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Algebra of Harmony: Transformations of Just Consonances

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Mathematics and Computation in Music (MCM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10527))

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Abstract

The paper focuses on mathematical aspects of harmonies in extended just intonation and their relations. The first part lays down a theoretical framework for the investigation of structural features of such harmonies. Among other aspects, it addresses symmetry, inversion, and multiplication of harmonies. The second part explores transformational relations among harmonies of the same type, while the approach is intrinsically dualistic. Riemann-Klumpenhouwer’s concepts of Schritts and Wechsels are generalized for ‘harmony spaces’ in extended just intonation. This enables a deeper analysis of harmonic ‘neighborhoods.’ Finally, a graphical representation of the complete neighborhood of a harmony, called ‘neighborhood network,’ is presented along with several simpler and more complex examples.

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Acknowledgement

This paper was supported by the scientific grant agency VEGA through the grant no. 1/0637/15.

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Authors and Affiliations

  1. Katedra muzikológie, Univerzita Komenského, Bratislava, Slovakia

    Marek Žabka

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  1. Marek Žabka
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Correspondence to Marek Žabka .

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Editors and Affiliations

  1. UNCA and UTM, Oaxaca, Mexico

    Octavio A. Agustín-Aquino

  2. UNAM, Mexico City, Mexico

    Emilio Lluis-Puebla

  3. Georgia State University, Atlanta, Georgia, USA

    Mariana Montiel

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Žabka, M. (2017). Algebra of Harmony: Transformations of Just Consonances. In: Agustín-Aquino, O., Lluis-Puebla, E., Montiel, M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science(), vol 10527. Springer, Cham. https://doi.org/10.1007/978-3-319-71827-9_7

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  • DOI: https://doi.org/10.1007/978-3-319-71827-9_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71826-2

  • Online ISBN: 978-3-319-71827-9

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