The latest Seed has a very interesting article on the complicated geometry underlying Western music, and the intuitive mathematical understanding demonstrated by composers:
The shapes of the space of chords we have described also reveal deep connections between a wide range of musical genres. It turns out that superficially different styles–Renaissance music, classical and Romantic music, jazz, rock, and other popular forms–all make remarkably similar use of the geometry of chord space. Traditional techniques for manipulating musical scales turn out to be closely analogous to those used to connect individual chords. And some composers have displayed a profound understanding of the higher-dimensional geometry of musical chords. In fact, one can argue that Romantic composers such as Chopin had an intuitive feel for non-Euclidean higher-dimensional spaces that exceeded the explicit understanding of their mathematical contemporaries.
Perhaps my next book should be Chopin was a Topologist. But the article also left me with lots of questions. Ultimately, music works not by mirroring complex mathematical functions but by titillating hair cells in the cochlea and exciting neural networks in the auditory cortex. Why does the mind find such obscure geometrical orders so alluring? One possibility is that mathematical music hijacks our brain’s penchant for patterns, allowing us to extract some predictable sequences from the cacophony of soundwaves. The equations might be obscure, but the cortex is still able to hear them. As I write in my book:
A work of music is not simply a set of individual notes arranged in time. Music really begins when the separate pitches are melted into a pattern. This is a consequence of the brain’s own limitations. Music is the pleasurable overflow of information. Whenever a noise exceeds our processing abilities – we can’t decipher all the different sound waves hitting our hair cells – the mind surrenders. It stops trying to understand the individual notes, and seeks instead to understand the relationships between the notes. Our auditory cortex pulls off this feat by using its short term memory for sound (in the left posterior hemisphere) to uncover patterns at the larger level of the phrase, motif and movement. This new approximation lets us extract order from all these notes haphazardly flying through space, and the brain is obsessed with order. We need our sensations to make sense.
It is this psychological instinct – this desperate neuronal search for a pattern, any pattern – that is the source of music. When we listen to a symphony, we hear a noise in motion, each note blurring into the next. The sound seems continuous. Of course, the physical reality is that each sound wave is really a separate thing, as discrete as the notes written in the score. But this isn’t the way we experience the music. We continually abstract on our own inputs, inventing patterns in order to keep pace with the onrush of noise. And once our brain finds a pattern, we immediately start to make predictions, imagining what notes will come next. We project our imaginary order into the future, transposing the melody we have just heard into the melody we expect. By listening for patterns, by interpreting every note in terms of our expectations, we turn the scraps of sound into the ebb and flow of a symphony.
PS. Nature has put a great collection of articles on neuroscience and music online.