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What could the Venice of the future look like? A project for an ideal city can take off from an allegory: arts and sciences inside a seed of Lodoicea maldivica, whose bipartition reminds us of a human brain. From the seeds left by the past, we derive the vision of the future. The ideal city certainly needs brains able to conceptualize images and develop ideas, and bridges to strengthen connections and in- teractions. A well-working brain needs “bridges” as connections between ideas and techniques. My vision for a future city contains a livable and stimulating space enhancing at one time creativity, enthusiasm, and scientific development. To this aim, I use the image and the metaphor of cerebral hemispheres, specialized in activities of different typologies yet interconnected. The image of the brain is one of the symbols of cognition. The right hemisphere deals with artistic expression, while the left hemisphere refers to logic thinking, mathematics, scientific attitude toward the world. Different cerebral areas, which contribute to the complex and rich life of an individual, are a metaphor for different places in a city, contributing to the completeness of a community life. Separation between hemispheres, also as a homage to Venice, is seen as a sort of Canal Grande, and connections are represented by bridges. The whole brain is seen as a giant seed of Lodoicea maldivica: the city of the future needs to develop from seeds, that is, knowledge inherited from the past and new ideas from our minds and thoughts. Thoughts that, in turn, are fed upon the Science of Complexity.

How can we relate quantum computing, robotics, and music? This is a position paper where we try to connect these fields. We discuss one main question regarding computational creativity and we imagine an experimental setup to test our hypothesis. Artificial intelligence and autonomous machines can accomplish simple creative tasks such as reorganizing given material. Because creativity is a human (and not only) resource to survive, we wonder if also artificial agents, such as robots, might develop creativity at a higher level to ensure self-survival. Then, we design an experimental setup with three robots, playing and dancing. If music and movement flow is regular (with the beat difference below a certain threshold), the voltage given to robots is constant. Otherwise, if there are inconsistencies, it drops, activating robots’ alert signal sensors, and triggering new musical activity. We use the paradigm of quantum computing to formalize our claims. This test might be performed in situ in a robotic lab.

The universal concept of a “music of the spheres” traverses the history of philosophy, science and art, from Pythagoras to Kepler and beyond. In modern times, a sphere in three dimensions—the Bloch sphere—is used to illustrate the state of a qubit, the basic unit of quantum information. On the same spherical surface, the fundamental components of voice production can be located, so that any utterance can be seen as the evolution of a unit two-dimensional vector having complex coefficients. Indeed, any sound can be analyzed and decomposed into overlapping sinusoidal components, broadband noises, and impulsive transients, which in turn can be associated to fundamental means of vocal sound production, such as phonation, turbulence, and slow myoelastic vibrations or pulses. The quantum sphere can be made to sing the universal sound.

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Arts can provide intuitive examples to enhance the understanding of complex mathematical concepts. Also, mathematics can give precise answers to musical questions that can be raised during the analysis of works of art including musical scores. If musical parameters and mathematical objects are seen as belonging to categories, analysis becomes a dialogue and a mapping between them. This paper develops an idea of “open” musicology that exploits suggestions and contaminations with other research areas, first of all from mathematics and the mathematical formalism behind physics. The focus is on the concept of entropy joined with Discrete Fourier Transforms (DFT), that can be extended to the definition of a musical entropy, as an abstract concept, as well as a computational paradigm. The entropy can be seen as the quantification of the degree of disorder throughout the temporal evolution of musical structures of an entire musical piece. Entropy can also be considered with respect to one or more musical parameters. Their temporal evolution acquires an artistic meaning in itself, as well as the variation of its degree. A method to quantitively evaluate the degree of entropy is presented: a new approach to the topic of entropy, that can also open new pedagogical scenarios. (Received September 16, 2019)

