the multiset, reduced to an octave. When we start from a multiset 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. |
| <h4>Original HTML content:</h4>
| | [[Category:math]] |
| <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, and <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a> the 7-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 />
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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> S of positive real numbers, we may define the corresponding genus Euler(S) to be the set of products of all the combinations of elements of the multiset, reduced to an octave. When we start from a multiset 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>
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As originally defined by Euler, an Euler genus consists of all divisors 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, and Adriaan Fokker the 7-limit.
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 multisets. If we start from any multiset S of positive real numbers, we may define the corresponding genus Euler(S) to be the set of products of all the combinations of elements of the multiset, reduced to an octave. When we start from a multiset 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.