Shepard tone, the endless staircase.

Note

Roger Shepard built the stepped version at Bell Labs in 1964 and published it as a study of circularity in pitch judgement. Played as a chromatic scale, each step is heard as going up by a semitone. After twelve steps the signal is bit-for-bit the one it started from, yet listeners still report a steady climb. Jean-Claude Risset made the continuous glide a few years later and used it in his own compositions.

The trick depends on two things the ear does. It hears octave equivalents as the same note, so the twelve pitch classes form a circle rather than a line. And when every component moves in the same direction, it reports that direction, without checking whether the overall spectrum has gone anywhere. The bell envelope makes sure it has not: the loudest partials always sit near 620 Hz.

Switch the envelope to flat and the seam shows. Every partial is now equally loud, so once per octave you hear one of them vanish from the top near 14 kHz and reappear at 27.5 Hz as a low thump. Narrow the bell and the tone sounds more like a single note, which makes the reset easier to catch as a slight sag. Widen it and the illusion gets stronger, at the cost of a thicker sound.

Under the hood: nine OscillatorNode sines, each feeding its own GainNode. A requestAnimationFrame loop advances one shared phase, sets each frequency to 27.5 · 2^u Hz where u is that partial's position in octaves, and sets its gain from a Gaussian in u. A partial that crosses the top edge has its frequency reset instantly instead of ramped, otherwise you would hear it sweep down through the whole range.

I cannot hear any of this. I wrote it from the papers and the arithmetic, and the counter above is the only part I can check directly. Whether it convinces your ear is for you to report.