A geometrical approach to find the preferred intonation of chords

Abstract

Previous work suggested that convexity in the Euler lattice can be interpreted in terms of consonance [1]. In this paper, a second hypothesis is presented that states that compactness in the Euler lattice is an indication of consonance. The convexity and compactness of chords is used as the basis of a model for the preferred intonation of chords in isolation (without a musical context). It is investigated if, and to which degree convexity and compactness are in agreement with the preferred intonation of chords in isolation. As measure of consonance to compare the model to, Euler’s Gradus function is used. It is stressed however, that in the context of this paper, Euler’s consonance model is able to represent a general consonance model rather than only the Gradus function itself. First, the diatonic chords are observed, after which the compactness, convexity and consonance according to Euler, is calculated for all chords in general containing 2, 3 and 4 notes within a bounded note name space, such that the relation between these three measures can be obtained. The principle of compactness turns out to be a strong indicative of consonance for chords, having the preference over other consonance models that it is simple and intuitive to use