Cos’è la simmetria in musica? Come investigare suoni e strutture musicali alla luce della simmetria? E quale può essere, in questa ricerca, il ruolo della matematica? Il premio Nobel per la Fisica F. Wilczek definisce la simmetria come un “cambiamento senza cambiamento”, un’invarianza rispetto a una trasforma- zione. La ricerca di simmetria e di bellezza può essere intesa come la ricerca di equilibrio fra regolarità e variazione. Se la regolarità aiuta la memoria, è invece la varietà a stimolare l’attenzione e la curiosità. Argomenti fisico-ma- tematici come la teoria dei gruppi, l’acustica e la teoria dei segnali fornisco- no strumenti di indagine anche all’estetica musicale. Il tema della simmetria fra matematica e musica è il filo conduttore dei testi di fisici, matematici, teorici della musica e compositori presenti in questa antologia, scelti e tradotti da Federico Favali e Maria Mannone. Il volume comprende scritti di D. Kempf, G. Rochberg, J. W. Bernard, D. Wilson, R. Donnini, D. Tymoczko, V. Hart, O. Fernández Herrero, G. W. Don - K. K. Muir - G. B. Volk - J. S. Walker. Prefazione di David Fontanesi; postfazione di Giovanni Albini.

“Welcome in the Mile High City”, benvenuti nella città a un miglio di altitudine, è la scritta che accoglie i visitatori a Denver, nello stato del Colorado, nella profonda America delle Montagne Rocciose. Denver è nota agli amanti di sport ed esplorazioni, nonché agli appassionati di paleontologia, per i numerosi fossili di dinosauri. Ed è nota agli studiosi per il gigantesco Convention Center, scelto come sede per il Joint Mathematics Meetings dello scorso gennaio 2020 (JMM 2020), vedi Figura 1.

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Il potere di astrazione della teoria delle categorie può essere utilizzato per descrivere procedure e strutture musicali. La stessa teoria matematica può essere applicata alle forme visuali e alle loro trasformazioni, comprese le applicazioni computazionali. Poiché è possibile associare musica e immagini attraverso apposite mappature (mappings), è anche possibile connettere reti (networks) di frammenti musicali e di immagini descrivendo progressivamente le modifiche apportate alle forme musicali e visuali. In questo articolo si intende proporre un nuovo approccio alla scrittura musicale secondo questi principi, ovvero componendo musica a partire da una rete di immagini in grado di “sonorizzare” ogni passaggio e creando una sequenza parallela di variazioni musicali e visuali. Powerful abstraction of mathematical category theory can be used to describe musical procedures and structures. The same mathematical theory can be applied to visual shapes and their transformations, including computational applications. Since we can connect music and images through mappings, we can also connect their networks step by step, describing progressive shape modifications. We propose a new approach to music composition based on these ideas, composing music from a network of images “sonifying” each step, as well as creating a parallel sequence of visual and musical variations.

The present paper presents a mechanism to catalyze that crucial step in a creative process, where the critical concept’s “walls” (referring to the common “box” metaphor) are identified and opened. We propose a concrete, but generic body of six concepts and the action of a group of transformations of this body, a toolbox that should offer a set of operational perspectives onto the critical concept’s walls. The conceptual body and the group action are deduced from the first paragraphs of Georg Wilhelm Friedrich Hegel’s Wissenschaft der Logik. On this body, a group, called the “Hegel group”, acts and reflects some of the relations and operations that are hidden in Hegel’s text. We then identify two technical themes from mathematical music theory, the Escher Theorem and the concept architecture of forms and denotators, as instances of this Hegel action. The Hegel action is applied to understand creative processes in two classical compositions — Beethoven’s Hammerklavier Sonata op. 106, and Liszt’s Mephisto Walzer — but also to the creation of a small model composition.

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 keys for creativity. Investigation of robotic models, including swarms of robots, might help shed new light on human creative interactions.

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 particular logic gate), different behaviors can be modeled. To show the global-behavior emergence, we propose a specific pairwise-interaction logic gate, embedding the corresponding quantum circuit in an ant-foraging-inspired algorithm. To implement a first application, we choose the foraging-ant behavior for its clarity and importance in nature, running experiments with toy swarms (3 and 10 robots). We consider ants’ individual and collective back-and-forth movements between the nest and the food source, analyzing the effect of entanglement. Our research can help shed light on quantum potentialities for swarms. The implications of our findings and results concern the future development of a decision-making system, based on the advantages of swarms and quantum computing. While an ant-foraging scenario is chosen as an example of application, our study is not focused on optimization. We present a new methodology, open to non-optimal solutions. Future developments can concern improvements toward optimization.

