Superpartient ratio: Difference between revisions

Line 1:

'''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]] 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.

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|~~ABC, High Quality Commas, and Epimericity]] [[Category:epimeric]]

See Also: [[ABC, High Quality Commas, and Epimericity]]

[[Category:epimeric]]

[[Category:greek]]

[[Category:ratio]]

[[Category:superpartient]]

@@ Line 1: / Line 1: @@
 '''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]] 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.
+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|ABC, High Quality Commas, and Epimericity]]      [[Category:epimeric]]
+See Also: [[ABC, High Quality Commas, and Epimericity]]
+[[Category:epimeric]]
 [[Category:greek]]
 [[Category:ratio]]
 [[Category:superpartient]]