Klein Concert: A Report from a Math-Musical Classroom Experience
Can music help mathematical education? Can some shapes, that challenge visual imagination, be translated into music? I present a pedagogical application of my interdisciplinary research. This concerns a cycle of seminars about mathematics and music I gave in Italy at the Music Conservatory of Palermo and that, in a shortened version, I also gave at Ca’ Foscari University in Venice, at the Conservatory of Livorno, and in the UK, at the University of Greenwich. Seminars included a theoretical section and a workshop. Mathematical concepts helped students formalize their musical knowledge, and music provided an intuitive understanding of mathematical concepts, such as the dualism continuous/dis…
Reversing arrows: Duality
What do you get reversing all arrows? The drawing ‘Duality’ is an homage to mirrors, classical art themes, and abstract mathematics.I’m looking for beauty in the arts and beauty in science. It’s a ...
Quantum Creativity and Cognition in Humans and Robots
In this research, we present a categorical framework to connect research on creativity and cognition for humans and robots, in light of the quantum paradigm. These fields and their relationships suggest a wider vision: modeling human creativity/cognition through quantum computing, and creating robots that can help us learn more about the humans themselves. We represent the human–robot comparison through functors (function generalization). Fundamental elements to understand human creativity are motivation and feedback as aesthetic pleasure. Is it possible to model it? Can the quantum paradigm help us in such an endeavor? We envisage the concept of emergence and quantum computing as decisive…
Modeling and designing a robotic swarm: A quantum computing approach
Nature is a neverending source of inspiration for technology. Quantum physics suggests applications to- ward quantum computing. Swarms’ self-organization leads to robotic swarm developments. Here, quantum computing is applied to swarm robotics. We model local interactions with a quantum circuit, testing it on simulators and quantum computers. To relate local with global behavior, we develop a block matrix-based model. Diagonal sub-matrices contain information on single robots; off-diagonal sub-matrices are the pairwise interaction terms. Comparing different swarms means comparing different block matrices. Choosing initial values and computation rules for off-diagonal blocks (with a particul…
The Topos of Music III: Gestures Musical Multiverse Ontologies
New element of the Topos book brings together progress in diverse field Presents gesture theory, including a gesture philosophy for music, the mathematics of gestures, concept architectures and software for musical gesture theory, and the multiverse perspective Presents applications of gesture theory, including counterpoint, modulation theory, and free jazz
Can Mathematical Theory of Music Be Easily Learnt and Also Be Fun?
[no abstract]
Quanta in Sound, the Sound of Quanta: A Voice-Informed Quantum Theoretical Perspective on Sound
Humans have a privileged, embodied way to explore the world of sounds, through vocal imitation. The Quantum Vocal Theory of Sounds (QVTS) starts from the assumption that any sound can be expressed and described as the evolution of a superposition of vocal states, i.e., phonation, turbulence, and supraglottal myoelastic vibrations. The postulates of quantum mechanics, with the notions of observable, measurement, and time evolution of state, provide a model that can be used for sound processing, in both directions of analysis and synthesis. QVTS can give a quantum-theoretic explanation to some auditory streaming phenomena, eventually leading to practical solutions of relevant sound-processing…
All About Music: The Complete Ontology: Realities, Semiotics, Communication, and Embodiment
[no abstract]
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.
Knots, Music and DNA
Musical gestures connect the symbolic layer of the score to the physical layer of sound. I focus here on the mathematical theory of musical gestures, and I propose its generalization to include braids and knots. In this way, it is possible to extend the formalism to cover more case studies, especially regarding conducting gestures. Moreover, recent developments involving comparisons and similarities between gestures of orchestral musicians can be contextualized in the frame of braided monoidal categories. Because knots and braids can be applied to both music and biology (they apply to knotted proteins, for example), I end the article with a new musical rendition of DNA.
CubeHarmonic: A new musical instrument based on Rubik{'}s cube with embedded motion sensor
A contemporary challenge involves scientific education and the connection between new technologies and the heritage of the past. CubeHarmonic (CH) joins novelty and tradition, creativity and edu- cation, science and art. It takes shape as a novel musical instrument where magnetic 3D motion tracking technology meets musical per- formance and composition. CH is a Rubik’s cube with a note on each facet, and a chord or chord sequence on each face. The posi- tion of each facet is detected through magnetic 3D motion tracking. While scrambling the cube, the performer gets new chords and new chord sequences. CH can be used to compose, improvise,1 and teach music and mathematics (group theory, permu…
A musical reading of a contemporary installation and back: mathematical investigations of patterns in Qwalala
Mathematical music theory helps us investigate musical compositions in mathematical terms. Some hints can be extended towards the visual arts. Mathematical approaches can also help formalize a "translation" from the visual domain to the auditory one and vice versa. Thus, a visual artwork can be mathematically investigated, then translated into music. The final, refined musical rendition can be compared to the initial visual idea. Can an artistic idea be preserved through these changes of media? Can a non-trivial pattern be envisaged in an artwork, and then still be identified after the change of medium? Here, we consider a contemporary installation and an ensemble musical piece derived from…
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…
Trajectory-based and Sound-based Medical Data Clustering
Challenges in medicine are often faced as interdisciplinary endeav- ors. In such an interdisciplinary view, sonification of medical data provides an additional sensory dimension to highlight often hard- to-find information and details. Some examples of sonification of medical data include Covid genome mapping [5], auditory repre- sentations of tridimensional objects as the brain [4], enhancement of medical imagery through the use of sound [1]. Here, we focus on kidney filtering-efficiency time-evolution data. We consider the estimated glomerular filtration rate (eGFR), the main indicator of kidney efficiency in diabetic kidney disease patients.1 We propose a technique to sonify the eGFR tra…
Categories, Musical Instruments, and Drawings: A Unification Dream
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 th…
"Mirrors"
[no abstract]
Shall We (Math and) Dance?
