Glarean’s Dodecachordon Revisited

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.