"Scientifically, I could never be made to understand-- Charles Lamb,

what a note in music is, or how one note differs

from another"

Essays

"Music has many resemblances to algebra"-- Novalis

"There is music wherever there is harmony,-- Sir Thomas Browne

order, or proportion"

"Check out Guitar George;-- Mark Knopfler, "Sultans of Swing"

He knowsallthe chords."

by Mike KeithFrom Polychords to Polya:Adventures in Musical Combinatorics

From the time of Pythagoras, who realized that the difficulty in constructing a musical scale is due to the fact that there are no integer solutions to the equation

(3/2)

^{m}= 2^{n},until the present time, when computer algorithms are used to compose musical pieces, there has been a lively interaction between the musical arts and the mathematical sciences.

This book explores the various connections between the basic musical building blocks - chords, scales, and rhythms - and the area of mathematics known as combinatorics, which is concerned with counting and classifying configurations of objects. We consider questions such as the following:

- How many essentially different musical chords are there?
- How many different scales are there? How many of certain types, such as 7-note diatonic?
- How many different rhythms can be constructed?
- Why are certain chords, scales and rhythms more prevalent than others in popular music?
In the process of examining these questions we find applications of many mathematical concepts: binomial coefficients, necklace counting, Pascal's triangle, the Fibonacci sequence, and Polya counting theory. Even the Catalan numbers make a brief appearance.

FPTP (as we like to call it) was first published in 1991 but is now out of print. However, a scan of the full book in PDF format is available here.