Superparticular number

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

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

Mathematics of musical scales
Moodswinger, experimental musical instrument with a superparticular number-scale for it's overtoning positions.

[edit] References

Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music". American Mathematical Monthly 79: 1096–1100. doi:10.2307/2317424. MR 0313189.