"Scientifically, I could never be made to understand
what a note in music is, or how one note differs
from another"
-- Charles Lamb, Essays"Music has many resemblances to algebra"
-- Novalis"There is music wherever there is harmony,
order, or proportion"
-- Sir Thomas Browne"Check out Guitar George;
He knows all the chords."
-- Mark Knopfler, "Sultans of Swing"
From Polychords to Polya: Adventures in Musical Combinatorics
by Mike Keith
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 = 2n,
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.
