Distant Neighbors and Interscalar Contiguities

Abstract

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.

Fishing for complements with chord, scale, and rhythm nets

“…Type (2) transitions, unlike type (1), might define minimal voice leadings of distance 2 and, thus, intermediate chromatic passing chords, which do not necessarily belong to CM 12,7,4 , and are reached and left through the movement of a single voice (Harasim and Rohrmeier (2019)). The process of adding such chords and connections as vertices and edges of the respective Douthett graph (and removing distance 2 edges) takes the name of chromatic saturation.…”

midiVERTO: A Web Application to Visualize Tonality in Real Time

“…A screenshot of the application is shown in Figure 1. 4 Apart from a welcome page ("Home") and a step-by-step tutorial ("Docs"), the application features an interface consisting of four parts ("Analysis"): an input-output menu on the left, two visualization panels in the center, and a control menu at the bottom.…”

“…For a general description of this method's mathematical details and music-theoretical interpretations see for instance [2]. This usage of the DFT essentially maps subsets of cyclic groups to six complex numbers (i.e., the Fourier coefficients 1 to 6) and many interesting properties, such as evenness [1,4], balancedness [7], and diatonicity [13], can be identified and studied with the Fourier coefficients.…”

The Tonal Diffusion Model

“…The infinite expansion of this graph is called the Tonnetz (Bigo and Andreatta, 2016;Cohn, 1997Cohn, , 2012Harasim et al, 2019) and has a number of historic precursors (e.g., Euler, 1739;Hostinský, 1879;von Oettingen, 1866;Riemann, 1896). Using the representation as a hexagonal graph (Douthett and Steinbach, 1998), PCDs can be plotted as color-coded heatmaps (Moss, 2019).…”