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

In the pitch class representation used for this network, each input unit stands for a
distinct musical note. As far as the hidden units are concerned, the “name” of each
note is provided by the connection weight between the input unit and the hidden
unit. Interestingly, Table 4-2 reveals that all three hidden units take input notes that
we would take as being different (because they have different names, as in the circle
of fifths in Figure 4-9) and treat them as being identical. That is, the hidden units
assign the same “name,” or connection weight, to input notes that we would give
different names to.
Furthermore, assigning the same “name” to different notes by the hidden units
is not done randomly. Notes are assigned according to strange circles, that is, circles
of major thirds and circles of major seconds. Let us briefly describe these circles,
and then return to an analysis of Table 4-2.
The circle of fifths (Figure 4-9) is not the only way in which notes can be
arranged geometrically. One can produce other circular arrangements by exploiting
other musical intervals. These are strange circles in the sense that they would very
rarely be taught to music students as part of a music theory curriculum. However,
these strange circles are formal devices that can be as easily defined as can be the
circle of fifths.
For instance, if one starts with the note C and moves up a major second (2
semitones) then one arrives at the note D. From here, moving up another major
second arrives at the note E. This can continue until one circles back to C but an
octave higher than the original, which is a major second higher than A#. This circle
of major seconds captures half of the notes in the chromatic scale, as is shown in the
top part of Figure 4-10. A complementary circle of major seconds can also be constructed
(bottom circle of Figure 4-10); this circle contains all the remaining notes
that are not part of the first circle.
C
F
G
A#
D
D#
A
G#
E
F#
Figure 4-10. The two circles of major seconds.
C# B
An alternative set of musical circles can be defined by exploiting a different musical
interval. In each circle depicted in Figure 4-11, adjacent notes are a major third (4
semitones) apart. As shown in Figure 4-11 four such circles are possible.
Elements of Connectionist Cognitive Science 167