Optimality in musical melodies and harmonic progressions: The travelling musician

摘要

A 'chord' is a collection of notes sounded simultaneously. A 'melody' is a path through a sequence of chords, and can be represented as a graph. The paper describes an algorithmic approach to the generation of tiles that represent chord sequences, and a method for evaluating remarkable paths through these sequences. Traditional harmony is based on a set of rules such as variety, connectivity, tiling and enumeration that eventually produces a "beautiful" and "interesting" melody/chord relationship. A set of rules and criteria are defined in this paper to evaluate this melody/chord relationship. The proposed compositional programme contains two algorithms: (1) The first generates a list of possible sets of chords fulfilling the optimality criteria. (2) The second connects them and evaluates the criteria for each path. The optimal connection is a case of minimal spanning tree. A selection of the best tiles is then made with the help of the composer. A beautiful tile might show - or not show - high symmetry, and remarkable connectivity. The best tiles given by the algorithms are compared to the ones in traditional use. Furthermore, a category of chord-sequences, which we shall call "karo', shows interesting mathematical properties.