Superparticular number

From Wikipedia, the free encyclopedia

Superparticular numbers, also called epimoric ratios, are improper vulgar fractions of the form

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

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 width, aspect ratios of 4:3, and 3:2 are common in digital photography, and aspect ratios of 7:6 and 5:4 are used in medium format and large format photography respectively.

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

Examples
Ratio Name Related musical interval
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
18:17 (super)sesquiseptimus decimus semitone

[edit] See also

[edit] References

[edit] External links