From Polychords to Polya: Adventures in Musical Combinatorics - ISBN:0963009702

From Polychords to Polya: Adventures in Musical Combinatorics

Date: 14 April 2011, 18:56

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How many different musical chords, or scales, or rhythms, are there, and why are some more popular than others? Questions such as these can be answered using the tools of mathematical combinatorics, as explained in this fascinating book accessible to the high-school or undergraduate student of mathematics, or the musician with some math background. Mathematical concepts which appear include binomial coefficients, necklace counting, Pascal's triangle, the Fibonacci sequence, and Polya counting theory.
Reader's review:
George Polya (1887-1985) was a Hungarian born mathematician who emigrated to the United States in 1940 out of concern over the spread of Nazism in Europe. He had a lifelong interest in combinatorics and problem solving, as well as mathematical education. His enumeration theorem tells you how to count complicated combinatorial objects where there is a symmetry group involved. A typical example of a problem to which Polya's method applies is the following. How many different necklaces can be made from three red beads, two sepia beads and five turquoise beads. The answer is given as the coefficient of r^3.s^2.t^5 (bollocks, this thing doesn't accept HTML) in a polynomial called the configuration counting series, and a general formula for calculating this polynomial.
Michael Keith's book describes applications of Polya's enumeration theorem to the combinatorics of chords, scales and keys. Throughout, the author deals with the cyclic group consisting of the twelve musical transpositions in the twelve tone equal tempered scale. Unfortunately, atonal music theorists such as Allen Forte and Elliott Carter all seem to use the dihedral group of order twenty-four obtained by allowing inversions. Nevertheless, the ideas described in the book can be applied just as easily in this case.
This is a nice little book, attractively presented, and with more mathematics than music in it.
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