Tone rows and tropes

Abstract

Applying various methods based on group actions we provide a complete classification of tone rows in the twelve-tone scale. The main objects of the present paper are the orbits of tone rows under the action of the direct product of two dihedral groups. This means that tone rows are considered to be equivalent if and only if they can be constructed by transposition, inversion, retrograde, and/or time shift (rotation) from a single row. We determine the orbit, the normal form, the stabilizer class of a tone row, its trope structure, diameter distance, and chord diagram. A database provides complete information on all pairwise non-equivalent tone rows. It can be accessed via http://www.uni-graz.at/∼fripert/db/. Bigger orbits of tone rows are studied when we allow further operations on tone rows such as the quart-circle (multiplication), the five-step (multiplication in the time domain), or the interchange of parameters.