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

Another way to view the two classes of notes revealed by this analysis is in terms
of the two circles of major seconds that were presented in Figure 4-10. The first
circle of major seconds contains only those notes that have positive coordinates in
Figure 4-16. The other circle of major seconds captures the set of notes that have
negative coordinates in Figure 4-16. In other words, the jazz progression network
acts as if it has classified notes in terms of the circles of major seconds!
The order in which the notes are arranged in the one-dimensional map is also
related to the four circles of major thirds that were presented in Figure 4-11. The
bars in Figure 4-16 have been coloured to reveal four sets of three notes each. Each
of these sets of notes defines a circle of major thirds. The MDS map places notes in
such a way that the notes of one such circle are listed in order, followed by the notes
of another circle of major thirds.
To summarize, one musical formalism is the II-V-I jazz progression. Interestingly,
this formalism can be learned by a network from Old Connectionism, the perceptron.
Even though this network is simple, interpreting its representations is not
straightforward and requires the use of multivariate statistics. However, when such
analysis is performed, it appears that the network captures the regularities of this
jazz progression using the strange circles that were encountered in the earlier section
on chord classification. That is, the connection weights of the perceptron reveal
circles of major seconds and circles of major thirds.
4.18 What Is Connectionist Cognitive Science?
The purpose of the current chapter was to introduce the elements of connectionist
cognitive science, the “flavour” of cognitive science that was seen first as Old
Connectionism in the 1940s (McCulloch & Pitts, 1943) and which peaked by the late
1950s (Rosenblatt, 1958, 1962; Widrow, 1962; Widrow & Hoff, 1960). Criticisms concerning
the limitations of such networks (Minsky & Papert, 1969) caused connectionist
research to almost completely disappear until the mid-1980s (Papert, 1988),
when New Connectionism arose in the form of techniques capable of training
powerful multilayered networks (McClelland & Rumelhart, 1986; Rumelhart &
McClelland, 1986c).
Connectionism is now well established as part of mainstream cognitive science,
although its relationship to classical cognitive science is far from clear. Artificial
neural networks have been used to model a dizzying variety of phenomena including
animal learning (Enquist & Ghirlanda, 2005; Schmajuk, 1997), cognitive development
(Elman et al., 1996), expert systems (Gallant, 1993), language (Mammone,
1993; Sharkey, 1992), pattern recognition and perception (Pao, 1989; Ripley, 1996;
Wechsler, 1992), and musical cognition (Griffith & Todd, 1999; Todd & Loy, 1991).
198 Chapter 4