The Topos of Music III: Gestures

This is the third part of The Topos of Music and deals with gestures. We summarize the trajectory gestures took from the first edition of The Topos of Music to the present second edition.

This short section recapitulates the global architecture of the ontology of music.

The French school of diagrammatic philosophers was inspired by Gilles Deleuze’s comments on Bacon [258] and then elaborated upon and deepened by gesture theorists and philosophers, such as Gilles Châtelet [190] and Charles Alunni [24]. This important French approach to gestures reveals a delicate aspect of embodiment in that gestures are conceived as being presemiotic. Gestures—except when ‘tamed’ by social codes—are not signs in a semiotic environment.

This chapter deals with gestural arguments in cognitive science. We first discuss the idea of a “science of embodiment”, then discuss the neurological vicinity of speech and manual gesture centers (Broca area), also referring to Merleau-Ponty’s linguistic philosophy and to the 3D Mental Rotation experiments by Shepard and Cooper. In this context, Donald’s gestural anthropology is mentioned, together with Valéry’s philosophy of dance and Taylor’s critique of disembodied music. The chapter terminates with a fascinating discussion of musical gesture from the perspective of disability studies. Its focus is on two hand-impaired jazz pianists, Horace Parlan and Oscar Peterson.

We discuss contributions from music theory, performance, and technology to gestural modeling.

This chapter introduces the definition, some basic propositions and first examples regarding the mathematical concept of a gesture for topological spaces. It also includes a short discussion of the topostheoretic logic that is implied by the topos of directed graphs.

We generalize the topological approach to gestures, and culminate in the construction of a gesture bicategory, which enriches the classical Yoneda embedding and could be a valid candidate for the conjectured space X in the diamond conjecture [720]; see also Section 61.12. We discuss first applications thereof for topological groups, and then more concretely gestures in modulation processes in Beethoven’s Hammerklavier sonata. The latter offers a first concretization of answers to Lewin’s big question from [605] concerning characteristic gestures. This research is a first step towards a replacement of Fregean functional abstraction by gestural dynamics.

In this chapter we interpret the basic cubic chain spaces of singular homology in terms of hypergestures in a topological space over a series of copies of the arrow digraph ò. This interpretation allows for a generalized homological setup. The generalization is (1) to topological categories instead of topological spaces, and (2) to any sequence of digraph pΓnqnPZ instead of the constant series of Ò. We then define the corresponding chain complexes, and prove the core boundary operator equation B2 “ 0, enabling the associated homology modules over a commutative ring R. We discuss some geometric examples and a musical one, interpreting contrapuntal rules in terms of singular homology.

As singular homology is strongly related to de Rham cohomology, in particular by Stokes’ classical theorem, it is natural to ask for such a theorem in our context of hypergestures. But there is a deeper reason for such a project, namely the idea that music theory of hypergestures could provide us with models of energy exchange in gestural interaction. In such a (still hypothetical) theory, Stokes’ theorem would play a crucial role regarding questions of energy conservation (integral invariants).

In this chapter we describe the mathematical framework for the three fundamental layers of musical ontology: facts, processes, and gestures. The layer of facts is described by the theory of local and global compositions, a major topic in American Set Theory [765] and in the European school developed by the author and his collaborators [682]. The second layer is captured by the American Transformational Theory [605, 538] and, again in Europe, by the author’s theory of categorical limits (and colimits) as embedded in topos theory [714]. The third layer has been the author’s main concern in the last ten years [720, 723, 727], also paralleled by American research such as Robert S. Hatten’s work [446].

We discuss global categories of compositions, processes, and gestures.

We claim that category theory is a mathematical theory, proceeding from the observation of mathematical activities and gestures, and constructing a mathematical theory as a kind of algebra of these gestures. Especially, categoricians observe their own activity, and so category theory is also constructing a mathematical theory of itself, of its own system of gestures. We imagine that this theory can be used to model any activity, by a parallel action with the categorical activity. This categorical modeling is what we need for a mathematical holding of mathematical creativity because every activity is in fact somehow an activity of modeling.

This chapter introduces the concept architecture of forms and denotators for gesture theory. It also discusses a Galois theory of concepts in the case of denotators over the category Mod.

In this chapter, we give a short presentation of the RUBATO® Composer software environment. It is the basis for the subsequent chapters about the BigBang rubette.

The BigBang rubette is a gestural music visualization and composition tool that was developed with the goal to reduce the distances between the user, the mathematical framework, and the musical result. In its early stages, described for instance in [1043, 1045], it enabled defining, manipulating, and transforming Score denotators using an intuitive visual and gestural interface. Later on, it was generalized for transformationtheoretical paradigms based on the ontological dimension of embodiment, consisting of facts, processes, and gestures, and the communication between these levels [1042].