Can we use mathematics, and in particular the abstract branch of category theory, to describe some basics of dance, and to highlight structural similarities between music and dance? We first summarize recent studies between mathematics and dance, and between music and categories. Then, we extend this formalism and diagrammatic thinking style to dance.
Comparison of non-Markovianity criteria in a qubit system under random external fields
We give the map representing the evolution of a qubit under the action of non-dissipative random external fields. From this map we construct the corresponding master equation that in turn allows us to phenomenologically introduce population damping of the qubit system. We then compare, in this system, the time-regions when non-Markovianity is present on the basis of different criteria both for the non-dissipative and dissipative case. We show that the adopted criteria agree both in the non-dissipative case and in the presence of population damping.
Color and Timbre Gestures: An Approach with Bicategories and Bigroupoids
White light can be decomposed into different colors, and a complex sound wave can be decomposed into its partials. While the physics behind transverse and longitudinal waves is quite different and several theories have been developed to investigate the complexity of colors and timbres, we can try to model their structural similarities through the language of categories. Then, we consider color mixing and color transition in painting, comparing them with timbre superposition and timbre morphing in orchestration and computer music in light of bicategories and bigroupoids. Colors and timbres can be a probe to investigate some relevant aspects of visual and auditory perception jointly with thei…
cARTegory Theory: Framing Aesthetics of Mathematics
Mathematics can help investigate hidden patterns and structures in music and visual arts. Also, math in and of itself possesses an intrinsic beauty. We can explore such a specific beauty through the comparison of objects and processes in math with objects and processes in the arts. Recent experimental studies investigate the aesthetics of mathematical proofs compared to those of music. We can contextualize these studies within the framework of category theory applied to the arts (cARTegory theory), thanks to the helpfulness of categories for the analysis of transformations and transformations of transformations. This approach can be effective for the pedagogy of mathematics, mathematical mu…
Classes of Colors and Timbres: A Clustering Approach
Similarities between different sensory dimensions can be addressed considering common “movements” as causes, and emotional responses as effects. An imaginary movement toward the “dark” produces “dark sounds” and “dark colors,” or, toward the “bright,” “brighter colors” and “brighter sounds.” Following this line of research, we draw upon the confluence of mathematics and cognition, extending to colors and timbres the gestural similarity conjecture, a development of the mathematical theory of musical gestures. Visual “gestures” are seen here as paths in the space of colors, compared with paths in the space of orchestral timbres. We present an approach based on clustering algorithm to evaluate…
Parametric Natura Morta
Parametric equations can also be used to draw fruits, shells, and a cornucopia of a mathematical still life. Simple mathematics allows the creation of a variety of shapes and visual artworks, and it can also constitute a pedagogical tool for students.
