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Triads as Modes within Scales as Modes

  • Thomas NollEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9110)

Abstract

The paper revisits results from scale theory through the study of modes. Point of departure is the nested hierarchy of triads embedded into diatonic modes embedded into the chromatic scale. Generic diatonic triads can be described as stacks of thirds or as stacks of triple-fifths. These two possibilities lead to different generalizations and it is argued that the latter possibility is better suited for the understanding of harmonic tonality. Three types of well-formed modes are investigated: Simple Modes, Chain Modes, and Diazeuctic Modes. Chain Modes and Diazeuctic modes are the modal counterparts of the two types of reduced scales with the Partitioning Property in Clough and Myerson (1985), as well as of Agmon’s (1989) types A and B of generalized diatonic systems.

Keywords

Diatonic modes Well-formed words Algebraic combinatorics on words 

References

  1. 1.
    Agmon, E.: A mathematical model of the diatonic system. J. Music Theory 33(1), 1–25 (1989)CrossRefGoogle Scholar
  2. 2.
    Babbitt, M.: The stucture and function of music theory: I. Coll. Music Symp. 5, 49–60 (1965)Google Scholar
  3. 3.
    de Berthé, V.A., Luca, C., Reutenauer, C.: On an involution of christoffel words and sturmian morphisms. Eur. J. Comb. 29(2), 535–553 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carey, N., Clampitt, D.: Aspects of well formed scales. Music Theory Spectr. 11(2), 187–206 (1989)CrossRefGoogle Scholar
  5. 5.
    Carey, N., Clampitt, D.: Self-similar pitch structures their duals, and rhythmic analogues. Perspect. New Music 34(2), 62–87 (1996)CrossRefGoogle Scholar
  6. 6.
    Clampitt, D., Noll, T.: Modes, the height-width duality, and handschin’s tone character. Music Theory Online, 17/1 (2011)Google Scholar
  7. 7.
    Clough, J., Myerson, G.: Variety and multiplicity in diatonic systems. J. Music Theory 29(2), 249–270 (1985)CrossRefGoogle Scholar
  8. 8.
    Clough, J., Douthett, J.: Maximally even sets. J. Music Theory 35, 93–173 (1991)CrossRefGoogle Scholar
  9. 9.
    Clough, J., Engebretsen, N., Kochavi, J.: Scales, sets, and interval cycles: a taxonomy. Music Theory Spectr. 21, 74–104 (1999)CrossRefGoogle Scholar
  10. 10.
    de Jong, K., Noll, T.: Fundamental passacaglia: harmonic functions and the modes of the musical tetractys. In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds.) MCM 2011. LNCS, vol. 6726, pp. 98–114. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  11. 11.
    Descartes, R.: 1656 = 1978. In: Descartes, R. (ed.) Musicae Compendium - Leitfaden der Musik. Wissenschaftliche Buchgesellschaft, Darmstadt (1978) Google Scholar
  12. 12.
    Douthett, J.: Filtered Point-Symmetry and Dynamical Voice-Leading. In: Douthett, J., et al. (eds.) Music Theory and Mathematics: Chords, Collections, and Transformations. University of Rochester Press, Rochester (2008) Google Scholar
  13. 13.
    Ganter, B., Wille, R.: Formal Concept Analysis Mathematical Foundations. Springer, Heidelberg (1999) CrossRefzbMATHGoogle Scholar
  14. 14.
    Handschin, J.: Der Toncharakter: Eine Einführung in die Tonpsychologie. Atlantis, Zurich (1948) Google Scholar
  15. 15.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002) CrossRefzbMATHGoogle Scholar
  16. 16.
    Žabka, M.: Editorial. J. Math. Music 7(2), 83–88 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Departament de Teoria, Composició i DireccióEscola Superior de Música de CatalunyaBarcelonaSpain

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