Related papers: Motivic Rhythms
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the dividing Nisnevich topology on log…
The perception of consonance/dissonance of musical harmonies is strongly correlated with their periodicity. This is shown in this article by consistently applying recent results from psychophysics and neuroacoustics, namely that the just…
Following the work of Brown, we can canonically associate a family of motivic periods -- called the motivic Feynman amplitude -- to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory…
Melody reduction, as an abstract representation of musical compositions, serves not only as a tool for music analysis but also as an intermediate representation for structured music generation. Prior computational theories, such as the…
Melodic similarity measurement is of key importance in music information retrieval. In this paper, we use geometric matching techniques to measure the similarity between two melodies. We represent music as sets of points or sets of…
This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on…
We survey over some recent applications of motivic homotopy theory in the definition and the study of $p$-adic cohomology theories. In particular, we revisit the proof of the $p$-adic weight-monodromy conjecture for smooth projective…
This essay recounts my personal journey towards a deeper understanding of the mathematical foundations of algorithmic music composition. I do not spend much time on specific mathematical algorithms used by composers; rather, I focus on…
We define a motive whose realizations afford modular forms (of arbitrary weight) on an indefinite division quaternion algebra. This generalizes work of Iovita--Spiess to odd weights in the spirit of Jordan--Livn\'e. It also generalizes a…
We explore computational tools that allow to compute the class on the Grothendieck ring of varieties of finite cyclic quotients in some interesting examples. As an main application, we determine the motive of low rank representation…
The aim of this work is to develop a theory parallel to that of motivic complexes based on cycles and correspondences with coefficients in quadratic forms. This framework is closer to the point of view of $\mathbb{A}^1$-homotopy than the…
We pose thirty conjectures on arithmetical sequences, most of which are about monotonicity of sequences of the form $(\root n\of{a_n})_{n\ge 1}$ or the form $(\root{n+1}\of{a_{n+1}}/\root n\of{a_n})_{n\ge1}$, where $(a_n)_{n\ge 1}$ is a…
We compute Benois $\mathscr{L}$-invariants of weight $1$ cuspforms and of their adjoint representations and show how this extends Gross' $p$-adic regulator to Artin motives which are not critical in the sense of Deligne. Benois'…
We give a formal definition of the musical concept of voice leading in mathematical terms, based on the idea of partial permutations of certain ordered multisets. Then we associate a partial permutation matrix with a voice leading in a…
Music is a repetition of patterns and rhythms. It can be composed by repeating a certain number of bars in a structured way. In this paper, the objective is to generate a loop of 8 bars that can be used as a building block of music. Even…
We have recently seen great progress in learning interpretable music representations, ranging from basic factors, such as pitch and timbre, to high-level concepts, such as chord and texture. However, most methods rely heavily on music…
We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula…
We use Google's MusicVAE, a Variational Auto-Encoder with a 512-dimensional latent space to represent a few bars of music, and organize the latent dimensions according to their relevance in describing music. We find that, on average, most…
Paraphrasing Luciano Pavarotti on music, let me say: "Learning mathematics by only reading about it is like making love by e-mail."
Structural segmentation of music refers to the task of finding a symbolic representation of the organisation of a song, reducing the musical flow to a partition of non-overlapping segments. Under this definition, the musical structure may…