APPLICATION OF GROUP THEORY TO THE PLR-GROUP IN MUSIC THEORY

When listening to music, some people think about whether the notes in a piece have a particular mathematical order. Some research observes that some pieces do have this mathematical order if we put the pitch classes on a clock and regard this clock as a dodecagon (12-gon). The symmetry group of a dodecagon can be defined as a dihedral group of order 24. A triad is a combination of three different pitches. Changes of triads can be represented as operations of triads: the Parallel operation (P), Leading tone exchange operation (L), and Relative operation (R). We will define the PLR-group as a subgroup of the symmetric group on S generated by the bijections P, L, and R, where S is the set of major and minor triads. We will show that the PLR-group can be generated by L and R, and that it is isomorphic to the dihedral group of order 24. At the end, we will also see the P, L, R operations in real life by looking at some compositions that have P, L, R operations.