The Use of Ratios in the Player Piano Studies of Conlon Nancarrow

The forty-nine Studies for player piano by Conlon Nancarrow are pervaded by the use of mathematical ratios, particularly ratios based on intervals found in the justly-tuned scale. Nancarrow found a remarkable variety of ways in which to deploy ratios in his Studies. The most obvious and pervasive use is in the establishment of different simultaneous tempos related by ratios as simple as 3:4 to as complex as e/π. Ratios were also used to establish relationships between pitch materials and other details, and in certain structural features. This paper is an introductory survey of the use of ratios in these Studies. Conlon Nancarrow (1912–1997) was a remarkable musical pioneer of the twentieth century who, while working in virtual isolation in Mexico, turned to the player piano as a means to realize complex rhythmic and metric structures that were unplayable by human performers. His often transparent compositional processes are heavily influenced by the use of ratios, and he deployed ratios in his compositions in remarkably inventive ways to control many of his musical materials. Over a span of almost 50 years, Nancarrow wrote 49 numbered “Studies” for the player piano. He used ratios in several different ways in these pieces, and many of his ideas—particularly the earlier ones—are obviously influenced by the ideas of Henry Cowell as set forth in New Musical Resources of 1930 [3]. It was Cowell who first presented the idea of relating simultaneous tempos to acoustical pitch ratios found in musical intervals and chords. For instance, the interval ratio for the purely-tuned perfect fifth, 3:2, could be represented rhythmically in a passage such as shown in Example 1. Example 1: Conjectural passage showing a 3:2 rhythmic relationship. Rhythmic relationships of 3-divisions in one voice against 2-divisions in another are very common in music going all the way back to medieval times. More complex rhythmic ratios, such as 4:3 (corresponding to the musical fourth), 5:4 (the major third), 6:5 (the minor third), and 5:3 (the major sixth) occur in much Romantic piano music such as that of Chopin (see Example 2); however, these uses are primarily decorative and not representative of large-scale polytempo as Cowell envisioned it. Cowell went much further in proposing a system in which correspondences are established between rhythm and interval ratios. The illustration in Example 3 shows one such relationship based on the second-inversion major triad, in which the ratio between the three chord members is 5:4:3. This experimental use of three different tempo values at one time, none related by powers of 2, was unprecedented in Western music. Nancarrow was especially inclined to use ratios found in the Fibonacci series 1, 1, 2, 3, 5, 8, 13, ..., with the ratios 3:5, 3:8, and 5:8 being particularly prevalent in his compositions. This is at least partially because it happens that each of these ratios is also representative of a purely-tuned musical interval, with the ratios 3:5 and 5:8 representing the pitch intervals major sixth and minor sixth, respectively, while the