A Study of Musical Pitch Distance Using a Self-Organized Hierarchical Linear Dynamical System on Acoustic Signals
The hierarchical linear dynamical system (HLDS) is a self-organizing architecture to cluster acoustic time series. The HLDS architecture is equivalent to a Kalman filter whose top-layer state learns to create subspaces that tessellate the acoustic signal in regions that correspond to different musical pitches. The observation layer of the HLDS is built from a biologically plausible gammatone filter bank that provides the representation space for the state assignments. An important characteristic of the methodology is that it is adaptive and self-organizing, i.e., previous exposure to the acoustic input is the only requirement for learning and recognition. In this article we show that the representation space that the algorithm learns from acoustic signals preserves the organization found in monophonic notes, and exhibits (for isolated pitches and triads) properties suggested in the theory of efficient chromatic voice leading and neo-Riemannian theories.