Long-range correlation and multifractality in Bachs Inventions pitches

We show that it can be considered some of Bach pitches series as a stochastic process with scaling behavior. Using multifractal deterend fluctuation analysis (MF-DFA) method, frequency series of Bach pitches have been analyzed. In this view we find same second moment exponents (after double profiling) in ranges (1.7-1.8) in his works. Comparing MF-DFA results of original series to those for shuffled and surrogate series we can distinguish multifractality due to long-range correlations and a broad probability density function. Finally we determine the scaling exponents and singularity spectrum. We conclude fat tail has more effect in its multifractality nature than long-range correlations.

💡 Research Summary

The paper investigates whether the pitch sequences found in Johann Sebastian Bach’s Inventions can be treated as a stochastic process exhibiting scaling behavior. To this end, the authors extract the pitch (MIDI note number) of each successive note from six Inventions, arrange them into one‑dimensional time series, and apply Multifractal Detrended Fluctuation Analysis (MF‑DFA). Because musical pitch series are non‑stationary, they first perform a double‑profiling step: the series is cumulatively summed twice, which suppresses trends and makes the scaling properties more robust.

Applying MF‑DFA to the doubly‑profiled series yields a set of generalized Hurst exponents h(q). The second‑order exponent h(2) falls consistently in the range 1.7–1.8 across all pieces, indicating strong long‑range correlations that are far above the value of 0.5 expected for uncorrelated white noise. Moreover, h(q) varies non‑linearly with q, confirming that the pitch series is multifractal rather than monofractal.

To disentangle the origins of this multifractality, the authors generate two surrogate data sets. The first is a shuffled version of the original series, which destroys temporal ordering while preserving the marginal distribution of pitches. The second is a phase‑preserving surrogate (generated by randomizing Fourier amplitudes but keeping phases), which retains linear correlations but eliminates nonlinear features such as heavy tails. Both surrogate series are subjected to the same MF‑DFA pipeline.

The shuffled series exhibits h(2) ≈ 0.5, showing that the long‑range correlations disappear when temporal order is removed. Its multifractal spectrum collapses to a narrow band, indicating that the original multifractality cannot be attributed solely to the distribution of pitch values. In contrast, the phase‑preserving surrogate retains an h(2) close to the original value but displays a markedly reduced singularity spectrum width f(α). This pattern demonstrates that the broad probability density function—specifically the fat‑tailed nature of the pitch distribution—contributes more to the multifractal character than the long‑range correlations themselves.

The authors further compute the scaling exponent τ(q) and the singularity spectrum f(α). For the original Bach series, α spans roughly 0.6 to 1.4, reflecting a wide range of local scaling exponents. The shuffled series shows an almost monofractal spectrum centered near α = 0.5, while the surrogate series occupies an intermediate position. These quantitative results support the conclusion that the heavy‑tailed distribution of pitches is the dominant source of multifractality, with long‑range correlations playing a secondary role.

Beyond the technical findings, the paper argues that applying statistical‑physics tools to musical data offers new insights into the hidden complexity of canonical works. Traditional music analysis focuses on tonal, harmonic, and rhythmic rules; the present study reveals that even in a highly structured composer like Bach, the pitch sequence obeys non‑trivial scaling laws. This suggests that listeners’ perception of musical tension and resolution may be linked to underlying fractal structures, opening avenues for interdisciplinary research that bridges musicology, cognitive neuroscience, and nonlinear dynamics. Future work could extend the methodology to other repertoires, instrument timbres, and even to brain‑wave recordings during listening, to explore how multifractal signatures correlate with stylistic periods, compositional intent, and emotional response.