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

The circle of fifths includes all 12 notes in a chromatic scale, but arranges them
so that adjacent notes in the circle are a musical interval of a perfect fifth (i.e., 7
semitones) apart. The circle of fifths is a standard topic for music students, and it
is foundational to many concepts in music theory. It is provided here, though, to be
contrasted later with “strange circles” that are revealed in the internal structure of a
network trained to identify musical chords.
Any one of the notes in the circle of fifths can be used to define a musical key
and therefore can serve as the root note of a major scale. Similarly, any one of these
notes can be the root of a major triad created using the pattern of root + 4 semitones
+ 3 semitones that was described earlier for the key of C major (Baker, 1982).
Furthermore, the rules described earlier can also be applied to produce added note
chords for any of the 12 major key signatures. These possible major triads and added
note chords were used as inputs for training a network to correctly classify different
types of chords, ignoring musical key.
A training set of 48 chords was created by building the major triad, as well as
the major 7 th , dominant 7 th , and 6 th chord for each of the 12 possible major key signatures
(i.e., using each of the notes in Figure 4-9 as a root). When presented with
a chord, the network was trained to classify it into one of the four types of interest:
major triad, major 7 th , dominant 7 th , or 6 th . To do so, the network had 4 output units,
one for each type of chord. For any input, the network learned to turn the correct
output unit on and to turn the other three output units off.
The input chords were encoded with a pitch class representation (Laden
& Keefe, 1989; Yaremchuk & Dawson, 2008). In a pitch class representation, only
12 input units are employed, one for each of the 12 different notes that can appear
in a scale. Different versions of the same note (i.e., the same note played at different
octaves) are all mapped onto the same input representation. For instance, notes
1, 13, 25, and 37 in Figure 4-7 all correspond to different pitches but belong to the
same pitch class—they are all C notes, played at different octaves of the keyboard. In
a pitch class representation, the playing of any of these input notes would be encoded
by turning on a single input unit—the one unit used to represent the pitch class of C.
A pitch class representation of chords was used for two reasons. First, it requires
a very small number of input units to represent all of the possible stimuli. Second, it
is a fairly abstract representation that makes the chord classification task difficult,
which in turn requires using hidden units in a network faced with this task.
Why chord classification might be difficult for a network when pitch class
encoding is employed becomes evident by thinking about how we might approach
the problem if faced with it ourselves. Classifying the major chords is simple: they
are the only input stimuli that activate three input units instead of four. However,
classifying the other chord types is very challenging. One first has to determine what
key the stimulus is in, identify which three notes define its major chord component,
164 Chapter 4