Minimum description length for selection of models of musical rhythm

Abstract

We apply the minimum description length (MDL) methodology from information theory in order to select mathematical models for musical rhythm. We consider six models proposed by David Temperley and mathematically formalize them, which allows for an MDL analysis. As a consequence, we find that two of the models are not suitable to apply this methodology, as their codification strategy does not represent all possible rhythm sequences. A slight modification of Temperley's hierarchical model provides an improvement in codification performance, robust across all the four corpora considered in the experiments: three classical corpora commonly used in music studies and one of tango songs recently released. Our study confirms the usefulness of the MDL approach to solve the classical trade-off between model complexity and its ability to fit the data.

Keywords:

• Musical rhythm
• probabilistic modelling
• information theory
• minimum description length
• model selection

2010 Mathematics Subject Classification:

• 00A65
• 68Q30
• 62B10

2012 Computing Classification Scheme:

• Applied computing

Acknowledgments

The authors would like to thank Dr. Ignacio Ramírez, Dr. Marcelo Fiori and Dr. Paola Bermolen for fruitful discussion about this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed online at http://dx.doi.org/10.1080/17459737.2023.2213471.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 Note that other metrical levels could be used as reference, such as the sixteenth-note level.

2 This is not the only reason. The models discussed in the following also treat different time signatures differently.

3 Mavromatis (2012) uses the term probabilistic musical grammar.

4 As usual in information theory we use $\log x={\log }_{2}x$.

5 To encode one element within a set of $d=2^{\ell }$ a binary stings, we need to use $\ell ={\log }_{2}d=\log d$ bits.

6 In both cases the description length is proportional to $\log N$. This fact was observed by Rissanen (1978).

7 https://github.com/mrocamora/mdlfit

8 Aesthetic preferences can result in a different rate of occurrence of different rhythmic patterns, see Figure 5.

Additional information

Funding

This work was partially supported by funding agencies Agencia Nacional de Investigación e Innovación (ANII) and Comisión Sectorial de Investigación Científica (CSIC), UdelaR.