Hearing the music of the primes: auditory complementarity and the siren song of zeta

Abstract

A counting function for the primes can be rendered as a sound signal whose harmonies, spanning the gamut of musical notes, are the Riemann zeros. But the individual primes cannot be discriminated as singularities in this 'music', because the intervals between them are too short. Conversely, if the prime singularities are detected as a series of clicks, the Riemann zeros correspond to frequencies too low to be heard. The sound generated by the Riemann zeta function itself is very different: a rising siren howl, which can be understood in detail from the Riemann-Siegel formula.

Hearing random matrices and random waves

“…One aspect not considered here is the possibility of hearing the complementary aspects of a function and its Fourier transform. One of us has explored this recently [5] in the context of the counting function for the prime number fluctuations, whose Fourier 'harmonies' are the zeros of the Riemann zeta function. This revealed an acoustic complementarity: it is possible to hear the singularities in the counting function, or the harmonies, but not both together.…”

“…We present six cases. In addition to the randommatrix and extreme functions, we include a stretch of the Riemann zeta function Z(t) on the critical line [5,12], the fluctuations of whose zeros are known to be the same as those of eigenvalues in the unitary ensemble (at least for not too distant zeros [13,14]). Thus figure 2 shows Q 2 (θ ) for the sequence uniform → CSE → CUE ≈ RiemannZeta → COE → Poisson.…”

“…The way we prefer here is to represent the functions as phase modulations of a carrier wave with a frequency in the range of comfortable human hearing. We think that this sounds clearer than representing them as sounds directly (and after considerable speeding up), as in a recent approach to rendering sounds related to arithmetic [5]. Undoubtedly, many other ways could be explored, and one of our aims in writing this paper is to encourage this.…”

“…Hearing mathematics is not a new idea. A variety of functions has been rendered as sound, including fractals [1,2], the 'music of the primes' and the Riemann zeta function [3][4][5]. The command Play in Mathematica [6] is a particularly flexible tool for creating a variety of musical and mathematical sounds.…”

Sonification of the Riemann Zeta Function

“…Such sonifications tend to be based on sinusoidal resynthesis following the gradual journey along the critical line where all known zeroes have been found, incorporating the contribution of each zero as it arises. In perhaps the most developed precedent, the distinguished physicist Michael V Berry explores a number of sonifications [4], including a sum of sinusoids corresponding to the Riemann zeroes, and direct synthesis of the zeta function along the critical line based on the Riemann-Siegel formula. We differ from this prior work in considering direct synthesis based on the naive approach of summing the zeta function, on exploring rhythm and scales, and in a greater willingness to accept any 'noisy' outputs as acceptable within the wider space of sound available in computer music.…”