The music is constructed by systematically taking all the combinations of something, but each movement does this in a quite different way. Commisioned by MärzMusik in Berlin, for the premier by the Bozzini Quartet in MärzMusik 2004. 25 minutes. Score $18, parts $18.
The piece contains five movements, and each of these contains all the combinations of something. As usual, I wanted the music to know what it was doing, to be correct and complete in a rigorous sense, and this is one way of achieving this. The theory of combinations is a totally explored mathematical discipline, as we have known for more than a century how to calculate all kinds of combinations and probabilities, and how to prove all of this. So what I say about my composition can not have fundamental significance for mathematics. I can, however, demonstrate that new questions arrive when one wants to go inside some set of combinations, to see how they come together, to observe the many symmetries within them, to find the best sequence for them, to consider how they might sound, to turn them into music.
Tilework for String Quartet is a compilation of all the ways one can tile a line of 12 points by overlapping a single six-note rhythm. The four musicians play these rhythms in canon for 10 minutes in a rapid music requiring great precision. The work was premiered in a KlangAktion concert in Münich in December, 2004. Score and parts $22.
“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. Tilework for Five Conductors and One Drummer, Tilework for Piano, and Tilework for String Quartet came later, in 2003.
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
The first Johnson string quartet, premiered by the Brindisi Quartet in 1994. Eight movements, each following a mathematical formula. 20 minutes, score $16, parts $24.
288 three-note chords with sums of 72 (middle C = 24), preferably for violin, viola, violoncello. Score and parts $16.
As in all the music in the Rational Harmonies series, I want to deduce my chords, rather than choosing them according to the usual musical and esthetic criteria. To compose the Trio I defined a chromatic scale with notes numbered 0 to 48 and counted all the combinations of three notes having sums of 72, permitting octave relationships, but not unisons. Then I found a chain connecting all 288 chords, requiring that each chord have one note in common with each subsequent chord, the remaining two voices moving by half steps in contrary motion with no crossing of voices. The performers may wish to make little glissandos as they move from one chord to the next.
The piece seems best as a trio for violin, viola and cello, with a tempo of about mm. 40 and a duration of about seven minutes, although interpreters may transpose and arrange the music in other ways if they wish. Faster tempos may be tried as well, although it is important that we hear harmonies and not melodic motion.
