Tonal harmony, the topology of dynamical score networks and the Chinese postman problem
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We introduce the concept of dynamical score networks for the representation and analysis of tonal compositions: a score can always be interpreted as a dynamical network where every chord is a node and each progression links successive chords. We demonstrate that in a tonal harmony context this network displays scale-free properties, and optimal (most economical) chord progressions can be found by solving a path optimization like the Chinese postman problem. Moreover, the dynamical network can be viewed as a time series of a non-stationary signal and as such can be partitioned for the automatic identification of key regions using well-established techniques for time series analysis and change point detection. Based on this interpretation we introduce a key-finding algorithm that does not rely on comparisons with pre-determined reference sets, as in the Krumhansl-Schmuckler model, or extensive corpora, as in machine-learning approaches. Finally, we show how the principles discussed in this work can be used to design a generative model of tonal compositional design.
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