Modelling the similarity of pitch collections with expectation tensors

Milne, Andrew J.; Sethares, William A.; Laney, Robin and Sharp, David B. (2011). Modelling the similarity of pitch collections with expectation tensors. Journal of Mathematics and Music: Mathematical and Comptational Approaches to Music Theory, Analysis, Composition and Performance, 5(1) pp. 1–20.



Models of the perceived distance between pairs of pitch collections are a core component of broader models of music cognition. Numerous distance measures have been proposed, including voice-leading [1], psychoacoustic [2–4], and pitch and interval class distances [5]; but, so far, there has been no attempt to bind these different measures into a single mathematical or conceptual framework, nor to incorporate the uncertain or probabilistic nature of pitch perception.

This paper embeds pitch collections in expectation tensors and shows how metrics between such tensors can model their perceived dissimilarity. Expectation tensors indicate the expected number of tones, ordered pairs of tones, ordered triples of tones, etc., that are heard as having any given pitch, dyad of pitches, triad of pitches, etc.. The pitches can be either absolute or relative (in which case the tensors are invariant with respect to transposition). Examples are given to show how the metrics accord with musical intuition.

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