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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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| == 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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| == Superpartient subcategories ==
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| Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.
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| These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in [[Delta-N|delta-N terminology]] as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.
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| === Examples ===
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| * Delta-2 (superbipartient) ratios: [[3/1]], [[5/3]], [[7/5]], [[9/7]], [[11/9]], [[13/11]], etc.
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| * Delta-3 (supertripartient) ratios: [[4/1]], [[5/2]], [[7/4]], [[8/5]], [[10/7]], [[11/8]], etc.
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| * Delta-4 (superquadripartient) ratios: [[5/1]], [[7/3]], [[9/5]], [[11/7]], [[13/9]], [[15/11]], etc.
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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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| [[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]] |