A Shepard tone, named after Roger Shepard, 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. It has been described as a "sonic barber's pole".
Each square in the figure indicates a tone, 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 C(4) (middle C) and a loud C(5) (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 D(4) and a still quieter D(5). The two frequencies would be equally loud at the middle of the octave (F#), and the eleventh tone would be a loud B(4) and an almost inaudible B(5) with the addition of an almost inaudible B(3). 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 a gaussian function of its separation in semitones from a peak frequency, which in the above example would be B(4).)
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The acoustical illusion can be constructed by creating a series of overlapping ascending or descending scales. Similar to the Penrose stairs optical illusion (as in M. C. Escher's lithograph Ascending and Descending) or a barber's pole, the basic concept is shown in figure 1.
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 instead of legato or portamento).
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 descend) 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.
The tritone paradox
A sequentially played pair of Shepard tones separated by an interval of a tritone (half an octave) produces the tritone paradox. In this auditory illusion, first reported by Diana Deutsch in 1986, the scales may be heard as either descending or ascending. Shepard had predicted that the two tones would constitute a bistable figure, the auditory equivalent of the Necker cube, that could be heard ascending or descending, but never both at the same time. Deutsch later found that perception of which tone was higher depended on the absolute frequencies involved, and that different listeners may perceive the same pattern as being either ascending or descending.
- In a film by Shepard and E. E. Zajac, a Shepard tone accompanies the ascent of an analogous Penrose stair.
- In his book Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter explains how Shepard scales can be used on the Canon a 2, per tonos in Bach's Musical Offering (called the Endlessly Rising Canon by Hofstadter) for making the modulation end in the same pitch instead of an octave higher.
- In Super Mario 64, a modified Shepard tone is incorporated into the music of the endless staircase.
- In the video game Elite: Dangerous, Shepard–Risset glissandi are used in the supercruise engine sound effects for acceleration to/deceleration from superluminal speeds.
- In the film The Dark Knight and its follow-up The Dark Knight Rises, a Shepard tone was used to create the sound of the Batpod, a motorcycle that the filmmakers didn't want to change gear and tone abruptly but to constantly accelerate.
- A constant Shepard tone plays throughout the Autechre song "Dial".
- The end of the Beatles song "I Am the Walrus".
- The start of the Queen song "Tie Your Mother Down" and at the end of "Teo Torriatte (Let Us Cling Together)" both on A Day At The Races.
- The end of the Beck song "Lonesome Tears".
- The end of the solo in the Led Zeppelin song "No Quarter".
- The end of the Pink Floyd song "Echoes" .
- The end of the Muse song "Ruled By Secrecy".
- Starting at the 7 minute mark in the Godspeed You! Black Emperor song "A Nervous, Sad, Poor...".
- The end of the Flaming Lips song "Moth in the Incubator".
- The Machinedrum song "Infinite Us" taken from the album "Vapor City" uses the Shepard tone throughout.
- The end of the Sonic Youth song "Sunday".
- The end of the Wilco song "Born Alone".
- Tom Middleton's song "Penrose Steps".
- Butric's (Ricardo Villalobos and Butch) song "Up".
- Throughout Phish's song "Foam".
- The end of the Sting song "Epilogue (Nothing 'Bout Me)" from Ten Summoner’s Tales.
- Used as a compositional technique for digital orchestral music on a piece entitled The Journey by composer Renaldo Ramai. 
- In the website fallingfalling.com, the tone is played along with an array of colors giving a trance-like audio-visual experience.
- Roger N. Shepard (December 1964). "Circularity in Judgements of Relative Pitch". Journal of the Acoustical Society of America 36 (12): 2346–53. doi:10.1121/1.1919362.
- "The Sonic Barber Pole: Shepard's Scale". at cycleback.com
- Risset rhythm
- Diana Deutsch (1986). "A musical paradox" (PDF). Music Perception 3: 275–280. doi:10.2307/40285337. Retrieved 26 May 2012.
- R.N. Shepard. Circularity in judgments of relative pitch. Journal of the Acoustical Society of America, 36(12):2346–2353, 1964.
- Deutsch, D. (1992). "Some New Pitch Paradoxes and their Implications". Philosophical Transactions of the Royal Society B: Biological Sciences 336 (1278): 391–397. doi:10.1098/rstb.1992.0073. PMID 1354379.
- Roger N. Shepard and Edward E. Zajac (1967). A Pair of Paradoxes. AT&T Bell Laboratories.
- Hofstadter, Douglas (1980). Gödel, Escher, Bach: An Eternal Golden Braid (1st ed.). Penguin Books. p. 10. ISBN 0-14-005579-7.
- Ibid. pp. 717–719.
- Get High Now (2009). "Shepard Tones". Get High Now. Retrieved 18 March 2013.
- Richard King (4 February 2009). "'The Dark Knight' sound effects". Los Angeles Times. Retrieved 12 September 2012.
- Pollack, Alan W. (1996). "Notes on "I Am the Walrus"". Retrieved 8 June 2007.
- The partials of a Shepard tone
- BBC science show, Bang Goes the Theory, explains the Shepard Tone
- Demonstration of discrete Shepard tone (requires Macromedia Shockwave)
- Visualization of the Shepard Effect using Java
- The Shepard Tone from Wikipedia in easier to access format
- A demonstration of a rising Shepard Scale as a ball bounces endlessly up a Penrose staircase (and down)