Superparticular ratio

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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:

where n is a positive integer.


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 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[2] and the history of mathematics.[3]

Mathematical properties[edit]

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 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.[4]

The Wallis product

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:[5]

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.[6]

Other applications[edit]

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.[7] 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.[2]

These ratios are also important in visual harmony. Aspect ratios of 4:3 and 3:2 are common in digital photography,[8] and aspect ratios of 7:6 and 5:4 are used in medium format and large format photography respectively.[9]

Ratio names and related intervals[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:

Ratio Cents Name/musical interval Letter
from C
2:1 1200 duplex:[a] octave C' About this sound Play 
3:2 701.96 sesquialterum:[a] perfect fifth G About this sound Play 
4:3 498.04 sesquitertium:[a] perfect fourth F About this sound Play 
5:4 386.31 sesquiquartum:[a] major third E About this sound Play 
6:5 315.64 sesquiquintum:[a] minor third E About this sound Play 
7:6 266.87 septimal minor third E7 About this sound Play 
8:7 231.17 septimal major second D7 upside-down- About this sound Play 
9:8 203.91 sesquioctavum:[a] major second D About this sound Play 
10:9 182.40 sesquinona:[a] minor tone D- About this sound Play 
11:10 165.00 greater undecimal neutral second D- About this sound Play 
12:11 150.64 lesser undecimal neutral second D About this sound Play 
15:14 119.44 septimal diatonic semitone C7 upside-down About this sound Play 
16:15 111.73 just diatonic semitone D- About this sound Play 
17:16 104.96 minor diatonic semitone C17 About this sound Play 
21:20 84.47 septimal chromatic semitone D7 About this sound Play 
25:24 70.67 just chromatic semitone C About this sound Play 
28:27 62.96 septimal third-tone D7- About this sound Play 
32:31 54.96 31st subharmonic,
inferior quarter tone
D31U- About this sound Play 
49:48 35.70 septimal diesis D77 About this sound Play 
50:49 34.98 septimal sixth-tone B7 upside-down7 upside-down- About this sound Play 
64:63 27.26 septimal comma,
63rd subharmonic
C7 upside-down- About this sound Play 
81:80 21.51 syntonic comma C+ About this sound Play 
126:125 13.79 septimal semicomma D7 upside-downdouble flat About this sound Play 
128:127 13.58 127th subharmonic About this sound Play 
225:224 7.71 septimal kleisma B7 upside-down About this sound Play 
256:255 6.78 255th subharmonic D17 upside downdouble flat- About this sound Play 
4375:4374 0.40 ragisma C7- About this sound Play 

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.


  1. ^ a b c d e f g Ancient name


  1. ^ Throop, Priscilla (2006). Isidore of Seville's Etymologies: Complete English Translation, Volume 1, p. III.6.12, n. 7. ISBN 978-1-4116-6523-1.
  2. ^ a b 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. MR 0313189.
  3. ^ Robson, Eleanor; Stedall, Jacqueline (2008), The Oxford Handbook of the History of Mathematics, Oxford University Press, ISBN 9780191607448. 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.
  4. ^ Leonhard Euler; translated into English by Myra F. Wyman and Bostwick F. Wyman (1985), "An essay on continued fractions" (PDF), Mathematical Systems Theory, 18: 295–328, doi:10.1007/bf01699475. See in particular p. 304.
  5. ^ Debnath, Lokenath (2010), The Legacy of Leonhard Euler: A Tricentennial Tribute, World Scientific, p. 214, ISBN 9781848165267.
  6. ^ Erdős, P.; Stone, A. H. (1946). "On the structure of linear graphs". Bulletin of the American Mathematical Society. 52 (12): 1087–1091. doi:10.1090/S0002-9904-1946-08715-7.
  7. ^ Barbour, James Murray (2004), Tuning and Temperament: A Historical Survey, Courier Dover Publications, p. 23, ISBN 9780486434063, The paramount principle in Ptolemy's tunings was the use of superparticular proportion..
  8. ^ Ang, Tom (2011), Digital Photography Essentials, Penguin, p. 107, ISBN 9780756685263. 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.
  9. ^ 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. Schaub, George (1999), How to Photograph the Outdoors in Black and White, How to Photograph Series, 9, Stackpole Books, p. 43, ISBN 9780811724500.

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