1. Gyorgy Ligeti, "On My Etudes for Piano," trans. Sid McLauchlan, Sonus 9/1 (1988): 4-5. An abbreviated version appears in the liner notes to Erato Compact Disc ECD 75555.

2. Hofstadter, Metamagical Themas: Questing for the Essence of Mind and Pattern (New York: Basic Books, 1985): 177-78. This article originally appeared in Scientific American, April 1982. (At the time, Hofstadter was apparently unaware of the many 20th-century composers who have radically experimented with rhythm.)

3. Trans. as African Polyphony and Polyrhythm: Musical Structure and Methodology by Martin Thom, Barbara Tuckett, and Raymond Boyd (New York: Cambridge University Press, 1991).

4. Ligeti speaks frequently of his studies of earlier music; see for example Peter Varnai, "Beszelgetesek Ligeti Gyorgyyel, translated by Gabor J. Schabert in Ligeti in Conversation (London: Eulenberg, 1983): 14-15.

5. For this point, the author is indebted to editorial readers of this essay.

6. Ligeti, "On My Etudes for Piano," 5. Complex additive rhythms also characterize the music of Eastern Europe, Ligeti's heritage. The fourth etude, "Fanfares," as well as the Horn Trio's second movement and the harpsichord piece Hungarian Rock (1978), pay homage to the rhythms of Ligeti's homeland. But Eastern European rhythms still tend to fall into meters--3+2 or 2+2+2+3 are common examples--while sub-Saharan African (and Ligetian) rhythms do not.

7. Ligeti, "On My Etudes for Piano," 5-6.

8. See the author's dissertation, The Lamento Motif: Metamorphosis in Ligeti's Late Style, (Cornell University, 1994). Also, Richard Steinitz has written at length about this motif in a recent series of three articles for the Musical Times: "Music, Maths, and Chaos," (March 1996): 14-20; "The Dynamics of Disorder," (May 1996): 7-14; and "Weeping and Wailing," (August 1996): 17-22. Other examples of the motif occur in the last movement of the Horn Trio (1982), the second and third movements of the Piano Concerto (1985-88), and the fourth and fifth movements of the Violin Concerto (1990-92). Faster descending scales have also played an important part in the ninth piano etude, "Vertige" (1989), and the third movement of the Violin Concerto.

9. Steinitz, "The Dynamics of Disorder," 13.

10. Nancarrow, another important influence on Ligeti's music of the past fifteen years, employs an even more dizzying variety of polyrhythms than Ligeti does, thanks to his use of the player piano. For Ligeti, one important goal was to produce a rhythmic complexity comparable to Nancarrow's using only a single, living interpreter (Ligeti, "On My Etudes for Piano," 6).

11. Gyorgy Ligeti, "Polyrhythmical Aspects in My Piano Etudes," lecture given at the International Bartok Seminar and Festival, Szombathely, Hungary, 26 July 1990, quoted in Lois Svard, "Illusion in Selected Keyboard Works of Gyorgy Ligeti" (D.M.A. dissertation, Peabody Conservatory of Music, 1990): 76.

12. See two analyses by Jonathan W. Bernard published in Music Analysis: "Inaudible Structures, Audible Music: Ligeti's Problem, and His Solution," 6/3 (1987): 207-36; "Voice Leading as a Spatial Function in the Music of Ligeti," 13/2-3 (1994): 227-253.

13. In the last movement of the Horn Trio (which marks the Lamento motif's first appearance in Ligeti's music), the underlying passacaglia is "hidden" by the motif in a similar way. See Ulrich Dibelius, "Ligetis Horntrio," Melos 46/1 (1984): 44-61; and Taylor, "The Lamento Motif: Metamorphosis in Ligeti's Late Style."

14. See Bernard, "Voice Leading as a Spatial Function in the Music of Ligeti," for additional discussion of perception-related issues.

15. Hofstadter 174.

16. See Jane Piper Clendenning, "The Pattern-Meccanico Compositions of Gyorgy Ligeti," Perspectives of New Music 31/1 (Winter 1993): 194-234. This article is drawn from her Yale dissertation, "Contrapuntal Techniques in the Music of Gyorgy Ligeti" (1989).

17. Ligeti, "Ma Position comme compositeur aujourd'hui," Contrechamps 12/13 (1990): 8-9. Also published as Ecrits in the collection Musique/Passe/Present (Paris: Christian Bourgeois Editeur, 1985).