Maria Mannone

Hypergestures in Complex Time: Creative Performance Between Symbolic and Physical Reality

Musical performance and composition imply hypergestural transformation from symbolic to physical reality and vice versa. But most scores require movements at infinite physical speed that can only be performed approximately by trained musicians. To formally solve this divide between symbolic notation and physical realization, we introduce complex time (\(\mathbb {C}\)-time) in music. In this way, infinite physical speed is “absorbed” by a finite imaginary speed. Gestures thus comprise thought (in imaginary time) and physical realization (in real time) as a world-sheet motion in space-time, corresponding to ideas from physical string theory. Transformation from imaginary to real time gives us a measure of artistic effort to pass from potentiality of thought to physical realization of artwork. Introducing \(\mathbb {C}\)-time we define a musical kinematics, calculate Euler-Lagrange equations, and, for the case of the elementary gesture of a pianist’s finger, solve corresponding Poisson equations that describe world-sheets which connect symbolic and physical reality.

Global functorial hypergestures over general skeleta for musical performance

Musical performance theory using Lagrangian formalism, inspired by physical string theory, has been described in previous research. That approach was restricted to zero-addressed hypergestures of local character, and also to digraph skeleta of simple arrow type. In this article, we extend the theory to hypergestures that are defined functorially over general topological categories as addresses, are global, and are also defined for general skeleta. We also prove several versions of the important Escher Theorem for this general setup. This extension is highly motivated by theoretical and practical musical performance requirements of which we give concrete examples.

