Modeling the Perception of Tonal Structure with Neural Nets

ly summarized by the circle of fifths. Analogously, a net exposed to Indian ragas fills in expected tones when presented with subsets of the raga tones. An auto-associative net trained on the scales of one culture can be tested with the scales of another, making predictions about tonal implications 46 Computer Music Journal This content downloaded from 157.55. 39.231 on Thu, 06 Oct 2016 04:43:25 UTC All use subject to http://about.jstor. org/terms generated in the minds of listeners hearing an unfamiliar form of music. A net trained on the Western major and minor scales seems to assimilate some Indian ragas to the Western scales, sometimes shifting the tonic (Bharucha and Olney 1989). Learning Hierarchical Representations Through Self-Organization Hierarchical relationships, such as between tones, chords, and keys, can be learned passively by algorithms for self-organization (Kohonen 1984; Linsker 1986; Rumelhart and McClelland 1986; Carpenter and Grossberg 1987). Most self-organization mechanisms assume the prior existence of abstract units into which the input units feed. These abstract units initially have no specialization, since the links from the input units are initially random. However, repeated exposure to commonly occurring patterns causes some of these abstract units to tune their responses to these patterns. One of the more straightforward, self-organization algorithms, called competitive learning (Rumelhart and McClelland 1986) accomplishes this as follows. For any given pattern some arbitrary abstract unit will respond more strongly than any other, simply because the weights are initially random. Of the links that feed into this unit, those that contributed to its activation are strengthened and the others are weakened. This unit's response will subsequently be even stronger in the presence of this pattern and weaker in the presence of other, dissimilar patterns. In similar fashion, other abstract units learn to specialize to other patterns. This process can be continued to even more abstract layers, at which units become tuned to patterns that commonly occur in the lower layer. The overwhelming preponderance of major and minor chords in the popular Western musical environment would drive such a net to form units that respond accordingly. Furthermore, the typical combinations in which these chords are used would drive units at a more abstract layer to register larger organizational units such as keys. The notion that individual neurons specialize to respond to complex auditory patterns has some preliminary empirical support from single-cell recording studies on animals (Weinberger and McKenna 1988). Once these chord and key units have organized themselves, the net models the implication of tones, chords, and keys given a set of tones. A hierarchical constraint satisfaction net built on this organization has been reported in earlier work (Bharucha 1987a; 1987b). In this net, called MUSACT, activation spreads from tone units to chord and key units and reverberates phasically through the net until a state of equilibrium is achieved. At equilibrium, all constraints inherent in the net have been satisfied. Given a key-instantiating context, the unit representing the tonic becomes the most highly activated. The other chord units are activated to lesser degrees the further they are from the tonic along the circle of fifths. Two behaviors of the net illustrate its emergent properties. First, the above activation pattern does not require the tonic chord to be played at all. An F major chord followed by a G major chord will cause the C major chord unit to be the most highly activated. Second, the circle of fifths implicit in the activation pattern cannot be accounted for on the basis of shared tones alone. If a C major context chord is played, the D major chord unit is more highly activated than the A major chord unit, even though the latter shares one tone with the sounded chord (C major) and the former shares none at all. A careful tracking of the net's behavior as activation reverberates and before it converges to an equilibrium state reveals a lower initial activiation of D major over A major, reflecting an initial bottom-up influence of shared tones. As activation has a chance to reverberate back from the key units (a top-down influence), this advantage is lost, and D major overtakes A major. So the circle of fifths is truly an emergent property of the simultaneous satisfaction of elementary associations between tones and clusters of tones. See Bharucha (1987a; 1987b) for details. Learning With Sequential Nets Some of the schematic expectancies that are essentially sequential, as in chord progressions, can be modeled with sequential nets. The architecture