In mathematics, a superpartient ratio, also called superpartient number or epimeric ratio, is a

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 (for example, The set of all ...

that is greater than one and is not superparticular. The term has fallen out of use in modern pure mathematics, but continues to be used in

music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...

and in the historical study of mathematics. Superpartient ratios were written about by

Nicomachus Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his ...

in his treatise ''

Introduction to Arithmetic Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagoreanism, Neopythagorean philosopher from Gerasa, in the Syria (Roman province), Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical pr ...

''.

Overview

Mathematically, a superpartient number is a

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 the form :

\frac\, ,

where ''a'' is greater than 1 (''a'' > 1) and is also

coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

to ''n''. Ratios of the form

\tfrac

are also greater than one and fully reduced, but are called superparticular ratios and are not superpartient.

Etymology

"Superpartient" comes from

Greek Greek may refer to: Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of all kno ...

ἐπιμερής ''epimeres'' "containing a whole and a fraction," literally "superpartient".

Overview

Etymology

See also

Further reading