mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a superparticular ratio, also called a superparticular number or epimoric ratio, is the

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

of two consecutive

integer number An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s. More particularly, the ratio takes the form: :

\frac = 1 + \frac

where is a

positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

. Thus: Superparticular ratios were written about by

Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...

in his treatise ''

Introduction to Arithmetic The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus (60–120 AD). Summary The work contains both philosophical prose and ...

''. Although these numbers have applications in modern

pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...

, the areas of study that most frequently refer to the superparticular ratios by this name are

music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...

and the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

Mathematical properties

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

observed, the superparticular numbers (including also the multiply superparticular ratios, numbers formed by adding an integer other than one to a

unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, et ...

) are exactly the

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s whose

continued fraction In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writ ...

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 In mathematics, the Wallis product for , published in 1656 by John Wallis, states that :\begin \frac & = \prod_^ \frac = \prod_^ \left(\frac \cdot \frac\right) \\ pt& = \Big(\frac \cdot \frac\Big) \cdot \Big(\frac \cdot \frac\Big) \cdot \Big(\fr ...

\prod_^ \left(\frac \cdot \frac\right) = \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots = \frac\cdot\frac\cdot\frac\cdots=2\cdot\frac\cdot\frac\cdot\frac\cdots=\frac

represents the

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...

in several ways as a product of superparticular ratios and their inverses. It is also possible to convert the

Leibniz formula for π In mathematics, the Leibniz formula for , named after Gottfried Leibniz, states that 1-\frac+\frac-\frac+\frac-\cdots=\frac, an alternating series. It is also called the Madhava–Leibniz series as it is a special case of a more general serie ...

into an

Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eul ...

of superparticular ratios in which each term has a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

as its numerator and the nearest multiple of four as its denominator: :

\frac = \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot\frac\cdots

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise via the

Erdős–Stone theorem In extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an ''H''-free graph for a non-complete graph ''H''. It is named after Paul Erdős and Arthur Stone, who pr ...

as the possible values of the

upper density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...

of an infinite graph.

Other applications

In the study of

harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...

, many musical intervals can be expressed as a superparticular ratio (for example, due to

octave equivalency In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

, 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 Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...

's formulation of musical harmony. In this application,

Størmer's theorem In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equ ...

can be used to list all possible superparticular numbers for a given

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

; that is, all ratios of this type in which both the numerator and denominator are

smooth number In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × ...

s. These ratios are also important in visual harmony. Aspect ratios of 4:3 and 3:2 are common in

digital photography Digital photography uses cameras containing arrays of electronic photodetectors interfaced to an analog-to-digital converter (ADC) to produce images focused by a lens, as opposed to an exposure on photographic film. The digitized image is sto ...

, and aspect ratios of 7:6 and 5:4 are used in

medium format Medium format has traditionally referred to a film format in photography and the related cameras and equipment that use film. Nowadays, the term applies to film and digital cameras that record images on media larger than the used in 35&nbs ...

and

large format Large format refers to any imaging format of or larger. Large format is larger than "medium format", the or size of Hasselblad, Mamiya, Rollei, Kowa, and Pentax cameras (using 120- and 220-roll film), and much larger than the frame o ...

photography respectively.The 7:6 medium format aspect ratio is one of several ratios possible using medium-format

120 film 120 is a film format for still photography introduced by Kodak for their '' Brownie No. 2'' in 1901. It was originally intended for amateur photography but was later superseded in this role by 135 film. 120 film survives to this day as the only ...

, 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. .

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) A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''. Pitched musical instruments are often based on an acoustic resonator su ...

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.

Notes

Citations

External links

Superparticular numbers
applied to construct

pentatonic scale A pentatonic scale is a musical scale with five notes per octave, in contrast to the heptatonic scale, which has seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many ancien ...

s b
David Canright

by

Anicius Manlius Severinus Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...

{{Rational numbers Rational numbers *