From music to mathematics and backwards: introducing algebra, topology and category theory into computational musicology

International audienceDespite a long historical relationship between mathematics and music, the interest of mathematicians is a recent phenomenon. In contrast to statistical methods and signal-based approaches currently employed in MIR (Music Information Research), the research project described in this paper stresses the necessity of introducing a structural multidisciplinary approach into computational musicology making use of advanced mathematics. It is based on the interplay between three main mathematical disciplines: algebra, topology and category theory. It therefore opens promising perspectives on important prevailing challenges, such as the automatic classification of musical styles or the solution of open mathematical conjectures, asking for new collaborations between mathematicians, computer scientists, musicologists, and composers. Music can in fact occupy a strategic place in the development of mathematics since music-theoretical constructions can be used to solve open mathematical problems. The SMIR project also differs from traditional applications of mathematics to music in aiming to build bridges between different musical genres, ranging from contemporary art music to popular music, including rock, pop, jazz and chanson. Beyond its academic ambition, the project carries an important societal dimension stressing the cultural component of 'mathemusical' research, that naturally resonates with the underlying philosophy of the “Imagine Maths”conference series. The article describes for a general public some of the most promising interdisciplinary research lines of this project

Diagrammatic approaches in Computational Musicology: Some theoretical and Philosophical Aspects

10th International Conference, Diagrams 2018, Edinburgh, UK, June 18-22, 2018, ProceedingsInternational audienceDespite a long historical relationship between mathematics and music, the use of diagrammatic approaches in computational musicology is a relatively recent phenomenon. Within the different branches of formal methods in music analysis, the so-called "transformational" paradigm has progressively shifted from an object-oriented to a graph-theoretical and categorical approach. Both graph theory and category theory make large use of diagrams which enable the description of the inner relationships of musical structures. In the categorical framework recently proposed by the authors, whose results are summarized and discussed in this abstract, musical transformations are viewed as natural transformations between chords represented as labelled graphs with vertices corresponding to the notes and arrows corresponding to musical transpositions and inversions operations. The diagrammatic approach also provides a very powerful conceptual tool that can have crucial theoretical implications for music cognition. We discuss this aspect by showing some deep connections between transforma-tional music analysis and some mathematically-oriented directions in developmental psychology and cognition (such as Halford and Wilson's neostructural-istic approach, Ehresmann and Vanbremeersch's Memory Evolutive Systems, Phillips and Wilson's Categorical Compositionality, Fauconnier and Turner's Conceptual Blending and its structural extension proposed by Goguen) and epis-temology (Gaston-Granger's "objectal" and "operational" duality)

From Dürer's Magic Square to Klumpenhouwer Tesseracts: On Melencolia (2013) by Philippe Manoury

Many Western art music composers have taken advantage of tabulated data for nourishing their creative practices, particularly since the early twentieth century. The arrival of atonality and serial techniques was crucial to this shift. Among the authors dealing with these kinds of tables, some have considered the singular mathematical properties of magic squares. This paper focuses on a particular case study in this sense: Philippe Manoury's Third String Quartet, entitled Melencolia. We mainly analyse mainly several strategies conceived by the French composer – through his own sketches – in order to manipulate pitches and pitch-classes over time. For that purpose, we take advantage of Klumpenhouwer networks as a way to settle wide and dense isographic relationships. Our hyper-K-nets sometimes reach a total of 32 arrows that allow geometrical arrangements as tesseracts in which their different dimensions cluster related families of isographies. In doing so, we aim to provide an instructive example of how to contextualise K-nets and isographies as powerful tools for the analysis of compositional practices

Meter networks: a categorical framework for metrical analysis

This paper develops a framework based on category theory which unifies the simultaneous consideration of timepoints, metrical relations, and meter inclusion founded on the category Rel of sets and binary relations. Metrical relations are defined as binary relations on the set of timepoints, and the subsequent use of the monoid they generate and of the corresponding functor to Rel allows us to define meter networks, i.e. networks of timepoints (or sets of timepoints) related by metrical relations. We compare this to existing theories of metrical conflict, such as those of Harald Krebs and Richard Cohn, and illustrate that these tools help to more effectively combine displacement and grouping dissonance and reflect analytical claims concerning nineteenth-century examples of complex hemiola and twentieth-century polymeter. We show that meter networks can be transformed into each other through meter network morphisms, which allows us to describe both meter displacements and meter inclusions. These networks are applied to various examples from the nineteenth and twentieth century.Accepted manuscrip

