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

the II chord if the latter is minor (Steedman, 1984). Thus the II-V-I progression is a
powerful tool for establishing the tonality of a musical piece.
Third, the II-V-I progression lends itself to a further set of chord progressions
that move from key to key, providing variety but also establishing tonality. After
playing the Cmaj7 chord to end the II-V-I progression for C major, one can change
two notes to transform Cmaj7 into Cm7, which is the II chord of a different musical
key, A# major. As a result, one can move from performing the II-V-I progression
in C major to performing the same progression in a major key one tone lower. This
process can be repeated; the full set of chord changes is provided in Table 4-6. Note
that this progression eventually returns to the starting key of C major, providing
another powerful cue of tonality.
Chord Progression For Key
Key II V I
C Dm7 G7 Cmaj7
A# Cm7 F7 A#maj7
G# A#m7 D#7 G#maj7
F# G#m7 C#7 F#maj7
E F#m7 B7 Emaj7
D Em7 A7 Dmaj7
C Dm7 G7 Cmaj7
Table 4-6. A progression of II-V-I progressions, descending from the key of C
major. The chords in each row are played in sequence, and after playing one
row, the next row is played.
A connectionist network can be taught the II-V-I chord progression. During training,
one presents, in pitch class format, a chord belonging to the progression. The
network learns to output the next chord to be played in the progression, again using
pitch class format. Surprisingly, this problem is very simple: it is linearly separable
and can be solved by a perceptron!
How does a perceptron represent this jazz progression? Because a perceptron
has no hidden units, its representation must be stored in the set of connection
weights between the input and output units. However, this matrix of connection
weights is too complex to reveal its musical representations simply by inspecting it.
Instead, multivariate statistics must be used.
First, one can convert the raw connection weights into a correlation matrix.
That is, one can compute the similarity of each pair of output units by computing
196 Chapter 4