Applications of Fractal Geometry to the Player Piano Music of Conlon Nancarrow

The relationship between music and geometry goes back thousands of years to the Greek quadrivium. Fractal structures have been explored in music and sound since at least 1978 (Gardner) and this work has recently been extended to specifically explore fractal structures in melodies (Mason and Saffle, 1994; Chesnut, 1996) and in musical forms and phrase structures (Solomon, 1998). Among the fractals that have been identified in musical structures are Sierpinski’s triangle, Peano curves, and the Koch snowflake. This paper is an effort to apply fractal observations to the player piano studies of American-Mexican composer Conlon Nancarrow. The most clearly mathematically-oriented of Nancarrow’s Studies are the canons that explore mathematical relationships as simple as two voices in the relationship 3:4 or as complex as twelve voices proportional to the pitches of the justly-tuned chromatic scale. In particular, those of the canons which are also “acceleration canons”—that is, using carefully controlled rates of acceleration and deceleration among the voices—offer compelling possibilities for study of fractal properties. Among the studies which will be examined here are Nos. 14 (two voices) and 19 and 27 (three voices).