Shepard tone

A Shepard tone, named after Roger Shepard, is a sound consisting of a superposition of sine waves separated by octaves. When played with the bass 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 seems to continually ascend or descend in pitch, yet which ultimately gets no higher or lower.
Construction.
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 C4 and a slightly quieter C5; 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 (F4 and F5), and the twelfth tone would be a loud B4 and an almost inaudible B5 with the addition of an almost inaudible B3. The thirteenth 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 raised cosine 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 curve actually employed."
The theory behind the illusion was demonstrated during an episode of the BBC's show "Bang Goes the Theory", where the effect was described as "a musical barber's pole."
The scale as described, with discrete steps between each tone, is known as the discrete Shepard scale. The illusion is more convincing if there is a short time between successive notes (staccato or marcato rather than legato or portamento).
Variants.
Shepard-Risset glissando.
Jean-Claude Risset subsequently created a version of the scale where the tones glide continuously, and it is appropriately called the continuous Risset scale or Shepard–Risset glissando. When done correctly, the tone appears to rise (or fall) continuously in pitch, yet return to its starting note. Risset has also created a similar effect with rhythm in which tempo seems to increase or decrease endlessly.
Tritone paradox.
A sequentially played pair of Shepard tones separated by an interval of a tritone (half an octave) produces the tritone paradox. Shepard had predicted that the two tones would constitute a bi-stable figure, the auditory equivalent of the Necker cube, that could be heard ascending or descending, but never both at the same time.
In 1986, Diana Deutsch discovered that the perception of which tone was higher depended on the absolute frequencies involved, and that an individual would usually hear the same pitch as the highest (this is determined by the absolute pitch of the notes). Interestingly, different listeners may perceive the same pattern as being either ascending or descending, depending on the language or dialect of the listener (Deutsch, Henthorn and Dolson found that native speakers of Vietnamese, a tonal language, heard the tritone paradox differently from Californians who were native speakers of English).
Perpetual melody.
Pedro Praticio observed in 2012 that by using a Shepard tone as a sound source and applying it to a melody he could reproduce the illusion of a continuously ascending or descending movement characteristic of the Shepard Scale. Regardless of the tempo and the envelope of the notes, the auditory illusion is effectively maintained. The uncertainty of the scale the Shepard tones pertain allows composers to experiment with deceiving and disconcerting melodies.