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| {{Wikipedia|Superpartient ratio}}
| | #REDIRECT [[Delta-N ratio]] |
| In mathematics, a '''superpartient ratio''', also called an '''epimeric ratio''' or a '''delta-''d'' ratio''' (''d'' > 1), is a rational number that is greater than 1 and is not [[superparticular]].
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| More particularly, the ratio takes the form:
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| :<math>\frac{n + d}{n} = 1 + \frac{d}{n}</math>,
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| where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is [[Wikipedia:Coprime|coprime]] to <math>n</math>.
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| == Etymology ==
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| In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
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| [[Kite Giedraitis]] has proposed the term delta-1 (where [[delta]] means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.
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| == Definitions ==
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| 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.
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| == Properties ==
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| All superpartient ratios can be constructed as products of 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]]s, 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. [[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''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 ==
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| * [[Abc, high quality commas, and epimericity|''abc'', high quality commas, and epimericity]]
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| [[Category:Ratio]]
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| [[Category:Terms]] | | [[Category:Terms]] |
| [[Category:Greek]] | | [[Category:Ancient Greek music]] |