The Topos of Music III: Gestures

ion & Generalization & Specialization. For example this means that, starting with a datum as a Latin square (or with the datum of a taquin), we abstract by forgetting some specifications, e.g. the fact that it is presented in a square, and keeping in mind other facts, e.g. the fact that we have permutations in the square and the fact that they are acting on Latin squares (or on states of the taquin). We abstract by forgetting: “penser c’est oublier” [1076], and this is an aspect of mathematical pulsation, its aspect of estrangement, as expressed in our conclusion (Section 67.7), which means that firstly if by any chance we do some combinations or substitutions, we forget some data within the problem—at the risk wasting time—and possibly recognize the rest as being essential when it is seen in a new context, and at last decide to choose that renewed rest to pursue our job. As in traditional Zen archery this forgetting or oversight is an initial moment of relaxation: see [414] for an explanation of the pulsation in relation to Zen. With the accent on permutations, and so on elements of groups of permutations, we reach the more abstract structure of a group. The notion of a group itself comes by abstraction from groups of permutations as well as groups of geometrical transformations, by keeping in mind only some properties of the composition law of these permutations or transformations. At this moment we have brought out the part played by the structure of a group into the configuration of a Latin square. But now a group itself is a new type of configuration, which can be studied and classified, and represented as itself. Of course in this general study of groups, it will be permissible and useful to look at special cases, etc. Groups, lattices, topologies, linear spaces, metric spaces, fields, rings, all these classic structures (the “structures-mères” according to Bourbaki), and also local structures, categories and topoi, arise in a historical process of “structuration” and are basically of the same nature as figures, formulas, configurations, designs or diagrams. However, with a very important precision: an object or a structure, or a type of structure, 67.3 Method and Objects, Summarily Explained: II—Data 1027 cannot just be an arbitrary formal free definition; it has to be fruitful, or already it is itself a theorem (a theorem of existence), or it will be quickly clear that it implies interesting facts about previously unsolved questions. Now, let us give an example. In a particular higher state of mathematics, namely in the mathematical area of algebraic topology, we consider the rather intuitive idea (“intuitive” in this area, of course) that projective modules over commutative rings are like vector bundles on compact spaces. This puzzle has two precise formulations and proofs by Jean-Pierre Serre [969] and then Richard Swan [1028]. We consider that these Serre-Swan theorems are not really different from a puzzle like “taquin”, and our effective recourses for solving are the same. The only difference is that we move at different levels, with different preliminary formation, into another “mathematical history” or personal historical scenery for mathematical activity. In both cases we have to move writings, letters of algebra and figures of geometry, and diagrams; to introduce abbreviations and rules of modifications,; to construct and de-construct; and to structure, by synthesis and analysis. Ultimately any theorem says that two paths of constructions and de-constructions arrive at the same result. In recreational mathematics or in higher mathematics in the construction of the “paths” we have the critical moment when simultaneously we have to invent the “space” in which the path is going now, usually a space of configurations or structures into another space used in a previous step of the path. Our tracks are the road. So creative mathematics are possible as well at the level of recreational mathematics, as in very abstract algebraic topology. The point is the pulsative imagination within analysis-and-synthesis; deeper creation in mathematics is exactly invention of pulsations as methods, that is to say invention of new calculus expressing the tension between analysis and synthesis. 67.3.1.6 Undirectness, Synthetic Thinking and Intuitions The gestures of folding, bending, cutting, erasing, gluing, etc. are our concrete accesses to any understanding or construction or de-construction of mathematical objects. In algebra it is known through factorizing and expanding expressions, in geometry it is realized in the set of geometrical constructions. But it is of a much more general scope, at any level with any model of any structure, presented by generators and relations. We can do these gestures within the system of elements or within the system of relations, or both. For example if we read Gaston Tarry [1036] on the problem of the 36 officers, we have to admit that his analysis proceeds by such gestures; in this way he can construct various coordinations of the hypothetical solution object, with various symmetries, modifications, insertions, quotients, group actions, etc., and this allows us to count data of different types, and to conclude when we find impossible values. This is a process of abstraction, combined with structuration, inventions of figures to observe in the given situation, as permutations, paths, or what he calls magic groups (already considered by Euler under the name of “formules directrices”). Also he reduces the analysis by classifying possible Latin squares in a solution up to isotopies or conjugations. The historical process of structuring by abstraction is parallel to a systematic movement of undirectness in the development of mathematics in the 19th century. For example, concerning the geometry of curves, from the abridged method of Gabriel Lamé to Etienne Bobillier and Julius Plücker, and to Max Noether, followed by Emanuel Lasker and then Emmy Noether, we start from classical problems on curves through the intersections of given curves, and we reach the invention of Noetherian rings. For the following generations, they think geometry with, the notion of a Noetherian ring at the beginning of their thoughts; they could forget concrete curves and Italian geometry, etc. The same “synthetic” fact arrives with groups: whatever would be the very good motivations for groups, once assimilated the explanations of geometries in terms of groups according to Klein, the new generations will think directly in terms of groups. We can perhaps consider that mathematical intuition, which is very transversal to the skill for application of logical rules, comes from the habits of working at a given level of abstraction (with numbers, figures, equations, polynomials, spaces, groups, rings, categories, topos, etc.), i.e., in a certain category of objects, which becomes familiar; objects are now very complex configurations which became simple for the mind; it 1028 67 Mathematical Models of Creativity is the skill to move into a given category with such formerly old data. For instance, someone can have or not have the intuition of the object R (as an object of a category C). Definitely we consider that mathematical activity and mathematical intuition are absolutely relative, starting from the datum of a category as an allegedly natural setting of a first organized intuitive world. 67.3.1.7 Categories, Sets, Groups, Lattices, Structures, out of Logical Concern An idea suggested by several logicians or philosophers [115, 653] is the opposition between sets and categories as a proposal for foundations. More precisely, with Jean-Yves Béziau [115] we can analyze the relations of set theory and category theory with respect to their relation to the question of foundations. Béziau remarks that at least we have three distinct problems: (A): axiomatic foundation of mathematics (to describe a set of axioms from which we can deduce mathematics); (C): conceptual foundation (to describe basic concepts and their links, in order to think mathematics); (L): logical foundation (to show that mathematics are not contradictory). We can add a fourth case: (F): functional foundation (description of a system of mathematical actions which are enough to develop mathematics). Roughly speaking, set theory is good for (A) and (L), and category theory is better for (C) and (F). So the opposition of categories and sets is wrong as an epistemological perspective. Category theory is a method of analysis of the production of mathematical works—and this method itself is mathematical, and a mathematical theory—whereas set theory (and logic) is possibly a ground for all mathematical developments and constructions of structures. We have to completely dissociate the question of observation, description and control of mathematical gestures, using categories, from the question of foundations and logic, using set theory. The control of the size of universes is interesting for questions on existence based on set theoretical constructions. Some mathematicians believe that categoricians are mainly interested in foundational questions, and consequently are unable to see that groups are everywhere, that geometrical thoughts innervate good mathematics. In fact the notion of category unified the two geometrical notions of lattice and group. In [1150, p.37] “category theory” and “structures” are opposed; Bernard Zarca said that the concept of category is a rival of the concept of structure; this is a serious mistake from a historical perspective; it is a wrong interpretation of the discussion in Bourbaki about the question of introduction of categories in Bourbaki’s Elements. In fact, the more significative example of category are the categories of fundamental structures (structures-mères), also because each structure-mère “is” a category. So a group is a category, a lattice is a category, and even, indirectly, any object C of a category C becomes a categor

Directed and Undirected Graphs

Up to now, we have been able to construct all basic number domains ℕ, ℤ, ℚ, ℝ, ℂ. But we have not considered geometric objects. This chapter begins to fill that gap. It introduces the most elementary geometric objects: graphs—systems of points and arrows connected by directed or undirected lines. We shall conclude part IV with the introduction of higher-dimension graphical objects that relate to coverings of sets by a system of subsets.

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

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.

Physical and Musical Multiverses

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.

Gestures: Gestural Interaction and Gesturalization

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.

Categories of Gestures over Topological Categories

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.

Models from Music

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

Mathematical Models of Creativity

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.