An algebra of chords for a non-degenerate Tonnetz

Abstract

For an n-TET tuning system, we propose a formalism to study the transformations of k-chords over a generalized non-degenerate Tonnetz generated by a given interval structure. Root and mode are the two components of a directed chord on which the algebra operates, so that chord transformations within one chord cell or towards other cells, and paths or simple circuits over the chord network can be determined without resorting to computational algorithms or geometrical representations. The one-step transformations over the edges of the chord network associated with the k − 1 drift operators generalize the basic operators P, R and L of the Neo-Riemmanian triadic progressions and the maximally smooth cycles of the 12-TET system to any higher dimensional space.