Superparticular ratio

In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is a ratio of the form

$\frac{n + 1}{n} = 1 + \frac{1}{n}$ where n is a positive integer.

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 ratios were written about by Nicomachus in his treatise "Introduction to Arithmetic". Although these numbers have applications in modern pure mathematics, the areas of study that most frequently refer to the superparticular ratios by this name are music theory and the history of mathematics.

Mathematical properties

As Euler observed, the superparticular numbers (including also the multiply superparticular ratios, numbers formed by adding an integer other than one to a unit fraction) are exactly the rational numbers whose continued fraction terminates after two terms. The numbers whose continued fraction terminates in one term are the integers, while the remaining numbers, with three or more terms in their continued fractions, are superpartient.

The Wallis product

$\prod_{n=1}^{\infty} \left(\frac{2n}{2n-1} \cdot \frac{2n}{2n+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots = \frac{4}{3}\cdot\frac{16}{15}\cdot\frac{36}{35}\cdots=2\cdot\frac{8}{9}\cdot\frac{24}{25}\cdot\frac{48}{49}\cdots=\frac{\pi}{2}$

represents the irrational number π in several ways as a product of superparticular ratios and their inverses. It is also possible to convert the Leibniz formula for π into an Euler product of superparticular ratios in which each term has a prime number as its numerator and the nearest multiple of four as its denominator:

$\pi/4=\frac{3}{4} \cdot \frac{5}{4} \cdot \frac{7}{8} \cdot \frac{11}{12} \cdot \frac{13}{12} \cdot\frac{17}{16}\cdots$

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.

Other applications

In the study of harmony, many musical intervals can be expressed as a superparticular ratio. Indeed, whether a ratio was superparticular was the most important criterion in Ptolemy's formulation of musical harmony. 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. 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.

Ratio names and related intervals

Every pair of adjacent positive integers represent a superparticular ratio, and similarly every pair of adjacent harmonics in the harmonic series (music) represent a superparticular ratio. Many individual superparticular ratios have their own names, either in historical mathematics or in music theory. These include the following:

Examples

Ratio	Name	Related musical interval	Audio
2:1	duplex	octave	<phonos file="Perfect octave on C.mid">Play</phonos>
3:2	sesquialterum	perfect fifth	<phonos file="Just perfect fifth on C.mid">Play</phonos>
4:3	sesquitertium	perfect fourth	<phonos file="Just perfect fourth on C.mid">Play</phonos>
5:4	sesquiquartum	major third	<phonos file="Just major third on C.mid">Play</phonos>
6:5	sesquiquintum	minor third	<phonos file="Just minor third on C.mid">Play</phonos>
7:6		septimal minor third	<phonos file="Septimal minor third on C.mid">Play</phonos>
8:7		septimal major second	<phonos file="Septimal major second on C.mid">Play</phonos>
9:8	sesquioctavum	major second	<phonos file="Major second on C.mid">Play</phonos>
10:9	sesquinona	minor tone	<phonos file="Minor tone on C.mid">Play</phonos>
11:10		greater undecimal neutral second	<phonos file="Greater undecimal neutral second on C.mid">Play</phonos>
12:11		lesser undecimal neutral second	<phonos file="Lesser undecimal neutral second on C.mid">Play</phonos>
15:14		septimal diatonic semitone	<phonos file="Septimal diatonic semitone on C.mid">Play</phonos>
16:15		just diatonic semitone	<phonos file="Just diatonic semitone on C.mid">Play</phonos>
21:20		Septimal chromatic semitone	<phonos file="septimal chromatic semitone on C.mid">Play</phonos>
25:24		just chromatic semitone	<phonos file="Just chromatic semitone on C.mid">Play</phonos>
28:27		septimal third-tone
49:48		septimal diesis	<phonos file="Septimal diesis on C.mid">Play</phonos>
50:49		Septimal sixth-tone
81:80		syntonic comma	<phonos file="Syntonic comma on C.mid">Play</phonos>
126:125		septimal semicomma	<phonos file="Septimal semicomma on C.mid">Play</phonos>
225:224		septimal kleisma
4375:4374		ragisma

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.

External links

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