Glarean’s Dodecachordon Revisited
Diatonic Modes can be modeled through automorphisms of the free group F 2 stemming from special Sturmian morphisms. Following  and  we associate special Sturmian morphisms f with linear maps E(f) on a vector space of lattice paths. According to  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.
KeywordsDiatonic Modes Well-Formed Words Twisted Adjoints Algebraic Combinatorics on Words Sturmian Morphisms Sturmian Involution Lattice Paths Adjoint Linear Maps
Unable to display preview. Download preview PDF.
- 3.Clampitt, D., Noll, T.: Modes, the height-width duality, and Handschin’s tone character. Music Theory Online 17(1) (2011)Google Scholar
- 6.Glarean, H.: Dodecachordon. Basel (1547); Reprint: Olms, Hildesheim (1969)Google Scholar
- 7.Megozzi, S.: The Renaissance Reform of Medieval Music Theory: Guido of Arezzo between Myth and History. Cambridge University Press (2011)Google Scholar