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

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Figure 4-11. The four circles of major thirds.
What do these strange circles have to do with the internal structure of the network
trained to classify the different types of chords? A close examination of Table
4-2 indicates that these strange circles are reflected in the connection weights that
feed into the network’s hidden units. For Hidden Units 1 and 2, if notes belong
to the same circle of major thirds (Figure 4-11), then they are assigned the same
connection weight. For Hidden Unit 3, if notes belong to the same circle of major
seconds (Figure 4-10), then they are assigned the same connection weight. In short,
each of the hidden units replaces the 12 possible different note names with a much
smaller set, which equates notes that belong to the same circle of intervals and differentiates
notes that belong to different circles.
Further inspection of Table 4-2 reveals additional regularities of interest.
Qualitatively, both Hidden Units 1 and 2 assign input notes to equivalence classes
based on circles of major thirds. They do so by using the same note “names”:
0.53, 0.12, –0.12, and –0.53. However, the two hidden units have an important difference:
they assign the same names to different sets of input notes. That is, notes
that are assigned one connection weight by Hidden Unit 1 are assigned a different
connection weight by Hidden Unit 2.
The reason that the difference in weight assignment between the two hidden
units is important is that the behaviour of each hidden unit is not governed by a
single incoming signal, but is instead governed by a combination of three or four
168 Chapter 4