Superpartient ratio: Difference between revisions
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Wikispaces>Sarzadoce **Imported revision 362669752 - Original comment: ** |
Wikispaces>guest **Imported revision 362905360 - 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:guest|guest]] and made on <tt>2012-09-07 15:06:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>362905360</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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">Superpartient numbers are ratios of the form p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and p - q is greater than 1. In ancient Greece they were called | <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 p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and p - q is greater than 1. In ancient Greece they were called epimeric (epimerēs) ratios, which is literally translated as "above a part." In ancient Greece, fractions like 3/1 and 7/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios. | ||
All epimeric ratios can be constructed as | All epimeric ratios can be constructed as products of [[superparticular|superparticular numbers]]. This is due to the following useful identity: | ||
[[math]] | |||
\prod_{i=1}^{P-1}\frac{i+1}{i}=P | |||
[[math]] | |||
When considering ratios, and particularly when they are ratios for [[comma|commas]], it can be useful to introduce the notion of the degree of epimericity. 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 ratios with degree of epimoricity less than or equal to n.</pre></div> | When considering ratios, and particularly when they are ratios for [[comma|commas]], it can be useful to introduce the notion of the **degree of epimericity**. 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 ratios with degree of epimoricity less than or equal to n. | ||
See Also: [[ABC, High Quality Commas, and Epimericity]]</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>Superpartient</title></head><body>Superpartient numbers are ratios of the form p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and p - q is greater than 1. In ancient Greece they were called | <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><strong>Superpartient</strong> numbers are ratios of the form p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and 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; In ancient Greece, fractions like 3/1 and 7/1 were not considered to be epimeric ratios because of their additional restriction that <a class="wiki_link" href="/Harmonic">multiples of the fundamental</a> cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.<br /> | ||
<br /> | |||
All epimeric ratios can be constructed as products of <a class="wiki_link" href="/superparticular">superparticular numbers</a>. This is due to the following useful identity:<br /> | |||
<!-- ws:start:WikiTextMathRule:0: | |||
[[math]]&lt;br/&gt; | |||
\prod_{i=1}^{P-1}\frac{i+1}{i}=P&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\prod_{i=1}^{P-1}\frac{i+1}{i}=P</script><!-- ws:end:WikiTextMathRule:0 --><br /> | |||
<br /> | <br /> | ||
When considering ratios, and particularly when they are ratios for <a class="wiki_link" href="/comma">commas</a>, it can be useful to introduce the notion of the <strong>degree of epimericity</strong>. 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 ratios with degree of epimoricity less than or equal to n.<br /> | |||
<br /> | <br /> | ||
See Also: <a class="wiki_link" href="/ABC%2C%20High%20Quality%20Commas%2C%20and%20Epimericity">ABC, High Quality Commas, and Epimericity</a></body></html></pre></div> | |||
Revision as of 15:06, 7 September 2012
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author guest and made on 2012-09-07 15:06:03 UTC.
- The original revision id was 362905360.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
**Superpartient** numbers are ratios of the form p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and p - q is greater than 1. In ancient Greece they were called epimeric (epimerēs) ratios, which is literally translated as "above a part." In ancient Greece, fractions like 3/1 and 7/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.
All epimeric ratios can be constructed as products of [[superparticular|superparticular numbers]]. This is due to the following useful identity:
[[math]]
\prod_{i=1}^{P-1}\frac{i+1}{i}=P
[[math]]
When considering ratios, and particularly when they are ratios for [[comma|commas]], it can be useful to introduce the notion of the **degree of epimericity**. 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 ratios with degree of epimoricity less than or equal to n.
See Also: [[ABC, High Quality Commas, and Epimericity]]Original HTML content:
<html><head><title>Superpartient</title></head><body><strong>Superpartient</strong> numbers are ratios of the form p/q, where p and q are relatively prime (so that the fraction is reduced to lowest terms), and p - q is greater than 1. In ancient Greece they were called epimeric (epimerēs) ratios, which is literally translated as "above a part." In ancient Greece, fractions like 3/1 and 7/1 were not considered to be epimeric ratios because of their additional restriction that <a class="wiki_link" href="/Harmonic">multiples of the fundamental</a> cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.<br />
<br />
All epimeric ratios can be constructed as products of <a class="wiki_link" href="/superparticular">superparticular numbers</a>. This is due to the following useful identity:<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
\prod_{i=1}^{P-1}\frac{i+1}{i}=P<br/>[[math]]
--><script type="math/tex">\prod_{i=1}^{P-1}\frac{i+1}{i}=P</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
When considering ratios, and particularly when they are ratios for <a class="wiki_link" href="/comma">commas</a>, it can be useful to introduce the notion of the <strong>degree of epimericity</strong>. 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 ratios with degree of epimoricity less than or equal to n.<br />
<br />
See Also: <a class="wiki_link" href="/ABC%2C%20High%20Quality%20Commas%2C%20and%20Epimericity">ABC, High Quality Commas, and Epimericity</a></body></html>