Geometric representation and algebraic formalization of musical structures

Abstract : This thesis presents a generalizations of the neo-Riemannian PLR-group, that acts on the set of 24 major and minor triads. The work begins with a reconstruction on the history of the Tonnetz, a graph associated with the three transformations that generate the PLR-group. The thesis presents two generalizations of the PLR-group for seventh chords. The first one acts on the set of dominant, minor, semi-diminished, major and diminished sevenths, the second one also includes minor major, augmented major, augmented, dominant seventh flat five. We considered the most parsimonious operations exchanging two types of sevenths, moving a single note by a semitone or a whole tone. We also classified the most parsimonious transformations among the 4 types of triads (major, minor,augmented and diminished) and studied the group generated by them. Finally, we have introduced a general approach to define parsimonious operations between sevenths and triads, but also the operations already known between triads and those between sevenths.
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Sonia Cannas. Geometric representation and algebraic formalization of musical structures. Musicology and performing arts. Université de Strasbourg; Università degli studi (Pavie, Italie), 2018. English. ⟨NNT : 2018STRAD047⟩. ⟨tel-02179522⟩

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