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Invariance in Perception

For example in the perception of melody, we might assume that the sensory signal begins with sensory detectors tuned to certain specific pitches, or tones, represented by the keyboard at left, below. A melody would then be represented by a sequence of tones recorded in some kind of time trace, as suggested here for the first part of the tune "Mary Had a Little Lamb". A detector tuned to that melody would have to have a connectivity pattern somewhat like the melody template suggested below.

But to extend this kind of model to recognize the melody independent of pitch we must replicate this melody detector across the full range of pitches, as shown in the center (above). To also recognize a melody independent of tempo, requires replication of all of these detectors at every different tempo, as suggested on the right (above). For full invariance to pitch and tempo requires separate templates for every tempo, each replicated for every pitch.

As Ehrenfels recognized at the dawn of the Gestalt era, a straightforward model of perceptual invariance results in a combinatorial explosion in the required number of pattern detectors.
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