Superpartient ratio: Difference between revisions

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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.

'''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]] 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>

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 epimoricity n, there are only finitely many p-limit ratios with degree of epimoricity less than or equal to n.

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 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]]

== See also ==

[[Category:~~epimeric~~]]

* [[ABC, High Quality Commas, and Epimericity]]

[[Category:~~greek~~]]

[[Category:~~ratio~~]]

[[Category:Terms]]

[[Category:~~superpartient~~]]

[[Category:Greek]]

[[Category:Ratio]]

[[Category:Superpartient]]

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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.
+'''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]] numbers. This is due to the following useful identity:
@@ Line 5: / Line 5: @@
 <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]]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 epimoricity n, there are only finitely many p-limit ratios with degree of epimoricity less than or equal to n.
+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 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]]
+== See also ==
-[[Category:epimeric]]
+* [[ABC, High Quality Commas, and Epimericity]]
-[[Category:greek]]
-[[Category:ratio]]
+[[Category:Terms]]
-[[Category:superpartient]]
+[[Category:Greek]]
+[[Category:Ratio]]
+[[Category:Superpartient]]