Cette thèse propose une collection des nouveaux outils pour la représentation musicale. Ces modèles ont deux caractéristiques principales. D'un côté, ils sont inspirés par la géométrie et la topologie. De l'autre côté, ils ont une basse dimensionnalité, afin de garantir une visualisation intuitive des caractéristiques musicales qu'ils représentent. On s'est attaqué au problème de l'analyse musicale à partir de trois points de vue. On a représenté le contrepoint en utilisant des séries temporelles multivariées de matrices de permutations partielles. On a visualisé la conduite des voix en utilisant une classe particulière des tresses partielles et singulières. On donne ensuite une interpretation du Tonnetz comme complex simplicial et on utilise l'homologie persistante, afin de classifier des formes obtenues en déformant les sommets du Tonnetz. Ces déformations sont induites soit par des fonctions qui prennent en compte la nature symbolique de la musique, soit l'interaction symbol/signal. Les modèles basés sur la persistence topologique ont été testés sur une collection hétérogène de bases de données. Ces deux approches sont finalement combinées pour donner un troisième point de vue, qui a donné deux applications. Premièrement, on utilise l'alignement multiple des sequences, pour comparer plusieurs structures harmoniques et sémantiques déduites du signal audio, afin de visualiser et quantifier la propagation d’idée musicales entre artistes, genres et différentes époques. Ensuite on développe la théorie nécessaire pour comparer deux systèmes qui varient dans le temps, en représentant leurs caractéristiques géométriques comme des séries temporelles de diagrammes de persistence. / In this work, we suggest a collection of novel models for the representation of music. These models are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. We tackle the problem of music representation following three non-orthogonal directions. First, we propose an interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids, and their main features are visualised. Thereafter, we give a topological interpretation of the Tonnetz (a graph commonly used in computational musicology), whose vertices are deformed by both a harmonic and a consonance-oriented function. The shapes derived from these deformations are classified using the formalism of persistent homology. Thus, this novel representation of music is evaluated on a collection of heterogenous musical datasets. Finally, a combination of the two approaches is proposed. A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time. Then, music is represented as a time series of topological fingerprints, allowing the comparison of pairs of time-varying shapes in both topological and musical terms.

Cette thèse présente des généralisations u groupe néo-riemannien PLR, que agit sur l'ensemble des 24 triades majeures et mineures. Le travail commence par une reconstruction de l'histoire de Tonnetz, un graphe associé aux trois transformations qui génèrent le groupe PLR. La thèse présente deux généralisations du groupe PLR pour les accords de septième. Le premier agit sur le tournage des septièmes de dominantes, mineure, semi-diminuée, majeure et diminuée, le second comprend également la septième mineure majeur, majeure augmenté, l'augmentée et la septième dedominante bémol. Nous avons également classé les transformations les plus parcimonieuses parmi les 4 triades (majeure, mineure, augmentée et diminuée) et avons étudié le groupe généré par celles-ci. Enfin, nous avons introduit une approche générale permettant de définir des opérations parcimonieuses entre les accords de septième et de triade, mais aussi les opérations déjà connues entre triades et celles entre septièmes. / This thesis presents a generalizations of the neo-Riemannian PLR-group, that acts on the set of 24 major and minor triads. The work begins with a reconstruction on the history of the Tonnetz, a graph associated with the three transformations that generate the PLR-group. The thesis presents two generalizations of the PLR-group for seventh chords. The first one acts on the set of dominant, minor, semi-diminished, major and diminished sevenths, the second one also includes minor major, augmented major, augmented, dominant seventh flat five. We considered the most parsimonious operations exchanging two types of sevenths, moving a single note by a semitone or a whole tone. We also classified the most parsimonious transformations among the 4 types of triads (major, minor,augmented and diminished) and studied the group generated by them. Finally, we have introduced a general approach to define parsimonious operations between sevenths and triads, but also the operations already known between triads and those between sevenths.

