Abstract
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.
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Notes
- 1.
Following the Roadmap described in [18], we prefer to consider MIR as the field of Music Information Research instead of limiting the scope to purely Music Information Retrieval. This approach constitutes the core of an ongoing research project entitled SMIR (Structural Music Information Research: Introducing Algebra, Topology and Category Theory into Computational Musicology). See http://repmus.ircam.fr/moreno/smir.
- 2.
See [21] for an interesting discussion on the mutual influences between the Darmstadt school on Formal Concept Analysis and the French tradition on Treillis de Galois.
- 3.
See the Mutabor language (http://www.math.tu-dresden.de/~mutabor/) for a music programming language making use of the FCA-based Standard Language for Music Theory [12] originally conceived by Rudolf Wille and currently developed at the University of Dresden.
- 4.
Note that, at this stage, the time information is not taken into account, and a musical excerpt is considered as an unordered set of chords.
- 5.
- 6.
Note that the lattice contains the nodes representing the harmonic forms, and additional nodes, labeled with an arbitrary number, arising from the completion to obtain a complete lattice [16]. These intermediate nodes are already present in Wille’s original lattice-based formalization of the diatonic scale.
- 7.
An interesting question, which still remains open, concerns the possible ways of generating chains which are musically relevant by carefully selecting the underlying equivalence classes.
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Agon, C., Andreatta, M., Atif, J., Bloch, I., Mascarade, P. (2018). Musical Descriptions Based on Formal Concept Analysis and Mathematical Morphology. In: Chapman, P., Endres, D., Pernelle, N. (eds) Graph-Based Representation and Reasoning. ICCS 2018. Lecture Notes in Computer Science(), vol 10872. Springer, Cham. https://doi.org/10.1007/978-3-319-91379-7_9
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