Introduction to gestural similarity in music. An application of category theory to the orchestra

Abstract

Mathematics, and more generally computational sciences, intervene in several aspects of music. Mathematics describes the acoustics of the sounds giving formal tools to physics, and the matter of music itself in terms of compositional structures and strategies. Mathematics can also be applied to the entire making of music, from the score to the performance, connecting compositional structures to the acoustical reality of sounds. Moreover, the precise concept of gesture has a decisive role in understanding musical performance. In this article, we apply some concepts of category theory to compare gestures of orchestral musicians, and to investigate the relationship between orchestra and conductor, as well as between listeners and conductor/orchestra. To this aim, we will introduce the concept of gestural similarity. The mathematical tools used can be applied to gesture classification, and to interdisciplinary comparisons between music and the visual arts.

Keywords:

• gesture
• performance
• orchestral conducting
• category theory
• similarity
• composition
• visual arts
• interdisciplinary studies
• fuzzy logic

2010 Mathematics Subject Classification:

• 18B05
• 18B10
• 16D90
• 03B52

Acknowledgments

I am grateful to Guerino Mazzola for fruitful discussions about the relations between art and science in the framework of category theory. I am also grateful to Peter beim Graben for reading my manuscript, and for insightful suggestions between physics, mathematics, and semiotics. Thank you to Luca Nobile for reading and suggesting related studies in the field of crossmodal correspondences and neuroscience. I would also like to give special thanks to Giuseppe Metere for mathematical insights. Finally, I am grateful to Marco Betta for musical advice, to Jason Yust for careful work as editor, and to Thomas Fiore for his careful corrections to the galley proofs. I also thank the referees.

Disclosure statement

The author has no conflict of interest.

ORCID

Maria Mannone (logo) http://orcid.org/0000-0003-3606-3436

ORCID

Maria Mannone http://orcid.org/0000-0003-3606-3436

Notes

1 In string theory, point-like particles are substituted by vibrating strings. In Mannone and Mazzola (2015), the paradigm of string theory is used to describe musical performance in terms of gestures, rather than notes and sounds as isolated events. The “particles” correspond to the notes. To understand music as unfolding in time, rather than as isolated “points in time,” we can study gestures to understand music. However, the “strings” here are not vibrating: thus, such a reference to string theory is just a general metaphor.

2 For a precise but non-mathematical approach to gesture classification in the framework of mixed music (electroacoustic and with traditional instruments), see Bachratá (2011); the term similarity is used in describing imitational gestures qualitatively, see Iazzetta (2000).

3 Of course, in the limit that such a thought can be inferred from the score.

4 We will not go deeper into detail with semiotics. The passage from a simple, instrumental movement to an expressive movement may raise issues about semiotics. For example, a “caressing” piano touch is not only finalized to get a sound, but a sound with a specific, soft timbre, that also carries a meaning, the meaning of a caressing gesture. Visualization of such a gesture is also relevant for the perception of the intended meaning. An accurate study of this field should require a separate and detailed description, as well as some perception experiments, to substantiate the connection between our mathematical approach and musicological studies in the field (Henrotte 1992; Iazzetta 2000).

5 A curve c in X is a continuous function $c:I\to X$, where $I=[0,1]$ is the real unit interval.

6 The space $\overrightarrow{X}$ is a topological space, e.g. space–time.

7 This is the name introduced in Mazzola and Andreatta (2007) and used in the related literature. However, as suggested by a reader of these works, we could perhaps use the term metagesture.

8 For example, a transformation $t^{\ast }$ modifies the skeleton $\mathrm{\Delta }=\cdot \to \cdot$, i.e. an arrow between two points, into another skeleton $\mathrm{\Gamma }=\cdot \to \cdot \to \cdot$, with one more point and arrow.

9 More precisely, m is a continuous functor of topological categories.

10 The category of gestures can be described as a comma category $(\mathit{i}\mathit{d},\overrightarrow{\cdot })$, where id is the identity on directed graphs, and $\overrightarrow{\cdot }$, defined from spaces to directed graphs, is the functor that associates to a space X the digraph $\overrightarrow{X}$ with the paths in X.

11 We will not delve into details about fuzzy logic. Whereas Definition 3.1 assesses a criterion of gestural similarity, we can more precisely talk about degrees of gestural similarities, with infinite intermediate values between perfect similarity of two identical gestures and two completely unrelated ones.

12 If restricted to the physical curve, the potential is related to the force as known in common physical situations. And, of course, the force strongly influences the touch and the final spectral result at the level of acoustics. A more general “artistic force” determines the shape of the entire surface of the world-sheet.