Représentations symboliques musicales et calcul spatial. La notion d'espace symbolique est fréquemment utilisée en théorie, analyse et composition musicale. La représentation de séquences dans des espaces de hauteurs, comme le Tonnetz, permet de capturer des propriétés mélodiques et harmoniques qui échappent aux systèmes de représentation traditionnels. Nous généralisons cette approche en reformulant d'un point de vue spatial différents problèmes musicaux (reconnaissance de style, transformations mélodiques et harmoniques, classification des séries tous-intervalles, etc.). Les espaces sont formalisés à l'aide de collections topologiques, une notion correspondant à la décoration d'un complexe cellulaire en topologie algébrique. Un complexe cellulaire per- met la représentation discrète d'un espace à travers un ensemble de cellules topologiques liées les unes aux autres par des relations de voisinage spécifiques. Nous représentons des objets musicaux élémentaires (par exemple des hauteurs ou des accords) par des cellules et construisons un complexe en les organisant suivant une relation de voisinage définie par une propriété musicale. Une séquence musicale est représentée dans un complexe par une trajectoire. L'aspect de la trajectoire révèle des informations sur le style de la pièce et les stratégies de composition employées. L'application d'opérations géométriques sur les trajectoires entraîne des transformations sur la pièce musicale initiale. Les espaces et les trajectoires sont construits à l'aide du langage MGS, un langage de programmation expérimental dédié au calcul spatial, qui vise à introduire la notion d'espace dans le calcul. Un outil, HexaChord, a été développé afin de faciliter l'utilisation de ces notions pour un ensemble prédéfinis d'espaces musicaux / Musical symbolic representations and spatial computing. The notion of symbolic space is frequently used in music theory, analysis and composition. Representing sequences in pitch (or chord) spaces, like the Tonnetz, enables to catch some harmonic and melodic properties that elude traditional representation systems. We generalize this approach by rephrasing in spatial terms different musical purposes (style recognition, melodic and harmonic transformations, all-interval series classification, etc.). Spaces are formalized as topological collections, a notion corresponding with the label- ling of a cellular complex in algebraic topology. A cellular complex enables the discrete representation of a space through a set of topological cells linked by specific neighborhood relationships. We represent simple musical objects (for example pitches or chords) by cells and build a complex by organizing them following a particular neighborhood relationship defined by a musical property. A musical sequence is represented in a complex by a trajectory. The look of the trajectory reveals some informations concerning the style of the piece, and musical strategies used by the composer. Spaces and trajectories are computed with MGS, an experimental programming language dedicated to spatial computing, that aims at introducing the notion of space in computation. A tool, HexaChord, has been developped in order to facilitate the use of these notions for a predefined set of musical spaces

Parsimonious voice leading is a term, first used by Richard Cohn, to describe non-diatonic motion among triads that will preserve as many common tones as possible, while limiting the distance traveled by the voice that does move to a tone or, better yet, a semitone. Some scholars have applied these principles to seventh chords, laying the groundwork for this study, which strives toward a reasonably comprehensive, usable model for musical analysis. Rather than emphasizing mathematical proofs, as a number of approaches have done, this study relies on two- and three-dimensional geometric visualizations and spatial analogies to describe pitch-class and harmonic relationships. These geometric realizations are based on the organization of the neo-Riemannian Tonnetz, but they expand and apply the organizational principles of the Tonnetz to seventh sonorities. It allows for the descriptive “mapping” or prescriptive “navigation” of harmonic paths through a defined space. The viability of the theoretical model is examined in analyses of passages from the repertoire of Frédéric Chopin. These passages exhibit a harmonic syntax that is often difficult to analyze as anything other than “tonally unstable” or “transitional.” This study seeks to analyze these passages in terms of what they are, rather than what they are not.

How have we come to hear melody as going “up” or “down”? Why does the Western world predominantly adopt spatial terms such as “high” and “low” to distinguish musical notes while other non-Western cultures use non-spatial terms such as “large” and “small” (Bali), or “clear” and “dull” (South Korea)? Have the changing concepts of motion and space in people’s everyday lives over history also changed our understanding of musical space? My dissertation investigates the Western concept of music space as it has been shaped by social change into the way we think about music today. In our understanding of music, the concept of the underlying space is so elemental that it is impossible for us to have any fruitful discourse about music without using inherently spatial terms. For example a term interval in music denotes the distance between two combined notes; but, in fact, two sonic objects are neither near nor far from each other. This shows that our experience of hearing interval as a combination of different notes is not inherent in the sound itself but constructed through cultural and social means. In Western culture, musical sound is often conceptualized through various metaphors whose source domains reflect the society that incubated these metaphorical understandings. My research investigates the historical formation of the conceptual metaphor of music. In particular, I focus on historical formation of the three underlying assumptions we bring to our hearing of music: (1) “high” and “low” notes and motion between them, (2) functionality of musical chords, and (3) reliance on music notation. In each chapter, I contextualize various music theoretical writings within the larger framework of philosophy and social theory to show that our current understanding of musical sound is embedded with the history of Western culture.

