tuning is an interesting issue that is beyond the scope of this chapter. For pres-
ent purposes, it suffices to think of the set of pitch or pitch-class detectors with
semitone spacing as a subset of the more dense array that we know to exist.
Semitone spacing is thus not an additional postulate in these models, because if
the dense array were used, the feature detectors between the semitones would
simply not play much of a role (Bharucha, 1991).
When modeling the learning of musical sequences that are invariant across
transposition, aninvariant pitch-class representationis appropriate (Bharucha,
1988, 1991). A complete invariant pitch-class representation would have 12
units corresponding to the 12 pitch-class intervals above the tonic, which may
be referred to as Units 0 through 11 for tonic through leading tone, respec-
tively. Note that in an invariant pitch-class representation, a melody is con-
ceived not as a series of melodic intervals but as a series of scale degrees (i.e.,
intervals between each note and the tonic). The mapping from pitch class to
invariant pitch class can be accomplished by a circuit as described in Section
II,D.
Gjerdingen (1989b) uses an invariant pitch-class representation that is re-
stricted to the major diatonic scale (do, re, me,etc.),withtwoextraunitsrepre-
senting sharp and flat, respectively. Although the scale degrees can be mapped
from pitch-class representations (Section II,D), it is not clear how units repre-
senting sharp and flat are acquired.
C. Activation
Precisely how a neuron responds to the features to which it is tuned varies.
Neurons in the auditory nerve with characteristic frequencies below 4000 Hz
tend to spike at preferred intervals of time that correspond to integer multiples
of one cycle of the characteristic frequency. The probability distributions of
these interspike intervals are extremely provocative, given the simple integer
ratios they generate quite naturally, and suggest a timing code for pitch (Car-
iani & Delgutte, 1992), harmony (Tramo, Cariani, & Delgutte, 1992), and possi-
bly rhythm.
Beyond the auditory nerve, little evidence exists for timing as a coding strat-
egy. In the cochlear nucleus (the first junction from the auditory nerve to the
brain) and beyond, a neuron typically fires more rapidly the more intense the
tone, or the closer the tone is to its characteristic frequency. The more rapidly a
neuron fires, the more pronounced is its effect on neurons to which it is con-
nected, by virtue of the temporal summation that occurs at the receiving neu-
ron. Firing rate is thus taken to be the measure of response strength for most
neurons, and frequencies are represented by aspatial code—which neurons are
firing and how strongly—rather than by a timing code.
Most neural net models of pitch and tonality use spatial codes, and time
enters the coding scheme by changing a spatial code over time. In a spatial
code, each neuron has some response strength oractivationat any given time.
Activation is an abstract term for response strength and entails no commitment
to an underlying mechanism, although firing rate and temporal summation
lend neurophysiological plausibility to the postulation of activation as a theo-
retical construct for modeling cognitive phenomena. The term activation is also
used in models in which the units are postulated as cognitive rather than neu-
458 Jamshed J. Bharucha