The facts, or objects, that the rubette BigBang in RUBATO® Composer deals with are denotators, which can be considered points in the spaces defined by their forms, as introduced in Chapter 6. So far, we have only seen a small portion of the variety of forms that can be defined in RUBATO® Composer. However, any conceivable musical or non-musical object can be expressed with forms and denotators, many of them just with the category of modules Mod@. The most recent version of BigBang was made compatible with as many forms as possible, even ones that the users may spontaneously choose to define at runtime. In order to handle this as smoothly as possible, we had to find a suitable way of representing denotators within the rubette, which we call BigBangObjects.1 In this chapter, we describe how this works.

The main idea behind the BigBang rubette is to give composers and improvisers a way to use the software RUBATO® Composer in a way that is more intuitively understandable, more spontaneous, and more focused on audible results than on the mathematical underpinnings. After discussing the types of facts available in BigBang we need to examine how we can create them and what we can do with them. From earlier in this book we now know that both of these activities, making and manipulating, are instances of processes. BigBang keeps track of these processes in a more sophisticated way than other musical software, especially ones dedicated to symbolic structures.

We have so far seen that the BigBang rubette allows users to visualize and sonify facts, and create and manipulate them using processes. In the previous chapter, we also discussed that the only structures that BigBang represents internally are processes, only one of which refers to facts in the form of denotators (InputComposition). All other facts are generated dynamically, whenever an operation is added or modified. In order to offer an intuitive way of interacting with the software, we need yet another level: gestures.

The new BigBang rubette offers many possibilities of creating music, due to the great variety of forms that can be defined. We already presented some simple ideas of forms in Section 71.5. In this section, we introduce some of innumerable slightly larger musical examples created in the course of writing the code of BigBang and the corresponding thesis. These examples illustrate a variety of composition techniques and types of musical results possible with BigBang. All examples are available for listening on SoundCloud, and some of the more performative ones can be found on YouTube, at the addresses indicated below.

This chapter introduces the creative relationship between mathematical gesture theory and physical string theory.

We shortly discuss the question of unicity in music and physics, a question that in physics has been virulent since the advent of string theory, but which in music has been relevant since the approach to music via individual compositions at the end of the Middle Ages.

A critical review of Hermann Hesse’s idea of a Glass Bead Game in the light of recent developments in mathematics, music theory, and theoretical physics is presented. The common denominator of these new dynamics is the shift from Wittgenstein’s world of rigid facts to an ocean of elastic gestures. In such a soft architecture of knowledge production, the ultimate principle of uniqueness as conceived in the idea of a singular universe breaks down to a multiverse, a multiplicity of worlds that terminates the historical breakdowns of uniqueness principles from geocentricity (Copernicus) to anthropocentricity (Darwin), chronocentricity (Einstein), and ratiocentricity (computers). We discuss contributions from eminent mathematicians Alexander Grothendieck and Yuri Manin, theoretical physicist Edward Witten, music theorist David Lewin, and philosophers Tommaso Campanella, Paul Valéry, Gilles Châtelet, Jean Cavaillès, and Charles Alunni. We complement their positions with our own contributions to topos-theoretical concept architectures and theories in gestural music theory, and offer realizations, both by means of gestural composition software and with examples from contemporary free jazz. The chapter concludes with a reconsideration of the game concept as a synthesis of artistic and scientific activity in the light of gestural fluidity.A critical review of Hermann Hesse’s idea of a Glass Bead Game in the light of recent developments in mathematics, music theory, and theoretical physics is presented. The common denominator of these new dynamics is the shift from Wittgenstein’s world of rigid facts to an ocean of elastic gestures. In such a soft architecture of knowledge production, the ultimate principle of uniqueness as conceived in the idea of a singular universe breaks down to a multiverse, a multiplicity of worlds that terminates the historical breakdowns of uniqueness principles from geocentricity (Copernicus) to anthropocentricity (Darwin), chronocentricity (Einstein), and ratiocentricity (computers). We discuss contributions from eminent mathematicians Alexander Grothendieck and Yuri Manin, theoretical physicist Edward Witten, music theorist David Lewin, and philosophers Tommaso Campanella, Paul Valéry, Gilles Châtelet, Jean Cavaillès, and Charles Alunni. We complement their positions with our own contributions to topos-theoretical concept architectures and theories in gestural music theory, and offer realizations, both by means of gestural composition software and with examples from contemporary free jazz. The chapter concludes with a reconsideration of the game concept as a synthesis of artistic and scientific activity in the light of gestural fluidity.

This chapter deals with a model from mathematical physics of string theory that describes the transition from symbolic reality to physical reality of musical gestures. We demonstrate, using multidimensional Fourier theory and Green functions, that the physical gesture can be viewed as a function of a potential and the symbolic gesture. The role of this potential is however not fully understood to date, but the idea is that it should encompass artistic rationales, together with physical components.

The purpose of this chapter is to review our contrapuntal model such that the group-theoretical contrapuntal symmetries are reinterpreted in the framework of topology, where continuity can be addressed. In particular, we shall interpret the set of intervals as being a topological category. We shall then develop a theory of hypergestures in such a category and investigate the first singular homology group associated with hypergestures. It will turn out that the above conditions defining contrapuntal symmetries can be restated in terms of topology and its associated homology of hypergestures.

