Generating Milton Babbitt’s All-Partition Arrays

Abstract

In most of Milton Babbitt’s (1916–2011) works written since the early 1960s, both the pitch and rhythmic content is organized according to a highly constrained structure known as the all-partition array. The all-partition array provides a framework that ensures that as many different forms of a tone row as possible (generated by any combination of transposition, inversion or reversal) are expressed ‘horizontally’ and that each integer partition of 12 whose cardinality is no greater than the number of lynes in a piece is expressed by exactly one ‘vertical’ aggregate. We present a greedy backtracking algorithm for generating a particular type of all-partition array found in Babbitt’s works, known as a Smalley array. Constructing such an array is a difficult task, and we present two heuristics for helping to generate this type of structure. We provide the parameter values required by this algorithm to generate the specific all-partition arrays used in three of Babbitt’s works. Finally, we evaluate the algorithm and the heuristics in terms of how well they predict the sequences of integer partitions used in two of Babbitt’s works. We also explore the effect of the heuristics on the performance of the algorithm when it is used in an attempt to generate a novel array.