Shepard Tone¶

A Shepard tone is an acoustic illusion — a sound that appears to continuously rise (or fall) in pitch without ever actually leaving its frequency range. Cognitive scientist Roger Shepard first described this effect in 1964. It mirrors an M.C. Escher staircase: listeners perceive endless ascending motion that never reaches its destination.

How it works¶

The illusion relies on two principles:

Octave equivalence — the human ear perceives tones one octave apart as the “same note” at a different pitch height.
Spectral envelope — a fixed Gaussian curve in log-frequency space controls loudness. Tones near the centre are loud; those at edges fade nearly silent.

Multiple sine waves — each separated by one octave — sound simultaneously while gliding upward. As tones fade at the upper edge, new tones enter at the bottom, fading in. The loudest tones always occupy the middle and move upward, so the sound seems to ascend perpetually.

Signal model¶

\[x[n] = A_\text{norm}\sum_k \exp\!\left(-\frac{d_k(t)^2}{2\sigma^2}\right) \sin(\varphi_k(n))\]

where the octave distance from the Gaussian centre is

\[d_k(t) = k - c + R\,t, \quad c = \frac{L-1}{2}, \quad \sigma = \frac{L}{4}\]

and the instantaneous frequency of layer k is \(f_k(t) = f_\text{base} \cdot 2^{d_k(t)}\). Phase is accumulated sample-by-sample for smooth glides.

Example¶

import pyminidsp as md

# 5 seconds of endlessly rising Shepard tone at 44.1 kHz
sig = md.shepard_tone(5 * 44100, amplitude=0.8, base_freq=440.0,
                       sample_rate=44100.0, rate_octaves_per_sec=0.5,
                       num_octaves=8)

Listen — rising Shepard tone (5 seconds):

Falling Shepard tone:

Static chord (rate = 0):

Key parameters¶

Glissando rate (rate_octaves_per_sec):

0.0 — static chord (no motion)
0.5 — moderate rise (default)
Negative values → falling Shepard tone

Number of octaves (num_octaves):

4–6 — narrow, organ-like quality
8 — balanced (default)
10–12 — ethereal, diffuse texture

Base frequency (base_freq): centres the Gaussian envelope. Typical values: 200–600 Hz.

# Slowly falling Shepard tone
falling = md.shepard_tone(44100 * 3, amplitude=0.8, base_freq=300.0,
                           sample_rate=44100.0, rate_octaves_per_sec=-0.3,
                           num_octaves=10)

md.shutdown()