In a recent book [16], we have opened the discussion of a hypergestural restatement of mathematical counterpoint theory. The present chapter aims at a discussion in the same vein of the classical mathematical modulation theory [682, 670]. The present approach to modulation theory is based on the idea that degrees in the start tonality are interpreted as gestures that move to degrees (qua gestures) of the target tonality by means of hypergestures. This means that the symmetries relating tonalities in the classical setup are replaced by hypergestures that connect gesturally interpreted degrees. The present hypergestural model solves the problem, but it opens more questions than it answers in the sense that the construction of hypergestures that replace the classical inversion symmetries is by no means unique.

In the context of performance stemmata, different hypergestures correspond to different strategies of deformation from mother to daughter performances. We discuss and classify types of such strategies using topological obstructions in terms of singular hypergesture homology. This gives hypergesture homology a nice interpretation in terms of human performance practice.

In this chapter, we tackle both analysis and composition as reciprocal processes to investigate the performer’s body. We argue that performers, more specifically, the performers’ bodily gestures, are key to the critical understanding and the creation of music. This chapter contains three parts: first, we will investigate the concept of embodied musical gestures through a range of inter-disciplinary scholars, ultimately defining a concept that is useful and fruitful in discussing performance. In the second part, we will use Toru Takemitsu’s Rain Tree Sketch II for Piano (1994) as a testing ground for analyzing with the performative body as the starting point. And lastly, we will discuss how composing with performative gestures in my composition Sheng (2016) for piano, audience’s smartphones, and fixed audio playback elicits the cross-modal, intersensory nature of embodied musical gestures. Indeed, the concept of embodied musical gesture has the potential to dissolve the artificial fractures between the activities of thinking, creating, and doing. Analyzing and composing with the performing body do away with this mind-body split, offering refreshing and generative insights that do justice to the physical nature of music making.

Gestures can be studied as the connection of discrete points by continuous paths. In the gesture of the orchestral conductor, the points connected by the gestural path correspond to metric movements of time represented in space. Here, we will study the gestures of the conductor referring to some concept of homotopy theory. The basic metric gesture is a regular and symmetric spanning of the space between points. Musical interpretation modifies the form of these regular gestures, changing their time, velocity, energy, amplitude and directionality. Thus, the most important information for performance contained in the orchestral score can be described by gestures. The conductor can also, through his gesture, add elements not explicitly contained in the score. The conductor’s gestures anticipate and continuously prepare the gestures of each musician in the orchestra in a hierarchical structure that corresponds to the structure of the score: from the general form to the articulation of each single note. In the first part of this chapter, we will discuss a case of study. In the second part, we will give some mathematical hints for a precise description of conducting gestures. In the third part, we will see an example of technology applied to conducting.

Reviewing the production of the video Imaginary Time, we claimthat the time that is created in free improvisation (let us take the purest type of improvisation here to deal with the unmixed phenomenon) is categorically different from score-generated time.

The curves traced by a drummer’s sticks, the various characteristic hand shapes adapted for various note clusters on a piano, the various ways that elbow and shoulder joints can support strokes on a violin all have sonic consequences. Indeed, if the previous chapters have taught us anything, it is that musicking is inherently (rather than incidentally) gestural. But there may yet be a lingering suspicion in some readers’ minds (particularly those who are accustomed only to playing from notation) that the graceful arc of a pianist’s hand is less like a dancer twirling across the stage and more like a blacksmith hammering a piece of metal into a horseshoe. The skeptical claim would be that gesture is a necessary practical step in the production of a finished, pre-figured sonic product, and no more.

Gesture Theory has been first developed using the pianist’s gesture as paradigm. However, the analytical techniques and the results found can be applied to other musical situations, once symbolic and physical gestures have been identified. For this reason it is possible to develop a gesture theory of voice. Voice teachers explain vocal technique also via gestures and references to imaginary movements of the voice through the resonant cavities of the body: we can think, as just a first example, of the passage from the di petto register to di testa register. For voice, there really are some inner movements, of larynx, vocal folds, tongue, as well of the diaphragm. Thinking of imaginary movements, the singer effectively changes the real shape of his or her phonatory system, obtaining the desired effect. These movements, connecting (imaginary once, and then embodied) points, are gestures. In fact, there are gestures that help one sing, but the singing is itself a gestural activity. Moreover, we can adapt to the modern gestural math-musical formalism a powerful instrument of the past, the neumes. The neumatic notation is the ancient way to notate the shape of the voice singing Gregorian melodies. This system successively evolved into a precise notation of pitches via points (square notes) in a four-line staff, and finally evolved to today’s notation of (round) notes in the five-line staff. Explicit reference to gestures are also used in textbooks about the didactics of the Gregorian chant. We end the chapter with the proposal of a new neumatic notation for voice didactics and composition that can complete the information given by the musical score.