Superparticular number

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". 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	Play (help·info)
3:2	sesquialterum	perfect fifth	Play (help·info)
4:3	sesquitertium	perfect fourth	Play (help·info)
5:4	sesquiquartum	major third	Play (help·info)
6:5	sesquiquintum	minor third	Play (help·info)
9:8	sesquioctavum	major second	Play (help·info)
10:9	sesquinona	minor tone	Play (help·info)
16:15		just diatonic semitone	Play (help·info)
25:24		just chromatic semitone	Play (help·info)
81:80		syntonic comma	Play (help·info)
4375:4374		ragisma	Play (help·info)

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.

See also

• Mathematics of musical scales

Sources

1. ^ Throop, Priscilla (2006). Isidore of Seville's Etymologies: Complete English Translation, Volume 1, p.III.6.12,n.7. ISBN 9781411665231.

Further reading

• 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. 

External links

• An Arithmetical Rubric by Siemen Terpstra, about the application of superparticular numbers to harmony.^{[dead link]}
• Superparticular numbers applied to construct pentatonic scales by David Canright.
• De Institutione Arithmetica, liber II by Anicius Manlius Severinus Boethius