Abstract
Journal of Mathematics and Music's recent special issue 7(2) reveals substantial common ground between mathematical theories of harmony advanced by Tymoczko, Hook, Plotkin, and Douthett. This paper develops a theory of scalar inflection as a kind of voice-leading distance using quantization in voice-leading geometries, which combines the best features of different approaches represented in the special issue: it is grounded in the concrete sense of voice-leading distance promoted by Tymoczko, invokes scalar contexts in a similar way as filtered point-symmetry, and abstracts the circle of fifths like Hook's signature transformations. The paper expands upon Tymoczko's ‘generalized signature transform’ showing the deep significance of generalized circles of fifths to voice-leading properties of all collections. Analysis of Schubert's Notturno for Piano Trio and ‘Nacht und Träume’ demonstrate the musical significance of inflection as a kind of voice leading, and the value of a robust geometrical understanding of it.
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