This thesis is concerned with cognitively-motivated representations of musical structure. Three problems are addressed, each related in terms of their focus on music as an object of perception, and in the application of geometrical methods of knowledge representation. The problem of pattern discovery in discrete representations of polyphonic music is first considered, and a heuristic proposed which seeks to assist musicological analysis by identifying patterns that may be salient in perception, from a large number of potential patterns. This work is based on geometric principles that are far removed from plausible psychological models of pattern induction, but the method is motivated by psychological evidence for the importance of invariance and repetition in perception. The second and third problems explicitly adopt a cognitive theory of representation, namely the conceptual space framework developed by Gärdenfors (2000). Within this framework, concepts can be represented geometrically within perceptually grounded quality dimensions, and where distance in the space corresponds to similarity. The second problem concerns the prediction of melodic similarity, and the theory of conceptual spaces is investigated in the novel context of point set representations of melodic structure, employing the Earth Mover's Distance metric (Rubner 2000). This work builds on the work of Typke (2007) concerning the application of Earth Mover's Distance to melodic similarity. Evaluation is performed with respect to published psychological data (Müllensiefen 2004), and the MIREX 2005 symbolic melodic similarity evaluation. The third problem concerns the conceptual representation of metrical structure, informed by the psychological theory of metre developed by London (2004). A symbolic formalisation of this theory is developed, alongside two geometrical models of metrical-rhythmic structure, which are evaluated within a genre classification task.