A Tonnetz Model for pentachords

TL;DR

This paper introduces a geometric model using surfaces and tessellations to represent pentachords in music theory, generalizing existing models for triads to five-note segments within the twelve-tone scale.

Contribution

It develops a novel surface-based framework for pentachords, extending the Ottingen-Riemann torus concept to higher pitch segments in neo-Riemannian theory.

Findings

Constructed surfaces for pentachords as coverings of tessellations

Described the surfaces in the context of the twelve-tone enharmonic scale

Established a method to classify pentachords via surface tilings

Abstract

This article deals with the construction of surfaces that are suitable for representing pentachords or 5-pitch segments that are in the same $T/I$ class. It is a generalization of the well known \"Ottingen-Riemann torus for triads of neo-Riemannian theories. Two pentachords are near if they differ by a particular set of contextual inversions and the whole contextual group of inversions produces a Tiling (Tessellation) by pentagons on the surfaces. A description of the surfaces as coverings of a particular Tiling is given in the twelve-tone enharmonic scale case.