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Lattice-based and topological representations of binary relations with an application to music

Anton Freund 1, 2 Moreno Andreatta 2 Jean-Louis Giavitto 2, 3
2 Repmus - Représentations musicales
STMS - Sciences et Technologies de la Musique et du Son
3 MuTant - Synchronous Realtime Processing and Programming of Music Signals
Inria Paris-Rocquencourt, UPMC - Université Pierre et Marie Curie - Paris 6, IRCAM, CNRS - Centre National de la Recherche Scientifique
Abstract : Formal concept analysis associates a lattice of formal concepts to a binary relation. The structure of the relation can then be described in terms of lattice theory. On the other hand Q-analysis associates a simplicial complex to a binary relation and studies its properties using topological methods. This paper investigates which mathematical invariants studied in one approach can be captured in the other. Our main result is that all homotopy invariant properties of the simplicial complex can be recovered from the structure of the concept lattice. This not only clarifies the relationships between two frameworks widely used in symbolic data analysis but also offers an effective new method to establish homotopy equivalence in the context of Q-analysis. As a musical application, we will investigate Olivier Messiaen's modes of limited transposition. We will use our theoretical result to show that the simplicial complex associated to a maximal mode with m transpositions is homotopy equivalent to the (m − 2)–dimensional sphere.
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Anton Freund, Moreno Andreatta, Jean-Louis Giavitto. Lattice-based and topological representations of binary relations with an application to music. Annals of Mathematics and Artificial Intelligence, Springer Verlag, 2015, 73 (3-4), pp.311-334. ⟨10.1007/s10472-014-9445-3⟩. ⟨hal-01256854⟩

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