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From Music to Mathematics and Backwards: Introducing Algebra, Topology and Category Theory into Computational Musicology

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Abstract

Despite a long historical relationship between mathematics and music, the interest of mathematicians is a recent phenomenon. In contrast to statistical methods and signal-based approaches currently employed in MIR (Music Information Research), the research project described in this paper stresses the necessity of introducing a structural multidisciplinary approach into computational musicology making use of advanced mathematics. It is based on the interplay between three main mathematical disciplines: algebra, topology and category theory. It therefore opens promising perspectives on important prevailing challenges, such as the automatic classification of musical styles or the solution of open mathematical conjectures, asking for new collaborations between mathematicians, computer scientists, musicologists, and composers. Music can in fact occupy a strategic place in the development of mathematics since music-theoretical constructions can be used to solve open mathematical problems. The SMIR project also differs from traditional applications of mathematics to music in aiming to build bridges between different musical genres, ranging from contemporary art music to popular music, including rock, pop, jazz and chanson. Beyond its academic ambition, the project carries an important societal dimension stressing the cultural component of ‘mathemusical’ research, that naturally resonates with the underlying philosophy of the “Imagine Maths” conference series. The article describes for a general public some of the most promising interdisciplinary research lines of this project.

This paper provides an overview of the ongoing SMIR (Structural Music Information Research) Project, supported by the University of Strasbourg Institute for Advanced Study and carried on at the Institut de Recherche Mathématique Avancée (IRMA) in collaboration with the GREAM (Groupe de Recherche Expérimentale sur l’Acte Musical) and the Institut de Recherche et Coordination Acoustique/Musique (IRCAM). For a description of the institutional aspects of the project, including the list of participants and past and future events, see the official webpage: http://repmus.ircam.fr/moreno/smir.

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Notes

  1. 1.

    See http://repmus.ircam.fr/moreno/production for the complete list of students’ work focusing on these aspects of the relations between music and mathematics.

  2. 2.

    This interplay also provides a further example of the duality between temporal and spatial constructions which are the two fundamental ingredients of music according to the field medalist Alain Connes. As he suggested in a conversation with composer Pierre Boulez on the analogy and the difference between the creative processes in mathematics and music: “Concerning music, it takes place in time, like algebra. In mathematics, there is this fundamental duality between, on the one hand, geometry—which corresponds to the visual arts, an immediate intuition—and on the other hand algebra. This is not visual, it has a temporality. This fits in time, it is a computation, something that is very close to the language, and which has its diabolical precision. […] And one only perceives the development of algebra through music” [15].

  3. 3.

    As largely documented, these three aspects are deeply interconnected, particularly in Twentieth-Century music and musicology. See [3] for a detailed account of music-theoretical, analytical and compositional applications of the algebraic methods in contemporary music research.

  4. 4.

    See [7] for a description of Tiling Canons as a key to approach open Mathematical Conjectures.

  5. 5.

    See http://www.smcm-net.info/.

  6. 6.

    See http://www.tandfonline.com/toc/tmam20/current.

  7. 7.

    This approach takes origin in the writings of the German musicologist Hugo Riemann who proposed a “dualistic” perspective of Euler’s Tone System [23] based on inversional relations between major and minor chords. After a first algebraic formalisation by David Lewin through the concept of GIS or Generalized Interval System [26], neo-Riemannian theory and analysis has progressively integrated mathematical concepts belonging to topology and algebraic geometry [11] and shown its relevance to the analysis of a popular music repertoire [13].

  8. 8.

    To the class of “musically-relevant” groups acting on the family of all possible chords belong groups such as the cyclic group of order 12 (or group of transpositions), the dihedral group of order 24 (or group of transposition and inversions) and the affine group of order 48 (or group of “augmentation”, i.e. applications \(f \) of the form \(f(x)=ax + b\), where \(a \) belongs to the set of invertible elements of \(\mathbf{Z} _{\mathbf{12}}\) and \(b\) is any possible transposition factor). By using a term which has a strong philosophical meaning [25], we suggested to call “paradigmatic” a classification approach of musical structures based on an underlying group action [3]. This provides an elegant formalization of the most common chord catalogues, from Anatol Vieru’s catalogue of transposition classes of chords [42] to Mazzola/Morris catalogue of affine orbits [27, 31], including Julio Estrada’s catalogue of “identities” [17].

  9. 9.

    Category theory was originally introduced in music theory by Guerino Mazzola in his dissertation Gruppen und Kategorien in der Musik [27] and further extended in Geometrie der Töne [28] and The Topos of Music [29]. For an alternative approach to the categorical formalization of music theory, see Fiore and Noll [18] and our series of papers dealing with the categorical interpretation of Klumpenhouwer Networks, initially within the framework of Topos of Music [30] and, successively, through the set-up of generalized K-nets called “Poly-Klumpenhouwer Networks” [35–37].

  10. 10.

    See, in particular, the article “Contenus formels et dualité”, reprinted in [21].

  11. 11.

    See [24] as well as Piaget’s posthumous Morphismes et catégories [34].

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Author information

Authors and Affiliations

  1. IRCAM, CNRS UMR 9912 STMS, Sorbonne University, 75004, Paris, France

    Moreno Andreatta

  2. CNRS IRMA UMR 7501, GREAM, USIAS, 67000, Strasbourg, France

    Moreno Andreatta

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  1. Moreno Andreatta
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Correspondence to Moreno Andreatta .

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  1. Roma, Italy

    Michele Emmer

  2. Mathematics Department, University of Pisa , Pisa, Italy

    Marco Abate

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Andreatta, M. (2018). From Music to Mathematics and Backwards: Introducing Algebra, Topology and Category Theory into Computational Musicology. In: Emmer, M., Abate, M. (eds) Imagine Math 6. Springer, Cham. https://doi.org/10.1007/978-3-319-93949-0_7

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