[Paper Review] The fractal dimension of music: Melodic contours and time series of pitch

Maria H. Niklasson, Gunnar A. Niklasson|arXiv (Cornell University)|Apr 6, 2020

Music and Audio Processing4 citations

TL;DR

This study compares fractal dimension calculations of melodic contours and pitch time series in classical and folk music using box counting and detrended fluctuation analysis (DFA). It finds that DFA is more reliable than box counting due to inaccuracies in the latter near dimension two and potential biases from contour line connections, with folk music tending toward lower fractal dimensions than classical music, though cutoff effects in short tunes require further study.

ABSTRACT

We analyze the fractal dimension of melodic contours and pitch time series of classical music and folk music tunes. The fractal dimensions obtained from box counting and detrended fluctuation analysis show significant differences. They are ascribed to the low accuracy of box counting for dimensions close to two as well as to a possible bias because the pitches in the time series are connected by lines to obtain the melodic contour used in the box counting analysis. We observe a tendency that folk music exhibits lower fractal dimensions than classical music, but further studies are needed in order to assess cutoff effects in the comparatively short folk music tunes. We conclude that detrended fluctuation analysis is the preferable method for fractal analysis of music, and this verifies previous studies of analysis of short time series.

Motivation & Objective

• To assess the fractal dimension of melodic contours and pitch time series in classical and folk music.
• To compare the reliability of box counting and detrended fluctuation analysis (DFA) for estimating fractal dimensions in musical data.
• To investigate whether folk music exhibits lower fractal dimensions than classical music due to structural differences.
• To evaluate the impact of short tune lengths on fractal dimension estimation, particularly cutoff effects.
• To validate the suitability of DFA for analyzing short time series in music, consistent with prior studies.

Proposed method

• Box counting method applied to melodic contours derived from pitch sequences in music.
• Detrended fluctuation analysis (DFA) used to compute fractal dimensions from raw pitch time series.
• Fractal dimensions calculated separately for classical music and folk music datasets.
• Comparative analysis of results from both methods to identify discrepancies and biases.
• Statistical evaluation of dimension differences between music genres, considering tune length and data resolution.
• Use of pitch sequences connected by lines to form melodic contours, which may affect box counting accuracy.

Experimental results

Research questions

• RQ1 How do the fractal dimensions of melodic contours compare to those of raw pitch time series in classical and folk music?
• RQ2 Why do box counting and DFA yield significantly different fractal dimension estimates in this context?
• RQ3 Does folk music exhibit lower fractal dimensions than classical music, and if so, is this due to structural or statistical factors?
• RQ4 To what extent do short tune lengths introduce cutoff effects in fractal dimension estimation?
• RQ5 Is detrended fluctuation analysis a more robust method than box counting for analyzing musical pitch time series?

Key findings

• Box counting produces less accurate fractal dimension estimates for dimensions close to two, particularly in musical pitch data.
• The connection of pitch points by lines to form melodic contours introduces a potential bias in box counting results.
• Detrended fluctuation analysis is found to be a more reliable and preferable method for fractal analysis of musical time series.
• Folk music tends to exhibit lower fractal dimensions than classical music, though this trend requires further validation due to short tune lengths.
• Cutoff effects in short folk music tunes may influence fractal dimension estimates, indicating a need for further study on data length limitations.