View source for Superparticular ratio

[[File:Just diatonic semitone on C.png|thumb|Just diatonic semitone on C: {{frac|16|15}} = {{frac|15+1|15}} = 1+{{frac|1|15}} {{audio|Just diatonic semitone on C.mid|Play}}]]

In mathematics, a '''superparticular ratio''', also called a '''superparticular number''' or '''epimoric ratio''', is the [[ratio]] of two consecutive [[integer number]]s.

More particularly, the ratio takes the form:
:<math>\frac{n + 1}{n} = 1 + \frac{1}{n}</math>  where {{mvar|n}} is a [[positive integer]].

Thus:
{{quote|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)|<ref>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}}.</ref>}}

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>{{citation|title=The Oxford Handbook of the History of Mathematics|first1=Eleanor|last1=Robson|author1-link=Eleanor Robson|first2=Jacqueline|last2=Stedall|publisher=Oxford University Press|year=2008|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.</ref>

==Mathematical properties==
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 number|superpartient]].<ref>{{citation|title=An essay on continued fractions|author=Leonhard Euler; translated into English by Myra F. Wyman and Bostwick F. Wyman |lastauthoramp=yes |journal=Mathematical Systems Theory|volume=18|year=1985|pages=295–328|url=https://kb.osu.edu/dspace/bitstream/handle/1811/32133/Wyman.pdf|doi=10.1007/bf01699475}}. See in particular p.&nbsp;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|{{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>{{citation|title=The Legacy of Leonhard Euler: A Tricentennial Tribute|first=Lokenath|last=Debnath|publisher=World Scientific|year=2010|isbn=9781848165267|page=214|url=https://books.google.com/books?id=K2liU-SHl6EC&pg=PA214}}.</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 [[dense graph|upper density]] of an infinite graph.<ref>{{cite journal |last=Erdős |first=P. |authorlink=Paul Erdős |author2=Stone, A. H. |authorlink2=Arthur Stone (mathematician)  |year=1946 |title=On the structure of linear graphs |journal=[[Bulletin of the American Mathematical Society]] |volume=52  |pages=1087–1091 |doi=10.1090/S0002-9904-1946-08715-7 |issue=12}}</ref>

==Other applications==
In the study of [[harmony]], many musical [[interval (music)|interval]]s 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>{{citation|title=Tuning and Temperament: A Historical Survey|first=James Murray|last=Barbour|publisher=Courier Dover Publications|year=2004|isbn=9780486434063|page=23|url=https://books.google.com/books?id=G-pG77pmlp4C&pg=PA23|quotation=The paramount principle in Ptolemy's tunings was the use of superparticular proportion.}}.</ref> In this application, [[Størmer's theorem]] can be used to list all possible superparticular numbers for a given [[limit (music)|limit]]; that is, all ratios of this type in which both the numerator and denominator are [[smooth number]]s.<ref name="hh">{{cite journal
 |author1=Halsey, G. D. |author2=Hewitt, Edwin | title = More on the superparticular ratios in music
 | journal = [[American Mathematical Monthly]]
 | volume = 79
 | year = 1972
 | pages = 1096–1100
 |mr=0313189
 | doi = 10.2307/2317424
 | issue = 10
 | jstor = 2317424}}</ref>

These ratios are also important in visual harmony.  [[aspect ratio (image)|Aspect ratios]] of 4:3 and 3:2 are common in [[digital photography]],<ref>{{citation|title=Digital Photography Essentials|first=Tom|last=Ang|publisher=Penguin|year=2011|isbn=9780756685263|page=107|url=https://books.google.com/books?id=2EF4t2qkd5cC&pg=PT108}}. 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 (film)|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&times;5 inches and 8&times;10 inches. See e.g. {{citation|title=How to Photograph the Outdoors in Black and White|volume=9|series=How to Photograph Series|first=George|last=Schaub|publisher=Stackpole Books|year=1999|isbn=9780811724500|page=43|url=https://books.google.com/books?id=3uv3iRGXjH4C&pg=PA43}}.</ref>

==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:

