# Delta-N ratio

**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]\frac{n + d}{n} = 1 + \frac{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]

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*.