Abstract
The mathematical formalism of category theory allows to investigate musical structures at both low and high levels, performance practice (with musical gestures) and music analysis. Mathematical formalism can also be used to connect music with other disciplines such as visual arts. In our analysis, we extend former studies on category theory applied to musical gestures, including musical instruments and playing techniques. Some basic concepts of categories may help navigate within the complexity of several branches of contemporary music research, giving it a unitarian character. Such a ‘unification dream,’ that we can call ‘cARTegory theory,’ also includes metaphorical references to topos theory.
M. Mannone is an alumna of the University of Minnesota, USA.
F. Favali—Independent researcher.
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
Similar content being viewed by others
Notes
- 1.
For example, we have an object A, and object B, and two transformations f, g such that \(f:A\rightarrow B\), \(g:A\rightarrow B\). A transformation between them is some \(\eta \) such that \(\eta :f\rightarrow g\).
- 2.
For example, a crescendo is seen as a transformation from a loudness level to another, and the comparison between a faster and a slower crescendo is described via a comparison between transformations.
- 3.
In commutative diagrams, different arrows and combinations of arrows starting from the same object A reach the same object B, and the two different paths are equivalent.
- 4.
With augmented instrument, we mean a musical instrument that has some sensors, electric connections or controllers, that enable the instrument play new sounds, enriched musical sequences, or even, in the case of the so-called smart instruments with embedded artificial intelligence [38], a ‘dialogue’ between a library of motifs and the played music, under the direct or indirect control of the performer. According to [38], smart instruments are more general than augmented ones. For an effective use of them, see [39].
- 5.
In particular, the connection between audience and performers via smart instruments and wearable devices allows the creation of isomorphisms in the diagrams of [25]. If also the conductor gets this kind of feedback through a smart device, categorically we could have a connection between colimit (the conductor) and limit (a listener in the audience). See [25] for details on the categorical description of the orchestra.
- 6.
See [25] for percussion and flute examples.
- 7.
We can still compare gestures of different spaces, with opportune changes: for example, a crescendo can be done with an increase of acceleration and pressure with hammer on a percussion, and with bow’s movements on a violin [25].
- 8.
Examples of robotic conductors have been created at the University of Pisa and by the Music Conservatory of Palermo/University of Palermo, Engineering Department.
- 9.
- 10.
In the words of Olivia Caramello, topoi (also called ‘Toposes’) “are mathematical objects which are built from a pair, called a site, consisting of a category and a generalized covering, called Grothendieck topology, on it in a certain canonical way (the process which produces a topos from a given site can be described as a sort of ‘completion’).” We can describe topoi as ‘enhanced’ categories, with a whiff of topology or a similarity to the category of sets.
- 11.
In fact, a ‘smart’ technology, inspired by smart instruments and applied to visual arts, may consist in a smart tablet that takes as input a drawing gesture and gives as output a variegated, enriched visual representation. Thus, we can extend our comparison between extended sounds and extended visuals/drawings, and techniques developed in a field can be translated into new techniques to be applied into another field.
- 12.
For example, impressionistic painting uses imprecise contours and evident brush traces, and impressionistic music uses a lot of suspended chords and pedal piano effects. Might the ‘imprecision’ of a sketch, of an instant representation be at the core of impressionistic art? This could open a productive discussion on aesthetics, that is however out of the aim of this paper. Here, the word ‘sketch’ is used with its everyday meaning, and the term is not referred to the homonymous categorical construction.
- 13.
- 14.
As suggested by a reviewer, we could restrict these categories to sub-cats to be endowed with topos structure.
- 15.
In mathematics and physics, an entity can be invariant under certain transformations, thus being symmetric under a specific change. While dealing with transitions from a specific artistic expression to another, invariants can be nuclei of meaning that remain substantially unchanged. For example, we can wonder if there is some unchanged inner core behind artistic expressions belonging to the same artistic current.
- 16.
Category theory has also been used to describe the general process from the artistic production to the aesthetic contemplation [22]. The process from composition to performance/conducting and listening, described categorically in [25], can be seen as one of its possible ‘concrete’ applications, featuring several references to sounds and spectrograms. Curiously, both papers have been submitted on the same day.