Quantum planning for swarm robotics
Computational resources of quantum computing can enhance robotic motion, decision making, and path planning. While the quantum paradigm is being applied to individual robots, its approach to swarms of simple and interacting robots remains largely unexplored. In this paper, we attempt to bridge the gap between swarm robotics and quantum computing, in the framework of a search and rescue mission. We focus on a decision-making and path-planning collective task. Thus, we present a quantum-based path-planning algorithm for a swarm of robots. Quantization enters position and reward information (measured as a robot’s proximity to the target) and path-planning decisions. Pairwise information-exchan…
Embryo of a Quantum Vocal Theory of Sound
Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbule…
Quantum RoboSound: Auditory Feedback of a Quantum-Driven Robotic Swarm
Data sonification enhance and enrich information understanding with an additional sensory dimension. Sonification also opens the way to more creative applications, joining arts and sciences. In this study, we present sequences of chords obtained as auditory feedback from the trajectories of a robotic swarm. The swarm behavior is an emerging effect from simple local interactions and autonomous decisions of each robot. The swarm effect can be identified through sonification outcomes in terms of voice leading patterns. Thus, chord patterns represent behavior patterns. The convergence to the target is represented by the convergence to a specific pitch. The swarm decision process is based upon q…
Quantum GestART: Identifying and Applying Correlations between Mathematics, Art, and Perceptual Organization
Mathematics can help analyze the arts and inspire new artwork. Mathematics can also help make transformations from one artistic medium to another, considering exceptions and choices, as well as artists' individual and unique contributions. We propose a method based on diagrammatic thinking and quantum formalism. We exploit decompositions of complex forms into a set of simple shapes, discretization of complex images, and Dirac notation, imagining a world of "prototypes" that can be connected to obtain a fine or coarse-graining approximation of a given visual image. Visual prototypes are exchanged with auditory ones, and the information (position, size) characterizing visual prototypes is con…
A quantum vocal theory of sound
Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbule…
Dense Geometry of Music and Visual Arts: Vanishing Points, Continuous Tonnetz, and Theremin Performance
The dualism between continuous and discrete is relevant in music theory as well as in performance practice of musical instruments. Geometry has been used since longtime to represent relationships between notes and chords in tonal system. Moreover, in the field of mathematics itself, it has been shown that the continuity of real numbers can arise from geometrical observations and reasoning. Here, we consider a geometrical approach to generalize representations used in music theory introducing continuous pitch. Such a theoretical framework can be applied to instrument playing where continuous pitch can be naturally performed. Geometry and visual representations of concepts of music theory and…
Some Mathematical and Computational Relations Between Timbre and Color
In physics, timbre is a complex phenomenon, like color. Musical timbres are given by the superposition of sinusoidal signals, corresponding to longitudinal acoustic waves. Colors are produced by the superposition of transverse electromagnetic waves in the domain of visible light. Regarding human perception, specific timbre variations provoke effects similar to color variations, for example, a rising tension or a relaxation effect. We aim to create a computational framework to modulate timbres and colors. To this end, we consider categorical groupoids, where colors (timbres) are objects and color variations (timbre variations) are morphisms, and functors between them, which are induced by co…
Musical pitch quantization as an eigenvalue problem
How can discrete pitches and chords emerge from the continuum of sound? Using a quantum cognition model of tonal music, we prove that the associated Schrödinger equation in Fourier space is invariant under continuous pitch transpositions. However, this symmetry is broken in the case of transpositions of chords, entailing a discrete cyclic group as transposition symmetry. Our research relates quantum mechanics with music and is consistent with music theory and seminal insights by Hermann von Helmholtz.
Temari Balls, Spheres, SphereHarmonic: From Japanese Folkcraft to Music
Temari balls are traditional Japanese toys and artworks. The variety of their geometries and tessellations can be investigated formally and computationally with the means of combinatorics. As a further step, we also propose a musical application of the core idea of Temari balls. In fact, inspired by the classical idea of music of spheres and by the CubeHarmonic, a musical application of the Rubik’s cube, we present the concept of a new musical instrument, the SphereHarmonic. The mathematical (and musical) description of Temari balls lies in the wide background of interactions between art and combinatorics. Concerning the methods, we present the tools of permutations and tessellations we ado…
Cool Math for Hot Music
[no abstract]
Exploring Heterogeneity with Category and Cluster Analyses for Mixed Data
Precision medicine aims to overcome the traditional one-model-fits-the-whole-population approach that is unable to detect heterogeneous disease patterns and make accurate personalized predictions. Heterogeneity is particularly relevant for patients with complications of type 2 diabetes, including diabetic kidney disease (DKD). We focus on a DKD longitudinal dataset, aiming to find specific subgroups of patients with characteristics that have a close response to the therapeutic treatment. We develop an approach based on some particular concepts of category theory and cluster analysis to explore individualized modelings and achieving insights onto disease evolution. This paper exploits the vi…
Theoretical Physics and Category Theory as Tools for Analysis of Musical Performance and Composition
(There is no abstract)
Have Fun with Math and Music!
If abstraction makes mathematics strong, it often makes it also hard to learn, if not discouraging. If math pedagogy suffers from the lack of engaging strategies, the pedagogy of mathematical music theory must deal with the additional difficulty of double fields and double vocabulary. However, games and interdisciplinary references in a STEAM framework can help the learner break down complex concepts into essential ideas, and gain interest and motivation to approach advanced topics. Here we present some general considerations, followed by two examples which may be applied in a high-school or early college level course. The first is a musical application of a Rubik’s cube, the CubeHarmonic, …
Categories, Quantum Computing, and Swarm Robotics: A Case Study
The swarms of robots are examples of artificial collective intelligence, with simple individual autonomous behavior and emerging swarm effect to accomplish even complex tasks. Modeling approaches for robotic swarm development is one of the main challenges in this field of research. Here, we present a robot-instantiated theoretical framework and a quantitative worked-out example. Aiming to build up a general model, we first sketch a diagrammatic classification of swarms relating ideal swarms to existing implementations, inspired by category theory. Then, we propose a matrix representation to relate local and global behaviors in a swarm, with diagonal sub-matrices describing individual featur…
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 acoustical reality of sounds. Moreover, the precise concept of gesture has a decisive role in understanding musical performance. In this paper, we apply some concepts of category theory to compare gestures of orchestral musicians, and to investigate the relationship between orchestra and conductor, a…