A Hard Day’s Night by numbers: The Beatles decoded


“TWANG! It’s been a …”

There is perhaps no song as quintessentially Beatle-ish as A Hard Day’s Night – it just bubbles with unbridled enthusiasm and joy. And in my mind, there’s no other opening chord of a rock song that is as instantly recognisable as that one.

I grew up grudgingly playing the piano, practising only the half-hour before my lesson each week. But as soon as heard my first Beatles’ record, I dropped the piano to teach myself guitar eight hours a day during my high school summers.

Something about the early Beatles’ music struck a chord, so to speak, deep down inside of me, and it hasn’t left.

At about the same time, my love for mathematics blossomed, and I played in a band while attending my undergraduate studies. It was a tough choice, but I gave up music for the safer gig, as a mathematician. But unbeknown to me, the music that lay dormant inside me would serendipitously mix with the math inside me.

In 2004 I heard it was the 40th anniversary of the Beatles’ first movie – A Hard Day’s Night – and the soundtrack of the same name. All of the media attention brought to mind that famous opening chord that opened the movie and title song.

While teaching myself guitar years earlier, I had invested in a lot of Beatles songbooks, only to find that every book had a different transcription for how the author thought George Harrison had coaxed that initial sound out of his brand new twelve-string Rickenbacker guitar. All were derived by some combination of listening and music theory, but to me none sounded quite right.

My mathematical outlook had me take a different approach in 2004 – was there a scientific way to decide how the chord was played? Indeed, I had read a math book for leisure (yes we mathematicians do that sort of thing!) about ten years earlier that described the mathematics of sound and music.

In particular, there was a process, called a Fourier transform, that could allow one to decompose a sound wave into its constitute pure tones (which were modelled by sine and cosine curves). I had also remembered that there were algorithms to do just that, so I embarked on some CSI-like musical forensics. I took a small part of the opening chord and ran it through a Fourier transform, and held my breath waiting for the output.

It was a bit daunting – there were thousands of frequencies in the opening chord. But all was not lost, as I could tell the amplitudes of the frequencies, and the amplitude corresponds roughly with the loudness. So I began to make mathematical deductions from the data, and I quickly came upon some interesting conclusions.

First, all of the transcriptions I had seen for the guitar chord were incorrect – they had a low G note present, and the mathematics clearly indicated that the frequency simply wasn’t present.

Musicians thought they had heard the note, and as the key of the song was G, they believed it to be there all the more strongly. But it wasn’t.

Furthermore, I could see that the frequencies often were not particularly close to notes, so that it would have behoved the Beatles’ producer, George Martin, to have knocked on the studio window before the final take of the song and said: “Better tune up again, boys.” The Beatles’ guitars were gloriously slightly out-of-tune, adding to the difficulty in reproducing the chord.

A much bigger problem loomed. There were three frequencies corresponding to a certain “F” note, with no corresponding note up the octave, and this meant that note couldn’t have been played on George Harrison’s twelve string, and further, there was no way for the Beatles’ guitars to cover the frequencies. The answer involved throwing out the assumption that only the Beatles played on the opening chord.

A solution lay with insertion of a piano into the mix, as pianos have, toward the top end of the keyboard, three identically tuned strings under each note. Upon this realisation, the remainder of the chord began to unravel fairly quickly, and I could deduce what instruments (guitars, bass and piano) played what notes. A little bit of math went a long way!

The greatest difficulty I encountered after the research was finding a public forum to publish the work. It was going to appear in a peer-reviewed journal, but I thought the story was interesting enough for everyone to read. One magazine refused to read it based on the fact that the article had mathematics in it! But Guitar Player magazine loved the work, and was happy to publish the article, and the rest is, as they say, history.

Over the ensuing years, I have applied mathematics in a variety of ways to analyse pop music. In a second article in Guitar Player magazine I deduced mathematically that George Harrison must have recorded his famous, brilliant solo in A Hard Day’s Night by slowing down the tape speed in half, and recording the solo at half-speed down the octave.

Some musicians I’ve spoken with have been upset at the research, as perhaps it showed George’s technical skills were not what they should have been, but the truth I think says more – it showed George was a musician first, doing what it took to play what was in his head rather than in his fingers, and he had to have an incredible amount of confidence to choose to record a solo at half-speed, knowing that all of the world would be watching for when he played it up to speed, live (which, of course, he did!).

I’ve also written about why the music to I Want To Hold Your Hand was so imaginative and clever that it brought America to its knees, and why Paul McCartney so correctly named Little Richard’s Long Tall Sally as perhaps one of the greatest rock songs ever (and more generally, a mathematical basis for why the blues chord progression is so damn good).

Finally, in recent work with Robert Dawson of St. Marys University, we explained mathematically why George Martin’s famous edit in Strawberry Fields Forever never quite satisfied Paul (and never could).

Moreover, the research continues to open doors for me, especially as an ambassador for mathematics. I’ve written a book for the general public called Our Days Are Numbered: How Mathematics Orders Our Lives and published my first CD, Songs in the Key of Pi, of my own songs.

In fact, the Wall Street Journal came a-knocking back in 2008, and shot a video of a song I wrote in the style of the early Beatles, using mathematical principles I gleaned from their music.

And I continue to travel worldwide, giving public lectures on mathematics and music, most often with a guitar slung over my shoulder and with a rockin’ band behind me. The Beatles, it seems, gave me a great ticket to ride!


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