Thinking Out of the Grid and Inside the Spiral – Geometric Interpretations of and Comparisons with the Spiral Array Model

This paper presents a geometric interpretation of the Spiral Array model and its comparison to Lerdahl's Tonal Pitch Space and Krumhansl's spatial representation of pitch relations. The Spiral Array model is derived from a three-dimensional configuration of the Harmonic Network. The fundamental idea underlying the model is the representing of higher-level objects in the spiral's interior as convex combinations of the representations of the lower level components. By using the interior of the spiral, the original discrete space is relaxed to one that is continuous. Geometric mappings are demonstrated among Lerdahl's Tonal Pitch Space, Krumhansl's and Krumhansl & Kessler’s spatial representations of pitch class and key relations, and the Spiral Array model. The interior-point approach of the Spiral Array model is shown to generate higher-level structures that are consistent with the results of these other approaches. The advantages of the interior-point approach are that it facilitates comparisons across different hierarchical levels, and problems that were previously combinatorial in nature can be modeled more efficiently and robustly, mathematically and computationally, using the continuous space in the interior. Author’s estimated word count (<5,000): 6090 Date of submission: 15 October, 2003 Date of revision: 12 September, 2007 2 COMPUTING IN MUSICOLOGY PREPRINT • PREPRINT • PREPRINT • PREPRINT • PREPRINT