  Transformational music theory is a recent field in music theory which studies
the possible transformations between musical objects, such as chords. In the
framework of the theory initiated by David Lewin, the set of all
transformations forms a group which acts simply transitively on the set of
musical objects. In particular, neo-Riemannian theory focuses on the action of
the T/I group or the PLR group, which are both isomorphic to the dihedral group
D24, on the set of major and minor triads. It has been shown recently that
generalized neo-Riemannian groups of transformations can be built as group
extensions. By generalizing this construction, groupoids of transformations
between different pitch-class sets can be obtained. The goal of this paper is
to give details concerning the calculation of musical group and groupoid
actions, both in the covariant and contravariant case. In doing so, we
introduce a concept of generalized inversion transformations for the partial
transformation between pitch-class sets.
