Superpartient ratio: Difference between revisions
Wikispaces>genewardsmith **Imported revision 245063411 - Original comment: ** |
Wikispaces>Sarzadoce **Imported revision 245088753 - Original comment: ** |
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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:Sarzadoce|Sarzadoce]] and made on <tt>2011-08-09 15:43:20 UTC</tt>.<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">Superpartient numbers are ratios of the form (n+m)/n, or 1+m/n, where m is not n or a whole-number multiple of n, and where n is a whole number other than 1. Another way to say it is that is that they are non-integer ratios p/q, where p is greater than q and p and q are relatively prime (so that the fraction is reduced to lowest terms) and where p - q is greater than 1. In ancient Greece they were called Epimeric (epimerēs) ratios, which is literally translated as "above a part." These ratios were considered to be inferior to Epimoric ratios. | <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">Superpartient numbers are ratios of the form (n+m)/n, or 1+m/n, where m is not n or a whole-number multiple of n, and where n is a whole number other than 1. Another way to say it is that is that they are non-integer ratios p/q, where p is greater than q and p and q are relatively prime (so that the fraction is reduced to lowest terms) and where p - q is greater than 1. In ancient Greece they were called Epimeric (epimerēs) ratios, which is literally translated as "above a part." These ratios were considered to be inferior to Epimoric ratios. | ||
All epimeric ratios can be constructed as combinations of [[superparticular|superparticular numbers]]. For example, 9/5 is 3/2 × 6/5. | All epimeric ratios can be constructed as combinations of [[superparticular|superparticular numbers]]. For example, 9/5 is 3/2 × 6/5. Generally numbers like 3/1 and 7/1 are not considered to be epimeric ratios because they are [[Harmonic|multiples of the fundamental]]. | ||
When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the degree of epimericity. For 1+m/n, this is m; in terms of p/q reduced to lowest terms it is p - q. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem|Størmer's theorem]] can be extended to the claim that for each prime limit p and each degree of epimoricity n, there are only finitely many p-limit commas with degree of epimoricity less than or equal to n.</pre></div> | When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the degree of epimericity. For 1+m/n, this is m; in terms of p/q reduced to lowest terms it is p - q. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem|Størmer's theorem]] can be extended to the claim that for each prime limit p and each degree of epimoricity n, there are only finitely many p-limit commas with degree of epimoricity less than or equal to n.</pre></div> | ||
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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>Superpartient</title></head><body>Superpartient numbers are ratios of the form (n+m)/n, or 1+m/n, where m is not n or a whole-number multiple of n, and where n is a whole number other than 1. Another way to say it is that is that they are non-integer ratios p/q, where p is greater than q and p and q are relatively prime (so that the fraction is reduced to lowest terms) and where p - q is greater than 1. In ancient Greece they were called Epimeric (epimerēs) ratios, which is literally translated as &quot;above a part.&quot; These ratios were considered to be inferior to Epimoric ratios.<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>Superpartient</title></head><body>Superpartient numbers are ratios of the form (n+m)/n, or 1+m/n, where m is not n or a whole-number multiple of n, and where n is a whole number other than 1. Another way to say it is that is that they are non-integer ratios p/q, where p is greater than q and p and q are relatively prime (so that the fraction is reduced to lowest terms) and where p - q is greater than 1. In ancient Greece they were called Epimeric (epimerēs) ratios, which is literally translated as &quot;above a part.&quot; These ratios were considered to be inferior to Epimoric ratios.<br /> | ||
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All epimeric ratios can be constructed as combinations of <a class="wiki_link" href="/superparticular">superparticular numbers</a>. For example, 9/5 is 3/2 × 6/5.<br /> | All epimeric ratios can be constructed as combinations of <a class="wiki_link" href="/superparticular">superparticular numbers</a>. For example, 9/5 is 3/2 × 6/5. Generally numbers like 3/1 and 7/1 are not considered to be epimeric ratios because they are <a class="wiki_link" href="/Harmonic">multiples of the fundamental</a>.<br /> | ||
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When considering ratios, and particularly when they are ratios for <a class="wiki_link" href="/comma">comma</a>s, it can be useful to introduce the notion of the degree of epimericity. For 1+m/n, this is m; in terms of p/q reduced to lowest terms it is p - q. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem" rel="nofollow">Størmer's theorem</a> can be extended to the claim that for each prime limit p and each degree of epimoricity n, there are only finitely many p-limit commas with degree of epimoricity less than or equal to n.</body></html></pre></div> | When considering ratios, and particularly when they are ratios for <a class="wiki_link" href="/comma">comma</a>s, it can be useful to introduce the notion of the degree of epimericity. For 1+m/n, this is m; in terms of p/q reduced to lowest terms it is p - q. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/St%C3%B8rmer's_theorem" rel="nofollow">Størmer's theorem</a> can be extended to the claim that for each prime limit p and each degree of epimoricity n, there are only finitely many p-limit commas with degree of epimoricity less than or equal to n.</body></html></pre></div> | ||