{| class="wikitable"
|+Examples
! Ratio
! Cents
! Name/musical interval
! [[Ben Johnston notation|Ben Johnston<br/>notation]] above C
! Audio
|-
| 2:1 || 1200 || duplex:{{efn|name=Ancient name|Ancient name}} [[octave]] || C' || {{audio|Perfect octave on C.mid|Play}}
|- 
| 3:2 || 701.96 || sesquialterum:{{efn|name=Ancient name}} [[perfect fifth]] || G || {{audio|Just perfect fifth on C.mid|Play}}
|-
| 4:3 || 498.04 || sesquitertium:{{efn|name=Ancient name}} [[perfect fourth]] || F || {{audio|Just perfect fourth on C.mid|Play}}
|-
| 5:4 || 386.31 || sesquiquartum:{{efn|name=Ancient name}} [[major third]] || E || {{audio|Just major third on C.mid|Play}}
|-
| 6:5 || 315.64 || sesquiquintum:{{efn|name=Ancient name}} [[minor third]] || E{{music|b}} || {{audio|Just minor third on C.mid|Play}}
|-
| 7:6 || 266.87 || [[septimal minor third]] || E{{music|7}}{{music|b}} || {{audio|Septimal minor third on C.mid|Play}}
|-
| 8:7 || 231.17 || [[septimal major second]] || D{{music|L}}{{music|minus}} || {{audio|Septimal major second on C.mid|Play}}
|-
| 9:8 || 203.91 || sesquioctavum:{{efn|name=Ancient name}} [[major second]] || D || {{audio|Major second on C.mid|Play}}
|-
| 10:9 || 182.40 || sesquinona:{{efn|name=Ancient name}} minor tone || D{{music|minus}} || {{audio|Minor tone on C.mid|Play}}
|-
| 11:10 || 165.00 || greater undecimal neutral second || D{{music|up}}{{music|b}}{{music|minus}} || {{audio|Greater undecimal neutral second on C.mid|Play}}
|-
| 12:11 || 150.64 || lesser undecimal neutral second || D{{music|11U}} || {{audio|Lesser undecimal neutral second on C.mid|Play}}
|-
| 15:14 || 119.44 || [[septimal diatonic semitone]] || C{{music|L}}{{music|#}} || {{audio|Septimal diatonic semitone on C.mid|Play}}
|-
| 16:15 || 111.73 || [[just intonation|just]] diatonic [[semitone]] || D{{music|b}}{{music|minus}} || {{audio|Just diatonic semitone on C.mid|Play}}
|-
| 17:16 || 104.96 || [[minor diatonic semitone]] || C{{music|17}}{{music|#}} || {{audio|Minor diatonic semitone on C.mid|Play}}
|-
| 21:20 || 84.47 || [[septimal chromatic semitone]] || D{{music|7}}{{music|b}} || {{audio|septimal chromatic semitone on C.mid|Play}}
|-
| 25:24 || 70.67 || just chromatic semitone || C{{music|#}} || {{audio|Just chromatic semitone on C.mid|Play}}
|-
| 28:27 || 62.96 || [[septimal third-tone]] || D{{music|7}}{{music|b}}{{music|minus}} || {{audio|Septimal third-tone on C.mid|Play}}
|-
| 32:31 || 54.96 || 31st [[subharmonic]],<br/> inferior quarter tone  || D{{music|31U}}{{music|b}}{{music|minus}} || {{audio|Thirty-first subharmonic on C.mid|Play}}
|-
| 49:48 || 35.70 || [[septimal diesis]] || D{{music|7}}{{music|7}}{{music|b}} || {{audio|Septimal diesis on C.mid|Play}}
|-
| 50:49 || 34.98 || [[septimal sixth-tone]] || B{{music|L}}{{music|L}}{{music|#}}{{music|minus}} || {{audio|Septimal sixth-tone on C.mid|Play}}
|-
| 64:63 || 27.26 || [[septimal comma]],<br/> 63rd subharmonic  || C{{music|L}}{{music|minus}} || {{audio|Septimal comma on C.mid|Play}}
|-
| 81:80 || 21.51 || [[syntonic comma]] || C{{music|plus}} || {{audio|Syntonic comma on C.mid|Play}}
|-
| 126:125 || 13.79 || [[septimal semicomma]] || D{{music|L}}{{music|bb}} || {{audio|Septimal semicomma on C.mid|Play}}
|-
| 128:127 || 13.58 || 127th subharmonic ||  || {{audio|127th subharmonic on C.mid|Play}}
|-
| 225:224 || 7.71 || [[septimal kleisma]] || B{{music|L}}{{music|#}} || {{audio|Septimal kleisma on C.mid|Play}}
|-
| 256:255 || 6.78 || 255th subharmonic || D{{music|17U}}{{music|bb}}{{music|minus}} || {{audio|255th subharmonic on C.mid|Play}}
|-
| 4375:4374 || 0.40 || [[ragisma]] || C{{music|7}}{{music|#}}{{music|minus}} || {{audio|Ragisma on C.mid|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.

==Notes==
{{notelist}}

==Citations==
{{reflist|2}}

==External links==
* [https://home.comcast.net/~dcanright/super/ Superparticular numbers] applied to construct [[pentatonic scale]]s by [https://home.comcast.net/~dcanright/ David Canright].
* [http://www.music.indiana.edu/tml/6th-8th/BOEARI2_TEXT.html ''De Institutione Arithmetica, liber II''] by [[Anicius Manlius Severinus Boethius]]

{{Rational numbers}}

[[Category:Rational numbers]]
[[Category:Superparticular intervals|*]]