In this work, a novel representation system for symbolic music is described. The proposed representation system is graph-based and could theoretically represent music both from a horizontal (contrapuntal) and from a vertical (harmonic) point of view, by keeping into account contextual and harmonic information. It could also include relationships between internal variations of motifs and themes. This is achieved by gradually simplifying the melodies and generating layers of reductions that include only the most important notes from a structural and harmonic viewpoint. This representation system has been tested in a music information retrieval task, namely melodic similarity, and compared to another system that performs the same task but does not consider any contextual or harmonic information, showing how the structural information is needed in order to find certain relations between musical pieces. Moreover, a new dataset consisting of more than 5000 leadsheets is presented, with additional meta-musical information taken from different web databases, including author, year of first performance, lyrics, genre and stylistic tags.