Some remarks on hypergestural homology of spaces and its relation to classical homology. (English) Zbl 1471.00006

The mathematical theory of musical gestures is an example of an emerging research field that starts from performance practice in the arts. Performers’ gestures are necessary to obtain sounds from musical instruments. A gesture is the embodiment of a digraph in a topological space. Starting from Mazzola’s definition of gestures and hypergestures (gestures of gestures), in this article Arias-Valero and Lluis-Puebla provide geometric and physical boundaries of hypergestures. The authors also investigate the relationship between “hypergestural homology and classical cubical homology and prove that in many cases […] hypergestural homologies is invariant under homotopy equivalence of spaces” (p. 245). The authors start with the definition of semi-cubical categories and semi-cubical modules as functors. Then, they construct the cubical homology and the standard cube functor from semi-cubical categories to topological spaces. The authors review topological gestures and hypergestures, presenting them as sequences to build semi-cubical modules for homology. Finally, the authors identify hypergestural homology as a cubical homology, distinguishing different notions of boundary for gestures (Mazzola’s, geometric, natural). According to the authors, “gestures and hypergestures are just sequences of singular cubes satisfying some conditions” (p, 262). The authors conclude the article by discussing how their computations could be applied to piano playing, conducting gestures (see Mannone’s research), or any other discipline involving performance.


00A65 Mathematics and music
18G99 Homological algebra in category theory, derived categories and functors


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