4.18: Perceptrons and Jazz Progressions

We have seen that a particular type of network from Old Connectionism, the perceptron, can be usefully applied in the studies of classical conditioning and reorientation. In the current section we see that it can also be used to explore musical regularities. Also illustrated is the interpretation of the internal structure of such a network, which demonstrates that even simple networks can reveal some interesting algorithmic properties.

Jazz progressions are sequences of chords. Consider the C major scale presented earlier, in Figure 4-8. If one takes the first note of the scale, C, as the root and adds every second note in the scale—E, G, and B)—the result is a four-note chord—a tetrachord—called the C major 7^th chord (Cmaj7). Because the root of this chord is the first note of the scale, this is identified as the I chord for C major. Other tetrachords can also be built for this key. Starting with the second note in the scale, D, and adding the notes F, A, and C produces D minor 7^th (Dm7). Because its root is the second note of the scale, this is identified as the II chord for the key of C major. Using G as the root and adding the notes B, D, and F creates the G dominant 7^th chord (G7). It is the V chord of the key of C major because its root is the fifth note of the C major scale.

The I, II, and V chords are the three most commonly played jazz chords, and in jazz they often appear in the context of the II-V-I progression (Levine, 1989). This chord progression involves playing these chords in a sequence that begins with the II chord, moves to the V chord, and ends on the I chord. The II-V-I progression is important for several reasons.

First, chord progressions are used to establish tonality, that is, to specify to the listener the musical key in which a piece is being played. They do so by setting up expectancies about what is to be played next. For any major key, the most stable tones are notes I, IV, and V (Krumhansl, 1990), and the most stable chords are the ones built on those three notes.

Second, in the perception of chord sequences there are definite preferences for the IV chord to resolve into the V chord and for the V chord to resolve into the I chord, producing the IV-V-I progression that is common in cadences in classical music (Bharucha, 1984; Jarvinen, 1995; Katz, 1995; Krumhansl, Bharucha, & Kessler, 1982; Rosner & Narmour, 1992). There is a similar relationship between the IV chord and 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 \(\PageIndex{1}\). 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 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 \(\PageIndex{1}\).

Figure \(\PageIndex{1}\). 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 \(\PageIndex{1}\) 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.

Another way to view the two classes of notes revealed by this analysis is in terms of the two circles of major seconds that were presented in Figure 4.10.4. The first circle of major seconds contains only those notes that have positive coordinates in Figure \(\PageIndex{1}\). The other circle of major seconds captures the set of notes that have negative coordinates in Figure \(\PageIndex{2}\). In other words, the jazz progression network acts as if it has classified notes in terms of the circles of major seconds!

The order in which the notes are arranged in the one-dimensional map is also related to the four circles of major thirds that were presented in Figure 4.10.5. The bars in Figure \(\PageIndex{1}\) have been colored to reveal four sets of three notes each. Each of these sets of notes defines a circle of major thirds. The MDS map places notes in such a way that the notes of one such circle are listed in order, followed by the notes of another circle of major thirds.

To summarize, one musical formalism is the II-V-I jazz progression. Interestingly, this formalism can be learned by a network from Old Connectionism, the perceptron. Even though this network is simple, interpreting its representations is not straightforward and requires the use of multivariate statistics. However, when such analysis is performed, it appears that the network captures the regularities of this jazz progression using the strange circles that were encountered in the earlier section on chord classification. That is, the connection weights of the perceptron reveal circles of major seconds and circles of major thirds.