Chordal and timbral morphologies using Hamiltonian cycles

Abstract

In this paper, we investigate several musical morphologies that can be represented as paths in abstract graphs. Our examples come from questions posed by composers in the compositional process. In particular, we focus on Hamiltonian paths and cycles, which are central to graph theory. Our results show the circumstances in which such a path exists in the graphs derived from these musical ideas.

Keywords:

• graph theory
• musical form
• musical shape
• musical structure
• morphology
• Hamiltonicity
• Tom Johnson

Acknowledgements

We are most grateful to the referees for reading the paper very carefully and for so many valuable remarks and suggestions. The use of Cartesian products of graphs was suggested by a referee which helped us to simplify the proof of Theorem 5.5 and correct a flawed argument in the proof of Lemma 5.2. We also would like to extend our gratitude to Thomas Fiore for so many helpful comments.