Superparticular numbers, also called epimoric ratios, are ratio

Ratio

In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

s of the form

Thus:
Superparticular numbers were written about by Nicomachus

Nicomachus

Nicomachus was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa, in the Roman province of Syria , and was strongly influenced by Aristotle...

 in his treatise "Introduction to Arithmetic

Introduction to Arithmetic

Introduction to Arithmetic was written by Nicomachus almost two thousand years ago, and contains both philosophical prose and very basic mathematical ideas. Nicomachus refers to Plato quite often, and wrote about how philosophy can only be possible if one knows enough about mathematics. This is...

". They are useful in the study of harmony

Harmony

In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...

: many musical interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

s can be expressed as a superparticular ratio. In this application, Størmer's theorem

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 equations...

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

Limit (music)

In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...

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

Smooth number

In number theory, a smooth number is an integer which factors completely into small prime numbers. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography relying on factorization.-Definition:...

s.

In graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise as the possible values of the upper density

Dense graph

In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a sparse graph...

 of an infinite graph.

These ratios are also important in visual harmony. Most flag

Flag

A flag is a piece of fabric with a distinctive design that is usually rectangular and used as a symbol, as a signaling device, or decoration. The term flag is also used to refer to the graphic design employed by a flag, or to its depiction in another medium.The first flags were used to assist...

s of the world's countries have a ratio of 3:2 between their length and height. Aspect ratios

Aspect ratio (image)

The aspect ratio of an image is the ratio of the width of the image to its height, expressed as two numbers separated by a colon. That is, for an x:y aspect ratio, no matter how big or small the image is, if the width is divided into x units of equal length and the height is measured using this...

 of 4:3 and 3:2 are common in digital photography

Digital photography

Digital photography is a form of photography that uses an array of light sensitive sensors to capture the image focused by the lens, as opposed to an exposure on light sensitive film...

. Aspect ratios of 7:6 and 5:4 are used in medium format and large format

Large format

Large format refers to any imaging format of 4×5 inches or larger. Large format is larger than "medium format", the 6×6 cm or 6×9 cm size of Hasselblad, Rollei, Kowa, Pentax etc cameras , and much larger than the 24×36 mm frame of 35 mm format.The main advantage...

 photography respectively.

Examples

Ratio	Name	Related musical interval	Audio
2:1	duplex	octave Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
3:2	sesquialterum	perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
4:3	sesquitertium	perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
5:4	sesquiquartum	major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
6:5	sesquiquintum	minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
9:8	sesquioctavum	major second Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions...
10:9	sesquinona	minor tone
16:15		just diatonic semitone Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
25:24		just chromatic semitone
81:80		syntonic comma Syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
4375:4374		ragisma Ragisma In music and tuning, the ragisma is an interval with the ratio of 4375:4374, ≈0.396 cents. It is usually defined as the difference between the septimal minor third and two Bohlen-Pierce small semitones . It is also the difference between minor Bohlen-Pierce diesis and septimal semicomma ....

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.

Further reading

| doi = 10.2307/2317424
| issue = 10
| publisher = Mathematical Association of America
| jstor = 2317424}}

External links

• An Arithmetical Rubric by Siemen Terpstra, about the application of superparticular numbers to harmony.
• Superparticular numbers applied to construct pentatonic scale
  Pentatonic scale
  A pentatonic scale is a musical scale with five notes per octave in contrast to a heptatonic scale such as the major scale and minor scale...
  
  s by David Canright.
• De Institutione Arithmetica, liber II by Anicius Manlius Severinus Boethius
  Anicius Manlius Severinus Boethius
  Anicius Manlius Severinus Boëthius, commonly called Boethius was a philosopher of the early 6th century. He was born in Rome to an ancient and important family which included emperors Petronius Maximus and Olybrius and many consuls. His father, Flavius Manlius Boethius, was consul in 487 after...