Graph Theory

Looking at Numbers by Tom Johnson

By Tom Johnson

Galileo Galilei stated he used to be “reading the publication of nature” as he saw pendulums swinging, yet he may additionally easily have attempted to attract the numbers themselves as they fall into networks of variations or shape loops that synchronize at varied speeds, or connect themselves to balls passing out and in of the fingers of excellent jugglers. Numbers are, in spite of everything, part of nature. As such, and puzzling over them is a manner of knowing our courting to nature. but if we accomplish that in a technical, expert approach, we have a tendency to forget their simple attributes, the issues we will comprehend by means of easily “looking at numbers.”

Tom Johnson is a composer who makes use of good judgment and mathematical versions, resembling combinatorics of numbers, in his track. The styles he reveals whereas “looking at numbers” can be explored in drawings. This e-book specializes in such drawings, their attractiveness and their mathematical that means. The accompanying reviews have been written in collaboration with the mathematician Franck Jedrzejewski.

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Extra resources for Looking at Numbers

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At the bottom is the tightest grouping, 12 þ 13 þ 14; and at the top is the most spread out grouping, 1 þ 13 þ 25. The highest of the three numbers are all the same if one reads descending diagonals, the central numbers are the same if one reads the vertical columns, and the low numbers are the same if one reads ascending diagonals. Curiously, a kite-like shape results, and this was not because I did any trimming. The form just emerges like that if we follow these rules. The subsets all connect in a single line, but this line is not as neat as one might expect, as the zigzags have to be interrupted by straight lines in order to turn around and go the other way.

In Fig. 6 we find three notes in each chord, beginning with 123 at the left and ending with 678, with the sums of 6–21. Now the beginning and ending chords have only one connection, but two in the middle have six. The result is a kind of hierarchy between ordinary chords and special ones. 7 is a somewhat more twisted network, because now there are four notes in each chord. In this case, however, they are not really notes, but rather rhythms in a cycle of 8 beats. With rhythms of four notes in the space of 8 beats, there are more possibilities, so the drawing is denser and the music lasts longer.

The music is quite a bit noisier though, since we hear 6 notes in every 8-beat measure. Since these rhythms can never be linked into a single line, the movements of Mocking are divided into several different sequences, in which the percussionists alternate rhythms and seem to be mocking one another. For me the network of Fig. 10 has quite an elegant look, since the point 246, with its six connections, becomes the focal point at the center of the system. The two subsets having five connections make secondary focal points left and right.

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