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| '''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.
| | #REDIRECT [[Delta-N ratio]] |
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| All epimeric ratios can be constructed as products of [[superparticular|superparticular numbers]]. This is due to the following useful identity:
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| <math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>
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| 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.
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| See Also: [[ABC,_High_Quality_Commas,_and_Epimericity|ABC, High Quality Commas, and Epimericity]] [[Category:epimeric]]
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| [[Category:greek]]
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| [[Category:superpartient]] | |