Mathematical Connections in Art, Music, and Science 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/pi. 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 pro-cesses are heavily influenced by the use of ratios, and he deployed ratios in his compositions in remark-ably 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 o