Abstract
This article provides a first introduction to some formal and computational models applied in the analysis and generation of popular music (including rock, jazz, and chanson). It summarizes the main philosophy underlying the project entitled “Modèles formels dans et pour la musique pop, le jazz et la chanson”, which constitutes one of the research axes of the GDR ESARS (Esthétique, Art & Science). Initially conceived as an extension of the MISA project carried on by the Music Representation Team at IRCAM, this research axis aims at bringing together researchers from different horizons, from the traditional MIR community of Music Information Retrieval to the most sophisticated approaches in mathematical music theory and computational musicology. It also includes an epistemological and critical evaluation of the relations between music and mathematics, together with some programmatic reflections on the possible cognitive and perceptual implications of this research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This paper summarizes some aspects of this project that have been described in details in Andreatta (2014a). For a pedagogical and large-public introduction to mathematical models in popular music, also see Andreatta (2014b). A more technical presentation of the main concepts described in this paper and addressed to the community of researchers working on computational musicology is given in Bigo and Andreatta (2015).
- 2.
According to the Field-medallist Alain Connes, “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” (Connes 2004). This duality constitutes a major common point between music and mathematics, allowing proposing a common basis for the creative processes in both fields of music and mathematics, as suggested by Alain Connes in his dialogue with Pierre Boulez on creativity in mathematics and music (Boulez and Connes 2011). See Andreatta (2010) for a detailed description of the “mathemusical” research that has been carried on in the last ten years within the MISA project (Modélisation Informatique des Structures Algébriques en musique), with a special emphasis on the interplay between algebra and geometry. See Andreatta et al. (2013) for a description of a category-oriented framework for describing the creative process in music and mathematics.
- 3.
This question has been explicitly addressed in the conference “Musique savante/musiques actuelles: articulations” (Contemporary art music/popular music: articulations), hosted by IRCAM and organised under the auspices of the French Society of Music Analysis, in collaboration and with the financial support of the IReMus (Institute of Research in Musicology, UMR 8223, Paris-Sorbonne) and the BPI of the Centre Georges Pompidou and with the participation of the French component of IASPM (International Association for the Study of Popular Music). The Proceedings are forthcoming in a special issue of the multimedia online journal Musimédiane (Andreatta 2016). For a first attempt at analysing the necessity of substituting this typology with a finer taxonomy based on computational models focusing on musical objects and making use of different theoretical approaches in order to carry on computer-aided music analysis, see Bergomi et al. (2015).
- 4.
This typology constitutes what Tagg calls an axiomatic triangle of musical genres, each of which being characterized by criteria such as the usual or unusual mass distribution, the existence of a circle of professionals or a circle of amateurs who produces and transmits it, the principle modality of storage and distribution (ranging from oral transmission, in the case of folk music, to the recorded sound, in the case of popular music), the anonymous versus authorial character of the underlying compositional process, and so on.
- 5.
For a recent analytical application of the formal tools discussed in this paper from the perspective of a geometric-based automatic classification, see Bergomi et al. (2015).
- 6.
Two main models, the “Polarized Tonnetz” and the “Spinnen Tonnetz”, originally conceived by Hugo Seress and Gilles Baroin, represent a very interesting way of integrating some tonality-based constructions within transformational music analysis. For a critical presentation of these two models and their comparison with other tools belonging to the transformational musical analysis tradition, see Seress and Baroin (2016).
- 7.
See Bigo (2013) and Bigo and Andreatta (2015) for a historical description of the main geometric representations in computational music analysis. Algebraic topology has provided a very elegant theoretical framework for describing all these representations, as shown by Bergomi (2015) in his recent doctoral thesis.
- 8.
The reader interested to learn more about the three main contributions of Leonhard Euler (as a mathematician, physicist and music theorist) can refer to Hascher and Papadopoulos (2015).
- 9.
