A Tonnetz model for pentachords
Luis A. Piovan · Journal of Mathematics and Music · 2013
This article deals with the construction of surfaces that are suitable for representing pentachords or five-pitch-class segments that are in the same T/I-class. It is a generalization of the well-known Oettingen–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 generated by these 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 12-tone enharmonic scale case. Section 6 contains the main theorem, which gives the construction of a tiled surface by pentagons so that all transforms of a given pentachord fit together. This surface is a covering of a genus 13 surface, which is in turn a 2-to-1 cover of the genus 4 Bring surface.