On Musical Self-Similarity : Intersemiosis as Synecdoche and Analogy

Self-similarity, a concept borrowed from mathematics, is gradually becoming a keyword in musicology. Although a polysemic term, self-similarity often refers to the multi-scalar feature repetition in a set of relationships, and it is commonly valued as an indication for musical ‘coherence’ and ‘consistency’. In this study, Gabriel Pareyon presents a theory of musical meaning formation in the context of intersemiosis, that is, the translation of meaning from one cognitive domain to another cognitive domain (e.g. from mathematics to music, or to speech or graphic forms). From this perspective, the degree of coherence of a musical system relies on a synecdochic intersemiosis: a system of related signs within other comparable and correlated systems. The author analyzes the modalities of such correlations, exploring their general and particular traits, and their operational bounds. Accordingly, the notion of analogy is used as a rich concept through its two definitions quoted by the Classical literature—proportion and paradigm, enormously valuable in establishing measurement, likeness and affinity criteria. At the same time, original arguments by Benoît B. Mandelbrot (1924–2010) are revised, alongside a systematic critique of the literature on the subject. In fact, connecting Charles S. Peirce’s ‘synechism’ with Mandelbrot’s ‘fractality’ is one of the main developments of the present study

Motivic Metamorphosis: Modelling Intervallic Transformations in Schoenberg’s Early Works

Composers can manipulate a basic musical idea in theoretically infinite ways. This concept of manipulating musical material was a central compositional philosophy of Arnold Schoenberg (1874 – 1951). As Schoenberg states, “whatever happens in a piece of music is nothing but the endless reshaping of a basic shape” (Schoenberg, [1935] 1975). It is the variety of ways in which these basic ideas, commonly termed motives, are manipulated that contributes to a work’s unique identity. According to Schoenberg, these varied basic shapes work dialogically to unify a musical piece. But how are these basic shapes varied? Utilizing ordered intervals of pitch and duration, we may examine the structural properties of motivic segments which develop throughout a work. Exploring an analytical model tracking developmental transformations of melodic musical motives (shapes), this dissertation defines a robust group of intervallic transformations, equipping the analyst with a toolkit of transformational mechanisms. Applications of set-theory and other mathematically-based methodologies to Schoenberg’s post-1908 works often account for structural and motivic process. However, this is not the case for Schoenberg’s early works (1898 – 1908), where scholars typically examine form and harmony. But, as Carl Dahlhaus posits, Schoenberg thought motivically, and only detailed analyses of intervals demonstrate how motives relate to one another (Dahlhaus, 1987). Tracking such processes in Schoenberg’s early works, we come closer to understanding how new forms are created and their interrelations¬––how developed musical ideas emerge and are woven together to create coherence. Defining a suite of transformational devices, this dissertation examines the treatment of varied motivic forms within two instrumental early works by Schoenberg, Pelleas und Melisande op. 5 (1903) and String Quartet no. 2, op. 10 (1908). The analyses reveal developmental paths via networks which connect musical statements and quantify how one object moves into the next. The results demonstrate specific transformational moves which account for the manipulation of a motivic object, thereby creating subsequent forms. Such investigations permit larger connections and qualified observations to be made within the work of Schoenberg and all composers manipulating motivic forms. The resultant work engages Schoenberg’s technique of musical development and investigates his motivic metamorphoses

