Acoustical Illusions

Consider the inverted cosine curve \(g(x) = 0.5 - 0.5 \cos x\) between 0 and 1. Let us work with the frequency range from 1Hz to 20000Hz. Then for a sinusoid of frequency \(\theta\), let its amplitude be \(g(\log_{20000} \theta)\).

Now we work with sounds consisting of sinusoidal components (with amplitudes as chosen above) whose frequencies have logarithms are evenly spaced. The most common example is octaves: i.e. we have sinusoids of frequencies \(f, 2f, 4f, …​\) (where all frequencies lie within the range we are working with), so for every pitch we have exactly one sound. With these sounds we can construct scales that endlessly rise or fall (Shepard scale), or even a continuously rising or falling tone that never ends (Risset tone).

Other illusions are possible. Shifting the amplitude envelope up or down the \(x\)-axis over time results in a tone that seems to rise or fall in pitch, yet stays the same. Shifting the amplitude envelope one way and moving the component frequencies the other results in a tone that seems to rise and fall at the same time.