Diagrammatic approaches in Computational Musicology: Some theoretical and Philosophical Aspects
Résumé
Despite a long historical relationship between mathematics and music, the use of diagrammatic approaches in computational musicology is a relatively recent phenomenon. Within the different branches of formal methods in music analysis, the so-called "transformational" paradigm has progressively shifted from an object-oriented to a graph-theoretical and categorical approach. Both graph theory and category theory make large use of diagrams which enable the description of the inner relationships of musical structures. In the categorical framework recently proposed by the authors, whose results are summarized and discussed in this abstract, musical transformations are viewed as natural transformations between chords represented as labelled graphs with vertices corresponding to the notes and arrows corresponding to musical transpositions and inversions operations. The diagrammatic approach also provides a very powerful conceptual tool that can have crucial theoretical implications for music cognition. We discuss this aspect by showing some deep connections between transforma-tional music analysis and some mathematically-oriented directions in developmental psychology and cognition (such as Halford and Wilson's neostructural-istic approach, Ehresmann and Vanbremeersch's Memory Evolutive Systems, Phillips and Wilson's Categorical Compositionality, Fauconnier and Turner's Conceptual Blending and its structural extension proposed by Goguen) and epis-temology (Gaston-Granger's "objectal" and "operational" duality).
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