A music box plays from a steel comb: a row of teeth, each one a tiny cantilever, plucked by pins on a turning cylinder. The high notes are short teeth. The low notes are not just long teeth. They can't be, and the fix is a lump of lead. Pluck one and find out.
Drag the length down and the pitch climbs fast, as the square of how much shorter you made it. Now try to reach a low bass note by length alone. In a palm-sized comb you run out of room long before you run out of pitch. That is what the lead weight is for.
Each tooth is clamped at one end and free at the other. Flick the free end and it springs back, overshoots, and rings at one dominant rate set entirely by its shape and its steel. For a beam like this the pitch is the stiffness it fights back with, divided by the mass that has to move, under a square root:
f = (1 / 2π) · √( k / meff )
The stiffness at the tip is k = 3EI / L³, where E is the stiffness of steel, L the length, and I = w·t³/12 the stiffness of the cross-section (width w, thickness t). The effective moving mass is the tooth's own mass counted once at the tip by Rayleigh's rule, (33/140)·mbeam, plus any weight you solder on: meff = mtip + (33/140) mbeam. That is the whole model, and at zero added weight it agrees with the exact beam equation to about two percent.
Read what it says. Length is in the denominator cubed inside the stiffness, so halving the length makes a tooth eight times stiffer against a quarter of the mass: pitch goes as roughly 1/L². That is why the treble teeth are so stubby, and why the pitch rockets up as you shorten one.
To go an octave lower you need a quarter of the frequency. By length alone that means doubling the tooth, and the next octave down doubles it again. A bottom note would want a tooth too long to fit the movement, and too floppy to pluck cleanly. So instead of lengthening the tooth you load its tip: mtip climbs, meff climbs, and the same tooth sings lower without growing. Slide the lead up on a mid-length tooth and watch the note fall through notes a longer tooth could never have reached in the space.
A plucked string's overtones are a tidy harmonic ladder, 2, 3, 4 times the fundamental, which is why it sounds like a clear musical pitch. A clamped beam is different: its higher modes ring at about 6.27 and 17.5 times the fundamental, ratios that are not musical multiples of anything. Those inharmonic partials are baked into the synthesiser above, and they are why a music box has that cool, glassy, slightly-unplaceable shimmer on top of the note. You are hearing the shape of a bending bar, not a vibrating string.
Where the model stops. This is a uniform-thickness, lumped tip-mass approximation (Rayleigh), a teaching stand-in for a real tooth whose root is tapered and whose true mode shapes are the exact Euler-Bernoulli functions; the pitch tracks a real comb's scaling and the lead-weight effect, not any one maker's exact tuning. Steel is taken at a nominal E = 200 GPa, ρ = 7850 kg/m³; tooth width is fixed at 2 mm (it cancels from the pitch when no weight is added). The on-screen bending is slowed about a hundredfold to be visible; the real motion happens at the frequency shown. The decay and pluck are idealised.
Sources. Comb tuning, lead weights and the lead-free successors: Muro Box, "The Soul of a Music Box: The Comb" and The Naked Watchmaker on Reuge combs. Mechanism overview: Wikipedia, "Music box." Cantilever frequency, the 33/140 tip-referred mass and the 1 : 6.27 : 17.5 modal ratios are standard Euler-Bernoulli beam theory. Built by Scout; everything runs locally in your browser, no network.