Exploring the Manifolds of Possibilities in John Cage's Number Pieces: A Statistical Analysis of Four2

EXPLORING THE MANIFOLDS OF POSSIBILITIES IN JOHN CAGE’S NUMBER PIECES: A STATISTICAL ANALYSIS OF FOUR2 ALEXANDRE POPOFF INTRODUCTION HE NUMBER PIECES ARE A SERIES of works composed by John Cage at the end of his life, which represent his vision of an “anarchic harmony.” This notion is best described by Cage himself: “I now think the simple togetherness of art—I mean of sounds—produces harmony. That harmony means that there are several sounds . . . being noticed at the same time, hmm?” (Cage and Retallack 1996, 108). To this end, Cage used in almost all the Number Pieces (except, for example, One3 and Two) a particular time-structure, called “time-bracket,” for determining the temporal location of sounds. A usual time-bracket is made of three parts: a fragment of one or many staves, lying under two time intervals—one on the left, and one on the right. The time intervals consist of two real-time values separated by a two-way arrow.1 The intersection between the starting time T 42 Perspectives of New Music interval and ending time interval of a time-bracket will be called the internal overlap. The staves contain one or more sound events without any duration indications. A typical time-bracket is shown in Example 1. The duration of the sound indicated in a time-bracket is left free to the performer, provided its beginning occurs within the first time interval on the left, and its end occurs within the second one on the right. Successive time-brackets can occur in a Number Piece with possible overlaps between each other; i.e., the ending time interval of one time-bracket may overlap the starting time interval of the next one (the intersection of these time intervals will be called the external overlap). In a Number Piece with multiple performers, the superposition of the various parts, each containing time-brackets, creates an ever changing polyphonic landscape. Cage’s vision of an anarchic harmony shows itself in these highly indeterminate works, through the individual freedom granted to each musician and the many sonic possibilities offered by the time-bracket system. As noticed by previous authors, this last aspect renders the analysis of the Number Pieces quite challenging. Taking the specific example of Five2, Haskins notes that “coping with the myriad possibilities of pitch combinations—partially ordered subsets—within each time-bracket of Five2 remains an important issue” (Haskins 2004, 207). Weisser faced the same problem in the analysis of Four and consequently focused mainly on triads (Weisser 2004, 82–83; 2003). Similarly, Gresser (2014) has singled out two particular realizations of Four2 for analysis. However , Haskins reminds that “(one should not) valorize a single analysis, because the brackets offer a flexibility that creates many possibilities.” Recently, a statistical approach to the analysis of the Number Pieces has been proposed (Popoff 2010; 2011). This was applied to the determination of the structure of Five5 (Popoff 2015), in which this particular Number Piece was turned into a stochastic process by selecting the starting and ending time of the sounds in each timebracket according to a given probability distribution. In this way, a 0'00" 0'30" 0'20" 0'55" EXAMPLE 1: A TIME-BRACKET CONTAINING A SINGLE SOUND Exploring the Manifolds of Possibilities in John Cage’s Number Pieces 43 global description of the sonic possibilities of Five5 is obtained, based on the estimated probabilities of occurrence of the various set classes throughout time. In order to do so, the starting and ending times for each time-bracket are determined at random according to a specific probability distribution over the given time intervals. This probability distribution is generally a uniform distribution, though more complicated examples have been introduced (Popoff 2011). As was noted in these papers, such a simple distribution is unlikely to model the realtime behavior of an actual musician. Instead, it rather models a situation wherein the score is prepared in advance by chance operations, prior to the actual performance. This latter approach, which apparently seems at odds with the intended spirit of the Number Pieces, will be discussed in the first section of this paper. One of the main limitations of the previous statistical...