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

Dmitri Tymoczko

in The Oxford Handbook of Neo-Riemannian Music Theories

Published in print December 2011 | ISBN: 9780195321333
Published online September 2012 | | DOI:

Series: Oxford Handbooks

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

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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 nineteenth-century chromaticism involves inversional symmetry. Section 4 provides a Riemannian understanding of dualism and Section 5 illustrates a contrapuntal approach by examining a Brahms intermezzo.

Keywords: inversional symmetry; chromatic music; concepts of symmetry; Rameau; tonal harmony; Weber; dualism; Brahms

Article.  9355 words. 

Subjects: Music ; Music Theory and Analysis

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