work develops redundant pathways for the high-probability transitions (e.g.,
moving to the tonic), leading to the rapid activation of some units. Whether
or not the veridical expectancies are also schematic is therefore revealed by
the time course of activation. When the unusual transition involves not just a
low-probability transition but a move to a low-probability event (such as a
secondary dominant), the expectancy violation does not require a cascading
explanation; it is trivially accounted for by the disparity between the expecta-
tion of the sequential net and the autoassociator.
D. Transpositional Invariance
Pitch-class representations can be transformed into invariant pitch-class repre-
sentations by a simple gating mechanism (Bharucha, 1988, 1991; McNellis,
1993). This mechanism, shown in figure 19.10, is similar to one developed
by Hinton (1981) for object recognition in vision. The units labeled ‘‘p’’ multiply
the activation that feeds into them, thereby serving as ‘‘AND’’ gates. The units
arrayed vertically on the left are key units from MUSACT, and the units
arrayed horizontally at the bottom are pitch-class units. The most active key
unit gates the activation from pitch-class units into the pitch invariant repre-
sentation at the top. If the key is Cg, then C is gated to 11 and Cgto 0. If the key
is C, then C is gated to 0 and Cgto 1. This can account for the invariance of
pitch sequences under transposition.
III. Discussion
Neural net models are not intended to be statements of fact about how the
brain is wired. Like all models, they are systematic hypotheses based on avail-
able data, and they represent attempts to account for known phenomena and
guide further research. Some neural net models may be sufficiently closely tied
to known physiology that they serve as hypotheses of actual neural circuitry.
For most examples of complex human behavior, however, the precise circuitry
Figure 19.10
A network for transforming a pitch-class representation into an invariant pitch-class representation.
Units labeledpgate activation received from pitch class units and tonal center (key) units. (From
Bharucha, 1988.)
Neural Nets, Temporal Composites, and Tonality 475