[Paper Review] Motivic Rhythms

Alain Connes|arXiv (Cornell University)|Dec 24, 2018

History and Theory of Mathematics4 citations

TL;DR

This paper proposes a novel connection between number theory and music by interpreting mathematical motives—specifically the $H^1$ cohomology of hyperelliptic curves reduced modulo primes—as rhythmic structures. By mapping the arguments of Frobenius eigenvalues to time onsets, it generates periodic, palindromic, and irrational rhythms that closely mirror the compositional techniques of Olivier Messiaen, revealing a deep correspondence between arithmetic geometry and musical rhythm.

ABSTRACT

In this article on mathematics and music, we explain how one can "listen to motives" as rhythmic interpreters. In the simplest instance which is the one we shall consider, the motive is simply the $H^1$ of the reduction modulo a prime $p$ of an hyperelliptic curve (defined over $\mathbb Q$). The corresponding { time onsets} are given by the arguments of the complex eigenvalues of the Frobenius. We find a surprising relation between mathematical properties of the motives and the ideas on rhythms developed by the composer Olivier Messiaen.

Motivation & Objective

• To establish a correspondence between mathematical motives and rhythmic structures in music, particularly through the cohomology of hyperelliptic curves.
• To demonstrate that the time onsets derived from the arguments of Frobenius eigenvalues produce rhythms with properties mirroring those in Olivier Messiaen’s compositions.
• To illustrate how the functional equation of L-functions leads to palindromic rhythmic patterns.
• To show that the tempo of these rhythms accelerates with $\log p$, linking arithmetic parameters to musical tempo.
• To provide concrete visual and auditory examples of these motivic rhythms using explicit curves and prime reductions.

Proposed method

• Use the $H^1$ cohomology of hyperelliptic curves defined over $\mathbb{Q}$, reduced modulo primes $p$, as the source of rhythmic data.
• Extract the complex eigenvalues of the Frobenius endomorphism acting on $H^1(C_p)$, and use their arguments as time onsets in a rhythmic sequence.
• Apply the Weil conjectures to ensure that all eigenvalues lie on the unit circle, guaranteeing periodicity and symmetry in the resulting rhythms.
• Construct palindromic rhythms by exploiting the symmetry $\alpha_{2g+1-j} = -\alpha_j$, which arises from the functional equation of the L-function.
• Synchronize visual and auditory representations by displaying the eigenvalues in color during playback, with tempo increasing as $\frac{2\pi}{\log p}$ for increasing $p$.
• Use explicit examples of genus $g=5$ hyperelliptic curves (e.g., $y^2 = x^{11} - x^{10} + \cdots$) to generate and visualize the rhythms for primes $7 \leq p \leq 67$.

Experimental results

Research questions

• RQ1 Can mathematical motives, specifically $H^1$ of hyperelliptic curves over $\mathbb{F}_p$, be interpreted as rhythmic structures with musically meaningful properties?
• RQ2 To what extent do the rhythmic patterns derived from Frobenius eigenvalues resemble the rhythmic techniques of Olivier Messiaen, such as non-retrogradable rhythms and irrational onsets?
• RQ3 How does the functional equation of the L-function manifest in the rhythmic structure, and what symmetry does it enforce?
• RQ4 What is the relationship between the prime $p$ and the tempo of the resulting rhythm, and how does this affect the perception of the musical structure?
• RQ5 Can the geometric sieve process (e.g., Eratosthenes' sieve) be used to visually and musically represent the emergence of prime numbers through rhythmic descent?

Key findings

• The arguments of the Frobenius eigenvalues on $H^1(C_p)$ produce time onsets that form periodic, palindromic rhythms with $\alpha_{2g+1-j} = -\alpha_j$, satisfying the symmetry of Messiaen’s 'non-retrogradable' rhythms.
• The resulting rhythms are generally irrational in onset timing, as the arguments $\alpha_j$ are typically irrational numbers, matching Messiaen’s use of irrational time signatures.
• The period of the rhythm is $\frac{2\pi}{\log p}$, and the tempo increases with $p$, reflecting the natural scaling of the $p$-adic parameter in the $L$-function framework.
• For each prime $p$, the $2g$ time onsets correspond to the $2g$ complex eigenvalues of the Frobenius, and their distribution is constrained by the Weil conjectures to lie on the circle of radius $p^{-1/2}$.
• The visual choreography of the 'dance of primes'—where multiples of each prime descend—results in the upper rectangle containing only primes less than 3600, with 503 such primes remaining after the process.
• The video and audio synchronization shows that each motive $H^1(C_j/p)$ generates a distinct yet coherent rhythmic interpretation, with eigenvalues highlighted in real-time during playback, illustrating the direct mapping from arithmetic to rhythm.