The convention of representing pitch and key relations within a geometric scheme has a long history. Such schemes often emphasize perceptual similarities or differences among tones or keys. In the present work, we focus on the dynamics of perception of pitch movement, within the framework of geometric models. In the first two experiments, perception of the pitch pattern of pairs of Shepard tones (R. N. Shepard, 1964) is examined in three different orderings: (1) random permutation of tone pairs, (2) sequential increases in the frequency components of the second tone of each pair, and (3) sequential decreases in the second tone's frequency components. Consistent with previous reports, when tone pairs are randomly permuted, the pitch pattern is equally likely to be judged as ascending or descending as the frequency difference between tones nears the half-octave. In the ordered conditions, the boundary between ascending and descending pitch is sensitive to the direction of frequency change such that hysteresis, or perceptual assimilation, is observed. In Experiment 3, we obtain pitch judgments of all two-tone permutations of Shepard tones of the chromatic scale, then map the judgments onto a toroidal stimulus space formed by the product of two pitch circles. Perceptual dynamics are explored by systematic excursions through the stimulus space. The results indicate that spatial models of pitch provide an incomplete description of the higher than/lower than pitch relationship in Shepard tones; also crucial is the path taken through the space defined by the stimuli.
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