Neo-Riemannian Transformations

Physics: Parsimonious voice leading: each PLR operation moves exactly one voice by one semitone. Hexatonic cycle: P→L repeated → cycles through 6 triads (C+, Am, F+, Dm, A♭+, Fm), returning to start. Octatonic cycle: P→R → 8 triads. John Williams and Elfman use chromatic mediants (triads a M3 apart) for film music. Schubert's late works are full of chromatic third relations. Lewin (1987) "Generalized Musical Intervals and Transformations" founded the mathematical framework. The dihedral group D₂₄ acts on the 24 major/minor triads.