Wednesday, 27 February 2013

Complex Network Application in Music: Uncovering Universal Properties in Favourable Human Perception of Music

Across cultures and between individuals ,certain musical pieces are consistently rated more favorably than the others and a mathematical analysis of musical perception has a long history. But recently,a data-driven transformation to represent a musical score as a complex network has been tried by network scientist. What has been found is that those musical scores which are widely perceived to be 'good' generate complex network with certain invariant properties: scale-free networks with strong clustering of nodes within the network.They have also tried to generate random musical compositions from these networks and surprisingly found that scores generated in this manner are also perceived to be 'good' and are qualitatively similar to the specific score from which the generating network was produced.

Section of Mozart's Sonata

The network generated from entire Mozart's Sonata
Building the Network:
A musical score , or even the performance of a particular musician can be represented as a MIDI(musical instrument digital interface) file. The MIDI file encodes a musical composition as a series of events ,where each event includes information describing both the pitch and timing of a note.If a MIDI file is transcribed from a musical score, this information will be precise; if the information is derived from an actual performance there will be more variability. The network is then constructed the following way: Each MIDI event (a single musical note) corresponds to a node in the complex network. Two nodes in the network are linked by an edge if they succeed one another in the musical score. Weight is ascribed to a given edge based on the relative frequency with which the two nodes are adjacent. Direction is based on temporal order.
             This transformation was applied to a wide variety of musical composition including the sonatas of Bach and Mozart, Bach's "Well-Tempered Clavier", Chopin's waltzes, Russian folk music and Cantonese pop. In all cases the networks generated by this algorithm exhibit a scale-free distribution (using the maximum likelihood procedure at the 95% confidence interval) on the node degree (that is, the probability of a node having degree k, p(k) = k-gamma ) with degree exponent gamma falling in a narrow range 1 < gamma < 1.7. As a natural consequence of variability in a human performance it was observed that for MIDI files derived from actual rendition, the clustering within the network was much lower (clustering coefficient 0.127 +/- 0.069) in comparison to MIDI derived from musical score (0.37 +/- 0.056). Nonetheless, in all cases the scale exponent falls within a fairly narrow range. Notably, the scale exponents for Russian folk music (1.18) and Cantonese pop (1.01) are significantly lower than those derived from classical music (1.29 ~ 1.67).

Figure on the right: Degree distribution for the same Mozart's sonata network (number of links k versus frequency p(k).

These complex networks therefore encapsulate some features of the underlying musical scores used to generate them. These networks were then used to generate new artificial scores.  A random initial node on the network was chosen and randomly followed one of the links from that node to another based on the relative weight of the links. This procedure was repeated and the sequence of nodes that were traversed were recorded. The corresponding notes (both pitch and duration) gave a random score. If one was to then generate a new network from that score it would (asymptotically) be equivalent to the original. The surprising and intriguing consequence of this procedure was that the random scores were both pleasing and qualitatively similar to the original score. That is, random scores generated from the network derived from Chopin's sonatas also sound like Chopin, those derived from the Cantonese pop network also sound like Cantopop . Of course, these scores lacks the large scale structure of the original . But, this does demonstrate that the complex network encapsulates enough information to quantify basic features of a particular composition or style. As with actual composition, large scale structure can be imposed aposteriori and selection from between many candidate random scores can be used to produce "optimal" compositions.


1. X. Liu, C.K. Tse and M. Small, "Composing music with complex networks," International Conference on Complex Sciences: Theory and Applications, (COMPLEX2009), Shanghai, pp. 2196-2205, February 2009.
2. X. Liu, C.K. Tse and M. Small, "Complex network structure of musical compositions: Algorithmic generation of appealing music," Physica A, vol. 389, no. 1, pp. 126-132, January 2010.

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