2013 | OriginalPaper | Buchkapitel
Glarean’s Dodecachordon Revisited
verfasst von : Thomas Noll, Mariana Montiel
Erschienen in: Mathematics and Computation in Music
Verlag: Springer Berlin Heidelberg
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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.