Mind, Body, World- Foundations of Cognitive Science, 2013a

missing their fundamental frequency.
Consider a sine wave whose frequency is f. When we hear a musical sound,
its pitch (i.e., its tonal height, or the note that we experience) is typically associated
with this fundamental frequency (Helmholtz & Ellis, 1954; Seashore, 1967).
The harmonics of this sine wave are other sine waves whose frequencies are integer
multiples of f (i.e., 2f, 3f, 4f and so on). The timbre of the sound (whether we can
identify a tone as coming from, for example, a piano versus a clarinet) is a function
of the amplitudes of the various harmonics that are also audible (Seashore, 1967).
Interestingly, when a complex sound is filtered so that its fundamental frequency
is removed, our perception of its pitch is not affected (Fletcher, 1924). It is as
if the presence of the other harmonics provides enough information for the auditory
system to fill in the missing fundamental, so that the correct pitch is heard—a phenomenon
Schumann exploited in Humoreske. Co-operative interactions amongst
neurons that detect the remaining harmonics are likely responsible for this effect
(Cedolin & Delgutte, 2010; Smith et al., 1978; Zatorre, 2005).
Artificial neural networks can easily model such co-operative processing and
complete the missing fundamental. For instance, one important connectionist
system is called a Hopfield network (Hopfield, 1982, 1984). It is an autoassociative
network that has only one set of processing units, which are all interconnected.
When a pattern of activity is presented to this type of network, signals spread rapidly
to all of the processors, producing dynamic interactions that cause the network’s
units to turn on or off over time. Eventually the network will stabilize in a
least-energy state; dynamic changes in processor activities will come to a halt.
Hopfield networks can be used to model virtual pitch, because they complete
the missing fundamental (Benuskova, 1994). In this network, each processor represents
a sine wave of a particular frequency; if the processor is on, then this represents
that the sine wave is present. If a subset of processors is activated to represent
a stimulus that is a set of harmonics with a missing fundamental, then when the
network stabilizes, the processor representing the missing fundamental will be also
activated. Other kinds of self-organizing networks are also capable of completing
the missing fundamental (Sano & Jenkins, 1989).
An artificial neural network’s ability to deal with noisy inputs allows it to
cope with other domains of musical cognition as well, such as assigning rhythm
and metre (Desain & Honing, 1989; Griffith & Todd, 1999). Classical models of this
type of processing hierarchically assign a structure of beats to different levels of
a piece, employing rules that take advantage of the fact that musical rhythm and
metre are associated with integer values (e.g., as defined by time signatures, or
in the definition of note durations such as whole notes, quarter notes, and so on)
(Lerdahl & Jackendoff, 1983; Temperley, 2001). However, in the actual performance
of a piece, beats will be noisy or imperfect, such that perfect integer ratios of beats
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