Integer Programming Formulation of the Problem of Generating Milton Babbitt's All-partition Arrays

Milton Babbitt (1916–2011) was a composer of twelve-tone serial music noted for creating the all-partition array. The problem of generating an all-partition array involves finding a rectangular array of pitch-class integers that can be partitioned into regions, each of which represents a distinct integer partition of 12. Integer programming (IP) has proven to be effective for solving such combinatorial prob- lems, however, it has never before been applied to the problem addressed in this paper. We introduce a new way of viewing this problem as one in which restricted overlaps between integer partition regions are allowed. This permits us to describe the problem using a set of linear constraints necessary for IP. In particular, we show that this problem can be defined as a special case of the well-known problem of set-covering (SCP), modified with additional constraints. Due to the difficulty of the problem, we have yet to discover a solution. However, we assess the potential practicality of our method by running it on smaller similar problems.