The enduring myth of music and maths

Is there really a link between melodic and mathematic ability? Think carefully before buying those 'Mozart effect' CDs, says Tim Gowers

An excellent way to kill a conversation is to tell the person you are talking to that you are a mathematician. The conversation may limp on for a minute or two, but nearly always it is doomed. However, there is a miracle cure: just tell your interlocutor that you are a musician as well as a mathematician. Even people who know nothing about mathematics have heard that mathematical ability is connected, in some fascinating and counter-intuitive way, to musical ability.

As a mathematician with strong musical interests who grew up in a family of musicians, I have been asked about this connection many times. And I have bad news: although there are some obvious similarities between mathematical and musical activity – and although many musical patterns can be fruitfully analysed mathematically – there is (as yet) no compelling evidence for the kind of mysterious, almost magical connection that many people seem to believe in. I'm partly referring here to the "Mozart effect", where children who have been played music by Mozart are supposedly more intelligent, including at mathematics, than children from a control group. It is not hard to see why such a theory would be popular: we would all like to become better at mathematics without putting in any effort. But the conclusions of the experiment that originally prompted the widespread belief in the Mozart effect were much more modest and have been grossly exaggerated. If you want your brain to work better, then not surprisingly, you have to put in some hard graft; there is no such thing as an intellectual perpetual-motion machine. Baby Mozart CDs and toys that combine maths and music might help, but not much, and the effects are temporary.

Of course, this does not show that there is no interesting connection between mathematics and music. It was always a little implausible that lazily listening to Eine kleine Nachtmusik would earn you extra marks on that maths test tomorrow, but what about learning to read music or spending hours practising the piano? That takes genuine effort. Could it be that the rewards for that effort spill over into other areas of intellectual life, and in particular into mathematics? Is there any evidence that people who have worked hard to become good at music are better at mathematics than people who are completely unmusical? And in the other direction, are mathematicians better than average at music?

Demonstrating a connection of this kind is not as easy as one might think. To begin with, there are plenty of innumerate musicians and tone-deaf mathematicians, so the best one could hope to demonstrate would be a significant positive correlation between aptitudes at the two disciplines. And then one would face all the usual challenges of establishing a statistical connection. For example, if you want to show that professional mathematicians are on average better at music than other people, then you have to decide quite carefully who those "other people" are. You might expect that the kind of person who becomes a professional mathematician is much more likely than average to come from the kind of family that would consider music to be an important part of a child's education, so for that reason alone one would expect at least some "background correlation" between the two. Therefore, not much will be proved if you compare professional mathematicians with the population at large. Identifying and controlling for these kinds of effects is difficult, and as far as I know (though I would be delighted to be corrected), there has been no truly convincing study that has shown that musical ability enhances mathematical ability or vice versa.

And yet, the belief that the two are interestingly related won't go away without a fight. I cannot help observing that among the mathematicians I know, there do seem to be a surprising number who are very good indeed at the piano. (Incidentally, that is a study waiting to be done: are mathematicians more drawn to the piano than to other instruments? Of the mathematicians I can think of who are superb instrumentalists, all but one are pianists.) While we wait for scientific evidence to back up the anecdotal evidence, can we at least argue that it is plausible that there should be a connection?

Indeed we can. For a start, both mathematics and music deal with abstract structures, so if you become good at one, then it is plausible that you become good at something more general – handling abstract structures – that helps you with the other. If this is correct, then it would show a connection between mathematical and musical ability, but not the kind of mysterious connection that people hope for. It would be more like the connection, such as it is, between ability at football and ability at cricket. To become better at one of those then you need to improve your fitness and co-ordination. That makes you better at sport in general and therefore probably helps with the other.

Of course, abstract structures are not confined to mathematics and music. If you are learning a foreign language then you need to understand its grammar and syntax, which are prime examples of abstract structures. And yet we don't hear people asking about a mysterious connection between mathematical ability and linguistic ability. My guess is that that is because the connection exists but not the mystery: grammar feels mathematical, so it would hardly be a surprise to learn that mathematicians are better than average at learning grammar. Music, by contrast, is strongly tied up with one's emotions and can be enjoyed even by people who know very little about it. As such, it seems very different from mathematics, so any connection between the two is appealingly paradoxical.