Neo-Riemannian music analysis is a formal methodology developed after the writings by the German music theorist Hugo Riemann (1849–1919). Following David Lewin’s transformational turn in music theory and analysis (Lewin 1987/2007; 1993/2007), which integrates neo-Riemannian techniques within a much more general approach, one may speak about neo-Riemannian transformational music analysis as a structural methodology combining the two independent approaches. See Gollin and Rehding (2014) for a comprehensive textbook on Neo-Riemannian analysis.
- 10.
Hamiltonian Songs are so-called after the Irish physicist, astronomer, and mathematician Sir William Rowan Hamilton (1805–1865). In graph theory, a Hamiltonian cycle is a path passing through all possible nodes of a graph and ending precisely where it started. It is well known that there are exactly 124 Hamiltonian cycles in the Tonnetz (Albini and Antonini 2009) which can be classified by using their inner symmetries (i.e. the possibility of decomposing a given cycle into sub-sequences that repeat identically in order to generate the entire cycle). The complete list of Hamiltonian cycles with some examples of Hamiltonian Songs is available at: http://repmus.ircam.fr/moreno/music.
- 11.
The interest of using Neo-Riemannian techniques to analyse this passage has been originally pointed out by Capuzzo (2004).
- 12.
The Hamiltonian trajectories of the song have been visualised by Gilles Baroin by mixing his Hypersphere of Chords representation and the traditional Tonnetz. It is available online at the address: (www.mathemusic.net).
- 13.
Note that “hamiltonicity” does not only concern popular music strategies, but it plays an important role in contemporary art music. The history of Twentieth-Century music shows that Hamiltonian properties have been implicitly used by composers such as Pierre Boulez or Milton Babbitt, who developed combinatorial strategies as natural extensions of the twelve-tone compositional system. Both composers and music theorists claimed the necessity of having a “maximal variety principles” in composition, in order to precisely question the notion of expectation in the musical listening process.
- 14.
We analysed the relations between mental and mathematical representations of music in Acotto and Andreatta (2012).
- 15.
A special issue of the Journal of Mathematics and Music has been devoted to this specific problem with precisely the aim of bridging the Gap between Computational/Mathematical and Cognitive Approaches in Music Research. See Volk and Honingh (2002).
References
Acotto E, Andreatta M (2012) Between mind and mathematics. Different kinds of computational representations of music. Math Soc Sci 199:9–26
Agon C (2004). Langages de programmation pour la composition musicale. Habilitation à Diriger des Recherches, Université de Paris 6
Albini G, Antonini S (2009) Hamiltonian cycles in the topological dual of the Tonnetz. In: Proceedings of the Yale MCM conference, Springer, LNCS
Andreatta M (2010) Mathematica est exercitium musicae. La recherche ‘mathémusicale’ et ses interactions avec les autres disciplines, Habilitation à Diriger des Recherches, IRMA/Université de Strasbourg
Andreatta M (2014a) Modèles formels dans et pour la musique pop, le jazz et la chanson: introduction et perspectives futures. In: Kapoula Z, Lestocart L-J, Allouche J-P (eds) Esthétique & Complexité II: Neurosciences, évolution, épistémologie, philosophie, éditions du CNRS, pp 69–88
Andreatta M (2014b) Math’n pop: géométrie et symétrie eu service de la chanson. Tangente. L’aventure mathématique, special issue devoted on the creative process in mathematics, pp 92–97
Andreatta M (2016) Musique savante/musiques actuelles: articulations, special issue of the journal Musimédiane, French Society of Music Analysis
Andreatta M, Ehresmann A, Guitart R, Mazzola G (2013) Towards a categorical theory of creativity. In: Yust J et al. (eds) Proceedings of the mathematics and computation in music conference 2013—Springer, Lecture notes in computer science, vol 7937
