German Baroque composer Johann Sebastian Bach created music that is so carefully structured that it is often compared to mathematics. Although few of us are emotionally moved by mathematics, Bach's works – and music in general –touches us. It's more than sound; this is the message. And now, thanks to the tools of information theory, researchers are beginning to understand how Bach's music conveys this message.
By representing scores as simple networks of points, called nodes, connected by lines, called edges, scientists have quantified the information conveyed by hundreds of Bach's compositions. Analysis of these music networks published February 2 in Physical Research Review found that many of Bach's musical styles, such as chorales and toccatas, varied markedly in the amount of information they conveyed, and that musical networks contained structures that could make their messages easier to understand for human listeners.
“I found this idea really cool,” says physicist Suman Kulkarni of the University of Pennsylvania, lead author of the new study. “We used the tools of physics without making assumptions about the pieces of music, but just started with this simple representation and saw what it could tell us about the information being conveyed.”
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Researchers have quantified the information content of everything from simple sequences to intricate networks using information entropyconcept introduced mathematician Claude Shannon in 1948.
As the name suggests, information entropy is mathematically and conceptually related to thermodynamic entropy. It can be thought of as a measure of how unexpected a message is – where a “message” can be anything that conveys information, from a sequence of numbers to a piece of music. This point of view may seem counterintuitive, given that in colloquial language information is often equated with credibility. But the key takeaway from information entropy is that learning what you already know is not learning at all.
A conversation with a person who can only say one thing, such as the character Hodor in the television series. Game of Thrones, someone who only says “Hodor” will be predictable but uninformative. Chatting with Pikachu would be a little better; That Pokemon can only pronounce the syllables of his name, but can rearrange them, unlike Hodor. Likewise, a piece of music consisting of just one note will be relatively easy for the brain to “learn” or accurately reproduce as a mental model, but the piece will have difficulty conveying any message. Watching a double-headed coin being tossed will not provide any information at all.
Of course, packaging a message full of information isn't much of a good thing if whoever or whatever is receiving it can't accurately understand that information. And when it comes to musical messages, researchers are still trying to figure out how we know what music is trying to tell us.
“There are several different theories,” says cognitive scientist Marcus Pearce of Queen Mary University of London, who was not involved in the recent study. Physical Research Review study. “The main one, I think at the moment, is based on probabilistic learning.”
In this context, “learning” music means creating accurate mental representations of the actual sounds we hear (what researchers call a model) through the interplay of anticipation and surprise. Our mental models predict how likely a given sound is to occur based on what has come before. Then, says Pearce, “you figure out whether the prediction was right or wrong, and then you can update your model accordingly.”
Kulkarni and her colleagues are physicists, not musicians. They wanted to use the tools of information theory to search for information structures in music that might have something to do with how people extract meaning from melody.
So Kulkarni arranged Bach's 337 works into a web of interconnected nodes and calculated the information entropy of the resulting networks. In these networks, each note of the original score is a node, and each transition between notes is an edge. For example, if a piece includes the note E followed by the notes C and G played together, the node representing E will be connected to the nodes representing C and G.
Note transition networks in Bach's music contain more information than randomly generated networks of the same size—a result of the greater variety of network node degrees, or the number of edges connected to each node. In addition, scholars have discovered differences in the information structure and content of Bach's many compositional styles. Chorales, a type of hymn designed to be sung, created networks that were relatively sparse in information but still more informative than randomly generated networks of the same size. Toccatas and preludes, musical styles often written for keyboard instruments such as organ, harpsichord and piano, had higher information entropy.
“I was particularly excited by the higher level of surprise in toccatas than in chorale works,” says study co-author and physicist Dani Bassett of the University of Pennsylvania. “The two types of pieces feel different in my bones, and I was interested in seeing how that difference manifested itself in compositional information.”
Network structures in Bach's compositions can also make it easier for listeners to study these networks accurately. People don't learn networks perfectly. “We have biases,” Bassett says. “We seem to be ignoring some of the local information in favor of looking at the larger information picture across the entire system,” they add. Modeling this bias in how we build our mental models of complex networks, the researchers compared the total information of each music network with the amount of information a human listener could glean from it.
The music networks contained clusters of note transitions that could help our biased brains “learn” the music—accurately reproduce the information structure of the music as a mental model—without sacrificing much information.
“The particular way in which they reflect learning ability is quite interesting,” says Peter Harrison of the University of Cambridge, who was not involved in the study. “In some ways it's very simplistic. But it's quite complementary to other theories we have, and learning is quite difficult to understand.”
This type of network analysis is not special to Bach – it could work for any composer. Pearce says it would be interesting to use this approach to compare different composers or look for informative trends in music history. For his part, Kulkarni is happy to analyze the informational properties of scores that go beyond the Western musical tradition.
But music is more than just a sequence of notes, Harrison notes. Rhythm, volume, timbre of instruments – these and many other elements are important aspects of the musical message that were not considered in this study. Kulkarni says she would be interested in incorporating these aspects of music into her networks. Harrison adds that this process could work another way: rather than reducing musical functions to a network, he is curious about how network functions are translated into things that a musician recognizes.
“A musician might ask, 'What are the real musical rules or musical characteristics that drive this?' Can I hear it on the piano?’” Harrison says.
Finally, it's not yet clear exactly how the network patterns identified in the new study translate into the lived experience of listening to a Bach piece or any music, Pearce says. The solution to this question will be a matter of music psychology, he continues. Experiments can show “whether such things are actually perceived by people, and what impact they have on the pleasure people get when they listen to music.” Likewise, Harrison says he would be interested in experiments testing whether the types of network learning errors modeled by the researchers in this study actually matter for how people learn music.
“The fact that people have such imperfect, biased perceptions of complex information systems is critical to understanding how we make music,” Bassett says. “Understanding the informational complexity of Bach's works opens new questions about the cognitive processes that underlie how each of us appreciates different kinds of music.”
