Geometry of Chords

This article provides an introduction to basic geometric investigations of the 12-tone sytem and its subsets, the chords. The various definitions and results are intended to lay a theoretical basis for 12-tone-based explorative and empirical research on occidental harmony. The entire approach is motivated by the assumption that the 12 tones do not constitute an arbitrary 12-element set with arbitrary paradigmatic relations, but rather a discrete geometrical space. Transpositions obviously are instances of paradigmatic relations, that are reflected by affine transformations. From the mathematical point of view, the article presents a straightforward elaboration of this observation. From the music theoretical point of view it is of speculative nature and asks for experimental studies.