Combinatoriality without the Aggregate
Robert Morris · Perspectives of New Music · 1982
The attraction of specific harmonic or linear sequences of pitches and/or pitch-classes is a motivating force in composition, but the problem of finding a meaningful context for such entities is an equally familiar situation. One is often frustrated by the fact that the way to generalize or develop a pitchpattern is often unknown or unclear. Even with the automated means to generate the members of certain interesting categories of compositional materials, a composer is all too often ironically confronted with a multitude of patterns with the desired properties but which do not satisfy his or her immediate appetite. When the return to a more ad hoc and less intuitive approach would be less fruitful, the decision either to strive for a more powerful generalization or to be content working within a more predictable zone becomes a necessary fact of creative life. The development of the formalization of the Twelve-Tone System [1] and its role in compositional structure is one example of how successive generalizations can lead to greater flexibility and utility. Thus, an increasingly wider spectrum of compositional proprieties and inclinations is served. Here I am thinking of the development of the concept of chromatic completion implicit in the idea of the twelve-tone row from Arnold Schoenberg's earliest row pieces through his 'hexachordal' compositions. In turn, the generalization of Milton Babbitt and others [2], as described by the theory of hexachordal combinatoriality and secondary sets with trichordal generators, has produced the formalization of generalized combinatoriality and linear aggregate formation [3]. In this paper I will continue this line of development still further-but not in exactly the same direction-so that the contents of the rows and columns in a CM [4] or LA [5] need not be the un-, partiallyor totally-ordered set of all pcs but any sub-set of the total chromatic. Specifically, the rows and columns will all be members of the same set-class (SC) [6], that is, related under Th and/or I, or of two different SCs. With the two-dimensional CM, as in tonal music, the vertical structures can be of a different SC and cardinality from the horizontal ones. As Babbitt has pointed out, Given a collection of available elements, the choice of a sub-collection of these as a referential norm provides a norm that is distinguished by content alone!' [7] Thus, I will be concerned primarily