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Non-Contextual JQZ Transformations

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Mathematics and Computation in Music (MCM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11502))

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Abstract

Initiated by David Lewin, the contextual PLR-transformations are well known from neo-Riemannian theory. As it has been noted, these transformations are only used for major and minor triads. In this paper, we introduce non-contextual bijections called JQZ transformations that could be used for any kind of chord. These transformations are pointwise, and the JQZ group that they generate acts on any type of n-chord. The properties of these groups are very similar, and the JQZ-group could extend the PLR-group in many situations. Moreover, the hexatonic and octatonic subgroups of JQZ and PLR groups are subdual.

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Notes

  1. 1.

    Observe, however, that applying the formulae (1) to chord inversions leads to: \(L\left\langle 0,4,7\right\rangle =\left\langle 11,7,4\right\rangle \) (Em), \(L\left\langle 7,4,0\right\rangle =\left\langle 9,0,4\right\rangle \) (Am) and \(L\left\langle 4,7,0\right\rangle =\left\langle 3,0,7\right\rangle \) (Cm). In this case, one can not use the equivalence of chord inversions. But these formulae can alternatively be interpreted as voicing transformations [10].

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Acknowledgements

We thank anonymous reviewers for valuable remarks and Thomas Noll for comments that greatly improved the manuscript.

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Authors and Affiliations

  1. Atomic Energy Commission, Université Paris Lumières (CEA-CIPh), Paris, France

    Franck Jedrzejewski

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  1. Franck Jedrzejewski
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Correspondence to Franck Jedrzejewski .

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Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technical University of Madrid, Madrid, Spain

    Francisco Gomez-Martin

  3. Technological University of the Mixteca, Oaxaca, Mexico

    Octavio A. Agustín-Aquino

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Jedrzejewski, F. (2019). Non-Contextual JQZ Transformations. 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_12

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  • DOI: https://doi.org/10.1007/978-3-030-21392-3_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21391-6

  • Online ISBN: 978-3-030-21392-3

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