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Pairwise Well-Formed Modes and Transformations

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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

One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of the ancient and medieval world and that of the early modern world, associated especially with Zarlino. We do this at an abstract level, in the language of algebraic combinatorics on words. In scale theory the juxtaposition is between well-formed and pairwise well-formed scales and modes, expressed in terms of Christoffel words or standard words and their conjugates, and the special Sturmian morphisms that generate them. Pairwise well-formed scales are encoded by words over a three-letter alphabet, and in our generalization we introduce special positive automorphisms of F3, the free group over three letters.

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References

  1. Berthé, V., de Luca, A., Reutenauer, C.: On an involution of Christoffel words and Sturmian morphisms. Eur. J. Comb. 29(2), 535–553 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Clampitt, D., Noll, T.: Modes, the height-width duality, and Handschin’s tone character. Music Theor. Online 17(1) (2011)

    Google Scholar 

  3. Clampitt, D.: Mathematical and musical properties of pairwise well-formed scales. In: Klouche, T., Noll, T. (eds.) MCM 2007. CCIS, vol. 37, pp. 464–468. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04579-0_46

    Chapter  Google Scholar 

  4. Clampitt, D.: Pairwise well-formed scales: structural and transformational properties. Ph.D. dissertaion, SUNY at Buffalo (1997)

    Google Scholar 

  5. Carey, N., Clampitt, D.: Self-similar pitch structures, their duals, and rhythmic analogues. Perspect. New Music 34(2), 62–87 (1996)

    Article  Google Scholar 

  6. Carey, N., Clampitt, D.: Aspects of well-formed scales. Music Theor. Spectr. 11(2), 187–206 (1989)

    Article  Google Scholar 

  7. Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  8. Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory. Dover Publications, New York (2004)

    MATH  Google Scholar 

  9. Nielsen, J.: Die Isomorphismengruppe der freien Gruppen. Math. Ann. 91, 169–209 (1924)

    Article  MathSciNet  MATH  Google Scholar 

  10. Noll, T.: Ionian theorem. J. Math. Music 3(3), 137–151 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    David Clampitt

  2. Escola Superior de Música de Catalunya, Departament de Teoria, Composició i Direcció, Barcelona, Catalonia, Spain

    Thomas Noll

Authors
  1. David Clampitt
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  2. Thomas Noll
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Correspondence to David Clampitt .

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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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Clampitt, D., Noll, T. (2017). Pairwise Well-Formed Modes and Transformations. 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_3

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  • DOI: https://doi.org/10.1007/978-3-319-71827-9_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71826-2

  • Online ISBN: 978-3-319-71827-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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