Abstract
How does the brain represent musical properties? Even with our growing understanding of the cognitive neuroscience of music, the answer to this question remains unclear. One method for conceiving possible representations is to use artificial neural networks, which can provide biologically plausible models of cognition. One could train networks to solve musical problems, and then study how these networks encode musical properties. However, researchers rarely examine network structure in detail because networks are difficult to interpret, and because many assume that networks capture informal or subsymbolic properties. Here we report very high correlations between network connection weights and discrete Fourier phase spaces used to represent musical sets. This is remarkable because there is no clear mathematical relationship between network learning rules and discrete Fourier analysis. That networks discover Fourier phase spaces indicates that these spaces have an important role to play outside of formal music theory. Finding phase spaces in networks raises the strong possibility that Fourier components are possible codes for musical cognition.
Highlights
A main goal of studying musical cognition is identifying musical representations
For the 60 hidden units trained to classify triad types, each unit had a high correlation between their connection weights and a single phase space
How do artificial neural networks (ANNs) solve musical problems with the phase spaces they discover? Networks create a hidden unit space that codes each stimulus as a point whose coordinates are the hidden unit activities that it causes[12]
Summary
A main goal of studying musical cognition is identifying musical representations. Researchers seek musical representations by conducting psychological experiments[1,2,3], or by using the methods of cognitive neuroscience[4,5,6,7]. Musical set theory provides a quite different, formal, approach to studying music[16,17,18] Musical entities such as intervals, scales, or triads are combinations of pitch-classes that can be represented as musical sets. Musical set theory converts the twelve different Western pitch-classes (C, C#, D, D#, E, F, F#, G, G#, A, A#, B) into integers using the convention C = 0, C# = 1, and so on. Using this scheme the C major triad (C, E and G) becomes the set (0, 4, 7).
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