Shepard Tone — Endless Illusion

A Shepard tone is an auditory illusion in which octave-spaced sine partials glide under a fixed bell-shaped spectrum, so the pitch seems to climb or fall forever without running out of range. This generator runs the Shepard-Risset glissando in your browser with six, eight or nine oscillators and no upload.

How to use it

1. Choose the direction. Ascending is the endless rise, Descending the endless fall.
2. Set the glide rate. The slider runs from 0.04 to 0.30 octaves per second; the 0.12 default is one octave every 8.3 seconds. Lower rates are easier to follow.
3. Pick the number of octaves. Buttons select 6, 8 (standard) or 9 partials; changing the count restarts the stack.
4. Set master volume. The slider covers 5 to 70 percent, defaulting to 35 percent; the output is normalised across partials to avoid clipping.
5. Press Play. Each partial is a sine whose frequency updates every animation frame with 30 ms smoothing, so the glide has no zipper noise.
6. Watch the visualizer. The canvas draws the raised-cosine envelope across a 20 Hz to 20 kHz log axis and marks the active partials.
7. Listen for the seam. At slow rates, try to catch a partial leaving the top of the ladder and re-entering at the bottom; when you cannot spot it, the illusion is working.

Why the tone seems to rise forever

Pitch has two independent dimensions: chroma, which identifies the note within an octave, and height, which places it in a register.

Octave equivalence is the circular part. A tone at 440 Hz and a tone at 880 Hz share the same chroma while sitting an octave apart in height, and that circularity is what Roger Shepard exploited in 1964 in “Circularity in Judgments of Relative Pitch” (Journal of the Acoustical Society of America, 36(12), 2346 to 2353). Superimpose octave-spaced partials and weight them with a fixed bell-shaped envelope that reaches zero at both ends of the audible range; the spectrum is then nearly unchanged when transposed by an octave, so relative pitch judgments become circular.

This page builds the envelope as a raised cosine in log2 frequency between 20 Hz and 18 kHz: gain = 0.5 x (1 - cos(2 x pi x u)), where u runs from 0 to 1 across that span. Gain is zero at both bounds and peaks at the geometric centre, the square root of 20 x 18000, about 600 Hz. Partial frequencies are 30 Hz x 2^k, so the stack consists of sines spaced exactly one octave apart, and every partial glides upward at the same rate; at the top of the ladder it wraps to the bottom, one octave below the next partial.

The ear accepts the wrap because the envelope never moves: the energy balance across the spectrum stays fixed while the partials inside the window change identity. As one partial fades at the top edge, an equivalent partial emerges at the bottom, and the auditory system tracks continuous chroma motion rather than the sine that left. Jean-Claude Risset turned this into a continuous glissando at Bell Labs in the late 1960s, and this tool implements that variant, where a semitone is a ratio of 2^(1/12), about 1.0595.

Parameters and pitch arithmetic

Control	Range	Default	What it changes
Illusion direction	Ascending / Descending	Ascending	Sign of the glide
Glide rate	0.04 to 0.30 octaves/s	0.12 octaves/s	25 s per octave at minimum, 3.3 s at maximum
Simultaneous octaves	6, 8 or 9 partials	8	Sine stack size, spaced one octave apart from 30 Hz
Master volume	5 to 70 percent	35 percent	Output level
Spectral envelope	Fixed raised cosine	20 Hz to 18 kHz	Weighting of each partial; peak near 600 Hz

The highest occupied frequency depends on glide position and stack size: at rest the ladders top out at 960 Hz (6 partials), 3.84 kHz (8) and 7.68 kHz (9).

Accuracy, limits and troubleshooting

• The drawn envelope spans 20 Hz to 18 kHz, but the sine stack never reaches that high, and the wrap point depends on the ladder size. With 9 partials the top of the ladder approaches 15.4 kHz, where the gain is under 0.01, so the seam is essentially silent; with 6 partials the wrap happens near 1.9 kHz at a gain of about 0.74, which can tick audibly every 8 seconds.
• If you hear a tick at the default settings, switch to 9 octaves and slow the glide rate; the 8-partial stack wraps near 7.7 kHz at a gain of roughly 0.15, a small step that is easier to notice on headphones.
• Low-frequency reproduction matters more than treble extension: laptop speakers and earbuds roll off below 60 to 100 Hz, so the bottom partials may be weak. The illusion still works, because the envelope’s middle carries it, but it sounds thinner.
• Hearing loss above 10 kHz has little effect here, because the default stack tops out below 7.7 kHz.
• The visualizer is frame-rate bound and may stutter on a busy machine; the glide is computed from elapsed time, so the audio stays smooth even when the canvas drops frames.
• Frequencies are updated with a 30 ms time constant, so fast slider moves glide rather than jump; Stop ramps the gain down over about 30 ms and stops the oscillators roughly 100 ms later.
• The illusion is strongest with continuous listening: focusing on one partial, or watching the visualizer closely, can break it. Look away for a few seconds and it usually returns.
• Long sessions at high volume are an exposure risk regardless of how pleasant the tone is; keep the level moderate, especially with 9 partials.

Related tools

• Tone Generator — compare a single fixed sine with the octave stack to hear what the envelope adds.
• Perceptual Sine Sweep — hear a sweep stepped by musical intervals instead of linear frequency.
• Binaural Beats — another perceptual illusion built from precise tone synthesis.