Groups II

Abstract

Abstract This chapter opens Part Two, devoted to the study of transformation theory, with some additional topics in group theory that arise in musical applications. Transformation groups on finite spaces may be regarded as permutation groups; permutation groups on pitch-class space include not only the groups of transpositions and inversions but also the multiplication group, the affine group, and the symmetric group. Another musical illustration of permutations involves the rearrangement of lines in invertible counterpoint. The structure of a finite group may be represented in the form of a group table or a Cayley diagram (a kind of graph). Other concepts discussed include homomorphisms and isomorphisms of groups, direct-product groups, normal subgroups, and quotient groups. Groups underlie many examples of symmetry in music, as formalized through the study of equivalence relations, orbits, and stabilizers.