Elaine Chew - Professor of Digital Media, Queen Mary University of London
“Music is known to provoke the senses, give pleasure and sometimes move people to tears. Surely this has little to do with mathematical models which are so frequently associated with cold and rational logic. So what can maths tell us about this powerful phenomenon closely connected to the emotions? Can mathematics help us measure what’s sublime or ineffable about a piece of music?
Music evokes strong emotions such as frisson (goose bumps), awe and laughter – and has been found to use the same reward pathways as food, drugs and sex to induce pleasure. A shiver down one’s spine or an uncontrollable guffaw when listening to music is most often a case of the music defying your expectations. Expectations can be defined in two ways: schematic – knowing how a genre of music is supposed to go – or veridical – knowing how a particular piece of music unfolds.
On one end of the spectrum, a performance or a piece of music that does just what you’d expect runs the risk of becoming banal. On the other end, music like that of PDQ Bach – which uses tongue-in-cheek egregious violations of known expectations – makes many people laugh.
The craving that comes from musical anticipation and the euphoria that follows the reward have both been found to be linked to dopamine release. As a result, performers and composers alike play with listeners’ expectations, often going to great lengths to carefully choreograph their expectations, and then sometimes breaking them, to provoke and heighten emotional responses.
Playing with expectations
In tonal music, which is almost all of the music that we hear and can be thought of as being based on a scale, the note sequence sets up expectations, then suspends, fulfils, or violates them. For a simple example, sing the first three phrases of “Happy Birthday” and stop at the end of the penultimate phrase.
Anticipation for the resolution to this musical cliffhanger creates a palpable knot in the gut. This hollow feeling can be further intensified by delaying the final phrase. The release is evident when the final phrase is heard and ends happily on the most stable tone.
Two things are at work here in this miniature example: tonality and time. Tonality provides a framework through which expectations are formed – and the play on time, the delaying of expectations, uses the framework to create a musical cliffhanger and titillate the senses.
Where maths comes in
Expectations can be modelled mathematically and time can be measured – so the shaping of both expectations and time can be described in numbers. Over the years, in my research lab, we have developed models and computer algorithms for quantifying tonal properties and expressive parameters in music. Many of the tonal analysis algorithms are based on what is known as a “spiral array model”.
The spiral array can be plotted in 3D to allow us to visualise the dynamic evolution of musical keys and spot when the notes and their timing combine to do something interesting to tug at our emotions.
As music is heard, the notes can be mapped to the model, duly weighted and summarised as points inside it. Movements in the space inside the model allow listeners to see deviations from expected tonal behaviour.” Read the original article.
Happy Sunday, wherever you are!