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This essay offers a vision of voice as relational and temporal, in contrast to figurations of voice as “one” (as a soliloquy that directly expresses a sovereign individual subjectivity) and “now” (as immediately present, in contrast to the spatial distance and temporal delay of writing or reflection). This abstract construal of pristine subjective oneness and atemporal objective presence underlies both Husserlian semiotics and the Derridean critique of the “metaphysics of presence” long associated with the voice. In practice, however, most vocal action (public singing and speaking, chatting and harmonizing with others, vocal uproar, protest, negotiation) is undertaken in relation to others and unfolds over time. Politically, the irrelational individual-expressivist figuration provides the metaphysical scaffolding for groupist-expressivist figurations of voice, in which homogenous collectivities are understood to speak in unison, cut off from interlocutors, response, or counterpoint—cut off, that is, from actual sociality. What political and ethical possibilities might open up by turning from irrelational soliloquy to relational colloquy, among others, in time?

This chapter suggests ways in which accounts of particular, situated voices might work in a dialectic with general theories of “the voice.” It begins from five geographically proximate but ontologically far-flung voice-s, each revealed as real in its own way by five different situated practices: a political demonstration in Mumbai, a samā in which poems are sung for members of the Chisti Sufi brotherhood, a staged “Sufi” event, the editing of a playback singer’s vocal track, and a diagnosis in a voice clinic. It goes on to reflect in general on the practices and infrastructures that allow particular voices to show up as real, demonstrating that voices not only are different, but are differently. Finally, it turns to cases in which voices show up indeterminately: as muddled and unresolvable, or binaurally, as two things as once—thus offering a way into the ambiguities and contestations that animate a vocal “politics of what.”

The influential sixteenth-century Sanskrit treatise Svara-mela-kalānidhi describes a novel system of naming tones, of organizing rāga -s by pitch content, and of reckoning svara -s on 12 fret positions rather than 22 śruti -s. Contrary to its common construal as a sudden rupture in tradition, we highlight the rhetorical means by which the treatise systematically grounds its authority (and that of its ambitious patron, Rāmarāya) in the canon of saṅgīta-śāstra . We also offer a new translation and a new (non-Pythagorean) interpretation of its svayambhu -based tuning system.

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 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.

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.

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.

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 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.

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].

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.

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.

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.

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].

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.

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 Valery, Gilles Châtelet, Jean Cavailles, 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.

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.

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.

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.

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.

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.

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.

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.

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).

The harmonium is both widely played and widely condemned in India. During the Indian independence movement, both British and Indian scholars condemned the harmonium for embodying an unwelcome foreign musical sensibility. It was consequently banned from All-India Radio from 1940 to 1971, and still is only provisionally accepted on the national airwaves. The debate over the harmonium hinged on putative sonic differences between India and the modern West, which were posited not by performers, but by a group of scholars, composers, and administrators, both British and Indian. The attempt to banish the sound of the harmonium was part of an attempt to define a national sound for India, distinct from the West. Its continued use in education served a somewhat different national project: to standardize Indian music practice. This paper examines the intertwined aesthetic and political ideals that underlie the harmonium controversy.

The gestures that accompany improvisation in Indian vocal music, like the gestures that accompany speech, are closely co-ordinated with vocalization. Though linked to what is being sung, these movements are not determined by vocal action; nor are they taught explicitly, deliberately rehearsed, or tied to specific meanings. Students tend to gesture recognizably like their teachers, producing lineage-based gesture dialects, but the gestural repertoire of every vocalist is nonetheless idiosyncratic. This paper aims to trace a brief history of song gesture in India, and to show some of the links between gesture and vocalization. It also adapts Katharine Young’s theory of the “family body” to the transmission of gesture dialects through teaching lineages. Gesture and sound are taken to be parallel channels for the expression of melody.