Comparison of Heuristics for Generating All-partition Arrays in the Style of Milton Babbitt

An all-partition array is a mathematical structure developed by Milton Babbitt (1916–2011) for organizing the pitch classes in many of his twelve-tone works. Constructing an all-partition array requires finding a sequence of l aggregate-forming subsets of a matrix of pitch classes called a PCMatrix. Each of these l subsets corresponds to a distinct in- teger partition and l is the total number of distinct integer partitions possible, given the number of rows in the PCMatrix. Collectively, the l subsets (or integer partitions) form a partition of the PCMatrix. In some types of all-partition arrays, where l = 58, the PCMatrix does not contain all 12l pitch classes. In these types, Babbitt developed a method for adding to the PCMatrix what we call outer-aggregate repeated pcs (OARPs). A self-contained sequence of integer partitions is one in which each integer partition either contains a complete aggregate or an incomplete one that can be made complete by adding OARPs. It is noteworthy that, when constructing an all-partition array, Babbitt started out with a non-self-contained sequence of partitions. In this paper, we use a known self-contained sequence as a basis for forming two heuristics that select integer partitions likely to have been chosen by Babbitt. We suggest these heuristics will select integer partitions more likely to produce a self-contained sequence and we present it as a means for efficiently searching the space of possible sequences. We apply our heuristics to both types of sequences and conclude with a simple method for evaluating the results.