Shepard tone
A Shepard tone, named after Roger Shepard
Roger Shepard
Roger Newland Shepard is a cognitive scientist and author of Toward a Universal Law of Generalization for Psychological Science. He is seen as a father of research on spatial relations....

, is a sound consisting of a superposition of sine waves separated by octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

s. When played with the base pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

 of the tone moving upward or downward, it is referred to as the Shepard scale. This creates the auditory illusion
Auditory illusion
An auditory illusion is an illusion of hearing, the aural equivalent of an optical illusion: the listener hears either sounds which are not present in the stimulus, or "impossible" sounds...

 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
Barber's pole
A barber's pole is a type of sign used by barbers to signify the place or shop where they perform their craft. The trade sign is, by a tradition dating back to the Middle Ages, a staff or pole with a helix of colored stripes...


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
Penrose stairs
The Penrose stairs or Penrose steps, also dubbed the impossible staircase, is an impossible object created by Lionel Penrose and his son Roger Penrose...

 optical illusion
Optical illusion
An optical illusion is characterized by visually perceived images that differ from objective reality. The information gathered by the eye is processed in the brain to give a perception that does not tally with a physical measurement of the stimulus source...

 (as in M. C. Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

's lithograph Ascending and Descending
Ascending and Descending
Ascending and Descending is a lithograph print by the Dutch artist M. C. Escher which was first printed in March 1960.The original print measures 14" x 11 1/4”. The lithograph depicts a large building roofed by a never-ending staircase. Two lines of identically dressed men appear on the staircase,...

) or a barber's pole
Barber's pole
A barber's pole is a type of sign used by barbers to signify the place or shop where they perform their craft. The trade sign is, by a tradition dating back to the Middle Ages, a staff or pole with a helix of colored stripes...

, 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
Loudness is the quality of a sound that is primarily a psychological correlate of physical strength . More formally, it is defined as "that attribute of auditory sensation in terms of which sounds can be ordered on a scale extending from quiet to loud."Loudness, a subjective measure, is often...

 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
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

) 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 semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....

s from a peak frequency, which in the above example would be B(4).)

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
Staccato is a form of musical articulation. In modern notation it signifies a note of shortened duration and separated from the note that may follow by silence...

 or marcato
Marcato is a musical instruction indicating a note, chord, or passage is to be played louder or more forcefully than surrounding music. The instruction may involve the word marcato itself written above or below the staff or it may take the form of an accent mark, ^ , an open vertical wedge...

 instead of legato
In musical notation the Italian word legato indicates that musical notes are played or sung smoothly and connected. That is, in transitioning from note to note, there should be no intervening silence...

 or portamento
Portamento is a musical term originated from the Italian expression "portamento della voce" , denoting from the beginning of the 17th century a vocal slide between two pitches and its emulation by members of the violin family and certain wind instruments, and is sometimes used...

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
Jean-Claude Risset
Jean-Claude Risset is a French composer, best known for his pioneering contributions to computer music. He is a former student of André Jolivet and former co-worker of Max Mathews at Bell Labs....

 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
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...

 in which tempo seems to increase or decrease endlessly.

Shepard scales in music

  • James Tenney
    James Tenney
    James Tenney was an American composer and influential music theorist.-Biography:Tenney was born in Silver City, New Mexico, and grew up in Arizona and Colorado. He attended the University of Denver, the Juilliard School of Music, Bennington College and the University of Illinois...

    's For Ann (rising)
    For Ann (rising)
    For Ann is a piece of electronic music created by James Tenney in 1969.Tenney is the author of Meta Hodos, one of, if not the, earliest applications of gestalt theory and cognitive science to music, and later "Hierarchical temporal gestalt perception in music: a metric space model" with Larry...

    , composed in 1969, is based upon the Shepard scale concept.
  • In his book Gödel, Escher, Bach: An Eternal Golden Braid
    Gödel, Escher, Bach
    Gödel, Escher, Bach: An Eternal Golden Braid is a book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll"....

    , Douglas Hofstadter
    Douglas Hofstadter
    Douglas Richard Hofstadter is an American academic whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics...

     explains how Shepard scales can be used on the Canon a 2, per tonos in Bach's
    Johann Sebastian Bach
    Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...

     Musical Offering (called the Endlessly Rising Canon by Hofstadter) for making the modulation
    Modulation (music)
    In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest...

     end in the same pitch instead of an octave higher.
  • Pink Floyd
    Pink Floyd
    Pink Floyd were an English rock band that achieved worldwide success with their progressive and psychedelic rock music. Their work is marked by the use of philosophical lyrics, sonic experimentation, innovative album art, and elaborate live shows. Pink Floyd are one of the most commercially...

    's 1970 composition "Echoes
    Echoes or echos may refer to:* the plural of Echo -Songs:* "Echoes" , a 1971 Pink Floyd song* "Echoes" by Bennie Benjamin and George David Weiss...

    " ends with a rising Shephard-Risset glissando.
  • Songwriter Stephin Merritt
    Stephin Merritt
    Stephin Merritt is an American singer-songwriter based in Los Angeles , best known as the principal singer and songwriter in the band The Magnetic Fields...

     makes use of a Shepard scale in his 2007 composition "A Man of a Million Faces," written for NPR Music
    NPR Music
    NPR Music launched in November 2007 to present public radio music programming and original editorial content for music discovery. The Web site is a project of National Public Radio, a privately and publicly funded non-profit membership media organization that serves as a national syndicator to 797...

    's Project Song.
  • The Shepard tone has also been used in the 2008 film The Dark Knight
    The Dark Knight (film)
    The Dark Knight is a 2008 superhero film directed, produced and co-written by Christopher Nolan. Based on the DC Comics character Batman, the film is part of Nolan's Batman film series and a sequel to 2005's Batman Begins...

    to give the Batpod its transmissionless sound.
  • Canadian post-rock
    Post-rock is a subgenre of rock music characterized by the influence and use of instruments commonly associated with rock, but using rhythms and "guitars as facilitators of timbre and textures" not traditionally found in rock...

     band Godspeed You! Black Emperor
    Godspeed You! Black Emperor
    Godspeed You! Black Emperor is a Canadian post-rock band which originated from Montreal, Quebec in 1994...

     also used the Shepard tone at various points in their 1997 album F♯ A♯ ∞.
  • Marcus Schmickler
    Marcus Schmickler
    Marcus Schmickler is a German composer, musician and producer. He is also known under the pseudonym Pluramon....

    's 2010 computer music album Palace of Marvels (Queered Pitch) includes several musical explorations of Shephard scales.
  • Wilco
    Wilco is an American alternative rock band based in Chicago, Illinois. The band was formed in 1994 by the remaining members of alternative country group Uncle Tupelo following singer Jay Farrar's departure. Wilco's lineup has changed frequently, with only singer Jeff Tweedy and bassist John...

    's song "Born Alone" from their 2011 album "The Whole Love
    The Whole Love
    The Whole Love is the eighth album by American alternative rock group Wilco, released on September 27, 2011. It is their first album on their own label dBpm. Attendees at Wilco's 2011 Solid Sound Festival at the Massachusetts Museum of Contemporary Art from June 24 to 26 could purchase the first...

    " 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."
  • Björk
    Björk Guðmundsdóttir , known as Björk , is an Icelandic singer-songwriter. Her eclectic musical style has achieved popular acknowledgement and popularity within many musical genres, such as rock, jazz, electronic dance music, classical and folk...

    's song "Cosmogony" from their 2011 album "Biophilia" ends with a descending and wonderful choir of voices.

External links

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