Abstract
This paper presents a generalization of the neo-Riemannian PLR group to the set of triads with inversions (major, minor, diminished and augmented). A second generalization is proposed, using an extended system of seventh chords with inversions. Both the sets of triads and seventh chords are defined with constraints on semitone separation of voices. In the case of triads, the set of parsimonious transformations is shown to have the structure of a semi-direct product of groups of the form \(S_{n} \ltimes \mathbb {Z}_{12}^{n-1}\), where n is the number of chord types in the set.
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Acknowledgements
We would like to thank the referee and the organizer Thomas Noll for their very helpful comments and suggested references.
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Klassen, M. (2019). Constraint-Based Systems of Triads and Seventh Chords, and Parsimonious Voice-Leading. 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_15
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DOI: https://doi.org/10.1007/978-3-030-21392-3_15
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