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), 

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
3:2	sesquialterum	perfect fifth	Play
4:3	sesquitertium	perfect fourth	Play
5:4	sesquiquartum	major third	Play
6:5	sesquiquintum	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
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.

See also

• Mathematics of musical scales
• Superpartient number
• Just intonation

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.
• Superparticular numbers applied to construct pentatonic scales by David Canright.
• De Institutione Arithmetica, liber II by Anicius Manlius Severinus Boethius