According to the delta rule, as adapted shortly, the weights are assumed to
be random initially, representing a naive network. A tonal composite of a key
as input will initially result in a random pattern of expectations. This random
pattern of expectations is compared with the input pattern, and the weights are
changed so as to reduce the disparity. The weight change is incremental; each
time the network generates expectations in response to a tonal composite, the
weights change slightly so that the next time that tonal composite (or one sim-
ilar to it) is encountered, the expectations will more closely approximate the
input.
Ifaiis the activation of input uniti,andaeis the activation of expectation unit
e, then the disparity is called theerror signalðdeÞ, and is simply the difference:
de¼aiae:Ifwieis the weight on the link fromitoe,thenthechangeðDwieÞin the weight
is simply the activation ofitimes the error signal ate, scaled by a constant,
eð 0 aea 1 Þthat represents the learning rate:
Dwie¼eaide:The learning rate determines the extent to which a single experience can have a
lasting effect.
The patterns are presented repeatedly to the network until the error signal is
smaller than some criterion amount for all expectation units in response to all
patterns. Rosenblatt (1962) proved that, with this learning rule, a network with
two sets of units, one feeding into the other, will eventually be able to find a
solution (to any given degree of precision) if one exists. For an autoassociator
using the delta rule, a solution exists for any set of input vectors that areline-
arly independentof each other. A vector is linearly independent of a set of vec-
tors if it cannot be obtained by any combination of scalar multiplication and
addition of the other vectors, that is, it is not acompositeof any of the others.
Tonal composites for the 12 major keys are linearly independent of each other,
and modal composites for the Church modes are linearly independent of each
other. This is indeed a powerful system, because it can learn all these patterns
inthesamesetoflinks.
Two modes that have the same invariant pitch classes, albeit with different
probability distributions (e.g., major and natural minor), are not linearly inde-
pendent; this network would learn them as one pattern that is a composite of
the two. This is not a limitation of this scheme for modeling music cognition,
however, because it has never been suggested that tonal or modal composites
capture all features of music. These models can be expanded to include any
number of features that may discriminate two modes by the way in which they
are used. We have restricted ourselves to pitch classes or invariant pitch classes
in the present chapter only because we must begin somewhere and it behooves
us to understand something well before bringing all possible factors into play.
After learning, the model can be tested for its vaunted ability to complete
degraded patterns or to assimilate similar patterns to learned ones. An autoas-
sociator was fed tonal composites, in each of the 12 major keys, representing
the average probability distributions of works by Schubert, Mendelssohn,
Schumann, Mozart, Hasse, and Strauss (as reported by Krumhansl, 1990, p. 68).
470 Jamshed J. Bharucha