Predicting Babbitt’s orderings of time-point classes from array aggregate partitions in None but the Lonely Flute and Around the Horn

Milton Babbitt (1916–2011) is credited with developing several techniques of 12-tone composition that extend beyond pitch. One such technique, the time-point system, figures prominently in his mature rhythmic practice. In many of his works based on the all-partition array, the possible orderings of pitch classes and time-point classes from an aggregate partition are the same. However, the number available is often large and his reasons for choosing one ordering over another are diverse and not clearly understood. In this article, we propose that, when constructing linear orderings of time-point classes from aggregate partitions in an all-partition array, Babbitt attempted to minimize both (1) their dissimilarity to the orderings of pitch classes from the same aggregate partitions and (2) the amount of counter-evidence they provide against a preselected beat. We first review two existing measures, Rothgeb's dissimilarity measure using order inversions and Povel and Essens' clock induction model. We then present a way to determine the exact number of possible orderings of time-point classes (and pitch classes) without repetitions in a particular aggregate partition. Next, we introduce a novel heuristic, based on the aforementioned measures, for predicting from the available orderings of time-point classes those particular orderings chosen by Babbitt. We conclude by evaluating how well this heuristic predicts the orderings of time-point classes found in two of Babbitt's works, None but the Lonely Flute (1991) and Around the Horn (1993).