Jonathan D. Kramer
Summary Linearity is defined as "the determination of some aspects of music in accordance with expectations that arise from preceding music"; nonlinearity is "the determination of some aspects of music in accordance with expectations that arise from principles that govern an entire piece or section."* All music uses both linear and nonlinear temporal structures.
The most pervasive linearity is found in tonal music, but even tonal music displays nonlinearity in, for example, the creation of consistent textures. Nonlinearity can also operate in the realm of formal proportions--the lengths of sections and/or amounts of time spent in various tonal areas interrelate in an atemporal, nonlinear manner.
The perception of formal balances across an entire piece depends on what might be called "cumulative listening." Overtly discontinuous atonal works invite a cumulative understanding of nonlinear proportional balances. Certain works of Stravinsky, for example, exhibit the consistent use of a single ratio to determine the lengths of all sections.
Atonal linearity implies motion toward goals that are established contextually, given the absence of tonality's a priori hierarchy of degrees of stability. Linear goals in atonal music may be predictable or not. Linearity and nonlinearity can coexist on different or even the same hierarchic levels.
Total linearity and total nonlinearity are both impossibilities, but certain experimental compositions of the last quarter century have approached both these extremes.
Linearity and nonlinearity are the two fundamental means by which a piece of music structures its time and by which time structures a piece of music. Thus, nonlinearity is not merely the absence of linearity. It is itself a structural force. Virtually all music utilizes a mixture of linearity and nonlinearity. Since they may appear to different degrees and in different combinations on each level of music's hierarchic structure, their interplay determines both the style and the form of a composition. My aim is to show how this interaction operates in different kinds of music.
My definition of linearity is "the determination of some aspect(s) of music in accordance with expectations that arise from earlier events in the piece." Nonlinearity is "the determination of some aspect(s) of music in accordance with expectations that arise from principles or tendencies governing an entire piece or section." While these definitions serve adequately as a point of departure, they do have potential problems: the idea of "determination" requires further explanation, the term "aspect of the music" is vague, and the____________________