Permutation Groups and Chord Tesselations

Analytical approaches to musical analysis have generally attempted to provide some kind of justification, and, in the last decades, to develop analytical elements based on tesselations of pitch class sets. When applied to atonal music, analytical approaches frequently fail to find internal coherence. Most of the time, tilings of chords are based on a torus representation, which follows the traditional principles of functional harmonic progression. A possible solution for determining a nice coherence of atonal chords progression is dependent on the topology of the tesselation involved. In the first section, we show that some tilings of chords could be related to the Klein bottle or to the projective plane. It is well known that tesselations are connected with groups. Consequently, possibilities of musical analyses can be suggested through a theoretical group approach. In the second section, we consider the neo-Riemannian theory, and suggest a number of revisions. We substitute the contextual transformations by three pointwise operations based on permutations, in order to link harmonic hierarchical or categorized structures with permutation groups. In the last section, we draw upon Olivier Messiaen’s works in order to show that some groups are more suitable than others for analytical approaches, but in any case, are related to some permutation groups.