Abstract
This paper contributes to the transformational study of progressions of seventh chords and generalizations thereof. PLR transformations are contextual transformations that originally apply only to consonant triads. These transformations were introduced by David Lewin and were inspired the works of musicologist Hugo Riemann. As an alternative to other attempts to define transformations on seventh chords, we define new groups in this article, called Rameau groups, which transform all types of seventh or ninth chords or more generally, any chords formed of stacks of major or minor thirds. These groups form a hierarchy for inclusion. We study on musical examples the ability of these operators to show symmetries in the progression of seventh chords.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Arnett, J., Barth, E.: Generalizations of the Tonnetz: tonality revisited. In: Proceedings of the 2011 Midstates Conference on Undergraduate Research in Computer Music and Mathematics (2011)
Cannas, S., Antonini, S., Pernazza, L.: On the group of transformations of classical types of seventh chords. In: Agustín-Aquino, O.A., Lluis-Puebla, E., Montiel, M. (eds.) MCM 2017. LNCS (LNAI), vol. 10527, pp. 13–25. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71827-9_2
Cannas, S.: Geometric representation and algebraic formalization of musical structures. Ph.D. Dissertation, University of Strasbourg and Università degli Studi di Pavia e di Milano-Biocca (2018)
Childs, A.: Moving beyond Neo-Riemannian triads: exploring a transformational model for seventh chords. J. Music Theory 42(2), 191–193 (1998)
Cohn, R.: Neo-Riemannian operations, parsimonious trichords, and their Tonnetz representation. J. Music Theory 41, 1–66 (1997)
Cohn, R.: Introduction to neo-riemannian theory: a survey and a historical perspective. J. Music Theory 42(2), 167–180 (1998)
Cohn, R.: Audacious Euphony. Chromaticism and the Triad’s Second Nature. Oxford University Press, Oxford (2012)
Douthett, J., Steinbach, P.: Parsimonious graphs: a study in parsimony, contextual transformation, and modes of limited transposition. J. Music Theory 42(2), 241–263 (1998)
Gollin, E.: Some aspects of three-dimensional Tonnetze. J. Music Theory 42(2), 195–206 (1998)
Fiore, T., Satyendra, R.: Generalized contextual groups. Music Theory. Online 11(3), 1–27 (2005)
Jedrzejewski, F.: Hétérotopies musicales. Modèles mathématiques de la musique, Paris, Hermann (2019)
Kerkez, B.: Extension of Neo-Riemannian PLR-group to Seventh Chords. In: Bridges, Mathematics, Music, Art, Architecture, Culture, pp. 485–488 (2012)
Rameau, J.-P.: Traité de l’harmonie réduite à ses principes naturels, Paris (1722)
Rameau, J.-P.: Nouveau système de musique théorique, Paris (1726)
Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)
Riemann, H.: Handbuch der Harmonielehre. Breitkopf, Leipzig (1887)
Schillinger, J.: The Schillinger System on Musical Compostion. C. Fischer, New York (1946)
Acknowledgements
We thank anonymous reviewers for valuable remarks and Thomas Noll for comments that greatly improved the manuscript.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Jedrzejewski, F. (2019). The Hierarchy of Rameau Groups. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-21392-3_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21391-6
Online ISBN: 978-3-030-21392-3
eBook Packages: Computer ScienceComputer Science (R0)