Superparticular number

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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.

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

[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.