Advertisement

From Schritte and Wechsel to Coxeter Groups

  • Markus SchmidmeierEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11502)

Abstract

The PLR-moves of neo-Riemannian theory, when considered as reflections on the edges of an equilateral triangle, define the Coxeter group \(\widetilde{S}_3\). The elements are in a natural one-to-one correspondence with the triangles in the infinite Tonnetz. The left action of \(\widetilde{S}_3\) on the Tonnetz gives rise to interesting chord sequences. We compare the system of transformations in \(\widetilde{S}_3\) with the system of Schritte and Wechsel introduced by Hugo Riemann in 1880. Finally, we consider the point reflection group as it captures well the transition from Riemann’s infinite Tonnetz to the finite Tonnetz of neo-Riemannian theory.

Keywords

Tonnetz neo-Riemannian theory Coxeter groups 

Notes

Acknowledgements

The author would like to thank Benjamin Brück from Bielefeld, Germany, for helful comments regarding the Coxeter group. He is particularly grateful to Thomas Noll from Barcelona since his thoughtful advice, in particular regarding the action of interesting elements and subgroups of the Coxeter group on the Tonnetz, has led to substantial improvements of the paper (which about doubled in size).

References

  1. 1.
    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. GTM, vol. 231. Springer, Heidelberg (2005).  https://doi.org/10.1007/3-540-27596-7CrossRefzbMATHGoogle Scholar
  2. 2.
    Cohn, R.: Maximally smooth cycles, hexatonic systems, and the analysis of late-romantic triadic progressions. Music Anal. 15(1), 9–40 (1996)CrossRefGoogle Scholar
  3. 3.
    Cohn, R.: Neo-Riemannian transformations, parsimonious trichords, and their Tonnetz representations. J. Music Theory 41(1), 1–66 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cohn, R.: Introduction to neo-Riemannian theory: a survey and a historical perspective. J. Music Theory 42(2), 167–198 (1998)CrossRefGoogle Scholar
  5. 5.
    Cohn, R.: Audacious Euphony: Chromaticism and the Triad’s Second Nature: Oxford Studies in Music Theory. Oxford University Press, Oxford (2012)Google Scholar
  6. 6.
    Crans, A.S., Fiore, T.M., Satyendra, R.: Musical actions of dihedral groups. Am. Math. Mon. 116(6), 479–495 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Klumpenhouwer, H.: Some remarks on the use of Riemann transformations. Music Theory Online 0(9), 1–34 (1994)Google Scholar
  8. 8.
    Riemann, H.: Skizze einer Neuen Methode der Harmonielehre. Breitkopf und Haertel, Leipzig (1880)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Florida Atlantic UniversityBoca RatonUSA

Personalised recommendations