In an effort to dispel this air of paradox, let me give one example of a general aptitude that is useful in both mathematics and music: the ability to solve problems of the "A is to B as C is to D" kind. These appear in intelligence tests (car is to garage as aircraft is to what?) but they are also absolutely central to both music and mathematics. Consider, for instance, the opening two phrases of Eine kleine Nachtmusik. (In the unlikely event that you don't know this piece, it may help if I tell you that it is the piece that is played in a maddening endless loop on Ryanair flights.) The second phrase is a clear answer to the first. But one can be more precise about what this means. If you try to imagine any other second phrase, nothing seems "right" in the way that Mozart's chosen phrase does. So what is the question to which that phrase is the right answer? It is something like, "The first phrase goes broadly upwards and uses the notes of a G major chord; what would be the corresponding phrase that goes broadly downwards and uses the notes of a D7?" Music is full of little puzzles like this. If you are good at them, then when you listen to a piece, expectations will constantly be set up in your mind. Of course, some of the best moments in music come when one's expectations are confounded, but if you don't have the expectations in the first place then you will miss out on the pleasure.

Here is a fairly simple example from mathematics, which I recommend trying to answer for yourself before reading on: what is to multiplication as zero is to addition? If you are not mathematically inclined, then you will probably be aware that zero is a special number without ever having thought hard about why. The question requires you to relate zero to the general operation of addition. In other words, it requires you to identify what it is about the role that zero plays in the game of adding numbers together that distinguishes zero from all other numbers. That role is the following: adding zero to a number makes no difference. If we now want to solve the puzzle, we need a corresponding statement concerning multiplication. And there is one: multiplying a number by one makes no difference. So the answer is one.

That is by no means the end of the story. If you pursue the analogy far enough, you will find yourself inventing the theory of logarithms and exponentials. For many people logarithms mark the point where they part company with mathematics. Those who are good at "A is to B as C is to D" puzzles are less likely to fall at that hurdle, and the same goes for many subsequent hurdles.

In my view, the general question of whether mathematical ability and musical ability are related is much less interesting than some similar but more specific questions. I have already mentioned the possibility that mathematicians are more drawn to the piano than to other instruments. Are they more drawn to certain composers (Bach, for instance)? Are musical mathematicians more drawn to certain areas of mathematics? Do mathematicians tend to listen to music in a more analytical "A is to B as C is to D" way rather than simply allowing themselves to be caught up in the emotion? One can imagine many interesting surveys and experiments that could be done, but for now this is uncharted territory and all we can do is speculate.

Cambridge University maths professor Tim Gowers will speak at Cheltenham Music Festival on Sunday 10 July: see cheltenhamfestivals.com/music for details

Arts and Entertainment
Don’t send in the clowns: masks and make-up conceal true facial expressions, thwarting our instinct to read people’s minds through their faces, as seen in ‘It’
film
Arts and Entertainment
Go figure: Matt Parker, wearing the binary code scarf knitted by his mother
comedy Mathematician is using comedy nights to teach and preach sums
Arts and Entertainment
Ryan Gosling in 'Drive'
filmReview: Ryan Gosling is still there, but it's a very different film
Arts and Entertainment
Urban explorer: Rose Rouse has documented her walks around Harlesden, and the people that she’s encountered along the way
books Rouse's new book discusses her four-year tour of Harlesden
Arts and Entertainment
Shock of the news: Jake Gyllenhaal in ‘Nightcrawler’
film
PROMOTED VIDEO
Arts and Entertainment
Orson Welles made Citizen Kane at 25, and battled with Hollywood film studios thereafter
film
Arts and Entertainment
A scene from Franco Zeffirelli's production of 'Aida' at Milan's famed La Scala opera house
operaLegendary opera director in battle with theatre over sale of one of his 'greatest' productions
Have you tried new the Independent Digital Edition apps?
Arts and Entertainment
Juergen Wolf won the Young Masters Art Prize 2014 with his mixed media painting on wood, 'Untitled'
art
Arts and Entertainment
Iron Man and Captain America in a scene from
filmThe upcoming 'Black Panther' film will feature a solo black male lead, while a female superhero will take centre stage in 'Captain Marvel'
Arts and Entertainment
The Imperial War Museum, pictured, has campaigned to display copyrighted works during the First World War centenary
art
Arts and Entertainment
American Horror Story veteran Sarah Paulson plays conjoined twins Dot and Bette Tattler
tvReview: Yes, it’s depraved for the most part but strangely enough it has heart to it
Arts and Entertainment
The mind behind Game of Thrones George R. R. Martin
books

Will explain back story to fictional kingdom Westeros

Arts and Entertainment
Dorothy in Return to Oz

film Unintentionally terrifying children's movies to get you howling (in fear, tears or laughter)
Arts and Entertainment

music
Arts and Entertainment

film
Arts and Entertainment
Robert James-Collier as under-butler Thomas

TVLady Edith and Thomas show sad signs of the time
Arts and Entertainment

music
Arts and Entertainment
The Dad's Army cast hit the big screen

film
Arts and Entertainment

music
Arts and Entertainment
JK Rowling is releasing a new Harry Potter story about Dolores Umbridge

books
Arts and Entertainment
On The Apprentice, “serious” left the room many moons ago and yet still we watch

