String Lab
Put a wall at the corner, x = 1, and let waves bounce off it. On the log-time line the surviving waves are ordinary standing waves tied down at the wall — and each zeta zero's wave in the explicit formula is exactly one of them. The shape is a string's. What is not a string's is the list of pitches: a second plain wall would space them evenly, and the zeros are not evenly spaced. Three tabs: one standing wave against one zero wave; the plain-string pitches against the real ones; and the prime drift rebuilt from string modes with the right pitches, then the wrong ones.
Pitch E — drag it, or snap to the nearest zero
In log-time u = ln x the wave of a zero ρ = ½ + iγ is −2 x½ [½ cos(γu) + γ sin(γu)] / (¼ + γ²). Divide out the √x envelope and what is left is a sinusoid in u that is almost exactly sin(γu) scaled by 2/γ — a standing wave that vanishes at the wall u = 0, that is, at x = 1. The cos part is the small correction ½/γ, which is why the amber curve does not vanish at the wall by a hair. Set E to anything else and the cyan curve is a perfectly good standing wave on the same string, tied at the same wall — the string does not care. The zeros are the particular pitches the string picks out, and nothing in this picture says why those.
If the far end were a plain wall at u = L
A string tied at both ends has pitches that are whole-number multiples of a fundamental: evenly spaced rungs. Slide the far wall to put the first pitch on 14.13 and the next rungs land on 28.3, 42.4, 56.5 — while the zeros sit at 21.0, 25.0, 30.4, 32.9, 37.6. No position of a plain second wall can reproduce them, and no fixed set of walls can either, because the zeros' spacing shrinks slowly forever (the T log T count) while any finite string's spacing is constant. So whatever closes the far end is not a wall. It has to act like a boundary whose "position" keeps moving as the pitch rises — which is exactly what a reflection rule that knows about the primes would do.
Sum of standing waves, with the zeros' pitches and with a plain string's
Same string, same wall, same amplitudes 2/|ρ| for each mode. With the zeros as pitches, the sum of standing waves is the prime drift, sawtooth and all. Replace the pitches with a plain string's evenly spaced ones and the sum is still a sawtooth — but its teeth fall at x = e2L, e4L, e6L …, a geometric sequence of fake primes, evenly spaced in log-time. Evenly spaced pitches manufacture evenly spaced primes; the real primes' irregular spacing is exactly the zeros' irregular spacing, read through the fold. That's the whole situation in one picture: the instrument is a string in log-time tied at the corner, and everything the primes know is stored in which pitches the far end allows.