Hearing the Tonality in Microtonality

In the late 1970s and 1980s, composer-pianist Easley Blackwood wrote a series of microtonal compositions exploring the tonal and modal behavior of a dozen non–twelve-tone equal temperaments, ranging from 13 to 24 tones per octave. This dissertation investigates a central paradox of Blackwood’s microtonal music: that despite being full of intervals most Western listeners have never heard before, it still seems to “make sense” in nontrivial ways. Much of this has to do with the music’s idiosyncratic approach to tonality, which I define as a regime of culturally conditioned expectations that guides one’s attentional processing of music’s gravitational qualities over time. More specifically, Blackwood configures each tuning’s unfamiliar elements in ways that correspond to certain schematic expectations Western listeners tend to have about how tonal music “works.” This is why it is still possible to hear the forest of tonality in this music, so to speak, despite the odd-sounding trees that comprise it. Because of its paradoxical blend of expectational conformance and expectational noncompliance, Blackwood’s microtonal music makes for a useful tool to snap most Western-enculturated listeners out of their ingrained modes of musical processing and reveal certain things about tonality that are often taken for granted. Accordingly, just as Blackwood writes conventional-sounding music in unconventional tunings, this dissertation rethinks several familiar music-theoretic terms and concepts through the defamiliarizing lens of microtonality. I use Blackwood’s microtonal music as a prism to shine a light on traditional theories of tonality, scale degrees, consonance and dissonance, and harmonic function, arguing that many of these theories rely on assumptions that are tacitly tied to twelve-tone equal temperament and common-practice major/minor music. By unhooking these terms and concepts from any one specific tuning or historical period, I build up a set of analytical tools that can allow one to engage more productively with the many modalities of tonality typically heard on a daily basis today. This dissertation proceeds in six chapters. The four interior chapters each center on one of the terms and concepts mentioned above: scale degrees (Chapter 2), consonance and dissonance (Chapter 3), harmonic function (Chapter 4), and tonality (Chapter 5). In Chapter 2, I propose a system for labeling scale degrees that can provide more nuance and flexibility when reckoning with music in any diatonic mode (and in any tuning). In Chapter 3, I advance an account of consonance and dissonance as expectational phenomena (rather than purely psychoacoustic ones), and I consider the ways that non-pitched elements such as meter and notation can act as “consonating” and/or “dissonating” forces. In Chapter 4, I characterize harmonic function as arising from the interaction of generic scalar position and metrical position, and I devise a system for labeling harmonic functions that is better attuned to affective differences across the diatonic modes. In Chapter 5, I synthesize these building blocks into a conception of fuzzy heptatonic diatonic tonality that links together not only all of Blackwood’s microtonal compositions but also more familiar musics that use a twelve-tone octave, from Euroclassical to popular styles. The outer chapters are less explicitly music-analytical in focus. Chapter 1 introduces readers to Blackwood’s compositional approach and notational system, considers the question of his intended audience, and discusses the ways that enculturation mediates the cognition of microtonality (and of unfamiliar music more generally). Chapter 6 draws upon archival documents to paint a more detailed picture of who Blackwood was as a person and how his idiosyncratic worldview colors his approach to composition, scholarship, and interpersonal interaction. While my nominal focus in these six chapters is Blackwood’s microtonal music, the repertorial purview of my project is far broader. One of my guiding claims throughout is that attending more closely to the paradoxes and contradictions of Blackwood’s microtonality can help one better understand the musics they are accustomed to hearing. For this reason, I frequently compare moments in Blackwood’s microtonal music to ones in more familiar styles to highlight unexpected analogies and point up common concerns. Sharing space with Blackwood in the pages that follow are Anita Baker, Ornette Coleman, Claude Debussy, and Richard Rodgers, among others—not to mention music from Curb Your Enthusiasm, Fortnite, Sesame Street, and Star Wars. Ultimately, this project is a testament to the value of stepping outside of one’s musical comfort zone. For not only can this reveal certain things about that comfort zone that would not be apparent otherwise, but it can also help one think with greater nuance, precision, and (self-)awareness when “stepping back in” to reflect upon the music they know and love

The Varieties of Tone Presence: On the Meanings of Musical Tone in Twentieth-Century Music

This dissertation is about tone presence, or how musical tone shows up for experience in twentieth-century music. In exploring the subject of tone presence, I rethink notions of “pitch structure” in post-tonal theory and offer an alternative that focuses on the question of what it is to be a musical interval for experience, drawing on a wide range of research from social theory, semiotics, theories of emotion, African American studies, literary theory, usage-based linguistics, post-colonial theory, and phenomenology. I begin by offering a critique of three basic assumptions that constrain understandings of what we mean by pitch structure in post-tonal theory: that pitch structure concerns “intrinsic” properties of collections, that pitch is an autonomous parameter, and that pitch structure is best analyzed at the “neutral level.” Following this critique, I offer an alternative account of musical intervals that suggests that intervals cannot be reduced to a discrete quantity measured in semitones. I argue instead, that what it is to be an interval are all those conditions (in terms of culture, expression, musical form, motivic behavior, etc.) under which an interval becomes intelligible as such for experience. Such conditions include our concerned involvement with holistic situations and, following ideas rooted in Bakhtinian dialogism, our responsive understanding of “alien” understandings of the “same” interval. The understanding of what I describe as the modes of being of musical intervals is illustrated in an extended analytical case study of what it is to be an “authentic” atonal tritone in tonal and modal environments. Building on this account of the modes of being of musical intervals, I reexamine semiotic approaches to musical meaning, exemplified by topic theory, that treat musical meaning as a represented entity (i.e., a sign) that associates the “music itself” with “extramusical” meaning. Specifically, I offer an account that treats musical meaning as a social process (rather than an entity) in which cultural forms of meaning act as the ground that helps make musical tones—the “figure” in this gestalt metaphor—intelligible as such. The last chapter features an extended analysis of tone presence in Messiaen’s “Demeurer dans l’Amour” from Éclairs sur l’au-delà

