Chords in musical harmony can be viewed as objects having shapes
(major/minor/etc.) attached to base sets (pitch class sets). The base set and
the shape set are usually given the structure of a group, more particularly a
cyclic group. In a more general setting, any object could be defined by its
position on a base set and by its internal shape or state. The goal of this
paper is to determine the structure of simply transitive groups of
transformations acting on such sets of objects with internal symmetries. In the
main proposition, we state that, under simple axioms, these groups can be built
as group extensions of the group associated to the base set by the group
associated to the shape set, or the other way. By doing so, interesting groups
of transformations are obtained, including the traditional ones such as the
dihedral groups. The knowledge of the group structure and product allows to
explicitly build group actions on the objects. In particular we differentiate
between left and right group actions and we show how they are related to
non-contextual and contextual transformations. Finally we show how group
extensions can be used to build transformational models of time-spans and
rhythms.Comment: 30 pages, 4 figures ; submitted to Journal of Mathematics and Music -
v.4: corrected many errors, clarified some proposition
