Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

6 Perspectivic Approach to Voice Leadings

The last two paragraphs were solely dedicated to the study chord selfperspectives. This final paragraph now focusses on an idea concerning the possible syntactic roles of chord perspectives in the study of chord successions and voice leading.

While chords are just sets of tones, one is often interested in the distribution of tones along an ensemble of $k$ voices. The simultaneous presence of tones within these voices can be represented in terms of vectors, which are called $k$ -voicings:

( ) t1. v = .. . tk

In order to describe a $k$ -voice leading, where, from an melodic point of view, $k$ voices make their individual tone-steps simultaneously, or where, from an harmonic point of view, one has a succession of two $k$ -voicings, one can conveniently switch between the 3 interpretations of $k × 2- matrices$ ,

( ) ( ) (( ) ( )) s1 t1 (s1 t1) s1 t1 s = ... ... , smelo = ... , sharmo = ... ... . sk tk (sk tk) sk tk

Explorative studies suggest to interpret voice-leadings $s$ as follows: Consider the functions $p : Tk --> |CH | k$ mapping voicings to the underlying chords $p ((t,...,t)) := {t,...,t} k 1 k 1 k$ and let $s1 harmo$ and $s2 harmo$ denote the first and the second voicing of $s harmo$ , respectively. We distinguish two interpretations:

In the causal interpretation of the voice leading $s$ one considers the chord perspectives $º 1 2 scaus := A(pk(sharmo),pk(sharmo))$ and attributes to each coordinate $simelo = (si,ti)$ the subset $sicaus$ of those chord perspectives $f (- scaus$ satisfying $f(si) = ti$ .
In the final interpretation of the voice leading $s$ one considers the chord perspectives $sfin := ºA(pk(s2harmo),pk(s1harmo))$ and attributes to each coordinate $i smelo = (si,ti)$ the subset $i sfin$ of those chord perspectives $f (- sfin$ satisfying $f (ti) = si$ .

The sets $sicaus$ and $si fin$ are never empty, because they always contain exactly one constant tone perspective $ti0$ or $si0$ . The following diagrams show three examples: