Music and mathematics

A spectrogram of a violin waveform, with linear frequency on the vertical axis and time on the horizontal axis. The bright lines show how the spectral components change over time. The intensity colouring is logarithmic 

Music theory has no axiomatic foundation in modern mathematics, yet the basis of musical sound can be described mathematically (in acoustics) and exhibits "a remarkable array of number properties".Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.

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.

Time, rhythm and meterEdit

Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetition, accent, phrase and duration – music would not be possible. Modern musical use of terms like meter and measure also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and periodicity that is fundamental to physics.

The elements of musical form often build strict proportions or hypermetric structures (powers of the numbers 2 and 3).

Connections to mathematicsEdit

Set theoryEdit

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 and 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.

Abstract algebraEdit

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.

Transformational theory is a branch of music theory developed by David Lewin. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves.

Theorists have also proposed musical applications of more sophisticated algebraic concepts. The theory of regular temperaments has been extensively developed with a wide range of sophisticated mathematics, for example by associating each regular temperament with a rational point on a Grassmannian.

The chromatic scale has a free and transitive action of the cyclic group ${\displaystyle \mathbb {Z} /12\mathbb {Z} }$, with the action being defined via transposition of notes. So the chromatic scale can be thought of as a torsor for the group ${\displaystyle \mathbb {Z} /12\mathbb {Z} }$.

Real and complex analysisEdit

Real and complex analysis have also been made use of, for instance by applying the theory of the Riemann zeta function to the study of equal divisions of the octave.