Performing the Irrational: Paul Usher’s Arrangement of Nancarrow’s Study No. 33, Canon 2: \(\sqrt{2}\)

1. IntroductionTable 1. Arrangements of Nancarrow's primarily canonic studies[1.1] Conlon Nancarrow turned to mechanical means of realizing his musical ideas in part due to the difficulties performers had in playing his rhythmically challenging music. It is therefore somewhat ironic that just over half of his nearly fifty Studies for Player Piano have been arranged for live human performance without, in many cases, mechanical assistance or even click tracks for coordination. Table 1 lists the arrangements made for those Studies that are primarily canonic; the column marked "ratios" indicates the tempo ratios between voices of the canon. (See the online lists of works and arrangements by Monika Furst-Heidtmann [2012], Kyle Gann [2012], and Jurgen Hocker [2012], and a recent concert program by the London Sinfonietta [2012].)Figure 1. 12:15:20 tempo ratio as a polyrhythm in a single notated tempo[1.2] Not surprisingly, most of these arrangements are of canons with relatively simple tempo ratios such as 3:4, 4:5, and 3:5, not to mention the somewhat tongue-in-cheek 1:1 ratio of Study 26. But there are also arrangements of canons with the more complex tempo ratios of 12:15:20 and 21:24:25. While the ratios of Studies 17, 19, and 31 are indeed complex, it is still possible to conceive of these ratios in simpler terms. For instance, in Helena Bugallo's (2004) arrangement of Study 19 for two pianos, the two performers share the middle tempo stream of eighth notes. Player 1 groups these eighths into threes and performs a 4:3 polyrhythm in the right hand, while player 2 groups these eighths into fives and performs a 4:5 polyrhythm in the left hand (see Figure 1). This is possible because 12:15:20 is comprised of the simpler ratios 3:4, 3:5, 4:5. This is not to say that performing this polyrhythm is easy, especially given the highly syncopated nature of Nancarrow's lines. Nonetheless, the numerous performances and recordings of Ivar Mikhashoff (Ensemble Modern 1993) and Bugallo's arrangements (Bugallo-Williams Piano Duo 2004) serve as evidence that these complex tempo ratios can be realized successfully by live performers.[1.3] The canonic studies that have not been arranged for human performers mostly fall into three groups. First are those canons with tempo ratios that are even more complex and difficult to simplify, such as 17:18:19:20, the twelve-fold ratio of Study 37, and especially the exceedingly close 60:61 ratio of Study 48. Second are the acceleration canons in which tempos vary independently of one another. These include Studies 22 and 27 in which voices accelerate or decelerate at different rates indicated by percentages. (For example, an acceleration by 5% means that each duration will be 95% as long as the previous duration. See Gann 1995 and Callender 2001 for more details on acceleration canons.) Study 21, "Canon X," also belongs to this group: it is a two-voice canon in which the lower voice continuously accelerates and the upper continuously decelerates over the course of the entire work. In these three acceleration studies, the tempo ratios between voices constantly and continuously change, making it essentially impossible to perform these rhythms without the aid of click tracks.[1.4] The third group of canons, only one of which has been arranged, consists of the four irrational tempo canons, Studies 33, 40, 41a, and 41b. Since these irrational tempo relations cannot be expressed as rational ratios, there is no hope of conceiving or even notating these rhythms accurately in terms of polyrhythms with a single referential tempo, no matter how complex. Nevertheless, the ratios of these four studies have certain advantages from the perspective of human performance. One advantage is that the ratios are fixed, unlike the constantly changing ratios of the acceleration canons. Another is that the tempos are not especially close, unlike the 60:61 ratio of Study 48 in which the tempos are nearly identical. …