Page 9 - Summer 2010

THE PARADOX OF PITCH CIRCULARITY
Diana Deutsch
Department of Psychology, University of California, San Diego La Jolla, California 92093
“The phenomenon of pitch circularity has implications for our understanding of pitch perception, as well as for musical composition and performance.”
Introduction
In viewing M. C. Escher’s lithograph
Ascending and Descending shown on
the front cover, we see monks plod-
ding up and down an endless staircase—
each monk will ultimately arrive at the
place where he began his impossible
journey. Our perceptual system insists
on this interpretation, even though we
know it to be incorrect. The lithograph
was inspired by the endless staircase
devised by Lionel Penrose and his son
Roger Penrose,1 a variant of which is
shown in Fig. 1. Our visual system opts for a simple interpre- tation based on local relationships within the figure, rather than choosing a complex, yet correct, interpretation that takes the entire figure into account. We observe that each stair that is one step clockwise from its neighbor is also one step downward, and so we perceive the staircase as eternally descending. In principle, we could instead perceive the figure correctly as depicting four sets of stairs that are discontinu- ous, and viewed from a unique perspective—however such a percept never occurs.
This paper explores an analogous set of auditory figures that are composed of patterns that appear to ascend or descend endlessly in pitch. Here also, our perceptual system opts for impossible but simple interpretations, based on our perception of local motion in a particular direction–either upward or downward. These sound patterns are not mere curiosities; rather they provide important information con- cerning general characteristics of pitch perception.
Pitch as a two-dimensional attribute
By analogy with real-world staircases, pitch is often viewed as extending along a one-dimensional continuum of pitch
Fig. 1. An impossible staircase, similar to one devised by Penrose and Penrose.1
height. For sine waves, any significant increase or decrease in frequency is indeed associated with a corresponding increase or decrease in pitch—this is con- sistent with a one-dimensional represen- tation. However, musicians have long acknowledged that pitch also has a circu- lar dimension, known as pitch class— tones that stand in octave relation have a certain perceptual equivalence. The sys- tem of notation for the Western musical scale accommodates this circular dimen- sion. Here a note is designated by a letter
which refers to its position within the octave, followed by a number which refers to the octave in which the tone occurs. So as we ascend the scale in semitone steps, we repeatedly traverse the pitch class circle in clockwise direction, so that we play C, C#, D, and so on around the circle, until we reach C again, but now the note is an octave higher. Similar schemes are used in Indian musical notation, and in those of other musical cultures.
To accommodate both the rectilinear and circular
dimensions of pitch, a number of theorists—going back at
least to Drobisch in the mid-nineteenth century—have
argued that this be represented as a helix having one com-
plete turn per octave, so that pairs of points that are separat-
ed by an octave stand in close spatial proximity (Fig. 2).
Based on such a representation, Roger Shepard, then at Bell
Telephone Laboratories, conjectured that it might be possible
to exaggerate the dimension of pitch class and minimize the
dimension of height, so that all tones that are related by
octaves would be mapped onto a single tone which would
have a well-defined pitch class but an indeterminate height.
Because the helix would then be collapsed into a circle, judg-
ments of relative pitch for such tones should be completely
2, 3
by Max Mathews,4 Shepard synthesized a bank of complex tones, each of which consisted of 10 partials that were sepa- rated by octaves. The amplitudes of the partials were scaled by a fixed, bell-shaped spectral envelope, so that those in the middle of the musical range were highest, while the ampli- tudes of the others fell off gradually along either side of the log frequency continuum, sinking below the threshold of audibility at the extremes (Fig. 3). Such tones are well defined in terms of pitch class (C, C#, D; and so on) but poorly defined in terms of height, since the other harmonics that would provide the usual cues for height attribution are miss- ing. Using such a bank of tones, one can then vary the dimen- sions of height and pitch class independently. To vary height alone one can keep the partials constant but rigidly shift the spectral envelope up or down in log frequency; to vary pitch class alone one can rigidly shift the partials in log frequency, while keeping the position of the spectral envelope constant.
8 Acoustics Today, July 2010
circular.
Using a software program for music synthesis generated