SCOTT SIMON, host:

That jangly opening chord in the Beatles' "A Hard Day's Night" is certainly one of the most recognizable in pop music. Sort of sounds like a guitarist telling his band mates, Hey, we're doing a song here, so listen up. But a math professor in Canada looked into just how the Beatles produced that sound, back in 1964, before there were synthesizers and studio electronics like they have today. So to take us through this Beatles mathematical mystery tour, we've called on our math guy, Keith Devlin. He joins us from the studios of Stanford University. Thanks so much for being with us.

Professor KEITH DEVLIN (Mathematics, Stanford University): Hi, Scott. Good to speak to you again.

SIMON: And nice to talk to you. Look, we know that music has math. But other than the tempo or number of verses, where is the math?

Prof. DEVLIN: Sounds themselves are very mathematical things, and that was the secret to unraveling this particular mystery of this sound. I'll just say, it's obviously guitarlike. But if you listen closely, it's not like any guitar that anyone's really ever had played live. And so...

SIMON: Well now, let's credit the mathematician who got involved in this investigation, Jason Brown at Dalhousie University.

Prof. DEVLIN: And what Brown did was he took some mathematics that's fundamental to modern musical theory and to modern musical technology, in fact. That was mathematics invented in the late 18th, early 19th century by a French mathematician called Joseph Fourier. And Fourier showed how you can analyze any kind of wave. In fact, Fourier was studying heat dissipation waves, but let's continue to talk in terms of sound waves.

Sound waves come in different kinds, and the simplest kind is the kind of sound you get when you sound a pure note, for example, with a tuning fork. And that's a beautifully symmetrical, simple wave that we call a sine wave. That's sine, spelled S-I-N-E. A sine wave, beautiful hills and valleys, that go on and on smoothly. At the other end of the spectrum, as it were, you've got the kind of sound wave caused by conversation or a typical musical instrument.

What Fourier showed - it's incredibly remarkable results. Take a sound wave. No matter how complicated, you can decompose it into individual sine waves, these simple sine waves, that are superimposed to give that. What Jason Brown did is known as Fourier analysis. He took the recording of this Beatles' opening chord from "A Hard Day's Night."

(Soundbite of first chord of song "A Hard Day's Night")

On a computer - he didn't have to write the software, you can now get the software off the shelf - was able to analyze that sound wave, and this took him a long time. And it turns out - I mean, first of all, well, you know George Harrison had a 12-string guitar. Maybe, it's that. No. You can't produce that same spectrum of sine waves with a 12-string guitar. Maybe the other Beatles came in with their guitars. No. That doesn't work either.

Eventually, he figured that what must have been happening was George Martin, the producer, at the same time, hit the piano keyboard.

SIMON: Ah!

So that opening chord isn't a pure guitar. There's a piano on there as well. And it was that mixture of guitars and piano that gives it that rather unique sound.

(Soundbite of first chord of song "A Hard Day's Night")

SIMON: Keith, I salute the scholarship involved. But wouldn't it have been possible to get this, you know, settled in just two minutes by asking Sir Paul McCartney, how did you guys do it?

Prof. DEVLIN: You know, you'd think that. But in fact, the thing - interestingly enough, the thing that Jason Brown, this mathematician, is now trying is there is actually some dispute between - well, it was a historical dispute between Lennon and McCartney. Lennon, of course, can no longer contribute to that. But there was a famous song that was recorded a year after "A Hard Day's Night" on the Rubber Soul album, called "In My Life."

(Soundbite of song "In My Life")

BEATLES: (Singing) There are places I remember all my life. Though some have changed...

Prof. DEVLIN: And ever since that was recorded, there's been this dispute as to who wrote that. John Lennon claimed that he wrote the whole thing except for a musical bridge that McCartney eventually wrote. McCartney has always claimed, no, he actually wrote the music for that as well. Jason Brown, he's now analyzing this particular song and all of the other Beatles music in order to see if he can figure it out.

In other words, what he was trying to do is by analyzing lots of Beatles songs, he wants to see if he can find a sort of an audio fingerprint, a sound printing if you like, of songs that were written by Lennon and songs that were written by McCartney. If he can find some mathematical pattern in the waveforms that distinguishes their two composing styles, then he'll be able to look at that and see which pattern "In My Life" fits.

SIMON: Weekend Edition's math guy, Keith Devlin. Thanks so much.

Prof. DEVLIN: Oh, my pleasure, Scott, and season's greetings to you.

(Soundbite of "Hard Day's Night")

THE BEATLES: It's been a hard day's night, and I've been working like a dog...

SIMON: You're listening to the Fab Four from Weekend Edition, NPR News.

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