Editing Music theory (section)

===Mathematics===
{{Main|Music and mathematics}}
Music theorists sometimes use mathematics to understand music, and although music has no [[axiomatic]] foundation in modern mathematics, mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".{{sfn|Smith Brindle|1987|loc=42–43}} The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of [[set theory]], [[abstract algebra]] and [[number theory]]. Some composers have incorporated the [[golden ratio]] and [[Fibonacci numbers]] into their work.{{sfn|Smith Brindle|1987|loc=chapter 6, ''passim''}}{{sfn|Garland and Kahn|1995|loc={{Page needed|date=July 2015}}}} There is a long history of examining the relationships between music and mathematics. Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,{{sfn|Smith Brindle|1987|loc=42}} the [[Pythagoreanism|Pythagoreans]] (in particular [[Philolaus]] and [[Archytas]]){{sfn|Purwins|2005|loc=22–24}} of ancient Greece were the first researchers known to have investigated the expression of [[musical scale]]s in terms of numerical [[ratio]]s.
[[File:HarmonicIdentities.Names.Frequencies.svg|thumb|right|400px|The first 16 harmonics, their names and frequencies, showing the exponential nature of the octave and the simple fractional nature of non-octave harmonics]]
In the modern era, musical [[set theory]] uses the language of mathematical set theory in an elementary way to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as [[transposition (music)|transposition]] and [[Melodic inversion|inversion]], one can discover deep structures in the music. Operations such as transposition and inversion are called [[isometries]] because they preserve the intervals between tones in a set. Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the pitch classes in an equally tempered octave form an [[abelian group]] with 12 elements. It is possible to describe [[just intonation]] in terms of a [[free abelian group]].{{sfn|Wohl|2005}}