Formalized accelerando: an extension of rhythmic techniques in Nancarrow's acceleration canons.

[A]cceleration canon does not have the kind of clean structural implications Nancarrow found in the straight tempo canon ... Perceptually fascinating, it did not lead to the kind of compositional richness he found elsewhere.

Kyle Gann, The Music of Conlon Nancarrow

OF HIS FIFTY-ONE STUDIES for player piano, Nancarrow wrote only three canons based on large-scale continuous acceleration: (1) Studies 21, 22, and 27. (2) In his book on Nancarrow's music, Kyle Gann offers the above statement as an explanation as to why Nancarrow never returned to the acceleration canon after the last of these. I am inclined to agree with Gann's statement, with the following modification: it was Nancarrow's conception of accelerando, focusing on duration rather than tempo, which limited its compositional value, rather than any limitations inherent in large-scale accelerandi. The purpose of this paper is to explore the nature of Nancarrow's conception of accelerando, and the manner in which his approach to acceleration canons limited the structural elements which proved so useful in the straight tempo canons. Having identified the problems with this approach, I will consider a generalized model of acceleration, borrowing techniques from calculus. The latter portion of the paper will consider e xamples which exploit the potentials of this more general approach.

NANCARROW'S CONCEPTION OF ACCELERANDO

Before considering Nancarrow's approach to acceleration, it will be useful to briefly summarize the primary structural elements of straight tempo canons--canons in which each voice remains in a constant tempo for large passages and the relationship of tempi between voices is not 1:1. Whereas in canons in which the voices are in a 1:1 tempo relationship the elapsed time between entries of the canonic line (which Gann refers to as the echo distance) remains constant, in tempo canons this distance is continuously varying. The echo distance increases when faster voices are ahead of slower voices in the canon line, and decreases when the situation is reversed. A special feature of tempo canons is the moment at which the echo distance is reduced to zero, referred to by Gann as the convergence point. Convergence points often serve as a structural climax, whether occurring at the beginning, ending, in the middle, or at numerous moments in between. By comparing the change in the echo distance over time, it is possible for the listener to determine if the canon is approaching or receding from a convergence point, and, thus, to locate any given moment within the overall formal design of the canon. As an aid to perceiving the echo distance, Nancarrow often provides what may be called salient markers--gestures which the listener can easily perceive and retain, and compare the elapsed time between their appearance in one voice and another. These gestures often take the form of texturally distinct figures such as trills, glissandi, staccato chord punctuations, etc. Convergence points are often emphasized by special markers. A noteworthy example is the rising chromatic glissandi in Study No. 36 which signal the approach of the convergence point, itself marked by trills in all voices, and the corresponding descending glissandi which mark the subsequent departure.

In the acceleration studies, Nancarrow employs two different types of accelerando, which Gann (1995, 146-8) refers to as arithmetic and geometric. In an arithmetic acceleration, a constant is subtracted (or added in the case of a ritardando) to each successive duration so that the duration for the nth beat (4) is:

[d.sub.n] = d - cn, (1)