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.