is largely a mystery. We have a large and rapidly growing body of knowledge
about the physiology of single neurons and about the functions, and con-
nections between, some macroscopic regions of the brain. We also have a pre-
liminary understanding of how different types of neurons are interconnected
within the parts of the brain that are thought to play major roles in cognition,
namely, the cerebral cortex and the cerebellum. However, we know little about
the specific circuits that underlie specific phenomena, and even when the cir-
cuits are known, it is sometimes not clear why the circuit behaves the way it
does. Neural net models are attempts to bridge the gap between what we do
and do not know.
In the opinion of some authors (e.g., Fodor & Pylyshyn, 1988), the con-
nectionist conception of cognition and the representations that underlie it con-
trast sharply with rule-based systems, the latter being the hallmark of computer
programming languages and the grammars of modern linguistic theory (Chom-
sky, 1980). Fodor (1975; Fodor & Pylyshyn, 1988) argues, among other things,
that the mind is a formal symbol-manipulating device, in which a fairly clear
distinction is made between syntactic form and semantic content. Although
many of his arguments are specific to the study of language, one incisive argu-
ment derives from his critique of connectionism (Fodor & Pylyshyn) together
with his theory of the relationship between mental and physical states (Fodor,
1975). Roughly, he contends that connectionist models are merely models of
implementation. Because there may be radically different implementations of
the same symbolic process (e.g., there may be radically different hardware
designs that can implement the same computer program), an understanding of
one implementation does not entail an understanding of the formal symbolic
process it implements, any more than an understanding of the electrical activity
in the circuits of a computer chip entails an understanding of the program it is
running.
This is a powerful argument, but a careful analysis is beyond the scope of
this chapter. If the argument is correct, connectionist modelers will have to
settle for trying to understand how the brain—amereimplementation, but
what an implementation it is!—implements the formal symbolic processes that
we call music cognition. I suspect, however, that although highly trained musi-
cians may use formal symbolic processes together with a host of other pro-
cesses, the passive processing of music by most listeners is minimally symbolic.
Whatthendoesonemakeofrule-basedtheoriesofmusic,suchasthatofLer-
dahl and Jackendoff (1983)? These can be construed as formalizations of con-
straints on neural processing of music. In other words, either neural nets are
implementations of grammars, or grammars are formal descriptions of neural
nets. Future research will need to bridge the gap either way.
Appendices
A. Collinearity of Vectors
Ifvis a vector andsis a positive scalar, then their product,sv,isavectorthatis
collinear, that is, will point in the same direction. The multiplication of a vector
by a scalar is the vector resulting from multiplying each component by the
476 Jamshed J. Bharucha