Superparticular ratio
In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers.
More particularly, the ratio takes the form:
- <math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math> where Template:Mvar is a positive integer.
Thus: Template:Quote
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<ref name="hh"/> and the history of mathematics.<ref>Template:Citation. On pp. 123–124 the book discusses the classification of ratios into various types including the superparticular ratios, and the tradition by which this classification was handed down from Nichomachus to Boethius, Campanus, Oresme, and Clavius.</ref>
Mathematical propertiesEdit
As Leonhard 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 simple 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.<ref>Template:Citation. See in particular p. 304.</ref>
The Wallis product
- <math> \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}</math>
represents the irrational number [[pi|Template:Pi]] 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:<ref>Template:Citation.</ref>
- <math>\frac{\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</math>
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.<ref>Template:Cite journal</ref>
Other applicationsEdit
In the study of harmony, many musical intervals can be expressed as a superparticular ratio (for example, due to octave equivalency, the ninth harmonic, 9/1, may be expressed as a superparticular ratio, 9/8). Indeed, whether a ratio was superparticular was the most important criterion in Ptolemy's formulation of musical harmony.<ref>Template:Citation.</ref> 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.<ref name="hh">Template:Cite journal</ref>
These ratios are also important in visual harmony. Aspect ratios of 4:3 and 3:2 are common in digital photography,<ref>Template:Citation. Ang also notes the 16:9 (widescreen) aspect ratio as another common choice for digital photography, but unlike 4:3 and 3:2 this ratio is not superparticular.</ref> and aspect ratios of 7:6 and 5:4 are used in medium format and large format photography respectively.<ref>The 7:6 medium format aspect ratio is one of several ratios possible using medium-format 120 film, and the 5:4 ratio is achieved by two common sizes for large format film, 4×5 inches and 8×10 inches. See e.g. Template:Citation.</ref>
Edit
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:
The root of some of these terms comes from Latin sesqui- "one and a half" (from semis "a half" and -que "and") describing the ratio 3:2.
NotesEdit
CitationsEdit
External linksEdit
- Superparticular numbers applied to construct pentatonic scales by David Canright.
- De Institutione Arithmetica, liber II by Anicius Manlius Severinus Boethius