Convergence Points in Conlon Nancarrow’s Tempo Canons

[1] Conlon Nancarrow (1912-97) wrote approximately fifty Studies for Player Piano, many of which represent a new compositional genre: the tempo canon, in which canonic voices proceed at different speeds. The player piano proved to be the perfect vehicle for Nancarrow to realize a variety of textures (from two to twelve voices or strata) and tempo relationships among canonic voices. A primary and characteristic structural feature of Nancarrow's tempo canons is the convergence point, or "the infinitesimal moment at which all lines have reached identical points in the material they are playing" (Gann 1995, 21). Convergence points (CPs) serve varying purposes in Nancarrow's studies: sometimes they seem to be the raison d'etre for an entire piece (particularly in his earlier brief tempo canons with just one CP, such as Nos. 14, 18, and 19); sometimes they vary considerably in significance within the same piece (such as the thirteen CPs in No. 24). They may also serve to mark the most structurally significant points in a piece (such as the three CPs in No. 27). As noted by Margaret Thomas, these "points of synchrony-whether at the beginning, ending, or middle of a piece or passage-operate as significant structural moments and thus help to articulate the design of the studies" (2000, 111). CPs are also important aids to the listener in following relationships between the voices or strata. Eric Drott observes: "Convergence points function as the seams that hold the layers together within a given study, marking the few moments when the listener can hear a direct and immediate connection between the otherwise uncoordinated layers" (2004, 541). Finally, convergence points in Nancarrow's Studies represent the most temporally consonant areas in an otherwise temporally dissonant environment (Thomas 2000/01).[2] The voices in Nancarrow's tempo canons are often related temporally by ratios stated in the study's title (e.g., 21/24/25).(1) Unlike conventional canons in which the voices proceed at the same tempo, tempo canons present the possibility for one or more CPs between the canonic voices because the voices are moving at different speeds and either converging or diverging. Thomas (1996) classified tempo canons into four basic types: (1) the converging canon, with one CP at the end; (2) the diverging canon, which begins with a CP; (3) the converging-diverging (arch) canon, in which a single CP is somewhere in the middle; and (4) the diverging-converging canon, which begins and ends with a CP. Nancarrow wrote examples of each type, although he did not write any canons that only diverge-there is, after all, not much perceptual interest in a piece that diverges immediately away from a beginning point of convergence, never to return.(2)[3] In this paper I will examine the presence and placement of CPs in Nancarrow's tempo canons and their significance to the overall structure. CPs will be examined in their more conventional uses in the canons, and other devices examined will include tempo switches and overlaps (which can be used to create additional CPs) and techniques for emphasizing and de-emphasizing CPs. Several of Nancarrow's more complex works feature many CPs, and further complexity is possible through the addition of more voices to a canon, thereby providing opportunities for interior CPs among smaller groups of voices. A special sub-type of Nancarrow's tempo canons is the "acceleration canon," in which voices accelerate or decelerate. The discussion concludes with an analysis of Nancarrow's final acceleration canon, Study No. 27 ("Canon 5%/6%/8%/11%").[4] The effect of convergence in Nancarrow's tempo canons is perceptually heightened because it involves not only temporal convergence, but convergence of canonic material. The choice of canon as a means of portraying temporal proportions is quite natural. Thomas notes that in Nancarrow's tempo canons "there is a palpable sense of the voices being in different places at the same time, of gradually moving closer together, of a brief moment of coordination, and then a departure" (Thomas 2000, 111-12). …