TILEWORK

Tilework” has to do with fitting together little tiles to fill lines and loops. One can think of this as making mosaics in one dimension, but it is also very much like stringing beads onto necklaces in various patterns. In musical terms, the Tilework series is a collection of compositions in which individual tiles/rhythms fit together into musical sequences without simultaneities, often filling all the available points of a line or a loop.
Since the notes of different motifs come at different moments, it is possible for a single melodic instrument to play several voices at once, so “tiling” was particularly appropriate for unaccompanied melodic instruments. I gradually found so many ways of tiling musical phrases that I could not stop until I had a piece for each instrument of the orchestra. The year of 2002 was devoted almost exclusively to 14 pieces for 14 solo wind and string instruments.
At first, interlocking one tile with another seemed obvious, sort of like fitting together the pieces of a jigsaw puzzle. As the work went on, however, it became clear that this was not so simple. Sometimes it is not at all obvious how rhythms/tiles link together, and sometimes one can easily see six ways of solving a certain problem, without being sure if some seventh solution might also be possible, and sometimes the discussion can go way beyond the comprehension of musicians. I knew that making a loop of 16 beats by repeating one eight-note rhythm had something to do with group theory, and I was dimly aware that some mathematicians know how to determine the number of unique necklaces of length n possible using beads of m colors, never allowing two beads of the same color to touch, but I certainly hadn’t studied such things. Gradually I was entering a world I didn’t know much about.
As the Tilework project continued, I sometimes stumbled across questions that were as new and interesting for mathematicians as for myself. I owe much to the regular meetings devoted to Mathematical Music Theory at IRCAM in Paris, where I met many mathematicians, particularly Emmanuel Amiot, and Andranik Tangian, who gave me solutions to some problems. In the case of Tilework for String Quartet, which involves ways of tiling the line with six-note rhythms, I am particularly indebted to Harald Fripertinger and his most useful computer list of “rhythmic canons.” But of course, composers, interpreters, and listeners do not need to know all this, just as we do not need to master counterpoint in order to appreciate a Bach fugue. As always, one of the wonderful things about music is that it allows us to perceive directly things that we would never understand intellectually.
- Tom Johnson, Paris, October 2003

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