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Glarean’s Dodecachordon Revisited

  • Thomas Noll
  • Mariana Montiel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7937)

Abstract

Diatonic Modes can be modeled through automorphisms of the free group F 2 stemming from special Sturmian morphisms. Following [1] and [2] we associate special Sturmian morphisms f with linear maps E(f) on a vector space of lattice paths. According to [2] the adjoint linear map E(f) ∗  is closely related to the linear map E(f  ∗ ), where f and f  ∗  are mutually related under Sturmian involution. The comparison of these maps is music-theoretically interesting, when an entire family of conjugates is considered. If one applies the linear maps E(f 1), ..., E(f 6) (for the six authentic modes) to a fixed path of length 2, one obtains six lattice paths, describing a family of authentic common finalis modes (tropes). The images of a certain path of length 2 under the application of the adjoint maps E(f 1) ∗ , ..., E(f 6) ∗  properly matches the desired folding patterns as a family, which, on the meta-level, forms the folding of Guido’s hexachord. And dually, if one applies the linear maps \(E(f_1^\ast), ..., E(f_6^\ast)\) (for the foldings of the six authentic modes) to a fixed path of length 2, one obtains six lattice paths, describing a family of authentic common origin modes (“white note” modes). The images of a certain path of length 2 under the application of the adjoint maps \(E(f_1^\ast)^\ast, ..., E(f_6^\ast)^\ast\) properly match the desired step interval patterns as a family, which, on the meta-level, forms the step interval pattern of Guido’s hexachord. This result conforms to Zarlino’s re-ordering of Glarean’s dodecachordon.

Keywords

Diatonic Modes Well-Formed Words Twisted Adjoints Algebraic Combinatorics on Words Sturmian Morphisms Sturmian Involution Lattice Paths Adjoint Linear Maps 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Thomas Noll
    • 1
  • Mariana Montiel
    • 2
  1. 1.Departament de Teoria, Composició i DireccióEscola Superior de Música de CatalunyaBarcelonaSpain
  2. 2.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA

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