Tonal prisms: iterated quantization in chromatic tonality and Ravel's 'Ondine'

The mathematics of second-order maximal evenness has far-reaching potential for application in music analysis. One of its assets is its foundation in an inherently continuous conception of pitch, a feature it shares with voice-leading geometries. This paper reformulates second-order maximal evenness as iterated quantization in voice-leading spaces, discusses the implications of viewing diatonic triads as second-order maximally even sets for the understanding of nineteenth-century modulatory schemes, and applies a second-order maximally even derivation of acoustic collections in an in-depth analysis of Ravel's ‘Ondine’. In the interaction between these two very different applications, the paper generalizes the concepts and analytical methods associated with iterated quantization and also pursues a broader argument about the mutual dependence of mathematical music theory and music analysis.Accepted manuscrip

Distant Neighbors and Interscalar Contiguities

This paper studies the "integration" problem of nineteenth- century harmony the question whether the novel chromatic chord transitions in this time are a radical break from or a natural extension of the conventional diatonic system. We examine the connections between the local behavior of voice leading among diatonic triads and their generalizations on one hand, and the global properties of voice-leading spaces on the other. In particular, we aim to identify those neo-Riemannian chord connections which can be integrated into the diatonic system and those which cannot. Starting from Jack Douthett's approach of filtered point symmetries, we generalize diatonic triads as second-order Clough-Myerson scales and compare the resulting Douthett graph to the respective Betweenness graph. This paper generally strengthens the integrationist position, for example by presenting a construction of the hexatonic and octatonic cycles that uses the principle of minimal voice leading in the diatonic system. At the same time it provides a method to detect chromatic wormholes, i.e. parsimonious connections between diatonic chords, which are not contiguous in the system of second order Clough-Myerson scales