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Galileo Galilei said he was “reading the book of nature” as he observed pendulums swinging, but he might also simply have tried to draw the numbers themselves as they fall into networks of permutations or form loops that synchronize at different speeds, or attach themselves to balls passing in and out of the hands of good jugglers. Numbers are, after all, a part of nature. As such, looking at and thinking about them is a way of understanding our relationship to nature. But when we do so in a technical, professional way, we tend to overlook their basic attributes, the things we can understand by simply “looking at numbers.”

Tom Johnson is a composer who uses logic and mathematical models, such as combinatorics of numbers, in his music. The patterns he finds while “looking at numbers” can also be explored in drawings. This book focuses on such drawings, their beauty and their mathematical meaning. The accompanying comments were written in collaboration with the mathematician Franck Jedrzejewski.



Chapter 1. Permutations

Despite the fact that I am a musician and composer, this is not a book about music. It is a book about “looking at numbers.” Sometimes a particular number in one of these drawings represents a particular note in a particular composition, but all the numbers here represent a particular point in some sort of logical sequence, in some system of permutations or combinations, in some network of sets and subsets.

Tom Johnson, Franck Jedrzejewski

Chapter 2. Sums

Another interesting way of looking at numbers is simply to put them together when they have the same sum. I first became interested in this when I wanted to construct groups of chords having the same average height, that is, when the sums of the notes would all be the same. That would permit me to write harmonies that would move a lot without ever really going up or down. To make the music even more immobile, I wanted to link these chords by minimal differences, so that with each move one voice would move up a notch and one would move down a notch, and the rest would not change. How does this work?

Tom Johnson, Franck Jedrzejewski

Chapter 3. Subsets

Consider the set of digits 1,2,3,4,5. Taking the elements two at a time, we have 10 subsets: 1,2, 1,3, 1,4, 1,5, 2,3, 2,4, 2,5, 3,4, 3,5, 4,5. This is quite obvious, and yet the forms that result when you really look at these pairs and try to make connections between them can be surprising.

Tom Johnson, Franck Jedrzejewski

Chapter 4. Kirkman’s Ladies, A Combinatorial Design

I found a surprising number of new musical patterns in formations as simple as the permutations, sums and subsets already discussed, and in the case of my “counting music”, even simpler ones, but I was always interested in finding new directions in all this. One new direction presented itself quite unexpectedly in 2003, when I heard a piece by a young Dutch composer, Samuel Vriezen. Using a scale of only 11 notes, Vriezen constructed 11 five-note chords in such a way that each chord had exactly two notes in common with each other chord. I asked the composer how he had ever found such a group of chords, and he told me it was not too complicated. He thought I could construct such a system myself, if I thought about it a bit.

Tom Johnson, Franck Jedrzejewski

Chapter 5. Twelve

The number 12 has a special significance for musicians, since that is the number of notes in the chromatic scale, and since 12-tone music, after Schoenberg, became an international style, with hundreds of theoretical essays and thousands of compositions (e.g. [3]).

Tom Johnson, Franck Jedrzejewski

Chapter 6. (9,4,3)

One combinatorial design that has particularly interested me is (9,4,3). With only nine numbers, and only 36 pairs of numbers, the system is small enough that one should be able to see how it works, and since by definition each pair occurs three times in three different blocks of four, all that is necessary is to write out the 36 pairs and connect each pair with the three other pairs that form one of the 18 blocks of the system. I could imagine all these three-armed pairs holding hands with one another in lovely triangular lattices, and since the

Handbook of Combinatorial Design

gives 11 completely different solutions to work with, I was sure that clearly spaced drawings would allow us to look inside each system and see how it all connects. I could imagine 11 drawings, all different, and all containing lovely symmetries.

Tom Johnson, Franck Jedrzejewski

Chapter 7. 55 Chords

“Kirkman’s Ladies” was a (15,3,1) combinatorial design with 15 elements divided into subgroups of three, each pair occurring once.

Tom Johnson, Franck Jedrzejewski

Chapter 8. Clarinet Trio


Clarinet Trio

(2012) is a special case, because here everything in the music is a reflection of one of seven drawings, and everything in the seven drawings corresponds to something in the music. At the same time, both the drawings and the music are derived rigorously from a (12,3,2) design: 12 notes in the scale, 3 notes in each chord, each pair of notes appearing together twice. In order to construct a system like that one must find 44 chords, or blocks, and one can do this in a number of ways. In fact, the

Handbook of Combinatorial Designs

[1] informs us that P. R. J. Ostergard has counted exactly 242,995,846 completely different ways to do this. If you don’t believe me, or if you want to know how this was calculated (and if you are not a specialist, you probably don’t), you can consult his article in the

Australasian Journal of Combinatorics

, No. 22 (2000) [5].

Tom Johnson, Franck Jedrzejewski

Chapter 9. Loops

Frequently minimal music, particularly the sub-species referred to as “repetitive music,” turns around in loops. I never really wrote repetitive music, but I’ve written an awful lot of musical loops, and there are a great many ways of doing this. Most of the loops we’ll be discussing here might better be called “rhythmic canons”, a term introduced in

Perspectives of New Music

in 1991-1992 in an article by the Rumanian mathematician and music theorist Dan Tudor Vuza. Basically this article has to do with rhythms that repeat canonically in such a way that every point in time is touched exactly once by one of the voices.

Tom Johnson, Franck Jedrzejewski

Chapter 10. Juggling

The art of juggling is the art of cycling several balls through the air in different ways, and for quite a few centuries jugglers were happy to do this in the most obvious ways. After learning to throw three balls, keeping each one in the air for three beats, jugglers went on to four balls, keeping each one in the air for four beats, and then on to higher and higher throws with more and more balls or plates or bowling pins or whatever. The result was spectacular, and very few people could ever learn to do it, but the arithmetic was pretty simple. This all changed in 1985, however, when a few smart guys in the Cambridge University amateur juggling club sat down and decided to analyze how many ways this could really be done if one looked at the problem mathematically. They devised a sort of flow chart known as “site-swap” and it became immediately obvious that balls could fly around in lots of new ways. For example, instead of throwing three balls always to the same height in continuous cycles of 333, as jugglers had been doing for centuries, they could throw them in a cycle of 441. This was quite a different rhythm, quite a different look, and not really harder than just doing 333 all the time. Soon hundreds of other new patterns became obvious, the new information circulated to all the continents, and people like the Australian, Konrad Polthier, even wrote books on the mathematics of juggling. Since I had been composing so many loops, and since many of my loops were quite juggleable, it seemed inevitable that I began to meet jugglers and wanted to collaborate with them.

Tom Johnson, Franck Jedrzejewski

Chapter 11. Unclassified

As this text and these illustrations came together, I realized I had lots of good drawings that were sort of left over. With some drawings I couldn’t really remember myself how I did them, though I liked them and was sure that they were correct.

Tom Johnson, Franck Jedrzejewski


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