Shepard tone

Spectrum view of ascending Shepard tones (linear frequency scale)

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.^[1] It has been described as a "sonic barber's pole".^[2]

Construction of a Shepard scale

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.

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 ten 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).)

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

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). As a more concrete example, consider a brass trio consisting of a trumpet, a horn, and a tuba. They all start to play a repeating C scale (C–D–E–F–G–A–B–C) in their respective ranges, i.e. they all start playing Cs, but their notes are all in different octaves. When they reach the G of the scale, the trumpet drops down an octave, but the horn and tuba continue climbing. They're all still playing the same pitch class, but at different octaves. When they reach the B, the horn similarly drops down an octave, but the trumpet and tuba continue to climb, and when they get to what would be the second D of the scale, the tuba drops down to repeat the last seven notes of the scale. So no instrument ever exceeds an octave range, and essentially keeps playing exactly the same seven notes over and over again. But because two of the instruments are always "covering" the one that drops down an octave, it seems that the scale never stops rising.

Jean-Claude Risset subsequently created a version of the scale where the steps between each tone are continuous, 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.^[3]

Shepard scales in music

• James Tenney's For Ann (rising), composed in 1969, is based upon the Shepard scale concept.
• 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^[4]) for making the modulation end in the same pitch instead of an octave higher.^[5]
• Pink Floyd's 1970 composition "Echoes" ends with a rising Shephard-Risset glissando.
• Songwriter Stephin Merritt makes use of a Shepard scale in his 2007 composition "A Man of a Million Faces," written for NPR Music's Project Song.^[6][7]
• The Shepard tone has also been used in the 2008 film The Dark Knight to give the Batpod its transmissionless sound.^[8]
• Canadian post-rock band Godspeed You! Black Emperor also used the Shepard tone at various points in their 1997 album F♯ A♯ ∞.
• Marcus Schmickler's 2010 computer music album Palace of Marvels (Queered Pitch) includes several musical explorations of Shephard scales.^[9]
• Wilco's song "Born Alone" from their 2011 album "The Whole Love" ends with a descending Shepard glissando, as described by lead singer Jeff Tweedy:

  "So I came up with the idea that we would end the song with a Shepard tone, which is a series of chords that when repeated continuously sounds like its descending or ascending. It's kind of a musical trick—it sounds like it's endlessly going deeper and deeper into the abyss."^[10]

• 23's dubstep song "The Fall" is based around the Shepard tone.

See also

• Barberpole illusion
• Pitch circularity
• Strange loop

References

1. ^ 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. 
2. ^ "The Sonic Barber Pole: Shepard's Scale". at cycleback.com
3. ^ Risset rhythm
4. ^ Hofstadter, Douglas (1980). Gödel, Escher, Bach: An Eternal Golden Braid (1st ed.). Penguin Books. p. 10. ISBN 0-14-00-5579-7. 
5. ^ Ibid. pp. 717–719. 
6. ^ NPR Music's All Songs Considered: Project Song "Stephin Merritt: Two Days, 'A Million Faces'"
7. ^ NPR Music's All Songs Considered: Project Song "Video: Stephin Merritt Creates His Song"
8. ^ LA Times: The Envelope "'The Dark Knight' sound effects". Accessed April 14, 2009
9. ^ http://editionsmego.com/release/eMEGO+113
10. ^ http://www.theatlantic.com/entertainment/archive/2011/09/watch-wilcos-new-born-alone-video-and-read-the-story-behind-its-lyrics/244656/

External links

• The partials of a Shepard tone
• BBC science show, Bang Goes the Theory, explains the Shepard Tone
• Demonstrating an audio example of a continuous "endlessly descending" tone
• Demonstration of discrete Shepard tone (requires Macromedia Shockwave)
• Visualization of the Shepard Effect using Java
• Demonstration of the Shepard Scale of the Infinite Staircase in Super Mario 64
• The Shepard Tone from Wikipedia in easier to access format