A Graphical Theory of Musical Pitch Spelling

Abstract

Based on the conventions of staff-based music notation, musical passages that sound alike can often be written down in multiple ways, with each representation being considered a different "pitch-spelling". In this thesis, the pitch spelling problem -- that is, the question of finding the "right" spelling for a given passage of music -- is addressed. The theory presented departs from an emphasis placed on tonal harmony that has been the standard in prior literature; in addition to the conventional 12-tone scale, the problem is addressed in microtonal contexts, where "in-between-tones" -- first quarter tones, and then eighth tones -- are introduced. A novel graphical model of relationships between the notes of a passage of music is presented that reduces the problem of pitch-spelling to the minimum cut problem from the study of flow networks. The minimum cut problem is known to be efficiently solvable, but is generally encountered as the dual of the maximum flow problem. Here, minimum cut is presented as a primal problem. An inverse formulation is developed as a means of extracting cost parameters from a large corpus of notated music through an application of duality. The solution to the inverse problem is used to set parameters of the forward (minimum cut) problem.