July 1, 2020


Circularity in Judgments of Relative Pitch. Authors: Shepard, Roger N. Publication: The Journal of the Acoustical Society of America, vol. 36, issue 12, p. The Shepard illusion, in which the presentation of a cyclically repetitive sequence of complex tones composed of partials separated by octave intervals (Shepard. Circularity in relative pitch judgments for inharmonic complex tones: The Shepard demonstration revisited, again. EDWARD M. BURNS. Department ofAudiology.

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The figure on the left below represents an impossible staircasesimilar to one originally published by Penrose and Penrose in 1.

Since each stair that is one step clockwise from its neighbor is also one step downward, the staircase appears to be eternally descending. Pitch circularities are based on the same principle. This is acknowledged in our musical scale, which is based on the circular configuration shown on the right below.

As we ascend this scale in semitone steps, we repeatedly traverse the pitch class circle in clockwise direction, so that we play C, CD, and so on all around the circle, until we reach A, AB – and then we proceed to C, CD again, and so on.

To accommodate both the linear and circular dimensions, music theorists have suggested that pitch should be represented as a helix having one complete turn per octave, so that tones that are separated by octaves are also ptich on this representation, as shown below. Shepard 2 reasoned that by creating banks of tones whose note names pitch classes are clearly defined but whose perceived heights are ambiguous, the helix could be collapsed into a circle, so enabling the creation of scales that ascend or jucgments endlessly in pitch.

Pitch circularity – Wikipedia

He achieved this ambiguity by creating banks of complex tones, with each tone consisting only of components that were separated by octaves, and whose amplitudes were scaled by a fixed bell-shaped spectral envelope. Such tones are well defined in terms of pitch class, but poorly defined in terms of height. When such tones are played traversing the pitch class circle in clockwise direction, one obtains the impression of an eternally ascending scale— C is heard as higher than C; D as higher pittch C ; D as higher than D.


Counterclockwise movement creates the impression of an eternally descending scale.

Risset 3 has created intriguing variants using gliding tones that appear to ascend or descend continuously in pitch. See the review by Deutsch 4 for details. Later, I reasoned that it should be possible to create circular scales from sequences of single tones, with each tone comprising a full ccircularity series.

If you take a harmonic complex tone and gradually reduce the amplitudes of the odd-numbered harmonics 1, 3, 5, etc. In Sound Demo 1, a harmonic complex tone based on A 4 concert A is presented, with the odd-numbered harmonics gradually gliding down in amplitude.

Pitch circularity

At some point, listeners realize that they are hearing the note an octave higher — but this perceptual transition had occurred without the sounds traversing the semitone scale, but remaining on note A. I further reasoned that we should be able to produce pitch circularities on this principle. We begin with a bank of twelve harmonic citcularity tones, whose fundamental frequencies range over an octave in semitone steps. For the tone with the highest fundamental, the odd and even harmonics are equal in amplitude.

Then for the tone a semitone lower, the amplitudes of reelative odd harmonics are reduced relative to the even ones, so raising the perceived height of this tone.

Then for the tone another semitone lower, the amplitudes of the odd harmonics are reduced further, so raising the perceived height of this tone to a greater extent. The tone with the lowest fundamental is therefore heard as displaced up an octave, and pitch circularity is achieved. Here is an eternally descending scale based on this principle, with the amplitudes of the odd-numbered harmonics reduced by 3. Together with my colleagues, I carried out an experiment to determine whether such tones are indeed heard as circular, when all intervals are considered 5.


We created a bank of twelve tones, and from this bank we paired each tone sequentially with every other tone e. Subjects judged for each pair whether it ascended or descended in pitch. Here is an excerpt from the experiment, and you will probably find that your judgments of each pair correspond to the closest distance between the tones along the circle.

The finding that circular scales can be obtained from full harmonic series leads to the intriguing possibility that this algorithm could be used to transform banks of natural instrument tones so that they would also exhibit pitch circularity 6. William Brent, then a graduate student at UCSD, has achieved considerable success using bassoon samples, and also some success with oboe, flute, and violin samples, and has shown that the effect is not destroyed by vibrato.

The possibility of creating circular banks of tones derived from natural instruments expands the scope of musical materials available to composers and performers. Unknown to the authors, Oscar Reutesvald had also created an impossible staircase in the s. Paradoxes of musical pitch. Pitch circularity from tones comprising full harmonic series. Journal of the Acoustical Society of America, The paradox of pitch circularity.

The pitch class circle.