A Shepard tone is a sound that appears to rise (or fall) indefinitely in pitch without ever actually getting higher. It works by stacking N sinusoids separated by octaves under a Gaussian (bell-curve) amplitude envelope centred in log-frequency space. As every partial rises, it fades out at the top and a new one fades in at the bottom — the sonic equivalent of a Penrose staircase.
f_k(t) = f₀(t) · 2^k
A_k = exp( −(log₂(f_k) − μ)² / (2σ²) )
x(t) = ∑ A_k · sin(2π ∫ f_k dt)
Hans Zimmer used a Shepard tone in the Batpod chase scene of The Dark Knight to create the feeling of relentless, never-peaking acceleration. The endless staircase theme in Super Mario 64 is another famous example — the music climbs forever, just like the stairs.
A Shepard tone is an auditory illusion created by superimposing a set of sinusoids separated by octaves, each weighted by a bell-shaped (Gaussian) amplitude envelope. As the base frequency slowly rises and wraps back an octave, the brain perceives continuous upward pitch motion that never ends.
Roger Shepard, a cognitive scientist at Stanford, introduced this illusion in his 1964 paper in the Journal of the Acoustical Society of America. He demonstrated how tonal chroma (pitch class) and tone height (absolute pitch) can be dissociated in perception.
Jean-Claude Risset extended Shepard's discrete steps into a continuous version called the Risset glissando. Instead of discrete frequency jumps, the glissando slides each sinusoid continuously, producing a seamless infinite sweep. This simulator implements the continuous Risset variant.
The Gaussian bell curve is centred in the logarithmic frequency spectrum. Sinusoids near the centre are loudest; those at the top fade out while simultaneously new ones fade in at the bottom. This crossfading makes the ear always hear dominant tones in the middle register, hiding the octave wrap and sustaining the perception of endless ascent.
With base frequency f(t) and N components, partial k has frequency f_k = f(t) · 2^k. Its amplitude is A_k = exp(-(log₂(f_k) - μ)² / (2σ²)) where μ is the Gaussian centre and σ controls bandwidth. Each frame f(t) is multiplied by a small factor and wrapped modulo one octave.
The auditory system separates pitch chroma (the note letter) from pitch height (register). The Shepard tone constantly raises chroma while the envelope keeps height perceptually constant. Because chroma cycles through the octave seamlessly, the brain extrapolates continuous ascent with no endpoint.
A Shepard scale consists of discrete semitone steps, each a separate Shepard tone. The Shepard tone itself is a single sustained chord of stacked octave sinusoids. The Risset glissando is the continuous-sweep generalisation — which this simulator implements.
Hans Zimmer used a rising Shepard tone in The Dark Knight Batpod chase. The Beatles' 'A Day in the Life' coda uses a similar stacking principle. Super Mario 64's endless staircase theme is perhaps the most famous video-game example.
Yes. By slowly decreasing the base frequency, the same Gaussian-envelope technique produces an endlessly descending pitch. Toggle the Direction button in the HUD to switch to descending mode.
Pitch chroma is the cyclical quality of a note (C, D, E, … repeating every octave) independent of register. Pitch height is the overall position in the spectrum. Shepard tones manipulate these independently: chroma ascends while height stays constant, exploiting the brain's tendency to track chroma when the two cues conflict.