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Transformations for Pairwise Well-Formed Modes

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Mathematics and Computation in Music (MCM 2022)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13267))

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Abstract

The paper extends the transformational approach to pairwise well-formed (PWWF) modes, represented as words over a 3-letter alphabet. In particular it addresses open problems arising from an earlier approach (Noll, T and D. Clampitt. 2018. “Kaleidoscope substitutions and pairwise well-formed modes: Major-Minor duality transformationally revisited”, Journal of Mathematics and Music. 12(3)), wherein kaleidoscope transformations were shown to generate PWWF modes, but it was not yet shown that all PWWF modes could be so generated. This gap is filled. A further problem is that the kaleidoscope transformations are not closed under composition. The final section introduces a new construction for the generation of PWWF words/modes, transformations of words on a 4-letter alphabet. A strongly-supported conjecture is that these transformations form a monoid.

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References

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Author information

Authors and Affiliations

  1. Departament de Creació i Teoria Musical, Escola Superior de Música de Catalunya, Barcelona, Spain

    Thomas Noll

  2. School of Music, Ohio State University, Columbus, USA

    David Clampitt

Authors
  1. Thomas Noll
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  2. David Clampitt
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Corresponding author

Correspondence to David Clampitt .

Editor information

Editors and Affiliations

  1. Georgia State University, Atlanta, GA, USA

    Mariana Montiel

  2. Technological University of the Mixteca, Huajuapan de León, Mexico

    Octavio A. Agustín-Aquino

  3. Universidad Politécnica de Madrid, Madrid, Spain

    Francisco Gómez

  4. Life University, Marietta, GA, USA

    Jeremy Kastine

  5. National Autonomous University of Mexico, Mexico City, Distrito Federal, Mexico

    Emilio Lluis-Puebla

  6. Georgia State University, Atlanta, GA, USA

    Brent Milam

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Noll, T., Clampitt, D. (2022). Transformations for Pairwise Well-Formed Modes. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_12

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  • DOI: https://doi.org/10.1007/978-3-031-07015-0_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-07014-3

  • Online ISBN: 978-3-031-07015-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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