Maximal translational equivalence classes of musical patterns in point-set representations

Representing musical notes as points in pitch-time space causes repeated motives and themes to appear as translationally related patterns that often correspond to maximal translatable patterns (MTPs). However, an MTP is also often the union of a salient pattern with one or two temporally isolated notes. This has been called the problem of isolated membership. Examining the MTPs in musical works reveals that salient patterns may correspond more often to the intersections of MTPs than to the MTPs themselves. We therefore explore patterns that are maximal with respect to their translational equivalence classes (MTEC). We prove that a pattern is MTEC if and only if it can be expressed as the intersection of MTPs. MTECs could therefore solve the problem of isolated membership. We also prove a relationship between MTECs and so-called conjugate patterns