Music and mathematics

There are arguments that mathematics can be used to analyse and understand music, and at its core, to compose the music itself.

Connections to group theory

For any given octave, the standard musical notes form a commutative group with 12 elements.

It is possible to describe just intonation in terms of free abelian group.

Connections to number theory

Time signatures are usually given in the form ${\frac {n}{2^{m}}}$ where n and m are positive whole numbers. The most natural and usual time signatures are those where n is divisible by 2 or 3. For example the most commonly used time signatures are 4/4 and 3/4. Increasing the value of n makes a time signature in some respects more complicated, however the most complicated time signatures are those for which n is a prime number. For example, 12/8 can be thought of as being broken down into 4 lots of 3/8, so really just a simple extension of the standard time signatures. However for 13/8, no such break down can occur as 13 is prime. So a piece in 13/8 is more complicated and unusual than the standard time signatures, both to listen to and to play.

Modern interpretation of just intonation is fully based on fundamental theorem of arithmetic.

The Golden Ratio and Fibonacci Numbers

It is believed that some composers wrote their music using the golden ratio and the Fibonacci numbers to assist them.

James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale.[24] In Bartok's Music for Strings, Percussion and Celesta the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose Croix. His use of the ratio gave his music an otherworldly symmetry.

The golden ratio is also apparent in the organization of the sections in the music of Debussy's Image, "Reflections in Water", in which the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the φ position.

This Binary Universe, an experimental album by Brian Transeau, includes a track entitled 1.618 in homage to the golden ratio. The track features musical versions of the ratio and the accompanying video displays various animated versions of the golden mean.

The math metal band Mudvayne have an atmospheric instrumental track called "Golden Ratio" on their first album, L.D. 50. Mathematical concepts are also explored in other songs by Mudvayne.

See also

• David Cope's EMI program -- composes music in the style of different composers
• Mathematics of musical scales

Books

• Loy, Gareth (2006). Musimathics: The Mathematical Foundations of Music, Vol. 1. The MIT Press. ISBN 0-262-12282-0.

External links

• Music, Mathematics, Philosophy and tuning
• Google Scholar Seach for 'music and mathematics'
• The method for transformation of music into an image through numbers and geometrical proportions
• Twelve-Tone Musical Scale.
• Sonantometry or music as math discipline.