Slicing It All Ways: Mathematical Models for Tonal Induction, Approximation, and Segmentation Using the Spiral Array

This paper presents the spiral array model and its associated algorithms for tonal induction, approximation, and segmentation. The spiral array is a geometric model for tonality that clusters perceptually similar tonal entities. The model summarizes music information as interior points inside an array of spirals. Distances in the spiral array space are used to quantify tonal similarity. The paper traces the evolution, and presents general forms, of the existing algorithms for key finding, pitch spelling, and segmentation, and proposes a new O(n) algorithm, Argus, for tonal segmentation. The proposed algorithm computes a value that quantifies the discrepancy between the local contexts in the future and past at each point in time. Discrepancy values exceeding control thresholds are shown to mark the segmentation boundaries of the test set that concur with expert analyses. A number of window sizes and threshold settings are investigated. The algorithm is demonstrated using Edward McDowell’s “To A Wild Rose” and tested on Franz Schubert’s Allegretto from Moment Musical D780 No. 6 and Thema from Impromptu D935 No. 4. The algorithm accurately locates tonal boundaries in all three case studies.