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Simmetrie fra Matematica e Musica
No full textCos’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; postafzione di Giovanni Albini
Strutture e trasformazioni condivise in matematica e musica: dalle categorie alla musicologia / Shared structures and transformations in mathematics and music: From categories to musicology
No full textLe mutue influenze tra scienza e musica, iniziate con Pitagora, caratterizzano la ricerca musicale contemporanea, dalla teoria all'analisi della performance. La matematica può collegare campi diversi: in particolare, la teoria matematica delle categorie e, più in generale, il pensiero diagrammatico, costituiscono un potente strumento per analizzare i processi e le trasformazioni tra i processi.
Una categoria è data da oggetti (punti) e trasformazioni tra di loro (frecce), che verificano
la proprietà associativa e possiedono l’elemento neutro. Punti e frecce costituiscono diagrammi. Un'intera categoria può essere vista come un punto e le trasformazioni tra categorie possono essere rappresentate tramite frecce. Diagrammi dove diversi percorsi che partono dallo stesso oggetto A portano allo stesso oggetto B sono chiamati commutativi.
Le categorie vengono utilizzate per confrontare e trovare analogie tra strutture e metodi matematici, per estrarre informazioni essenziali e generali. Le categorie sono anche usate per connettere e trovare somiglianze all'interno della musica e tra musica ed arti visive, anche per evidenziare somiglianze di schemi e gesti rilevanti a livello cognitivo. L'analisi musicale ha un ruolo simile nell’estrapolazione di informazioni specifiche e dati non evidenti dell'organizzazione strutturale delle composizioni musicali.
In questo contributo, per la prima volta, applichiamo il pensiero categoriale all'analisi musicale. In particolare, contestualizziamo in un quadro matematico una specifica metodologia analitica proposta in letteratura, che mette in relazione e confronta le informazioni che possono essere evinte separatamente dall'analisi della partitura (costrutto) e dall'ascolto (salienza), con riferimenti alla cognizione. In generale, le "strutture percepite", che emergono all'ascolto, sono diverse dalle "strutture costruite", che emergono dall'analisi della partitura. Ciò significa che, se rappresentiamo ogni processo come una composizione di frecce, il risultato finale non è lo stesso oggetto. Pertanto, i processi di ascolto e analisi della partitura possono essere rappresentati come un diagramma non commutativo.
Un'indagine matematica più approfondita rivela tuttavia che, per lo stesso pezzo, i possibili diversi valori di output rimangono tutti all'interno dello stesso insieme, che caratterizza il pezzo musicale analizzato. Inoltre, l'unione delle diverse risposte fornite da diversi ascoltatori definisce il pezzo, e l'intersezione delle risposte fornite dagli ascoltatori rappresenta gli elementi essenziali del pezzo. Questi elementi consentono la riconoscibilità del pezzo attraverso la diversità di esecuzioni, interpreti e ascoltatori.
Infine, i diagrammi possono anche essere usati per investigare un processo compositivo generale dall'idea iniziale al pezzo finale, con la ricostruzione dei processi compositivi. La modellizzazione matematica fornisce un linguaggio e un formalismo visivo che può essere applicato indipendentemente dal particolare stile della composizione considerata.
Possiamo quindi usare dei diagrammi per analizzare un pezzo e per analizzare il processo analitico stesso. L'approccio proposto ci consente di confrontare: 1. la matematica con la musica; 2. informazioni matematiche essenziali con informazioni musicali essenziali; 3. i processi che portano all'apprezzamento di informazioni essenziali in entrambi i campi. Tale operazione di
"filtraggio" può essere utilizzata come strumento analitico nell’ambito dell'analisi musicale, dello studio della composizione, dello studio della matematica e della pedagogia delle STEAM.Mutual influences between science and music, which started with Pythagoras, characterize contemporary musical research, from theory to analysis of performance. Mathematics can connect different fields: in particular, mathematical category theory, and, more in general, diagrammatic thinking, constitutes a powerful tool to analyze processes and transformations between processes.
