Circular Tones

Circular tones are artificial complex tones typically comprised of several octave-spaced frequencies. By definition, all frequency components of octave-spaced circular tones belong to the same pitch class. The relative intensity of the octave-spaced partials in a circular tone is determined by a spectral (amplitude) envelope, such that the distribution of energy across the log frequency scale approximates a Gaussian (bell curve) shape. Although circular tones generate a perceived pitch, the pitch height of circular tones (the octave in which each tone occurs) is usually ambiguous, because of the shape of the spectral envelope and the absence of a complete harmonic series.

Originally designed in 1964 by Roger Shepard to illustrate the psychological separation of pitch height and pitch class, circular tones are sometimes referred to as Shepard tones. Because the spectral envelope remains fixed independently of any changes in pitch class, ascending or descending sequences of circular tones recycle every octave. When listening to multiple cycles of such scalar sequences, listeners may report an experience of tones that are endlessly ascending or descending in a manner that seems illusory. However, the perceptual effects of circular tones are still not entirely understood, and some researchers have questioned whether they demonstrate a true dissociation between pitch class and pitch height. Octave-spaced circular tones often generate multiple pitch sensations, and the perceptual effects can be easily manipulated by context. Moreover, it is not even necessary for frequency components to be octave-spaced in order to generate circular tones. Any constant spacing between components (such as eight semitones, 13 semitones) can be used to generate circularity as long as the amplitudes are constrained by a fixed spectral envelope. However, circular tones that are not octave-spaced tend to generate unclear or multiple pitch sensations.

Figure 1 shows the spectral envelope and the relative energy of 10 partials of two different pitch classes (say C and E). Although E is “higher” than C (in a given octave), it has less energy at its higher partials and more at the lower partials because of the shape of the spectral envelope that offsets this upward shift in pitch class. Thus, the aggregate physical pitch height stays constant, while pitch [Page 178]class changes. Other spectral envelopes are also effective, as long as they respect the principle of manipulating the intensity of partials in different octaves to keep the average pitch height the same.

Figure 1The Gaussian spectral envelope

The only remaining cue for pitch height is proximity—the above example would sound like an ascending interval (i.e., moving to the right) because the interval between C and E within an octave is smaller than the descending alternative of a minor sixth. A chromatic ascending/descending sequence of circular tones will give an illusion of a perpetually rising/falling pitch, although in reality it simply repeats the same 12 tones, in a circular fashion.

This illusion is conceptually reminiscent in the visual domain—the Penrose stairs, popularized as M. C. Escher's impossible staircase. The pitches need not be limited to the 12 pitch classes of Western music—Jean-Claude Risset created a version using a glissando instead. The distinctive timbre of such tones resembles an organ, and many examples are available online by searching for circular tones or Shepard tones.

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