The study of song cycles poses difficulties for both analysts and performers. These challenges stem largely from two qualities intrinsic to the genre: (1) the inclusion of two semiotic systems, language and music, and (2) the use of multi-movement structures. Several scholars have addressed these issues; however, a model built on a balanced consideration of both text-based/dramatic events and purely musical elements, has yet to be offered. This study proposes such a model with separate applications for both performers and analysts. Focusing on the identification of features connecting song cycles in their entirety, deep voice-leading associations and movements in key paths are examined in the application for analysts, whereas the performers’ application concentrates on recognizing underlying “atmospheres” and forms of acceleration. Each application is applied to Schumann’s Frauenliebe und –leben and Dichterliebe song cycles, demonstrating the benefits of employing this model in the development of both performative and analytical interpretations.

The study of song cycles poses difficulties for both analysts and performers. These challenges stem largely from two qualities intrinsic to the genre: (1) the inclusion of two semiotic systems, language and music, and (2) the use of multi-movement structures. Several scholars have addressed these issues; however, a model built on a balanced consideration of both text-based/dramatic events and purely musical elements, has yet to be offered. This study proposes such a model with separate applications for both performers and analysts. Focusing on the identification of features connecting song cycles in their entirety, deep voice-leading associations and movements in key paths are examined in the application for analysts, whereas the performers’ application concentrates on recognizing underlying “atmospheres” and forms of acceleration. Each application is applied to Schumann’s Frauenliebe und –leben and Dichterliebe song cycles, demonstrating the benefits of employing this model in the development of both performative and analytical interpretations.

One goal of music theory is to describe the resources of a pitch system. Traditionally, the study of pitch intervals was done using frequency ratios of the powers of small integers. Modern mathematical music theory offers an independent way of understanding the pitch system by considering intervals as transformations. This thesis takes advantage of the historical emergence of algebraic structures in musicology and, in the spirit of transformational theory, treats operations that form mathematical groups. Aspects of Neo-Riemannian theory are explored and developed, in particular the T/I and PLR groups as dual. Pitch class spaces, such as 12, can also be defined as torsors. In addition to surveying the group theoretical tools for music analysis, this thesis provides detailed proofs of many claims that are proposed but seldom supported.

The study of song cycles poses difficulties for both analysts and performers. These challenges stem largely from two qualities intrinsic to the genre: (1) the inclusion of two semiotic systems, language and music, and (2) the use of multi-movement structures. Several scholars have addressed these issues; however, a model built on a balanced consideration of both text-based/dramatic events and purely musical elements, has yet to be offered. This study proposes such a model with separate applications for both performers and analysts. Focusing on the identification of features connecting song cycles in their entirety, deep voice-leading associations and movements in key paths are examined in the application for analysts, whereas the performers’ application concentrates on recognizing underlying “atmospheres” and forms of acceleration. Each application is applied to Schumann’s Frauenliebe und –leben and Dichterliebe song cycles, demonstrating the benefits of employing this model in the development of both performative and analytical interpretations.

The study of song cycles poses difficulties for both analysts and performers. These challenges stem largely from two qualities intrinsic to the genre: (1) the inclusion of two semiotic systems, language and music, and (2) the use of multi-movement structures. Several scholars have addressed these issues; however, a model built on a balanced consideration of both text-based/dramatic events and purely musical elements, has yet to be offered. This study proposes such a model with separate applications for both performers and analysts. Focusing on the identification of features connecting song cycles in their entirety, deep voice-leading associations and movements in key paths are examined in the application for analysts, whereas the performers’ application concentrates on recognizing underlying “atmospheres” and forms of acceleration. Each application is applied to Schumann’s Frauenliebe und –leben and Dichterliebe song cycles, demonstrating the benefits of employing this model in the development of both performative and analytical interpretations.