PlaygroundPerception · 09
The Shepard tone
Frequency (Hz, octave lines) · bar height = loudness · dashed: the fixed loudness curve
Direction
Motion
What's going on
Every component of this sound is an octave apart from the next, and all of them move together. A fixed loudness curve fades them in at the bottom and out at the top, so the highest tones quietly disappear while new ones slip in underneath. Your brain follows the local motion, which is always upward, and never notices the reset.
Pitch has two sides: height, from low to high, and chroma, which note of the scale it is. The Shepard tone keeps the chroma moving while the height goes nowhere, like M. C. Escher's endless staircase.
Why it matters: composers use it to build tension that never resolves (Hans Zimmer's score for Dunkirk is built around it), and psychologists use it to study how the brain separates pitch height from pitch class.
The maths behind itFor the curious · formulas and details
The sound is a sum of octave-spaced sines whose loudness follows a fixed Gaussian over log-frequency. With p the distance travelled, in octaves:
x(t) = Σₖ A(fₖ) · sin(φₖ(t)) fₖ = f_min · 2^(k + p) A(f) = exp(−(log₂ f − log₂ f_c)² / 2w²)
Here f_min = 27.5 Hz, nine components, the curve centred on f_c = 440 Hz with width w = 1.2 octaves. When p grows by 1, every component has moved to where its neighbour was, so the spectrum, and the sound, is exactly the same: the tone is periodic in pitch. Components leave the top and re-enter at the bottom where the curve is practically zero, which hides the wrap.
Roger Shepard described the stepwise version in 1964; Jean-Claude Risset made the continuous glissando. Played as a sequence of pitch classes, the illusion shows that the auditory system tracks pitch as a helix: chroma around the circle, height along the axis.