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

chords. For instance, none of the hidden units by themselves pick out a set of properties
that uniquely define a particular type of chord. Instead, hidden units generate
some activity to different chord types, suggesting the existence of a coarse code.
In order to see how the activities of the hidden units serve as a distributed
representation that mediates chord classification, we must examine the hidden unit
space. The hidden unit space plots each input pattern as a point in a space whose
dimensionality is determined by the number of hidden units. The coordinates of
the point in the hidden unit space are the activities produced by an input pattern in
each hidden unit. The three-dimensional hidden unit space for the chord classification
network is illustrated in Figure 4-12.
1.0
6th
Major7
H3
0.5
1.0
H2 0.5
Dom7
Major
0.0
1.0
0.5
H1
0.0
Figure 4-12. The hidden unit space for the chord classification network. H1, H2,
and H3 provide the activity of hidden units 1, 2, and 3 respectively.
Because the hidden units generate identical responses to many of the chords,
instead of 48 different visible points in this graph (one for each input pattern), there
are only 8. Each point represents 6 different chords that fall in exactly the same
location in the hidden unit space.
The hidden unit space reveals that each chord type is represented by two different
points. That these points capture the same class is represented in Figure 4-12
by joining a chord type’s points with a dashed line. Two points are involved in defining
a chord class in this space because, as already discussed, each hidden unit is
sensitive to the organization of notes according to the two circles of major seconds.
For each chord type, chords whose root belongs to one of these circles are mapped
Elements of Connectionist Cognitive Science 171