The Note That Never Lands

I The staircase you can hear

Some tricks of the eye have an exact twin in the ear. The Penrose stairs that climb back to where they began; the barber's pole that rises endlessly while staying still — both have a sound.

Press play. The tone can seem to climb, step after step, octave after octave. The partials fade towards the edges of a fixed bell and recycle, keeping the comb in the same frequency band. This is a real signal, built right now in your browser from a stack of pure tones. The next panel shows the frequencies and gains being scheduled.

Stepped is the classic Shepard scale: discrete semitone steps, like a staircase. Gliding is Jean-Claude Risset's continuous version (1969), the same trick with the steps filed off. Set it descending and the cycling tone can seem to fall without end.

II The lie, in the spectrum

Here is the whole machine. A Shepard tone is not one frequency — it is a comb of pure tones spaced exactly one octave apart (20 Hz, 40, 80, 160, … each double the last). Over that comb sits a fixed bell-shaped envelope — the glowing curve below — that decides how loud each partial is: loudest in the middle of your hearing, fading to silence at the top and bottom.

When the tone "rises," the partials move up together: by semitone steps or in a glide. At the upper edge a voice returns to the bottom of the bell; descending reverses that journey. The comb keeps cycling; the bell never moves.

The number on the right is a window into the construction. It is the amplitude-weighted mean log-frequency of the scheduled partials, converted back to Hz: a weighted geometric mean. The broad Gaussian bell keeps this statistic near its centre, with a small variation as the finite comb cycles. Pitch height is a perception; this number does not measure what your brain hears. The display follows the audio targets, before the browser smooths their transitions. The check reports the numerical variation ↗ over a full cycle.

III Why an ear can be cheated this way

Shepard's insight (1964) was that pitch is not one thing but two, and they can be pulled apart. There is height — the plain "low-to-high" dimension, a straight line. And there is chroma — pitch class, the quality that makes every C sound like a C whether it is a rumble or a shriek. Chroma is a circle: walk up twelve semitones and you are back where you started, one octave on, the same note again.

Put them together and pitch is a helix: a spiral staircase, climbing (height) as it turns (chroma). Ordinary notes move along the helix, so the two cues often agree. A Shepard tone makes its octave placement ambiguous by spreading energy across octaves under a fixed bell. The circular motion of chroma can then give the impression of a climb with no destination.

IV The tritone that has no answer

Now the strangest turn. Play two Shepard tones in a row, a tritone apart: exactly half an octave, six semitones, the interval old theorists called diabolus in musica. On the chroma circle the two tones sit at opposite poles. Their octave placement is ambiguous; sharing a bell does not guarantee identical perceived height or exactly equal spectral statistics. So which way does the pair move: up, or down?

You may hear a definite direction. The catch, found by Diana Deutsch in 1986: different listeners hear the same pair differently, and a listener's judgments can depend systematically on the pitch classes played. Perceived direction also correlates with the language and dialect a listener grew up speaking (Deutsch, 1991).

Run it a few times. Each pair is a fresh, randomly chosen tritone, so the direction you hear may change with the pair. Deutsch's internal map of pitch classes concerns that dependence on which tones are played. A total of rises and falls across random pairs cannot establish your map. The octave ambiguity is the paradox.

V Where it gets out into the world

The illusion is not a lab curiosity. The endless-rising tone is a working tool of dread: Hans Zimmer built it through the whole score of Christopher Nolan's Dunkirk (2017), fusing it to a ticking watch so the music seems to tighten without ever resolving — Zimmer has described the device explicitly in interviews, and the same trick threads his earlier Nolan scores. Game composers have used the rising-forever tone for staircases that can't be climbed; sound designers reach for the falling version when something must seem to drop without end.

The honest edge

• It is an illusion, and the cues can compete. Octave ambiguity lets chroma play a strong role in the direction you hear. Adding a height cue can change that judgment, but does not always overturn it: a study with conflicting cues found that listeners often followed chroma.
• The check is numerical. It samples the Gaussian model and the finite playback targets. The wide, uncut model has a nearly constant mean log-frequency; the finite comb varies more. The verifier gives the figures and tolerances for each. It checks the Gaussian, not a comparison with a windowed bell, and it does not measure perceived pitch height or audibility.
• The tritone result is statistical and personal. It is a population effect associated with listeners' linguistic backgrounds. Your answer depends on the pair as well as on you.
• Equipment matters. Tiny speakers that can't reproduce the low and high partials weaken the effect; the climb is most convincing when the whole comb is audible at once.

Sources

• Shepard, R. N. (1964). "Circularity in judgments of relative pitch." Journal of the Acoustical Society of America 36(12): 2346–2353. doi:10.1121/1.1919362
• Risset, J.-C. (1969). "Pitch control and pitch paradoxes demonstrated with computer-synthesized sounds." J. Acoust. Soc. Am. 46(1A_Suppl): 88 — the continuous Shepard–Risset glissando.
• Deutsch, D. (1986). "A musical paradox." Music Perception 3(3): 275–280. ucpress
• Deutsch, D. (1991). "The tritone paradox: an influence of language on music perception." Music Perception 8(4): 335–347. pdf
• Zimmer on the Shepard tone in Dunkirk: interview, Deadline (2018); CBC Radio q (2017).