Who says maths is all cold logic and music all emotion? That's nonsense, writes Marcus du Sautoy ? the two are intimately connected
I used to do a lot of counting as a trumpeter in my local youth orchestra. Sitting in the brass section, counting out rests so I didn't crash in early with a fanfare, I began to realise that mathematics and music had even deeper links. It is certainly a connection people have commented on throughout the ages.
"Music," wrote the great 17th-century German mathematician Gottfried Leibniz, "is the sensation of counting without being aware you were counting." But there is more to this connection than counting. As the French baroque composer Rameau declared in 1722: "I must confess that only with the aid of mathematics did my ideas become clear."
So is there really a link? Or is it crazy to try to connect the creative art of music with the steely logic of mathematics? Certainly the grammar of music ? rhythm and pitch ? has mathematical foundations. When we hear two notes an octave apart, we feel we're hearing the same note, so much so that we give them the same name. (This is because the frequencies of the two notes are in an exact 1:2 ratio.)
Yet, while the combinations of notes we have been drawn to over the centuries can all be explained through numbers, music is more than just notes and beats ? just as Shakespeare is more than just words from a dictionary. And it is in putting the notes together to create, say, the Goldberg Variations or Don Giovanni that I believe the true connection between mathematics and music reveals itself.
Many people react angrily to such a claim, believing music to be so much richer and more emotional than mathematics, and that to make such a comparison is to misunderstand what music is truly about. But, as a professor of mathematics, I think this argument misunderstands what mathematics is truly about.
Just as notes and rhythms are not all there is to music, so arithmetic and counting are not all there is to mathematics. Mathematics is about structure and pattern. As we've explored the universe of numbers, we've discovered strange connections and stories about numbers that excite and surprise us. Take the discovery by Fermat, the 17th-century French mathematician, that a prime number that has a remainder of 1 after division by 4 (like 41) can always be written as the sum of two square numbers (41=16+25). It was a realisation that linked the seemingly separate worlds of primes and squares.
Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth "listening to".
I can get a computer to churn out endless true statements about numbers, just as a computer can be programmed to create music. The art of the mathematician lies in picking out what mathematics will excite the soul. Most mathematicians are driven to create not for utilitarian goals, but by a sense of aesthetics. The 19th-century French mathematician Henri Poincar� summed up this creative role thus: "To create consists precisely in not making useless combinations. Invention is discernment, choice . . . The sterile combinations do not even present themselves to the mind of the inventor."
But for me, what really binds our two worlds is that composers and mathematicians are often drawn to the same structures for their compositions. Bach's Goldberg Variations depend on games of symmetry to create the progression from theme to variation. Messiaen is drawn to prime numbers to create a sense of unease and timelessness in his famous Quartet for the End of Time. Schoenberg's 12-tone system, which influenced so many of the major composers of the 20th century, including Webern, Berg and Stravinsky, is underpinned by mathematical structure. The organic sense of growth found in the Fibonacci sequence of numbers 1,2,3,5,8,13 . . . has been an appealing framework for many composers, from Bart�k to Debussy.
Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry. As Stravinsky once said: "The musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface."
? Marcus du Sautoy is speaking at the Cheltenham music festival on Saturday. Details: cheltenhamfestivals.com
Source: http://www.guardian.co.uk/music/2011/jun/27/music-mathematics-fibonacci
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