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Harmonic structures of Beethoven quartets: a complex network approach

Abstract

We propose a complex network approach to the harmonic structure underpinning western tonal music. From a database of Beethoven’s string quartets, we construct a directed network whose nodes are musical chords and edges connect chords following each other. We show that the network is scale-free and has specific properties when ranking algorithms are applied. We explore the community structure and its musical interpretation, and propose statistical measures stemming from network theory allowing to distinguish stylistically between periods of composition. Our work opens the way to a network approach of structural properties of tonal harmony.

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Data Availibility Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.]

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Acknowledgements

We thank A. Ouali, who was involved in a preliminary study. We thank Calcul en Midi-Pyrénées (CalMiP) for access to its supercomputers.

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Correspondence to Olivier Giraud.

Appendix A Cleaning the database

Appendix A Cleaning the database

We found several issues in the database, such as missing data or faulty labels, most of which are listed at the database website [28]. We made the following corrections to the file all\(\_\)annotations.tsv available at [28]. Global keys labeled ‘nothing’ or ‘false’ have been replaced by their correct value, given in the first column of the database. For some entries, the local key was labeled ‘Ab’ instead of ‘VI’: they were restored to their correct value. The local keys labeled ‘I’ at the beginning of some minor segments were relabeled ‘i’.

We also chose to consider chords within a pedal segment to be treated without reference to the pedal (although the first chord of a pedal segment is treated as distinct). As for entries labeled ’none’, i.e. chords for which no consensual harmonic analysis could be extracted from the score, we chose to treat them as a chord on its own.

In order to check the consistency of the corrections we made to the database, we compared our results with the ones obtained in [21]. In particular, for both major and minor segments, we calculated the list of frequencies of each chord type and the heatmaps (frequency of each sequence of pairs of chords), following [21]; the numerical outcomes we obtain is close to the ones obtained in [21]. The main difference is the frequency of ’I’ in minor segments, which ranks 14 in frequency order in our database but 2 in Ref. [21]. It is very likely that the corrections listed in [28] have been performed after [21] was published, which would explain this discrepancy.

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Frottier, T., Georgeot, B. & Giraud, O. Harmonic structures of Beethoven quartets: a complex network approach. Eur. Phys. J. B 95, 103 (2022). https://doi.org/10.1140/epjb/s10051-022-00368-z

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