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

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

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". As well as having applications within mathematics, they are useful in the study of music theory and visual harmony.

## Applications

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.[2]

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

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.[citation needed] 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.[citation needed]

## Ratio names and related intervals

Examples
Ratio Name Related musical interval Audio
2:1 duplex octave  Play
3:2 sesquialterum perfect fifth  Play
4:3 sesquitertium perfect fourth  Play
5:4 sesquiquartum major third  Play
6:5 sesquiquintum minor third  Play
7:6 septimal minor third  Play
9:8 sesquioctavum major second  Play
10:9 sesquinona minor tone  Play
16:15 just diatonic semitone  Play
25:24 just chromatic semitone  Play
81:80 syntonic comma  Play
126:125 septimal semicomma  Play
4375:4374 ragisma  Play

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.

## Sources

1. ^ Throop, Priscilla (2006). Isidore of Seville's Etymologies: Complete English Translation, Volume 1, p.III.6.12,n.7. ISBN 978-1-4116-6523-1.
2. ^ Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music". American Mathematical Monthly (Mathematical Association of America) 79 (10): 1096–1100. doi:10.2307/2317424. JSTOR 2317424. MR0313189.
3. ^ Erdős, P.; Stone, A. H. (1946). "On the structure of linear graphs". Bulletin of the American Mathematical Society 52 (12): 1087–1091. doi:10.1090/S0002-9904-1946-08715-7.
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