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Looking at Numbers

Tom Johnson and Franck Jedrzejewski
Publication Date: 
Number of Pages: 
[Reviewed by
Charles Ashbacher
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Tom Johnson is a composer that uses mathematics, most specifically combinatorics, in his music; Franck Jedrzejewski is a mathematician. This book contains a series of undirected graph-like drawings that Johnson uses in his composition. Most of them represent chords and transitions between cords where operations such as transpositions and permutations were done. The vertices of the graphs are labeled with integers representing notes of the chord. For example, the nodes 2134 and 1234 are connected by a transposition of the first two numbers.

First Johnson describes how he created and then used the structure in the graph. Jedrzejewski then gives a mathematical explanation of the operations inherent in the graph. Sets, subsets, groups, group transformations, partitions and the terminology of combinatorics are used to give the mathematical descriptions of what Johnson has done.

The complexity of the mathematics is generally not that advanced; it is pretty easy to understand transpositions, permutation groups and the different ways one can select r from n when both are small. Reducing music composition to a graph that reminds one of a soccer ball is one of the most effective ways to demonstrate the links between mathematics and music at a level that most musicians and beginning mathematicians can understand. A chord is simply the selection of several notes to play simultaneously, an act that mathematicians know as a combination.

The title of the book is derived from the large number of undirected graphical drawings used to illustrate the concepts. With a little background explanation of the notation, both mathematicians and musicians will have little difficulty in reading them, a confluence that does not happen all that often.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.


1. Permutations
1.1 Symmetric Group
1.2 Bruhat Order
1.3 Euler Characteristic
1.4 Group Action
1.5 Permutohedra and Cayley Graphs
1.6 Coxeter Groups
1.7 Homometric Sets

2. Sums
2.1 Integer Partitions

3. Subsets
3.1 Combinatorial Designs

4 Kirkman’s Ladies, a Combinatorial Design
4.1 Steiner and Kirkman Systems

5. Twelve
5.1 (12,4,3)

6. (9,4,3)
6.1 Decomposition of Block Designs

7. 55 Chords
7.1 Chords and Designs

8. Clarinet Trio
8.1 Strange Fractal Sequences

9. Loops
9.1 Self-Replicating Melodies
9.2 Rhythmic Canons

10. Juggling
10.1 Juggling, Groups, and Braids

11. Unclassified
11.1 Some Other Designs

A Figures