Shepard tones

A Shepard tone, named after Roger Shepard (born 1929), is a sound consisting of a superposition of sine waves separated by octaves. When played with the base pitch of the tone moving upward or downward, it is referred to as the Shepard scale. This creates the auditory illusion of a tone that continually ascends or descends in pitch, yet which ultimately seems to get no higher or lower.

Each square in the figure indicates a tone, with any set of squares in vertical alignment together making one Shepard tone. The color of each square indicates the loudness of the note, with purple being the quietest and green the loudest. Overlapping notes that play at the same time are exactly one octave apart, and each scale fades in and fades out so that hearing the beginning or end of any given scale is impossible. As a conceptual example of an ascending Shepard scale, the first tone could be an almost inaudible C₄ (middle C) and a loud C₅ (an octave higher). The next would be a slightly louder C♯₄ and a slightly quieter C♯₅; the next would be a still louder D₄ and a still quieter D₅. The two frequencies would be equally loud at the middle of the octave (F♯₄ and F♯₅), and the eleventh tone would be a loud B₄ and an almost inaudible B₅ with the addition of an almost inaudible B₃. The twelfth tone would then be the same as the first, and the cycle could continue indefinitely. (In other words, each tone consists of two sine waves with frequencies separated by octaves; the intensity of each is e.g. a gaussian function of its separation in semitones from a peak frequency, which in the above example would be B₄. According to Shepard, "(...) almost any smooth distribution that tapers off to subthreshold levels at low and high frequencies would have done as well as the cosine [gaussian] curve actually employed.")

The acoustical illusion can be constructed by creating a series of overlapping ascending or descending scales. Similar to a barber's pole, the basic concept is shown in Figure 1.

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