What, exactly, makes music to the ears? Time will tell, according to a new study of five centuries’ worth of compositions.
Using techniques derived from statistical mechanics—typically used to study large groups of particles—a team of physicists has mathematically measured the “time irreversibility” of more than 8,000 pieces of Western classical music. Published in Physical Review Research in July, their study quantifies what many listeners intuit: noise can sound the same played forwards or backward in time, but composed music sounds dramatically different in those two time directions.
Time irreversibility—the existence of an “arrow of time”—is a concept drawn from fundamental physics, first formulated in 1927 by the British astronomer Arthur Eddington. But it is meaningful in many contexts, says Lucas Lacasa, a physicist at Queen Mary University of London and a co-author of the study. One can see it in action over breakfast: think of the implausibility of unscrambling an egg and returning it to a pristinely pieced-back-together shell. But until now, Lacasa says, time irreversibility “hasn’t been measured at all in music.” Lacasa became interested in analyzing music through conversations with co-authors Gustavo Martínez-Mekler of the National Autonomous University of Mexico and Alfredo González-Espinoza of the University of Pennsylvania, both of whom are physicists and musicians. By finding patterns across large bodies of composed music, they were hoping to find hints as to what makes a successful composer.
Compared with systems made of millions of particles, a typical musical composition consisting of thousands of notes is relatively short. Counterintuitively, that brevity makes statistically studying most music much harder, akin to determining the precise trajectory of a massive landslide based solely on the motions of a few tumbling grains of sand. For this study, however, Lacasa and his co-authors exploited and enhanced novel methods particularly successful at extracting patterns from small samples. By translating sequences of sounds from any given composition into a specific type of diagrams or graphs, the researchers were able to marshal the power of graph theory to calculate time irreversibility.
This is far from the first statistical study of music. In his 1963 book Formalized Music, composer and music theorist Iannis Xenakis used matrices and differential equations to buttress arguments about the nature of music and musical composition. Boldly, he posed that “much like a god, a composer may … invert Eddington’s ‘arrow of time.’” But confirming this contention proved elusive. The new paper, however, validates the claim: most compositions the researchers studied were found to follow an arrow of time.
Systems that are time-reversible, under statistical analysis, seem the same when the arrow of time is flipped. The unstructured static hiss of white noise is one example. A different kind of noise prevalent in biological systems, dubbed “pink noise,” is also time-reversible. And by certain statistical measures, it is almost indistinguishable from music. Specifically, when analyzing how much power each frequency component within a musical piece tends to have, scientists find the same distribution as in pink noise. Consequently, music has been accepted to be a type of pink noise.
The new study challenges this association, demonstrating that despite such basic similarities music has more structure than pink noise, and that this structure is meaningful. “Irreversibility gives you an idea of change in time; it approaches the idea of a narrative,” Martínez-Mekler says. Music being time-irreversible, then, might reflect a composer’s effort to tell a story through the progression of notes.
Time irreversibility is related to a measure of disorder that, in physics, is called entropy. The composition having the most entropy would be a strictly random shuffle of sounds. It would also look the same—fully disordered—in all time directions, thus displaying no arrow of time. Conversely, the most time-irreversible composition would be the one that is the least random, possessing the least amount of entropy and the most structure. In this sense, measuring time irreversibility might reflect how singular a particular composer’s style is—the difference, say, between the gaudy violinist Nicolo Paganini and the melancholy lutenist John Dowland.