Delta-N ratio
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This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. The corresponding expert page for this topic is Abc, high quality commas, and epimericity. |
The delta of a ratio is simply the difference between its numerator and its denominator. (Delta is also known as degree of epimoricity.) A ratio with a delta of N is called a delta-N ratio.
| Delta-1 ratios | 2/1 | 3/2 | 4/3 | 5/4 | 6/5 | 7/6 | etc. |
|---|---|---|---|---|---|---|---|
| Delta-2 ratios | 3/1 | 5/3 | 7/5 | 9/7 | 11/9 | 13/11 | etc. |
| Delta-3 ratios | 4/1 | 5/2 | 7/4 | 8/5 | 10/7 | 11/8 | etc. |
| Delta-4 ratios | 5/1 | 7/3 | 9/5 | 11/7 | 13/9 | 15/11 | etc. |
Thus superparticular ratios are delta-1 ratios, and superpartient ratios are all ratios except delta-1 ratios. The delta-N terminology was coined by Kite Giedraitis.
More particularly, a superpartient ratio takes the form:
- [math]\dfrac{n + d}{n} = 1 + \dfrac{d}{n}[/math],
where [math]n[/math] and [math]d[/math] are positive integers, [math]d \gt 1[/math] and [math]d[/math] is coprime to [math]n[/math].
Etymology
In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
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.
Superpartient subcategories
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 commas.
These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in delta-N terminology as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.
Examples
- Delta-2 (superbipartient) ratios: 3/1, 5/3, 7/5, 9/7, 11/9, 13/11, etc.
- Delta-3 (supertripartient) ratios: 4/1, 5/2, 7/4, 8/5, 10/7, 11/8, etc.
- Delta-4 (superquadripartient) ratios: 5/1, 7/3, 9/5, 11/7, 13/9, 15/11, etc.
Properties
All superpartient 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]
Likewise, you can also express any delta-N ratio as a product of any number of delta-M ratios with M being a divisor of N. For example,
- 5/3 = (10/8) (8/6) = (5/4) (4/3) — can’t get delta-2 because we have an even number of factors.
- 5/3 = (15/13) (13/11) (11/9).
- 5/3 = (20/18) (18/16) (16/14) (14/12) = (10/9) (9/8) (8/7) (7/6) — again only delta-1.
- 5/3 = (25/23) (23/21) (21/19) (19/17) (17/15).
- 4/1 = (8/5) (5/2).
- 4/1 = (12/9) (9/6) (6/3) = (4/3) (3/2) (2/1) — now we can’t get delta-3 because there are 3 factors.
- 4/1 = (16/13) (13/10) (10/7) (7/4).
Also, if you factorize like this into K factors, then each of them into L factors, you get the same as if you directly factored into K L factors (including their order).
- The general formula for this factorization is [math]\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A[/math]. Here you can see more clearly that actual delta of factors will be [math]N / \operatorname{gcd}(K, N)[/math].
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.