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
'The Archers' has an audience of about five million
radioA growing number of listeners are voicing their discontent; so loudly that even the BBC's director-general seems worried
Arts and Entertainment
Ready to open the Baftas, rockers Kasabian are also ‘great film fans’
musicExclusive: Rockers promise an explosive opening to the evening
Arts and Entertainment
Henry VIII played by Damien Lewis
tvReview: Scheming queens-in-waiting, tangled lines of succession and men of lowly birth rising to power – sound familiar?
Arts and Entertainment
tv
Arts and Entertainment
Hell, yeah: members of the 369th Infantry arrive back in New York
booksWorld War Z author Max Brooks honours WW1's Harlem Hellfighters in new graphic novel
PROMOTED VIDEO
Arts and Entertainment
Beer as folk: Vincent Franklin and Cyril Nri (centre) in ‘Cucumber’
tvReview: This slice of gay life in Manchester has universal appeal
Arts and Entertainment
‘A Day at the Races’ still stands up well today
film
Have you tried new the Independent Digital Edition apps?
Arts and Entertainment

ebooksNow available in paperback
Arts and Entertainment

ebooks
Arts and Entertainment
‘The Royals’ – a ‘twisted, soapy take on England’s first family’
tvAnd its producers have already announced a second season...
Arts and Entertainment
Kraftwerk performing at the Neue Nationalgalerie (New National Gallery) museum in Berlin earlier this month
musicWhy a bunch of academics consider German electropoppers Kraftwerk worthy of their own symposium
Arts and Entertainment
Icelandic singer Bjork has been forced to release her album early after an online leak

music
Arts and Entertainment
Colin Firth as Harry Hart in Kingsman: The Secret Service

film
Arts and Entertainment
Brian Blessed as King Lear in the Guildford Shakespeare Company's performance of the play

theatre
Arts and Entertainment
In the picture: Anthony LaPaglia and Martin Freeman in 'The Eichmann Show'

tv
Arts and Entertainment
Anne Kirkbride and Bill Roache as Deirdre and Ken Barlow in Coronation Street

tvThe actress has died aged 60
Arts and Entertainment
Marianne Jean-Baptiste defends Joe Miller in Broadchurch series two

tv
Arts and Entertainment
The frill of it all: Hattie Morahan in 'The Changeling'

theatre
Arts and Entertainment
Gillian Anderson and David Duchovny may reunite for The X Files

tv
Arts and Entertainment
Jeremy Clarkson, left, and Richard Hammond upset the locals in South America
TV
News
A young woman punched a police officer after attending a gig by US rapper Snoop Dogg
people
Arts and Entertainment
Reese Witherspoon starring in 'Wild'

It's hard not to warm to Reese Witherspoon's heroismfilm
Arts and Entertainment
Word up: Robbie Coltrane as dictionary guru Doctor Johnson in the classic sitcom Blackadder the Third
books

Arts and Entertainment
The Oscar nominations are due to be announced today

Oscars 2015
Arts and Entertainment
Hacked off: Maisie Williams in ‘Cyberbully’

Maisie Williams single-handedly rises to the challenge

TV
Arts and Entertainment
Eddie Redmayne in The Theory of Everything and Benedict Cumberbatch in The Imitation Game are both nominated at the Bafta Film Awards
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.

    Isis hostage crisis: Militant group stands strong as its numerous enemies fail to find a common plan to defeat it

    Isis stands strong as its numerous enemies fail to find a common plan to defeat it

    The jihadis are being squeezed militarily and economically, but there is no sign of an implosion, says Patrick Cockburn
    Virtual reality thrusts viewers into the frontline of global events - and puts film-goers at the heart of the action

    Virtual reality: Seeing is believing

    Virtual reality thrusts viewers into the frontline of global events - and puts film-goers at the heart of the action
    Homeless Veterans appeal: MP says Coalition ‘not doing enough’

    Homeless Veterans appeal

    MP says Coalition ‘not doing enough’ to help
    Larry David, Steve Coogan and other comedians share stories of depression in new documentary

    Comedians share stories of depression

    The director of the new documentary, Kevin Pollak, tells Jessica Barrett how he got them to talk
    Has The Archers lost the plot with it's spicy storylines?

    Has The Archers lost the plot?

    A growing number of listeners are voicing their discontent over the rural soap's spicy storylines; so loudly that even the BBC's director-general seems worried, says Simon Kelner
    English Heritage adds 14 post-war office buildings to its protected lists

    14 office buildings added to protected lists

    Christopher Beanland explores the underrated appeal of these palaces of pen-pushing
    Human skull discovery in Israel proves humans lived side-by-side with Neanderthals

    Human skull discovery in Israel proves humans lived side-by-side with Neanderthals

    Scientists unearthed the cranial fragments from Manot Cave in West Galilee
    World War Z author Max Brooks honours WW1's Harlem Hellfighters in new graphic novel

    Max Brooks honours Harlem Hellfighters

    The author talks about race, legacy and his Will Smith film option to Tim Walker
    Why the league system no longer measures up

    League system no longer measures up

    Jon Coles, former head of standards at the Department of Education, used to be in charge of school performance rankings. He explains how he would reform the system
    Valentine's Day cards: 5 best online card shops

    Don't leave it to the petrol station: The best online card shops for Valentine's Day

    Can't find a card you like on the high street? Try one of these sites for individual, personalised options, whatever your taste
    Diego Costa: Devil in blue who upsets defences is a reminder of what Liverpool have lost

    Devil in blue Costa is a reminder of what Liverpool have lost

    The Reds are desperately missing Luis Suarez, says Ian Herbert
    Ashley Giles: 'I'll watch England – but not as a fan'

    Ashley Giles: 'I'll watch England – but not as a fan'

    Former one-day coach says he will ‘observe’ their World Cup games – but ‘won’t be jumping up and down’
    Greece elections: In times like these, the EU has far more dangerous adversaries than Syriza

    Greece elections

    In times like these, the EU has far more dangerous adversaries than Syriza, says Patrick Cockburn
    Holocaust Memorial Day: Nazi victims remembered as spectre of prejudice reappears

    Holocaust Memorial Day

    Nazi victims remembered as spectre of prejudice reappears over Europe
    Fortitude and the Arctic attraction: Our fascination with the last great wilderness

    Magnetic north

    The Arctic has always exerted a pull, from Greek myth to new thriller Fortitude. Gerard Gilbert considers what's behind our fascination with the last great wilderness