Maximal Translational Equivalence Classes of Musical Patterns in Point-Set Representations
- Cite this paper as:
- Collins T., Meredith D. (2013) Maximal Translational Equivalence Classes of Musical Patterns in Point-Set Representations. In: Yust J., Wild J., Burgoyne J.A. (eds) Mathematics and Computation in Music. MCM 2013. Lecture Notes in Computer Science, vol 7937. Springer, Berlin, Heidelberg
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 suggests that salient patterns may correspond more often to the intersections of MTPs than to the MTPs themselves. This paper makes a theoretical contribution, by exploring properties of 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. We also prove a relationship between MTECs and so-called conjugate patterns.
KeywordsPattern Discovery Motivic Analysis Repetition in Music Point-Set Patterns Music Analysis Geometric Music Representations
Unable to display preview. Download preview PDF.