Connectionism, Music and Fourier Phase Spaces

Abstract

How does the brain represent musical properties? Even with our growing understanding of the cognitive neuroscience of music (Abbott, 2002; Peretz and Zatorre, 2003; Peretz and Zatorre, 2005; Zatorre and McGill, 2005), 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 (Rumelhart and McClelland, 1986; Bechtel and Abrahamsen, 2002; Enquist and Ghirlanda, 2005). One could train networks to solve musical problems, (Todd and Loy, 1991; Griffith and Todd, 1999) and then study how these networks encode musical properties. However, researchers rarely conduct detailed examinations of network structure(Dawson, 2009, 2013, 2018) because networks are difficult to interpret, and because it is assumed that networks capture informal or subsymbolic properties (Smolensky, 1988; McCloskey, 1991; Bharucha, 1999). Within this thesis, we report very high correlations between network connection weights and discrete Fourier phase spaces used to represent musical sets (Amiot, 2016; Callender, 2007; Quinn, 2006, 2007; Yust, 2016). This is remarkable because there is no clear mathematical relationship between network learning rules and discrete Fourier analysis (Rumelhart, Hinton et al., 1986; Dawson and Schopflocher, 1992; Amiot, 2016). 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.