Superpartient ratio
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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 5/1 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]]
\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {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** (not to be confused with plain //epimericity// - see link below). 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 5/1 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/>
\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P<br/>[[math]]
--><script type="math/tex">\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {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> (not to be confused with plain <em>epimericity</em> - see link below). 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>