Tiling in My Music

Abstract

TILING IN MY MUSIC TOM JOHNSON T MUST HAVE BEEN 1999 when Moreno Andreatta gave me a copy of Dan Tudor Vuza’s landmark essay “Supplementary Sets and Regular Complementary Unending Canons” (1991–93), because already in 2000 its influence on me became clear with a Canon in 3, 6 or 9 voices, written for an installation of Martin Riches in Berlin. When the MaMuX meetings at IRCAM began in 2001, I was able to hear regularly the mathematical music theories of Guerino Mazzola, Thomas Noll, Emmanuel Amiot, Franck Jedrzejewski and Harald Fripertinger, as well as Andreatta himself and many occasional visitors, and this information stimulated me more and more. My composing time during the year 2002 was devoted almost exclusively to fitting together little rhythmic tiles, and early in 2003 I brought out the edition Tilework: 14 Pieces for 14 Solo Instruments. Later that year I wrote Tilework for String Quartet, and Tilework for Piano, though by the end of that year my concentration was already turning more toward combinatorial designs, which is quite a different topic. I 10 Perspectives of New Music I recently reread the Vuza essay, and it seems all the more clear to me that this text has been important for all the mathematicians and music theorists cited above, not to mention for Jon Wild and other North American music theorists, and for quite a few composers. I now consider this work the most important music theory treatise of the last 20 years, particularly since it is one of those rare cases where music theory has preceded musical practice. Harmony and counterpoint books, essays on serial techniques, manuals for figured bass, and music analysis texts have generally dealt exclusively with musical procedures already practiced by composers. Only in rare cases like Vuza, Leonhard Euler, and Hugo Riemann have theorists preceded composers, though Henry Cowell’s 1930 book New Musical Resources should also be mentioned. Cowell’s highly original text never circulated much in Europe, but it was widely read in the U.S. The author never followed up on these theories much in his own music, but John Cage and Conlon Nancarrow both cited this as a seminal influence on their music, and younger American composers such as Kyle Gann, Larry Polansky, David First, and John Luther Adams also acknowledge the influence. Cowell’s book has left a significant mark on music history, as Vuza’s ideas are just beginning to do. As I was rereading Vuza, it also became clear that later researchers have tended to pay attention mostly to the stimulating problems in Vuza’s maximal category, and to the classical linear tilings, where every point is filled exactly once. We often forget that Vuza also considered many tilings where some points are not filled, or are filled with more than one note. In Part III of the article, for example, there is a case on page 112 where every sixth point is left empty, and another case on pages 109–110 where every fifth and sixth point is silent. In Part II, page 195, there is a lovely example of an eight-beat cycle where a simple two-note rhythm (0,5) enters four times (at points 0, 3, 4 and 7), resulting in an accumulative pattern of two notes, one note, pause, one note, two notes, one note, pause, one note, two notes . . . I suspect Vuza realized that long streams of eighth notes can make dull music, and that he should consider variations if he wanted his concepts to be useful for composers. In any case, irregular tiling patterns have been compositionally very rich for me, so the first section here is about “Tiling with Holes.” The following section, “Tiling in Different Tempos,” explores possibilities implied by Vuza but not specifically developed by him. Tiling in My Music 11 TILING WITH HOLES Perhaps the clearest example for showing the advantages of tiling with holes is Tilework for Oboe (Example 1), one of the fourteen pieces from the 2003 collection. In the excerpt shown here we see one rhythm moving closer and closer to another rhythm until they finally mesh together. Gradually filling holes is the central idea of this movement. One...

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