Dualism and the Beholder's Eye: Inversional Symmetry in Chromatic Tonal Music

This article considers the theme of inversional symmetry as it manifests itself in Riemann's theoretical writings and in late-nineteenth-century chromatic music. It examines the mathematical properties of the concepts of symmetry underlying musical systems and explores how these symmetrical properties can be brought to bear in innovative ways on musical structures. Section 1 of the article provides a historical background by examining Rameau and his proposed laws of tonal harmony which are invariant under four basic operations: reordering, octave shift, note duplication, and chromatic disposition. It also discusses Weber's Roman numeral notation which develops and fulfills Rameau's ideas. Section 2 discusses Riemann's "dualism" as an attempt to incorporate inversion into the Rameau/Weber collection of symmetries. Section 3 examines whether the "second practice" of the nineteenthcentury chromaticism involves inversional symmetry. Section 4 provides a Riemannian understanding of dualism and Section 5 illustrates a contrapuntal approach by examining a Brahms intermezzo.