# Superparticular number

Superparticular numbers, also called epimoric ratios, are ratios of the form

${\displaystyle {n+1 \over n}=1+{1 \over n}.}$

Thus:

A superparticular number is when a great number contains a lesser number, to which it is compared, and at the same time one part of it. For example, when 3 and 2 are compared, they contain 2, plus the 3 has another 1, which is half of two. When 3 and 4 are compared, they each contain a 3, and the 4 has another 1, which is a third apart of 3. Again, when 5, and 4 are compared, they contain the number 4, and the 5 has another 1, which is the fourth part of the number 4, etc.

—Throop (2006), [1]

Superparticular numbers were written about by Nicomachus in his treatise "Introduction to Arithmetic". They are useful in the study of harmony: many musical intervals can be expressed as a superparticular ratio. In this application, Størmer's theorem can be used to list all possible superparticular numbers for a given limit; that is, all ratios of this type in which both the numerator and denominator are smooth numbers.

In graph theory, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise as the possible values of the upper density of an infinite graph.

These ratios are also important in visual harmony. Most flags of the world's countries have a ratio of 3:2 between their length and height. Aspect ratios of 4:3 and 3:2 are common in digital photography. Aspect ratios of 7:6 and 5:4 are used in medium format and large format photography respectively.

Examples
Ratio Name Related musical interval Audio
2:1 duplex octave
3:2 sesquialterum perfect fifth
4:3 sesquitertium perfect fourth
5:4 sesquiquartum major third
6:5 sesquiquintum minor third
9:8 sesquioctavum major second
10:9 sesquinona minor tone
16:15 just diatonic semitone
25:24 just chromatic semitone
81:80 syntonic comma
4375:4374 ragisma

The root of some of these terms comes from Latin sesqui- "one and a half" (from semis "a half" + -que "and") describing the ratio 3:2.