Technical Reports

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

IMSC-03-002 PDF Version of Full Report
      Elaine Chew

Abstract:
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 based on 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 continous. Geometric mappings are demonstrated among Lerdahl's Tonal Pitch Space, Krumhansl's spatial representation of pitch 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 advantage of the interior point approach is that it facilitates comparisons across different hierarchical levels and problems that were previously combinatorial in nature can be modeled more efficiently using the continuous space in the interior.