The Note That Isn't There
Play a chord of upper harmonics with no root, and your ear invents the root anyway.
What it is
Five sine tones at 220, 330, 440, 550 and 660 Hz. Nothing at 110. You’ll hear 110, a bass note that isn’t being played by anything.
Turn the root on and off: the pitch doesn’t change, only the tone colour. This is why a phone speaker that can’t physically move enough air to make a low note still sounds like it has bass.
How it works
Those five frequencies are the 2nd to 6th harmonics of 110 Hz. The auditory system doesn’t measure pitch by finding the lowest frequency present, it looks for the pattern the partials fit, and the only fundamental that explains 220-330-440-550-660 is 110. So it reports 110, and turning the actual 110 on adds warmth without changing the note.
That raises the question of how it decides, and there are two candidate answers that usually agree.
One: take the greatest common divisor of the partials. Two: take the spacing between neighbouring partials. For a harmonic stack both give 110 and there’s nothing to choose between them.
What surprised me
They come apart violently, and the test that shows it is two lines.
Shift every partial up by the same 25 Hz: 245, 355, 465, 575, 685. The spacing is untouched, still 110. BUT the greatest common divisor collapses to 5, because those numbers share almost nothing. If the ear used the common divisor it would now hear a note four and a half octaves lower.
It doesn’t. It hears something near 110, a little sharp. Press the shift button and listen. The pitch barely moves while the timbre goes strange and bell-like.
So the spacing model wins, and the shifted residue is exactly the experiment that settles it. The tests assert both halves, that the spacing survives a shift and that the common divisor doesn’t, which turns a psychoacoustic claim into arithmetic I can check.
What I would do next
Stretch the partials instead of shifting them, which is what a real piano string does, and see how far the illusion holds.