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.
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- 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.
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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
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