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POINTS AND LINES TO PLANE: THE INFLUENCE OF THE SUPPORT IN JOHN CAGE’S VARIATIONS II ALEXANDRE POPOFF ARIATIONS II IS AN INDETERMINATE score by John Cage written in 1961, which is scored “For any number of players and any sound producing means” (Cage 1961). Its materials consist of eleven transparencies , five of them having a single point each and the other six having a single line each. The performance notes are instructions for constructing a score using these sheets: the transparencies should be laid out “on a suitable surface” (Cage 1961), superimposed or not. The distances of perpendiculars dropped from the points to the lines serve to determine the characteristics of the sounds, namely “(1) frequency (2) amplitude (3) timbre (4) duration (5) point of occurrence in an established period of time (6) structure of the event (number of sounds making up an aggregate or constellation).” A construction V 38 Perspectives of New Music example using two lines and two points is given in Example 1. More detailed examples are given in Christensen (2004). As noticed by James Pritchett, Variations II is not a score but rather a tool, in the sense that it gives instructions “capable of producing an infinite number of scores.” (1993). In a previous analysis of Cage’s Number Pieces (Popoff 2010), the notion of “meta-structure” was introduced (i.e, the structure of all possible structures). The same notion can be applied to Variations II as well: Cage’s instructions define a framework for all possible realizations, and the structure of this framework is precisely the meta-structure of the piece. Moreover, Pritchett also remarks that since Cage leaves an open possibility for additional measurements (“If questions arise regarding other matters or details . . . put the question in such a way that it can be answered by measurement” [Cage 1961]), the interpretation of these very measurements is left open. What counts is, therefore, “The basic unit of Variations II (which) is the measurement of a point to a line” (Pritchett 1993). The analysis of the structure of Variations II can thus be reduced to a purely geometrical problem, in which one studies the possible distances between points and lines and their interrelations , notwithstanding the sound characteristics to which these distances are associated. From a mathematical point of view, the notion of measurement of perpendiculars implies that the space on which lines EXAMPLE 1: AN EXAMPLE OF CONSTRUCTION FOR VARIATIONS II USING TWO POINTS AND TWO LINES, WITH THE CORRESPONDING MEASUREMENTS (PERPENDICULARS) The Influence of the Support in John Cage’s Variations II 39 and points can be drawn is endowed with a metric: i.e., a way to determine distances. The common Euclidean metric is a natural choice, but one could wonder whether other metrics could be used. Cage composed a similar piece, Variations I, in 1958. The main difference however is that Variations I uses transparencies having multiple lines, or points, each. Therefore, each configuration of lines (angles, positions, intersections, etc.) is fixed, and the same goes for the configuration of points. By removing these constraints, the materials of Variations II allow flexibility and freedom in determining the sound characteristics. Moreover, a fundamental difference regarding space exists between Variations I and Variations II. In Variations I, the transparencies contain multiple lines. Since transparencies (mylar sheets) are (tangible ) subsets of the infinite euclidean plane, they generally cannot be isometrically mapped to arbitrary curved surfaces, as per Carl Gauss’s famous Theorema Egregium. For example, the configuration of lines cannot be projected onto a sphere without modifying the distances and intersection points of these lines. In comparison, Variations II allows projection onto any surface since the transparencies only contain a single point or line. While a transparency is still a sheet of mylar in Variations II, it is merely a concretization of the point or the line as a geometrical concept that should be used according to the instructions. Therefore, what counts is the mathematical projection of the lines or the points in a given space, and less the actual process of laying down the mylar sheets. In this view, and since Cage’s instructions speak about “a suitable surface” without defining its particulars, a score...


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