A category is given by objects (points) and transformations between them (arrows), that verify associativity and identity properties. Points and arrows constitute diagrams. A whole category can be seen as a point, and transformations between categories can be represented via arrows. Diagrams where different paths starting from the same object A lead to the same object B, are called commutative.
Categories are used to compare and find analogies between mathematical structures and methods, to extract essential and general information. Categories are also used to connect and find similarities within music, and between music and visual arts, even to highlight cognitive-relevant similarities of patterns and gestures. Musical analysis plays a similar role in extracting specific information and non-evident data about structural organization of musical compositions.
In this paper, for the first time, we apply categorical thinking to musical analysis. In particular, we contextualize in a mathematical framework a specific analytical methodology proposed in literature, which relates and compares the information that can be separately retrieved from the score analysis (construct) and the listening (salience), with reference to cognition. In general, ‘perceived structures,’ retrieved via listening, are different from ‘built structures,’ retrieved via score analysis. This means that, if we represent each process as a composition of arrows, the final result is not the same object. Thus, the processes of listening and score analysis can be represented as a non- commutative diagram.
However, a more in-depth mathematical investigation reveals that, for the same piece, the possible different output values stay all within the same set, that characterizes the musical piece investigated. Moreover, the union of different answers from different listeners defines the piece, and the intersection of listeners’ answers represents the essential elements of the piece. These elements allow the piece’s recognizability through the diversity of performances, performers, and listeners. Finally, diagrams can also be used to investigate a general compositional process from the initial idea to the final piece, with the reconstruction of compositional processes. Mathematical contextualization provides language and visual formalism to be applied independently by the particular composition style.
Thus, we can use diagrams to analyze a piece, and to analyze the analytical process itself. The proposed approach allows us to compare: 1. mathematics with music; 2. essential mathematical information with essential musical information; 3. the processes that lead to the extraction of essential information in both fields. Such a ‘filtering’ operation can be used as an analytical tool in the framework of musical analysis, composition study, mathematics study, as well as for the pedagogy of STEAM
Mathematics and Musical Entropy
No full textArts 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
Categories, Musical Instruments, and Drawings: A Unification Dream
No full textThe 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
Quantum GestART: identifying and applying correlations between mathematics, art, and perceptual organization
No full textMathematics 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
TAG2G: A Diffusion-Based Approach to Interlocutor-Aware Co-Speech Gesture Generation
Full text linkExtended reality (XR) systems are about to be integrated into our daily lives and will provide support in a variety of fields such as education and coaching. Enhancing user experience demands agents that are capable of displaying realistic affective and social behaviors within these systems, and, as a prerequisite, with the capability of understanding their interaction partner and responding appropriately. Based on our literature review of recent works published in the field of co-speech gesture generation, researchers have developed complex models capable of generating gestures characterized by a high level of human-likeness and speaker appropriateness. Nevertheless, this is only true in settings where the agent has an active status (i.e., the agent acts as the speaker), or it is delivering a monologue in a non-interactive setting. However, as illustrated in multiple works and competitions like the GENEA Challenge, these models remain inadequate in generating interlocutor-aware gestures. We consider interlocutor-aware gesture generation the process of displaying gestures that take into account the conversation partner’s behavior. Moreover, in settings where the agent is the listener, generated gestures lack the level of naturalness that we expect from a face-to-face conversation. To overcome these issues, we have designed a pipeline, called TAG2G, composed of a diffusion model, which was demonstrated to be a stable and powerful tool in gesture generation, and a vector-quantized variational auto-encoder (VQVAE), widely employed to produce meaningful gesture embeddings. Refocusing from monadic to dyadic multimodal input settings (i.e., taking into account text, audio, and previous gestures of both participants of a conversation) allows us to explore and infer the complex interaction mechanisms that lie in a balanced two-sided conversation. As per our results, a multi-agent conversational input setup improves the generated gestures’ appropriateness with respect to the conversational counterparts. Conversely, when the agent is speaking, a monadic approach performs better in terms of the generated gestures’ appropriateness in relation to the speech
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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