MC 5501
Speaker
William Sethares (University of Wisconsin–Madison)
Title
Invariances in the Structure of Musical Spaces and Applications to Keyboard Design
Abstract
This talk describes the general principles underlying two-dimensional keyboard layouts that are invariant in both transposition and tuning. In these layouts, there is a continuous parameter that generates a continuum of tunings that can be mapped to a button field (the "keyboard") so that the geometric shape of each musical interval is the same within a key, across all keys, and throughout all tunings in the continuum. For example, the fingering of a major chord in 7-tone equal temperament, the major chord in 12-tone equal temperament and the major chord in 19-tone equal temperament are all the same (along with many others). A concrete description of these invariances are given by a row-reduced matrix of the temperament-mapping (the "musical spaces" of the title) and the keyboard geometry. The range of values over which consistent fingering can be maintained is described by the valid tuning range, and the overtones of the sound can be retuned via the same mapping in order to minimize sensory dissonance. A number of concrete examples illustrate the generality of the methods and their applicability to a wide variety of commas and temperaments, tuning continua, and keyboard layouts. A prototype software-synthesizer demonstrates the feasibility of the designs. Work in collaboration with Andrew Milne of Western Sydney University.