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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.

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.

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 it. We first mathematically investigate the installation, finding its patterns and structure, and then we compare them with structure and patterns of the musical composition. In particular, we apply two concepts of mathematical music theory, the Quantum GestART and the gestural similarity conjecture, to the analysis of Qwalala, realized for the Venice Biennale by Pae White, comparing it to its musical rendition in the homonymous piece for harp and ensemble composed by Federico Favali. Some sketches of generalizations follow, with the "Souvenir Theorem" and the "Art Conjecture."

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.

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.

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, turbulence, and supraglottal myoelastic vibrations. The intermingling of these states, and their temporal evolution, can still be interpreted in the Fourier/Gabor plane, and effective extractors can be implemented. The bases for a Quantum Vocal Theory of Sound, with implications in sound analysis and design, are presented., 32 pages, 11 figures

“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.

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

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 connected with the information (onset, duration, loudness, pitch range) characterizing auditory prototypes. The topic is contextualized within a philosophical debate (discreteness and comparison of apparently unrelated objects), it develops through mathematical formalism, and it leads to programming, to spark interdisciplinary thinking and ignite creativity within STEAM., Comment: Accepted for publication, Journal of Mathematics and Music. New references added in this version

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., preprint

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.

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, 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 visual arts., Comment: The final version of this paper has been published by the Journal of Mathematics and Music

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.

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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 performance strengthen the relationship between music and images: in this way, we can connect a theremin or violin performance with a study on perspective, always through mathematical ideas and paradigms. So can math explain musical concepts, and, on the contrary, can music explain mathematical concepts? Can music and math together give rise to visual arts in ever more innovative ways? In this paper, we try to connect different topics such as musical performance on instruments with continuous pitch, and the paradigm of geometry and continuity.

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.

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. Reti di musica e immagini 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.

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.

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