Tunes create context like language
Maths shows why tonal music is easy listening.
Ever felt as though a piece of music is speaking to you? You could be right: musical notes are strung together in the same patterns as words in a piece of literature, according to an Argentinian physicist.
His analysis also reveals a key difference between tonal compositions, which are written in a particular key, and atonal ones, which are not. This sheds light on why many people find it so hard to make sense of atonal works.
In both written text and speech, the frequency with which different words are used follows a striking pattern. In the 1930s, American social scientist George Kingsley Zipf discovered that if he ranked words in literary texts according to the number of times they appeared, a word's rank was roughly proportional to the inverse of its frequency. In other words, a graph of one plotted against the other appeared as a straight line.
The economist and sociologist Herbert Simon later offered an explanation for this mathematical relationship. He argued that as a text progresses, it creates a meaningful context within which words that have been used already are more likely to appear than other, random words. For example, it is more likely that the rest of this article will contain the word "music" than the word "sausage".
Physicist Damian Zanette of the Balseiro Institute in Bariloche, Argentina, used this idea to test whether different types of music create a semantic context in a similar fashion.
The key in which a piece of music is written is one factor that influences which notes are more or less likely to come next. The repetition and elaboration of particular melodic phrases is another.
From Bach to Schoenberg
To measure these effects, Zanette analysed four different compositions: J. S. Bach's Prelude Number 6 in D; Mozart's first movement from his Sonata in C (K545); Debussy's Menuet from the Suite Bergamasque; and the first piece from Schoenberg's Three Piano Pieces, Opus 11. Each is a solo piano piece, but they all differ in style and period.
Zanette counted the frequency of different notes in each piece (taking into account both the pitch and the length of the note), and plotted that against their rank, as Zipf did with texts.
All of the pieces showed a text-like distribution, especially for the higher-ranking notes. But the strength of the relationship varied, as indicated by the slope of each graph, published on the preprint server arXiv1.
The pieces by Bach, Mozart and Debussy all produced a relatively steep graph, suggesting a strong relationship between rank and frequency, and therefore a high level of meaningful context. In other words, if you have heard part of the piece, it is relatively easy to predict what kind of thing will come next. Zanette adds that jazz pieces he tested showed a similar pattern.
But the Schoenberg piece, one of the first truly atonal works, had a much flatter graph. This means that the piece does not have a set vocabulary of commonly used words that keep appearing. Instead, the size of the vocabulary increases at about the same rate as the length of the piece; new "words" are constantly introduced, while earlier ones are seldom repeated.
Although all of the piano pieces have a text-like property, the atonal composition has less structure and less context; it is like a story whose characters are constantly changing.
Zanette says the finding implies that the reason many people find it unsatisfying to listen to atonal music is not simply because its harmonic and melodic structures are unfamiliar, but because the meaning or context of the piece is constantly changing.
"That doesn't mean Schoenberg's music is not comprehensible," Zanette cautions. Indeed, Schoenberg himself wrote that the goal of the composer is to produce comprehensibility. Zanette points out that the sequence of notes is only one of the ways to create context in music. It could also be produced rhythmically, for example.
He suggests that to appreciate atonality, we may need to look for coherence in different aspects of the composition.
"It's very good to start having these scientific bases for understanding music", says Brazil-based composer Heather Jennings. "They provide a fresh perspective on musical theory."
- Zanette, D. H.. Preprint, http://xxx.arxiv.org/abs/cs.CL/0406015 (2004).