Superpartient 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.
More particularly, the ratio takes the form:
- [math]\displaystyle{ \frac{n + d}{n} = 1 + \frac{d}{n} }[/math],
where [math]\displaystyle{ n }[/math] and [math]\displaystyle{ d }[/math] are positive integers, [math]\displaystyle{ d \gt 1 }[/math] and [math]\displaystyle{ d }[/math] is coprime to [math]\displaystyle{ n }[/math].
Etymology
In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
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]\displaystyle{ \frac{n+2}{n} }[/math], likewise delta-3, delta-4, etc.
Definitions
In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that multiples of the fundamental cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.
Properties
All superpartient ratios can be constructed as products of superparticular numbers. This is due to the following useful identity:
[math]\displaystyle{ \displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P }[/math]
When considering ratios, and particularly when they are ratios for commas, it can be useful to introduce the notion of the degree of epimoricity (not to be confused with epimericity – see link below), or delta(proposed by Kite Giedraitis). 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. 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.