13 More generally, we can define a gesture via a function f containing such a $V\left(t\right)$ function:$f(g_{\sigma },V(t\left)\right):=g_{\phi }\left(t\right):=\left\{\begin{array}{ll}{g}_{\phi }^0,& \mathrm{for}t=0\\ g_{\phi }^F,& \mathrm{for}t=1.\end{array}\right.$

14 We denote by ↑ the digraph having two vertices that are connected by one arrow.

15 We can observe that, in the framework of 2-category formalism, vertical and horizontal composition properties are intuitively verified. Vertical composition means, in our orchestral context, transformation of loudness: a piano gesture can be deformed into a forte gesture, which can be deformed into a fortissimo gesture. Horizontal composition, given the same skeleton, means here transition from the space of gestures for a musical instrument, for example piano, to the space of gestures of another instrument, such as percussion, and then to another instrument, such as violin.

16 Two homotopic curves can be continuously transformed from one into the other. A homotopy transformation continuously transforms curves in curves.

17 Similar deformations in their respective spaces lead to similar effects in their resulting sound spectra. See Section A.1 in Appendix A for two graphic representations, Figures A1 and A2.

18 We discussed loudness, but gestural similarity can also involve articulation, and in some cases also rhythmic–melodic profiles. Even specific harmonic sequences can suggest particular gestural solutions. We can think of a deceptive cadence, highlighted by the performer with a fermata or a forte. In fact, elements from musical analysis can act as weights for gestures.

19 Intuitively, limits and colimits are generalizations of products and coproducts, respectively. The product is a special case of the limit, with discrete indexing category. The coproduct (also called sum) is the dual of the product, obtained by reversing the arrows. Given an object P and two maps $p_1:P\to B_1,p_2:P\to B_2$, P is a product of $B_1,B_2$ if for each object X and for each pair of arrows $f_1,f_2$ we have one and only one arrow $f:X\to P$ such that $f_1=p_{1}f,f_2=p_{2}f$ (Lawvere and Schanuel 2009), see diagram (17). (17)

20 This is not shown in the diagram.

21 This happens if we already have a musical score. If, in such a description, we include the composer of the orchestral score, then the colimit role may be envisaged into the composing activity. We may say that, in this extended description, the initial and final points are both in the mind: the mind of the composer, and the mind of the listener.

22 The connection of single curves from (symbolic) systems of continuous curves to single curves from other (physical) systems of continuous curves, their skeleta being the same, has been investigated via branched graphs and branched world-sheets (Mannone and Mazzola 2015; Mazzola and Mannone 2016).

23 Category theory includes functors: so, we could provocatively talk about functorial aesthetics.

24 In future developments of gestural similarity analysis, we may try to find analogies between a painting and the music inspired from it. This may be part of a more general approach to artistic movements, finding the connections between music, visual art, and poetry within a specific movement, in terms of basic “shared gestures.”

25 Two gestures are formally called similar if they satisfy the conditions of Definition 3.1.

26 Curves in $\overrightarrow{X}$ are mapped into points of X, for example with functions taking the tail and the head of an arrow. We can interpret these functions as directed graphs of the set of arrows and points. In fact, we indicate as $\overrightarrow{X}$ a directed graph, identified by the curves $c:\mathrm{\nabla }\to X$, whose head and tail functions project to X the final and initial points, respectively, of continuous curves in $\overrightarrow{X}$. This action of projecting a curve into points can be generalized not only for final and initial points, but also for every other point belonging to the curve.

27 Parametrized gestures have been used to describe mathematically the mechanism of the voice in singing (Mazzola et al. 2017, Chapter 37). The dimensions of the hand of the pianist, the anatomy of the vocal tract for the singer, and the values of diaphragm pressure and position of the larynx are examples of parameter choices.

28 We can define an A-parametrized gesture as a continuous function from the Cartesian product of the parameter space ∇ with the category A, with values in the topological space X, that is, $A\times \mathrm{\nabla }\to X$. According to what is called the Currying Theorem in informatics, for a category C we have $C(X\times Y,Z)\tilde{\to }C(X,Z^Y)$, where $Z^Y$ are the curves from Y to Z, and we can write that $q:A\times \mathrm{\nabla }\to X\sim q:A\to \mathrm{\nabla }@X$. As described in Mazzola and Mannone (2016), an A-addressed gesture with skeleton digraph Δ and body X is a digraph morphism $g:\mathrm{\Delta }\to A@\overrightarrow{X}$ into the spatial digraph of $A@\overrightarrow{X}$.

29 In Mazzola et al. (2017), complex time is introduced to extend the formalism of physics to the mental reality of the score, where music flows in imaginary time. The reality of the physical music performance flows in real time (Primas 2007).

30 The question of such an artistic potential is currently under research. It has an analogy at the level of notes with the performance operators (Mazzola 2002, 2011), both mathematically and conceptually.