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

the case that Austro-German music is (for example) formal while modern music
is not; instead, it may be more appropriate to claim that the former is more formal
(or more centrally controlled, or less improvised, or hotter) than the latter. The
possibility of quantitative distinctions raises the possibility that different types of
theories can be applied to the same kind of music, and it also suggests that one
approach to musical cognition may benefit by paying attention to the concerns of
another.
The likelihood that one approach to musical cognition can benefit by heeding
the concerns of another is easily demonstrated. For instance, it was earlier argued
that musical Romanticism was reflected in connectionism’s assumption that artificial
neural networks could capture regularities that cannot be formalized. One
consequence of this assumption was shown to be a strong preference for the use of
unsupervised networks.
However, unsupervised networks impose their own tacit restrictions upon what
connectionist models can accomplish. One popular architecture used to study musical
cognition is the Kohonen network (Kohonen, 1984, 2001), which assigns input
patterns to winning (most-active) output units, and which in essence arranges these
output units (by modifying weights) such that units that capture similar regularities
are near one another in a two-dimensional map. One study that presented such a
network with 115 different chords found that its output units arranged tonal centres
in a pattern that reflected a noisy version of the circle of fifths (Leman, 1991).
A limitation of this kind of research is revealed by relating it to classical work
on tonal organization (Krumhansl, 1990). As we saw earlier, Krumhansl found two
circles of fifths (one for major keys, the other for minor keys) represented in a spiral
representation wrapped around a toroidal surface. In order to capture this elegant
representation, four dimensions were required (Krumhansl & Kessler, 1982). By
restricting networks to representations of smaller dimensionality (such as a twodimensional
Kohonen feature map), one prevents them from detecting or representing
higher-dimensional regularities. In this case, knowledge gleaned from classical
research could be used to explore more sophisticated network architectures
(e.g., higher-dimensional self-organized maps).
Of course, connectionist research can also be used to inform classical models,
particularly if one abandons “gee whiz” connectionism and interprets the internal
structure of musical networks (Dawson, 2009). When supervised networks are trained
on tasks involving the recognition of musical chords (Yaremchuk & Dawson, 2005;
Yaremchuk & Dawson, 2008), they organize notes into hierarchies that capture circles
of major seconds and circles of major thirds, as we saw in the network analyses
presented in Chapter 4. As noted previously, these so-called strange circles are
rarely mentioned in accounts of music theory. However, once discovered, they are
just as formal and as powerful as more traditional representations such as the circle
310 Chapter 6