Is Bach Time Reversible?

In tonal music a leading tone or passing dissonance implies an expected resolution. Such expectations whether realized or not, constitute in our imagination goals toward which the music is directed (C. F. Hastey)

The quantitative description of the structure in musical compositions has a long history of interdisciplinary research, with contributions from musical theory, information theory, and mathematics to physics.

In 1975, Voss and Clark showed that the power spectrum for intensity fluctuations in a recording of Bach’s Brandenburg Concerto No. 1, and in many other instances of recorded music and human voices heard over the radio, was approximately 1/f over about 3 decades of frequency.

1 / f noise or pink noise is a special example of a large family of reversible stochastic processes widely found in nature. Is that what makes music to our ears? The fingerprint of “appealing sound”? Recent evidence challenges this vision.

In a paper published in Physical Review Research past July, a team of physicists measures the “time irreversibility” of more than 8,000 pieces of Western classical music and show that most of them display clear signatures of time irreversibility. Since by construction stochastic processes with a linear correlation structure (such as 1/ f noise) are time reversible, they conclude that musical compositions have a considerably richer structure:

we used tools from statistical physics, nonlinear dynamics, and graph theory to characterize a large database of 8856 classical music compositions ranging over five centuries, encompassing pieces from 77 composers from the Renaissance up to the early Modern period. We showed that, indeed, time irreversibility can not only be measured, but is also pervasive, as we found that over two-thirds of these compositions indeed display this signature, leading us to conclude that this is a common trait of tonal music,

González-Espinoza, A., Martínez-Mekler, G., and Lacasa, L. (2020). Arrow of time across five centuries of classical music. Phys. Rev. Research 2, 033166.

They also show that musical compositions display strong signs of nonlinear correlations, and that nonlinearity is correlated to irreversibility.

The fact that a signal is time irreversible points to an emergent time Arrow.

I can poetically conjecture that that’s the arrow which goes deep into our “heart”, and that if one day we come to understand consciousness, memory and such, then we’ll be able to test whether the “wound” of a fugue or a canon is also an indelible imprint on our mind.

Music is complex. Enjoy it!!


Featured Image: Georges Braque, Homage to J. S. Bach winter 1911-12


  1. A very interesting new perspective. The reference (Scholarpedia) is most enlightening and shows the relation of music, and many other mental and natural process, to a self-organized critical process. I was not aware of it.

    Incidentally, as you know, this is also the case in literature: a story is normally composed of introduction, conflict, climax and denouement, a non reversible process as well, but also controlled in a certain manner.

    The example you mention, a fugue, seems to point out to a more complex (or complicated) and somewhat periodic and repetitive structure with variations. (More) popular music seems to have a simpler, but also repetitive, structure. Less controlled and coming faster to the conclusion if you like.

    Thank you for the entry!

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