The two main artistic procedures of expressionism are the primordial utterance . . . and the imposition of an abstract structure, often specifically geometric, on reality. . . . [(Mittner, “L’Espressionismo fra l’Impressionismo.” Presented at the Convegno Internazionale di Studi sull’Espressionismo of the Maggio Fiorentino of 1964.)]

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. . . In the American period under consideration many kinds of “geometrical” schemata were applied to music, as they were also in Europe and Russia. The rhythmic experiments of Ives partly come out of this thinking, as do those of Varèse, while Ruggles, Ives, and Varèse seem to have experimented with pitch organization in comparative isolation. Ruth Crawford, in particular, developed all kinds of patterns of this sort. Her Piano Study in Mixed Accents (1930) uses variable meters and a retrograde pitch plan that reminds one of similar methods of Boris Blacher, while her String Quartet (1931), especially the last movement, juggles with quite a number of different “geometric” systems, one governing pitch, another dynamics, and still another the number of consecutive notes before a rest in any given passage. Besides, the whole movement is divided into two parts, the second a retrograde of the first a semitone higher. Cowell’s book, New Musical Resources has a chapter dealing with the association of pitch-interval ratios with speed ratios after the manner “discovered” later by certain Europeans. During the late 1920s and ’30s, Joseph Schillinger, who had come to America from Russia, bringing with him the fruits of similar thinking there, taught here. After his death, his The Schillinger System of Musical Composition was published (1946) with an introduction by Cowell; although it attempts to be an all-embracing method of explaining the technique of music of all types, it is, ultimately, simply another example of this aspect of expressionist “geometry” in that it applies “extrinsic” patterns derived from other fields of systematization and theoretical description to music, often without sufficiently taking into account the “intrinsic” patterns of musical discourse. As Mittner points out in this connection, “geometry” can be a way of building an entirely new world or a way of deforming or dissolving the old. It is possible that an illogical, disorganized geometry or a totally irrelevant one can be just as much of a deforming or even constructive pattern as one more obviously relevant and logical (although the chances are obviously higher that the latter will be more fruitful) in the hands of an imaginative composer. The history of the canon in all its phases is a clear demonstration of this.

Elliott Carter