|
|
| Line 1: |
Line 1: |
| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | '''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. |
| 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>2012-09-10 11:46:47 UTC</tt>.<br>
| |
| : The original revision id was <tt>363440596</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>
| |
| <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 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: | | 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 epimoricity** (not to be confused with //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.
| | <math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math> |
|
| |
|
| See Also: [[ABC, High Quality Commas, and Epimericity]]</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 epimoricity''' (not to be confused with ''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. |
| <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><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 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 />
| | See Also: [[ABC,_High_Quality_Commas,_and_Epimericity|ABC, High Quality Commas, and Epimericity]] [[Category:epimeric]] |
| <br />
| | [[Category:greek]] |
| 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 />
| | [[Category:ratio]] |
| <!-- ws:start:WikiTextMathRule:0:
| | [[Category:superpartient]] |
| [[math]]&lt;br/&gt;
| |
| \displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P&lt;br/&gt;[[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 epimoricity</strong> (not to be confused with <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></pre></div> | |
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 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 numbers. This is due to the following useful identity:
[math]\displaystyle{ \displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P }[/math]
When considering ratios, and particularly when they are ratios for commas, it can be useful to introduce the notion of the degree of epimoricity (not to be confused with 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. 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