John Cage's Number Pieces: The Meta-Structure of Time-Brackets and the Notion of Time
Alexandre Popoff · Perspectives of New Music · 2010
John Cage's Number Pieces: The Meta-Structure of Time Brackets and the Notion of Time Alexandre Popoff A striking feature of contemporary music from 1945 to nowadays is the use of so-called "open forms" by many composers. This refers to scores which rely in part or totally on the performer to determine the performance materials of the music. Thus, the performer is actively involved, having to choose among multiple possibilities concerning the characteristics of the sounds he or she is to produce. The choice may impact only few perceptual qualities of sounds, like pitch in Morton Feldman's Projections series, or temporal order of fixed sequences as in Stockhausen's Klavierstück XI. It may also involve the totality of sound parameters as in Cage's Variations II. These pieces differ from the scores written on the basis of outcomes of random events such as 66 Perspectives of New Music John Cage's Music of Changes, in which the performance materials are entirely determined by the composer. In those scores, the performer has no personal choice to make. Although "open form" scores may rely on the outcome of random choices from the performer, this does not mean that the structure of the resulting performance, i.e. the perceptual qualities of the realization, are absolutely random. Among the possible values for the parameters of the sounds being produced, some might appear more often than others. For example, following the instructions given by the composer, one might end up with a realization of a piece containing more long sounds than short ones. Thus, the score determines a "meta-structure" of the performance which encompasses all the structures of the possible realizations. Let us give a detailed example of a meta-structure and how to find it. Suppose a score with a definite set of instructions is given to a performer who realizes it. For a given performance, a structure for this realization can be determined by examining all the characteristics of the occurring sounds and their correlations. Once a structure has been found, corresponding to a single performance, the study is brought to a higher level of analysis, by establishing relationships and variations among structures coming from different realizations. The "meta structure," or global structure of the score is therefore accessed. It should be noted that if the creation process is random in nature, this study will naturally be considered from a statistical point of view. In a detailed analysis,1 Thomas DeLio thus described the underlying meta-structure of John Cage's Variations II in terms of a correlation of several statistical distributions. Consequently, he concludes his analysis by stating that Variations II is "one large comprehensive system which itself represents the total accumulation of its many constituents realizations," a statement obviously similar to the definition of a meta structure. In this analysis, the late "number pieces" of John Cage will be discussed from a statistical point of view. Through a mathematical study of Five we will show that the system of time-brackets used in these pieces determines a specific statistical meta-structure in terms of position and length of the sounds, which is distinct from complete randomness. Moreover, we will show that this meta-structure is dependent upon the conception of time used for the analysis, a statement with strong consequences on the interpretation of the number pieces. John Cage's Number Pieces 67 The Number Pieces In 1987, John Cage began to write a series of scores, the titles of which refer to the number of performers involved, hence the name "number pieces." 2 All in all, Cage produced 47 pieces in the last five years of his life. This output arose in order to face the many commissions he received. In almost all number pieces, John Cage used a particular time structure for ordering the sounds, named time-bracket. These time brackets already appeared in earlier works such as Thirty Pieces for Five Orchestras and Music for .... However, in the number pieces, and especially in his late works, Cage simplified the contents of the time brackets, most of which containing only a single tone. A time-bracket is basically made of three parts: a fragment...