The Shepard Tone

The partials of harmonic sounds are related to each other by an arithmetical relationship. The partials of a Shepard Tone are related to each other by a geometrical relationship. Shepard used such signals to prove his hypothesis of the circularity of pitch perception[45].

Schroeder[43, page 96] shows that the auditory paradox created by the Shepard Tone, which is generated according to a Weierstrass function, has become possible due to the self-similarity of the signal. A Weierstrass function is constructed as follows:

$$w(t) = \sum_{k=0}^{\infty}\alpha^k\cos(\beta^{k}t)$$ (4.1)

where $\alpha$ is real and $\beta$ is odd. Weierstrass showed that under certain conditions of $\alpha$ and $\beta$, this function is everywhere continuous but nowhere differentiable. For creating a Shepard tone we can drop the term since we are only going to be dealing with a finite number of partials. Therefore, we have:

$$w(t) = \sum_{k=0}^{M}\cos(\beta^{k}t)$$ (4.2)

where $M$ is the number of partials and $\beta$ is the geometrical relationship between two adjacent partials. Although Shepard applies a formant-like envelope to the frequency domain representation of the signal, this is done for smoothing the perceptual transition and sustaining the paradox effect. The paradox is created from the fact that the ear attempts to extract a one-dimensional signal (the variable being pitch) out of a multidimensional signal (timbre). We can think of pitch as a value which identifies a relationship between the partials of a signal in a one-dimensional way. If we view the frequency domain representation of the signal, then time scaling according to the same geometrical relationship does not change the ``body'' of the signal but only its boundary conditions; therefore we hear the same pitch and not a pitch scaled according to the scale factor. Scaling the function in time by a factor of gives : substituting into equation 4.2, we get:

$$w(\beta t) = \sum_{k=0}^{M}\cos(\beta^{k+1}t) = \sum_{k=1}^{M+1}\cos(\beta^{k}t)$$ (4.3)

which is the same as except for the boundary conditions of and .