## Saturday, October 08, 2016

### [wdgqdnby] Riemann zeta sound

In "Prime Number Races" by Andrew Granville and Greg Martin is an interesting formula involving the Riemann zeta function, reexpressed below in Haskell-style notation for mathematics:

\x -> 1 + 2 * (sum \$ do { s <- rootOf zeta ; guard \$ real_part s == 0.5 ; let { gamma = imag_part s} ; guard \$ gamma > 0 ; return \$ sin (gamma * log x) / gamma})

It is the sum of sine functions, like a Fourier transform.  This suggests synthesizing a sound using the same coefficients, the imaginary part of the roots of zeta along the critical line.  The first root is at 14.13, so we scale down every root by that factor yielding the sequence 1, 1.49, 1.77, 2.15,...  Then, take logarithms base 2^(1/12) to express the frequency ratios of the higher harmonics in musical semitones.  Here are the first 100 intervals in semitones:

0 6.87 9.88 13.27 14.64 16.93 18.40 19.39 21.17 21.79 22.87 23.97 24.84 25.27 26.44 26.96 27.58 28.20 29.05 29.38 29.87 30.63 31.00 31.55 31.82 32.52 32.92 33.14 33.67 34.10 34.51 34.79 35.07 35.68 35.81 36.19 36.48 36.85 37.23 37.45 37.63 38.08 38.36 38.56 38.87 39.04 39.46 39.66 39.84 40.08 40.42 40.59 40.90 41.00 41.24 41.58 41.75 41.88 42.14 42.33 42.60 42.77 42.97 43.05 43.40 43.54 43.70 43.89 44.04 44.26 44.51 44.58 44.73 44.93 45.17 45.26 45.46 45.60 45.70 45.98 46.09 46.23 46.33 46.54 46.68 46.86 46.99 47.09 47.22 47.45 47.58 47.63 47.83 47.91 48.10 48.24 48.39 48.44 48.57 48.78

Two potential issues:

The amplitude of each harmonic properly should be scaled by the reciprocal of gamma, as in the formula above.  However, the human ear is has variable sensitivity across the frequency spectrum, so it might be better to allow the listener to individually control the amplitude of each harmonic to explore different aspects of the music of the primes.  Let the listener also control the fundamental frequency and apply a band-pass filter, in order to hear different segments of harmonics.

The harmonics get more and more densely packed the higher one goes, in contrast to regular Fourier transforms, in which the harmonics are equally spaced in frequency.  For example, there are 5888 harmonics at 147 semitones above the fundamental (rounded to the nearest semitone), corresponding to the 88075th through 93962nd zeros.  (We used this table of Riemann zeta zeros.)

One idea is not to worry and synthesize the sound anyway.  Harmonics get somewhat densely packed even for regular Fourier transforms when viewed in the logarithmic semitone scale: there are 281 harmonics at 147 semitones above the fundamental, corresponding to the 4733rd through 5013th harmonics.

Another idea is to rescale the coefficients (roots) so that they are roughly equally spaced in frequency.  The spacing of the zeros along the critical line is related to Gram points and the Riemann-Siegel theta function.

Previous work on the sound of the Riemann zeta function.