Musical Descriptions Based on Formal Concept Analysis and Mathematical Morphology

In the context of mathematical and computational representations of musical structures, we propose algebraic models for formalizing and understanding the harmonic forms underlying musical compositions. These models make use of ideas and notions belonging to two algebraic approaches: Formal Concept Analysis (FCA) and Mathematical Morphology (MM). Concept lattices are built from interval structures whereas mathematical morphology operators are subsequently defined upon them. Special equivalence relations preserving the ordering structure of the lattice are introduced in order to define musically relevant quotient lattices modulo congruences. We show that the derived descriptors are well adapted for music analysis by taking as a case study Ligeti’s String Quartet No. 2.

[1]  Isabelle Bloch,et al.  Mathematical Morphology Operators over Concept Lattices , 2013, ICFCA.

[2]  Isabelle Bloch,et al.  Mathematical Morphology , 2007, Handbook of Spatial Logics.

[3]  Hugues Talbot,et al.  Mathematical Morphology: from theory to applications , 2013 .

[4]  H. Heijmans.  Morphological image operators , 1994 .

[5]  Bruno Leclerc,et al.  Lattice valuations, medians and majorities , 1993, Discret. Math..

[6]  Bernard Monjardet,et al.  Metrics on partially ordered sets - A survey , 1981, Discret. Math..

[7]  H. Heijmans,et al.  The algebraic basis of mathematical morphology , 1988 .

[8]  Moreno Andreatta,et al.  Using Formal Concept Analysisto Represent Chroma Systems , 2013, MCM.

[9]  Xavier Serra,et al.  Roadmap for Music Information ReSearch , 2013 .

[10]  R. Wille,et al.  Musiktheorie und Mathematik , 1985 .

[11]  Stefan E. Schmidt,et al.  A formal concept analysis of harmonic forms and interval structures , 2010, Annals of Mathematics and Artificial Intelligence.

[12]  Rudolf Wille.  Sur la fusion des contextes individuels , 1984 .

[13]  Bernhard Ganter,et al.  Formal Concept Analysis , 2013 .

[14]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[15]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[16]  Bernhard Ganter,et al.  Formal Concept Analysis, 6th International Conference, ICFCA 2008, Montreal, Canada, February 25-28, 2008, Proceedings , 2008, International Conference on Formal Concept Analysis.

[17]  Henk J. A. M. Heijmans,et al.  The algebraic basis of mathematical morphology. I Dilations and erosions , 1990, Comput. Vis. Graph. Image Process..

[18]  Isabelle Bloch,et al.  Some Relationships Between Fuzzy Sets, Mathematical Morphology, Rough Sets, F-Transforms, and Formal Concept Analysis , 2016, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[19]  Christian Ronse,et al.  Adjunctions on the lattices of partitions and of partial partitions , 2010, Applicable Algebra in Engineering, Communication and Computing.