Mozart’s Sonata No. 16 (KV 545)

Three different representations of Mozart’s Sonata No. 16 (KV 545) first bar: (a) A piece of music can be modelled as a dynamic system with a set of musical notes evolving over time. (b) Traditional musical score. (c) A musical network that takes chords (harmony) into account. The roman numerals represent the order in which each edge was created. In that particular case, all edges have weight equal to 1, because none of them were used more than once.

Previous work suggests that musical networks often present the scale-free and the small-world properties. From a musician's perspective, the most important aspect missing in those studies was harmony. In addition to that, the previous work made use of outdated statistical methods. Traditionally, least-squares linear regression is utilised to fit a power law to a given data set. However, according to Clauset et al. such a traditional method can produce inaccurate estimates for the power law exponent. In this paper, we present an analysis of musical networks which considers the existence of chords (an essential element of harmony). Here we show that only 52.5% of music in our database presents the scale-free property, while 62.5% of those pieces present the small-world property. Previous work argues that music is highly scale-free; consequently, it sounds appealing and coherent. In contrast, our results show that not all pieces of music present the scale-free and the small-world properties. In summary, this research is focused on the relationship between musical notes (Do, Re, Mi, Fa, Sol, La, Ti, and their sharps) and accompaniment in classical music compositions.

Main References

1. X. F. Liu, C. K. Tse, M. Small, Complex network structure of musical compositions: Algorithmic generation of appealing music, Physica A 389 (2010) 126-132.

2. T. J. Perkins, E. Foxall, L. Glass, R. Edwards, A scaling law for random walks on networks, Nat Commun 5 (2014).

3. A. Clauset, C. R. Shalizi, M. E. J. Newman, Power-law distributions in empirical data, SIAM Rev. 51 (2009) 661-703.