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

the correlation between the connection weights that feed into them. Once the
weights have been converted into correlations, further analyses are then available
to interpret network representations. Multidimensional scaling (MDS) can summarize
the relationships within a correlation matrix made visible by creating a
map (Kruskal & Wish, 1978; Romney, Shepard, & Nerlove, 1972; Shepard, Romney,
& Nerlove, 1972). Items are positioned in the map in such a way that the more similar
items are, the closer together they are in the map.
The MDS of the jazz progression network’s correlations produced a one-dimensional
map that provided a striking representation of musical relationships amongst
the notes. In a one-dimensional MDS solution, each data point is assigned a single
number, which is its coordinate on the single axis that is the map. The coordinate
for each note is presented in a bar chart in Figure 4-16.
2
D
E
A# F#
Position On Dimension 1
1
0
-1
B
D#
G
A
C#
F
C
G#
-2
Figure 4-16. Coordinates associated with each output note, taken from an MDS of
the Table 4-8 correlations. Shading reflects groupings of notes as circles of
major thirds.
The first regularity evident from Figure 4-16 is that half of the notes have negative
coordinates, while the other half have positive coordinates. That is, the perceptron’s
connection weights separate musical notes into two equal-sized classes. These
classes reflect a basic property of the chord progressions learned by the network: all
of the notes that have positive coordinates were also used as major keys in which the
II-V-I progression was defined, while none of the notes with negative coordinates
were used in this fashion.
Elements of Connectionist Cognitive Science 197