Candidate:
Diogo Miguel Filipe Cocharro
Date, time and location:
22nd of July 2025, 15:00, Sala de Atos of the Faculty of Engineering of University of Porto.
President of the Jury:
António Fernando Vasconcelos Cunha Castro Coelho (PhD), Associate Professor in the Department of Informatics Engineering at the Faculty of Engineering of the University of Porto.
Members:
Matt Chiu (PhD), Assistant Professor of Music Theory at the Conservatory of Performing Arts at the Baldwin Wallace University, EUA;
Daniel Gómez-Marín (PhD), Profesor del Departamento de Diseño e Innovación de la Escuela de Tecnología, Diseño e Innovación de la Facultad Barberi de Ingeniería, Diseño y Ciencias Aplicadas de la Universidad Icesi, Colombia;
Sofia Carmen Faria Maia Cavaco (PhD), Assistant Professor in the Department of Computer Science at the Faculty of Science and Technology of Universidade Nova de Lisboa;
Sérgio Reis Cunha (PhD), Assistant Professor in the Department of Electrical and Computer Engineering at the Faculty of Engineering of the University of Porto;
Gilberto Bernardes de Almeida (PhD), Assistant Professor in the Department of Informatics Engineering at the Faculty of Engineering of the University of Porto (Supervisor).
The thesis was co-supervised by Rui Luis Nogueira Penha (PhD), Coordinating Professor of ESMAE – School of Music and Performing Arts.
Abstract:
Music is inherently a temporal manifestation, and rhythm is a crucial component. While rhythm can exist without melody or harmony, the latter cannot exist without rhythm. However, rhythm is often understudied compared to harmony. Rhythmic affinity is a musical concept that describes the natural and pleasing relationship between two or more rhythmic patterns. This happens when these patterns, no matter how complex or seemingly unrelated, come together to create a sense of cohesion and flow rather than dissonance or conflict.
This affinity can arise from various factors, such as shared rhythmic motives, complementary and interlocking rhythmic structures, or a strong underlying pulse that unifies the different layers. For example, two complementary patterns that completely occupy the set of pulses in a cycle by filling each other’s silent pulses with their own active pulses are called interlocking rhythms. These interlocking rhythms are not limited to just the complementary nature of rhythms; we believe they can also be observed in patterns that feature coincident onsets or different underlying pulse grids. This diversity in rhythmic structures represents some of the musical properties we aim to explore in this study.
Music scholars have recently begun to explore affinity-related musical phenomena, particularly building on Harald Krebs’s seminal work on rhythmic dissonance, which offers a comprehensive framework for understanding and categorizing metric dissonance within music. Similarly, Godfried Toussaint’s research examines various methods for measuring rhythmic similarity and for analyzing and generating complementary and interlocking rhythms, providing insights into the structural interrelationships between different rhythmic patterns. Additionally, Clarence Barlow’s work on metrical affinities—often overlooked—contributes important perspectives on the relational characteristics between different meters.
We conducted preliminary experiments to assess the behavior of typical rhythmic similarity metrics across genres. Key findings revealed that similarity varies within a limited range across genres and instruments, which we identify as affinity space. This systematic analysis motivates the discussion and research on the concept of rhythmic affinity, emphasizing the need to understand it as a distinct concept from rhythmic similarity. Furthermore, we identified several limitations that draw this thesis’s main objectives and methodologies, namely the lack of metrics for multi-meter corpus analysis in the context of rhythmic cycles, e.g., loops.
In this context, this study focuses on preprocessing multi-meter representations of rhythmic patterns in the time domain specifically designed for projection in the Discrete Fourier Transform (DFT) space with the goal of exploring rhythmic affinities. We aimed to study the DFT of rhythmic loops towards a mathematical space that reflects metrical levels of alignment (or misalignment), which closely relates to Krebs definition of metric dissonance. This phenomenon relates to practices commonly found in musical composition, such as poly-meter and poly-rhythms, which enable the superimposing of rhythmic patterns that, in principle, show low similarity between each other but that are perceptually pleasing as a combined dissonance, the most known example is the hemiola of a three against two.
Our research follows and extends the body of music theory literature on applying the DFT of pitch classes to distances that reflect human perception and music-theoretical principles. Its application to rhythmic structures is currently limited to particular contexts of a musical piece, not encompassing strategies for multi-meter rhythmic analysis. The main contribution lies in a methodology for multi-meter analysis in the DFT space. Our findings demonstrated that up-sampling the grid of pivotal metrical levels underlying rhythmic pattern representations enables the simultaneous depiction of meters with simple and compound subdivisions. This approach highlights structural relationships within the DFT space, reflected by close distances between related simple and compound metrical templates—for instance, between $4/4$ and $12/8$ or $3/4$ and $6/8$. We implemented this methodology in a prototype system capable of generating rhythmic patterns based on metrical templates and sorting them according to their similarity to a user-defined pattern.