TV
Arts and Entertainment
The Doctor finds himself in a forest version of London in Doctor Who episode 'In the Forest of the Night'
TVReview: Is the Doctor ever going stop frowning?
Arts and Entertainment
A scene from David Ayer's 'Fury'

film
Arts and Entertainment
Taylor Swift performs at the 2014 iHeart Radio Music Festival
music review
Arts and Entertainment
Paul Anderson plays Arthur Shelby in Peaky Blinders series two
tvReview: Arthur Shelby Jr seems to be losing his mind as his younger brother lets him run riot in London
Arts and Entertainment
Miranda Hart has called time on her award-winning BBC sitcom, Miranda
tv
Arts and Entertainment
Sir Nicholas Serota has been a feature in the Power 100 top ten since its 2002 launch
art
Latest stories from i100
Have you tried new the Independent Digital Edition apps?

ES Rentals

    Independent Dating
    and  

    By clicking 'Search' you
    are agreeing to our
    Terms of Use.

    The drugs revolution starts now as MPs agree its high time for change

    The drugs revolution starts now as MPs agree its high time for change

    Commons debate highlights growing cross-party consensus on softening UK drugs legislation, unchanged for 43 years
    The camera is turned on tabloid editors in Richard Peppiatt's 'One Rogue Reporter'

    Gotcha! The camera is turned on tabloid editors

    Hugh Grant says Richard Peppiatt's 'One Rogue Reporter' documentary will highlight issues raised by Leveson
    Fall of the Berlin Wall: It was thanks to Mikhail Gorbachev that this symbol of division fell

    Fall of the Berlin Wall

    It was thanks to Gorbachev that this symbol of division fell
    Halloween 2014: What makes Ouija boards, demon dolls, and evil clowns so frightening?

    What makes ouija boards and demon dolls scary?

    Ouija boards, demon dolls, evil children and clowns are all classic tropes of horror, and this year’s Halloween releases feature them all. What makes them so frightening, decade after decade?
    A safari in modern Britain: Rose Rouse reveals how her four-year tour of Harlesden taught her as much about the UK as it did about NW10

    Rose Rouse's safari in modern Britain

    Rouse decided to walk and talk with as many different people as possible in her neighbourhood of Harlesden and her experiences have been published in a new book
    Welcome to my world of no smell and odd tastes: How a bike accident left one woman living with unwanted food mash-ups

    'My world of no smell and odd tastes'

    A head injury from a bicycle accident had the surprising effect of robbing Nell Frizzell of two of her senses

    Matt Parker is proud of his square roots

    The "stand-up mathematician" is using comedy nights to preach maths to big audiences
    Paul Scholes column: Beating Manchester City is vital part of life at Manchester United. This is first major test for Luke Shaw, Angel Di Maria and Radamel Falcao – it’s not a game to lose

    Paul Scholes column

    Beating City is vital part of life at United. This is first major test for Shaw, Di Maria and Falcao – it’s not a game to lose
    Frank Warren: Call me an old git, but I just can't see that there's a place for women’s boxing

    Frank Warren column

    Call me an old git, but I just can't see that there's a place for women’s boxing
    Adrian Heath interview: Former Everton striker prepares his Orlando City side for the MLS - and having Kaka in the dressing room

    Adrian Heath's American dream...

    Former Everton striker prepares his Orlando City side for the MLS - and having Kaka in the dressing room
    Simon Hart: Manchester City will rise again but they need to change their attitude

    Manchester City will rise again but they need to change their attitude

    Manuel Pellegrini’s side are too good to fail and derby allows them to start again, says Simon Hart
    Isis in Syria: A general reveals the lack of communication with the US - and his country's awkward relationship with their allies-by-default

    A Syrian general speaks

    A senior officer of Bashar al-Assad’s regime talks to Robert Fisk about his army’s brutal struggle with Isis, in a dirty war whose challenges include widespread atrocities
    ‘A bit of a shock...’ Cambridge economist with Glasgow roots becomes Zambia’s acting President

    ‘A bit of a shock...’ Economist with Glasgow roots becomes Zambia’s acting President

    Guy Scott's predecessor, Michael Sata, died in a London hospital this week after a lengthy illness
    Fall of the Berlin Wall: History catches up with Erich Honecker - the East German leader who praised the Iron Curtain and claimed it prevented a Third World War

    Fall of the Berlin Wall

    History catches up with Erich Honecker - the East German leader who praised the Iron Curtain and claimed it prevented a Third World War
    How to turn your mobile phone into easy money

    Turn your mobile phone into easy money

    There are 90 million unused mobiles in the UK, which would be worth £7bn if we cashed them in, says David Crookes