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.[1]

A spectrum view of ascending Shepard tones on a linear frequency scale.


Figure 1: Shepard tones forming a Shepard scale, illustrated in a sequencer.

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."[1]

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."[2]

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).[citation needed]


Shepard-Risset glissandoEdit

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.[3] 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.[4]

Tritone paradoxEdit

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 bistable figure, the auditory equivalent of the Necker cube, that could be heard ascending or descending, but never both at the same time.[1]

In 1986, Diana Deutsch discovered the paradoxical auditory illusion where scales may be heard as either descending or ascending.[5] 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.[6]


  • In a 1967 film by Shepard and E. E. Zajac, a Shepard tone accompanies the ascent of an analogous Penrose stair.[7]
  • The ending of the 1971 song "Echoes" from the album Meddle by Pink Floyd features a Shepard tone that fades out to a wind sound (actually a white noise processed through a tape echo unit).[8]
  • In his 1980 book Gödel, Escher, Bach: An Eternal Golden Braid, Douglas Hofstadter explained how Shepard scales could be used on the Canon a 2, per tonos in Bach's Musical Offering (called the Endlessly Rising Canon by Hofstadter[9]:10) for making the modulation end in the same pitch instead of an octave higher.[9]:717–719
  • In 1995, Braus argued the last notes of the 1885 piano work Bagatelle sans tonalité, composed by Franz Liszt could be continued to produce a Shepard scale using Hofstadter’s technique.[10]
  • In Godspeed You! Black Emperor's 1997 debut F♯ A♯ ∞, the opening song The Dead Flag Blues has a section mainly consisting of tornado sirens using a Shephard tone to create its unique apocalyptic atmosphere.
  • In Stephin Merritt's 2007 song "Man of a Million Faces", composed for NPR's "Project Song", the Shepard tone is a key aspect.[11]
  • In the 2008 film The Dark Knight and its 2012 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 accelerate constantly.[12]
  • In the 2017 film Dunkirk, a Shepard tone is used to create the illusion of an ever increasing moment of intensity across intertwined storylines.[13]
  • The 2018 track "Always Ascending" by Franz Ferdinand from the album of the same name features a rising Shepard tone throughout the song. The video for the song echoes the effect, with the camera apparently rising continually throughout.[14]

See alsoEdit


  1. ^ a b c Shepard, Roger N. (December 1964). "Circularity in Judgements of Relative Pitch". Journal of the Acoustical Society of America. 36 (12): 2346–53. doi:10.1121/1.1919362.
  2. ^ "Clip from Series 4, Episode 6". Bang Goes the Theory. 18 April 2011. BBC. It's like a barber's pole of sound.
  3. ^ "Jean-Claude Risset, who reimagined digital synthesis, has died - CDM Create Digital Music". CDM Create Digital Music. 22 November 2016. Retrieved 30 December 2019. The sound for which Risset is best known is perhaps the most emblematic of his contributions. Creating a sonic illusion much like M.C. Escher’s optical ones, the Shepherd-Risset glissando / Risset scale, in its present form invented by the French composer, seems to ascend forever.
  4. ^ Risset rhythm
  5. ^ Deutsch, Diana (1986). "A musical paradox" (PDF). Music Perception. 3: 275–280. doi:10.2307/40285337.
  6. ^ 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.
  7. ^ Shepard, Roger N.; Zajac, Edward E. (1967). A Pair of Paradoxes. AT&T Bell Laboratories.
  8. ^ Hurskiy, S. (2018). "Auditory illusions in modern film industry and music" (PDF). Zhytomyr State University Library. Retrieved 25 September 2020.
  9. ^ a b Hofstadter, Douglas (1980). Gödel, Escher, Bach: An Eternal Golden Braid (1st ed.). Penguin Books. ISBN 0-14-005579-7.
  10. ^ Braus, I. (1995). "Retracing one's steps: An overview of pitch circularity and Shepard tones in European music, 1550–1990". Music Perception: An Interdisciplinary Journal. pp. 323–351.
  11. ^ Stephin Merritt: Two Days, 'A Million Faces' (video). NPR. 4 November 2007. Retrieved 9 October 2015. It turns out I was thinking about a Shepard tone, the illusion of ever-ascending pitches.
  12. ^ King, Richard (4 February 2009). "'The Dark Knight' sound effects". Los Angeles Times.
  13. ^ Haubursin, Christopher (26 July 2017). "The sound illusion that makes Dunkirk so intense".
  14. ^ McCormick, Neil (9 February 2018). "Franz Ferdinand are still operating on an elevated plateau – Always Ascending, review". The Telegraph.

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