The Computational Attitude in Music Theory

Music studies’s turn to computation during the twentieth century has engendered particular habits of thought about music, habits that remain in operation long after the music scholar has stepped away from the computer. The computational attitude is a way of thinking about music that is learned at the computer but can be applied away from it. It may be manifest in actual computer use, or in invocations of computationalism, a theory of mind whose influence on twentieth-century music theory is palpable. It may also be manifest in more informal discussions about music, which make liberal use of computational metaphors. In Chapter 1, I describe this attitude, the stakes for considering the computer as one of its instruments, and the kinds of historical sources and methodologies we might draw on to chart its ascendance. The remainder of this dissertation considers distinct and varied cases from the mid-twentieth century in which computers or computationalist musical ideas were used to pursue new musical objects, to quantify and classify musical scores as data, and to instantiate a generally music-structuralist mode of analysis. I present an account of the decades-long effort to prepare an exhaustive and accurate catalog of the all-interval twelve-tone series (Chapter 2). This problem was first posed in the 1920s but was not solved until 1959, when the composer Hanns Jelinek collaborated with the computer engineer Heinz Zemanek to jointly develop and run a computer program. Recognizing the transformation wrought on modern statistics and communications technology by information theory, I revisit Abraham Moles’s book Information Theory and Esthetic Perception (orig. 1958) and use its vocabulary to contextualize contemporary information-theoretic work on music that various evokes the computational mind by John. R. Pierce and Mary Shannon, Wilhelm Fucks, and Henry Quastler (Chapter 3). I conclude with a detailed look into a score-segmentation algorithm of the influential American music theorist Allen Forte (Chapter 4). Forte was a skilled programmer who spent several years at MIT in the 1960s, with cutting-edge computers and the company of first-rank figures in the nascent fields of computer science and artificial intelligence. Each one of the researchers whose work is treated in these case studies—at some stage in their relationship with music—adopted what I call the computational attitude to music, to varying degrees and for diverse ends. Of the many questions this dissertation seeks to answer: what was gained by adopting such an attitude? What was lost? Having understood these past explorations of the computational attitude to music, we are better suited ask of ourselves the same questions today

Musicology and Mediation: an examination of the cultural materialisms of Raymond Williams and Pierre Bourdieu in relation to the fields of contemporary music and musicology, with a case study of Arvo Part and ECM

Merged with duplicate record 10026.1/2795 on 06.20.2017 by CS (TIS)This thesis examines the usefulness of the work of Pierre Bourdieu and Raymond Williams for discussions of material mediation in musicology. In part I, I focus on Bourdieu's discussions of cultural production as set out in The Rules of Art and "The Field of Cultural Production", and reconstruct the terms of Williams's late theoretical project. In establishing the terms of these projects, I draw a parallel between their attempts to materialize the categories of Marx's superstructure - noting in Williams's subsequent use of a revised Marxist production paradigm a proximity to the work of Adorno - before noting the differences imposed by the pressures and limits of their respective intellectual cultures. The tensions between these two models are therefore identified as the opportunity for dialogue between theoretical traditions. In part 2, these reflections are tested through a discussion of Arvo Pärt's music and the record label Edition of Contemporary Music (ECM). Using data from musical scores, CDs, reviews, critical essays, magazine articles, interviews, and so on, Part's emergent field position in the late 1970s and early 1980s is reconstructed and ECM's function as both institution and artistic formation is argued. These instances of musical practice remain rhetorically committed to the ideals of autonomy while spanning the opposition of autonomy and heteronomy. This ambiguity puts strain on Williams's and Bourdieu's readings of cultural production, allowing for a critical approach to this range of debate. In this sense, the method becomes part of the subject matter, and the discussions combine both theoretical and musical reflection