Baroin G (2011) The planet-4D model: an original hypersymmetric music space based on graph theory. In: Agon C, Andreatta M, Assayag G, Amiot E, Bresson J, Mandereau J (eds) Proceedings of the mathematics and computation in music conference 2011, Springer, Lecture notes in computer science, vol 6726, pp 326–329
Bergomi M (2015) Dynamical systems and musical structures, PhD, UPMC/LIM Milan/IRCAM
Bergomi M, Andreatta M (2015) Math’n pop versus Math’n folk? A computational (ethno)-musicological approach. In: Proceedings international folk music analysis conference, Paris, pp 32–34
Bergomi M, Fabbri F, Andreatta M (2015) Hey maths! Modèles formels et computationnels au service des Beatles. Volume! La revue des musiques populaires (eds by Grégoire Tosser and Olivier Julien, special issue devoted to the Beatles)
Bigo L (2013) Représentations symboliques musicales et calcul spatial, PhD, University of Paris Est/IRCAM
Bigo L, Andreatta M (2014) A geometrical model for the analysis of pop music. Sonus 35(1):36–48
Bigo L, Andreatta M (2015) Topological structures in computer-aided music analysis. In: Meredith D (ed) Computational music analysisy. Springer, pp 57–80
Bigo L, Andreatta M, Giavitto J-L, Michel O, Spicher A (2013) Computation and visualization of musical structures in chord-based simplicial complexes. In: Yust J et al (eds) Proceedings of the mathematics and computation in music conference 2013, Springer, lecture notes in computer science, vol 7937, pp 38–51
Boulez P, Connes A (2011) Creativity in mathematics and music. Mathematics and computation in music conference, IRCAM. Video available online at the address http://agora2011.ircam.fr
Briginshaw S (2012) A neo-riemannian approach to jazz analysis. Nota Bene Can Undergraduate J Musicol 5(1, Article 5). Available online at http://ir.lib.uwo.ca/notabene/vol5/iss1/5
Capuzzo G (2004) Neo-Riemannian theory and the analysis of pop-rock music. Music Theory Spectrum 26(2):177–199
Connes A (2004) CNRS images, Vidéothèque du CNRS, 2004. Available online at http://videotheque.cnrs.fr/
Cohn R (2012) Audacious euphony: chromatic harmony and the triad’s second nature. Oxford University Press
Euler L (1774) De harmoniae veris principiis per speculum musicum repraesentatis. In: Novi Commentarii academiae scientiarum Petropolitanae 18:330–353
Gollin E, Rehding A (2014) The Oxford handbook of Neo-Riemannian music theories. Oxford
Hascher X, Papadopoulos A (2015) Leonhard Euler. Mathématicien, physicien et théoricien de la musique, CNRS éditions
Hascher X (2007) A harmonic investigation into three songs of the beach boys: all summer long, help me Rhonda, California Girls. SONUS 27(2):27–52
Lewin D (1987/2007) Generalized musical intervals and transformations. Yale University Press (orig. Yale University Press. Reprint Oxford University Press, 2007)
Lewin D (1993/2007) Musical form and transformation. Yale University Press (orig. Yale University Press. Reprint Oxford University Press, 2007)
Mersenne M (1636) Harmonie universelle, contenant la théorie et la pratique de la musique. Paris
Seress H, Baroin G (2016) Le Tonnetz en musique savante et en musique populaire. In: Andreatta M (ed) Musique savante/musiques actuelles: articulations, Musimédiane (forthcoming)
Tagg PH (1982) Analysing popular music: theory, method and practice. Popular Music 2:37–67
Volk A, Honingh A (2002) Mathematical and computational approaches to music: three methodological reflections. Spec Issue J Math Music 6(2)
Zatorre RJ, Krumhansl CL (2002) Mental models and musical minds. Science 298(5601):2138–2139. (13 December)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Andreatta, M., Baroin, G. (2016). An Introduction on Formal and Computational Models in Popular Music Analysis and Generation. In: Kapoula, Z., Vernet, M. (eds) Aesthetics and Neuroscience. Springer, Cham. https://doi.org/10.1007/978-3-319-46233-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-46233-2_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46232-5
Online ISBN: 978-3-319-46233-2
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)