The maths behind the music of Johann Sebastian Bach

A good piece of music should bring a certain amount of surprise without overwhelming the listener. Researchers led by physicist Suman Kulkarni from the University of Pennsylvania have now attempted to express this intuitive concept of good music using a mathematical model. In a study published in February 2024 in "Physical Review Research", they investigated how the structure of a piece of music is related to human perception: What leads people to remember certain passages, and which moments of a song seem predictable? Kulkarni and her team analysed works by the composer Johann Sebastian Bach using methods from information theory - and were thus able to explain why his music remains popular after so many centuries.

"Bach seemed to be an ideal starting point for this study, as his work has a strong mathematical structure," Kulkarni told the American Physical Society. His pieces also feature many different compositional forms, from preludes to fugues to chorales. Kulkarni and her team have analysed hundreds of them.

The physicists initially visualised the pieces of music as networks. Each node corresponded to a note of the song; two notes were connected by arrows if they followed each other. The experts also assigned a "weight" to the arrows: the more often one note followed another, the more weighty the connecting arrow became. For each network obtained in this way, Kulkarni and her colleagues were able to calculate a mathematical value, the so-called Shannon entropy, which quantifies the information content of the network.

How much information is there in a piece of music?

The researchers recognised a pattern: the same compositional forms had similar amounts of entropy. For example, chorales that are sung in churches usually contain little information, in contrast to toccatas and fugues, which have entertaining and surprising passages. According to Kulkarni, the differences in entropy thus reflect the functions of the different compositional forms.

Kulkarni and her team also wanted to find out whether this complexity was actually received by listeners. To do this, they used a computer model that simulates the average perception of a person. To train the model, they showed people different sequences of images and measured how surprising the impressions were for the test subjects. Using this programme, the team created a new network: in this case, the connections between two nodes indicate how obvious - or surprising - a person perceives the sequence of the corresponding notes to be. Because people do not have perfect powers of comprehension, the network created in this way usually differs from the one obtained directly from the piece of music.

As the researchers found, the difference between the networks was small for pieces of music by Bach - in contrast to randomly generated networks, which they also analysed with their computer model. They conclude that musical compositions apparently minimise discrepancies between the structure of a piece and human perceptions of it.

To confirm this thesis, however, more careful analyses are needed that also take into account other elements of a composition, such as rhythm or the presence of chords. This would also allow pieces by different performers and musical styles to be compared. "I would like to carry out the same analyses for other composers and non-Western music," Kulkarni told New Scientist.

Spectrum of Science

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