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 ...

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 adopted here, and the core idea for the SphereHarmonic. As the results, we first describe a classification of structures according to the theory of groups. Then, we summarize the main passages implemented in our code, to make the SphereHarmonic play on a laptop. Our study explores an aspect of the deep connections between the mutually inspiring scientific and artistic thinking.

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 their connections. Thus, the use of categories proposed here aims to investigate color/timbre perception, influencing the computer science developments in this area.

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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 continuous maps. We also sketch some gestural variations of this scheme. Thus, we try to soften the differences and focus on the similarity of structures

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 quantum computing. We first present logic gates and their implementations as quantum circuits, describing examples with 2- and 3-dimensional motion of a 3-robot toy swarm. The considered scenarios are ant foraging (2D) and underwater search and rescue (3D). Then, we provide and discuss some examples of the harmonic sequences that can be obtained as feedback from robotic motion.

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.

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., Comment: 23 pages, 16 figures

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, to approach group theory and combinatorics jointly with musical chords; the second is an application of category theory to investigate simple musical variations together with transformations on a visual shape.

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 theory.

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, permutations) with colors and physical manipulation supporting abstract thinking. Further- more, CH allows visually impaired people to enjoy Rubik’s cube manipulation by using sounds instead of colors.

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We developed a new musical interface, CubeHarmonic, with the magnetic 3D motion tracking system IM3D. This sys- tem precisely tracks positions of tiny, wireless, battery-less, and identifiable markers (LC coils) in real time. The Cube- Harmonic is a musical application of the Rubik’s cube, with notes on each little piece. Scrambling the cube, we get dif- ferent chords and chord sequences. Positions of the pieces which contain LC coils are detected through IM3D, and transmitted to the computer to recognize the status of the Rubik’s cube, that plays sounds. The central position of the cube is also measured by the LC coils located into the corners of Rubik’s cube, and, depending on the position, we can manipulate overall loudness and pitch changes, as in theremin playing. This new instrument, whose first idea comes from mathematical theory of music, can be used as a teaching tool both for math (group theory) and music (music theory, mathematical music theory), as well as a composition device, a new instrument for avant-garde per- formances, and a recreational tool.

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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

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In una città che da più secoli viveva ai margini del set internazionale, era ghiotta l’occasione dell’Unità d’Italia per compiere quel salto che avrebbe permesso a Palermo di confrontarsi con le maggiori città del Regno. Insieme ai problemi sociali, economici e sanitari di una città priva dei servizi di base, i progetti utopici e “grandiosi” spingevano a realizzare quelle attrezzature in grado di proiettarla in un panorama più vasto. Tra queste emergevano i teatri: il Politeama per spettacoli diurni e popolari, ma ancora più il Massimo V. E. che avrebbe dovuto rappresentare un riferimento internazionale per l’attività lirica. Nonostante lo sfasamento di circa 30 anni tra la prima ideazione e l’ultimazione, il teatro non fu giudicato inadeguato rispetto alle “moderne” esigenze. Esso presenta aspetti costruttivi e tecnologici di rilevante interesse sia perché testimonia il massimo livello qualitativo dell'epoca, sia perché contiene soluzioni costruttive audaci ed originali, sia ancora per l'impulso che alla scienza, alla tecnica ed all'industria locali ha dato quest'opera, fucina di continue sperimentazioni.

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., 8 pages, 1 figure. Some changes made. In press on Physica Scripta T (special issue)

L'articolo introduce i risultati del lavoro di laboratorio svolto dagli studenti che hanno partecipato attivamente al seminario e workshop ComposizioneMusicaArchitettura.

L'articolo introduce gli atti del seminario e workshop "Composizione Musica Architettura" ed i materiali prodotti durante il laboratorio ad esso legato. Vengono riassunte le attività svolte all’interno del seminario e descritte tematiche trattate. L'articolo riassume ed inquadra il contesto in cui si genera il dibattito e l'interesse per la tematica trattata.