References
Antoniadis, P., Bevilacqua, F.: Processing of symbolic music notation via multimodal performance data: Brian Ferneyhough’s Lemma-Icon-Epigram for solo piano, phase 1. In: Hoadley, R., Nash, C., Fober, D. (eds.) Proceedings of TENOR 2016, pp. 127–136 (2016)
Antoniadis, P.: Embodied navigation of complex piano notation: rethinking musical interaction from a performer’s perspective. Ph.D. thesis, Université de Strasbourg, pp. 274–278 (2018)
Arias, J.S.: Spaces of gestures are function spaces. J. Math. Music 12(2), 89–105 (2018)
Arias, J.S.: Gestures on locales and localic topoi. In: Pareyon, G., Pina-Romero, S., Agustin-Aquino, O.A., Lluis-Puebla, E. (eds.) The Musical-Mathematical Mind. CMS, pp. 29–39. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-47337-6_4
Caramello, O.: Theories, Sites, Toposes. Oxford University Press, New York (2018)
Caramello, O.: The theory of topos-theoretic ‘bridges’-a conceptual introduction. Glass-bead (2016). http://www.glass-bead.org/article/the-theory-of-topos-theoretic-bridges-a-conceptual-introduction/?lang=enview
Caramello, O.: The unification of mathematics via topos theory. ArXiv arXiv:1006.3930 (2010)
Collins, T., Mannone, M., Hsu, D., Papageorgiou, D.: Psychological validation of mathematical gesture theory, Submitted (2018)
Delle Monache, S., Rocchesso, D., Bevilacqua, F., Lemaitre, G., Baldan, S., Cera, A.: Embodied sound design. Int. J. Hum.-Comput. Stud. 118, 47–59 (2018)
Dick, R.: The Other Flute. Lauren Keiser Music Publishing, Oxford (1989)
Di Donato, B., Bullock, J., Tanaka, A.: Myo Mapper: a Myo armband to OSC mapper. In: Dahl, L., Bowman, D., Martin, T. (eds.) Proceedings of NIME Conference, Blacksburg, USA, pp. 138–143 (2018)
Ehresmann, A., Gomez-Ramirez, J.: Conciliating neuroscience and phenomenology via category theory. Prog. Biophys. Mol. Biol. (JPMB) 19(2), 347–359 (2016)
Fiore, T., Noll, T., Satyendra, R.: Morphisms of generalized interval systems and PR-groups. J. Math. Music 7(1), 3–27 (2012)
Fish, S., L’Huillier, N.: Telemetron: a musical instrument for performance in zero gravity. In: Dahl, L., Bowman, D., Martin, T. (eds.) Proceedings of NIME Conference, Blacksburg, USA, pp. 315–317 (2018)
Genette, G.: L’opera dell’arte, two volumes. Clueb, Bologna (1999)
Guo, C., Song-Chun, Z., Wu, Y. N.: Towards a mathematical theory of primal sketch and sketchability. In: 9th IEEE Conference on Computer Vision (2003)
Heller, F., Cheung Ruiz, I.M., Borchers, J.: An augmented flute for beginners. In: Cumhur Erkut, C., De Götzen, A. (eds.) Proceedings of NIME Conference, Copenhagen, Denmark, pp. 34–37 (2017)
Isherwood, N.: The Techniques of Singing. Bärenreiter, Germany (2013)
Isnard, V., Taffou, M., Viaud-Delmon, I., Suied, C: Auditory sketches: very sparse representations of sounds are still recognizable. PloS One 11(3) (2016). https://doi.org/10.1371/journal.pone.0150313
Jedrzejewski, F.: Hétérotopies musicales. Modéles mathématiques de la musique. Éditions Hermann, Paris (2019)
Kelkar, T., Jensenius, A.R.: Analyzing free-hand sound-tracing of melodic phrases. Appl. Sci. 8(135), 1–21 (2018)
Kubota, A., Hori, H., Naruse, M., Akiba, F.: A new kind of aesthetics - the mathematical structure of the aesthetic. Philosophies 3(14), 1–10 (2017)
Kubovy, M., Schutz, M.: Audio-visual objects. Rev. Philos. Psychol. 1(1), 41–61 (2010)
Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1978). https://doi.org/10.1007/978-1-4757-4721-8
Mannone, M.: Introduction to gestural similarity. An application of category theory to the orchestra. J. Math. Music 12(3), 63–87 (2018)
Mannone, M.: Knots, music and DNA. J. Creat. Music Syst. 2(2), 1–23 (2018). https://www.jcms.org.uk/article/id/523/
Mannone, M., Favali, F.: Shared structures and transformations in mathematics and music: from categories to musicology. In: ESCOM-Italy Primo Meeting, Roma, Italy (2018)
Mannone, M.: Mathematics, Music, Nature (in Progress)
Mannone, M., Kitamura, E., Huang, J., Sugawara, R., Kitamura, Y.: CubeHarmonic: a new interface from a magnetic 3D motion tracking system to music performance. In: Dahl, L., Bowman, D., Martin, T. (eds.) Proceedings of NIME Conference, Blacksburg, USA, pp. 350–351 (2018)
Mastropasqua, M.: L’evoluzione della tonalità nel XX secolo. L’atonalità in Schoenberg. Clueb, Bologna (2004)
Mazzola, G., et al.: The Topos of Music, 2nd edn. Springer, Heidelberg (2018)
Mazzola, G., Andreatta, M.: Diagrams, gestures and formulae in music. J. Math. Music 1(1), 23–46 (2007)
Mazzola, G., Mannone, M.: Global functorial hypergestures over general skeleta for musical performance. J. Math. Music 10(7), 227–243 (2016)
Popoff, A.: Using monoidal categories in the transformational study of musical time-spans and rhythms. arXiv:1305.7192v3 (2013)
Russolo, L.: L’arte dei rumori. Edizioni futuriste di poesia, Milano (1910)
Sloboda, J.: Musical Mind. Oxford Science Publication, New York (2008)
Spence, C.: Crossmodal correspondences: a tutorial review. Atten. Percept. Psychophys. 73(4), 971–995 (2011)
Turchet, L.: Smart musical instruments: vision, design principles, and future directions. IEEE Access 7(1), 8944–8963 (2019)
Turchet, L.: Smart Mandolin: autobiographical design, implementation, use cases, and lessons learned. In: Cunningham, S., Picking, R. (eds.) Proceedings of the Audio Mostly Conference 2018, pp. 13:1–13:7, Wrexham Glyndwr University, Wrexham (2018). http://doi.acm.org/10.1145/3243274.3243280
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Mannone, M., Favali, F. (2019). Categories, Musical Instruments, and Drawings: A Unification Dream. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-21392-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-21391-6
Online ISBN: 978-3-030-21392-3
eBook Packages: Computer ScienceComputer Science (R0)