It's probably the most recognisable sound in popular music.

"This is the one chord that everyone around the world knows," says Randy Bachman, a rock star in his own right from The Guess Who and Bachman Turner Overdrive.

It dates to July 1964 — the height of Beatlemania. The band was about to release its third album.

For the first time, it was all original music. Plus, the Beatles were shifting away from their rock 'n' roll roots to a more poppy sound, and this album was to be the soundtrack for their first feature film.

They needed to make a statement.

After a lot of experimentation, the band come up with a wondrous, jangling cacophony of sound: the opening chord to the song, album, and film A Hard Day's Night.

"To me, it's like the feeling I might get if I jumped into a lake on a hot summer's day," Beatles superfan Jason Brown says.

"I don't think there's another song with an opening like that. It's just an indescribable chord of beauty."

Indescribable and — as it turns out — really rather mysterious.

Because for decades, no-one could figure out exactly how those two seconds of music were made.

Share The Beatles brave the rain in Sydney on their Australian tour in 1964.

The magical mystery

You might be wondering how a chord could be a mystery — especially one of the most well-known chords played by one of history's most famous bands.

Surely anyone trained in music theory could figure out what notes fit together to make a particular sound?

It's not that easy.

For starters: we're humans, with human flaws. Even the most well-trained musicians will sometimes disagree when transcribing chords by ear, and nonstandard chords are obviously even harder.

"It's hard to tell exactly what is being played," says Jason Brown, who also happens to be a professor of mathematics at Dalhousie University in Canada, and a musician himself.

"People hear notes that aren't in chords, and they miss notes that are there. Every musician, no matter how schooled, will hear different things."

The physics of sound isn't on your side, either.

"When a string is plucked, you not only get the main frequency, but you get harmonics, which are higher multiples of that original frequency."

This shifting mish-mash of sound swamps your brain, along with incidental noise. "Anything else rattling in the room" will produce its own frequencies, Dr Brown says.

So it's not entirely surprising that the particular notes in this chord remained a bit of a puzzle.

But why not just ask a Beatle?

Share The Beatles arriving at London Airport after a trip to Paris in 1964.

Ask Paul

Well, somebody tried that back in February 2001. In an online chat, someone asked George Harrison how he played the opening chord to A Hard Day's Night — and he could only vaguely remember his own part in the chord, let alone what any of the other Beatles were playing.

"You'll have to ask Paul about the bass note to get the proper story," he wrote.

So there appears to be no first-hand account on record. And even if there were, it might not be reliable.

"In the absence of having anyone there transcribe what actually was being played, it is a mystery," says Dr Brown.

1964 was a chaotic time for the Beatles. They were constantly busy recording, playing live, filming movies and just generally being the biggest band on the planet.

The song A Hard Day's Night was recorded in a single day.

"I think it's hard [for them] to remember exactly what was played," Dr Brown explains. "They remember big-picture things … but not all the details."

beatles secret comments teaser We asked if mathematics had helped your musical abilities. Read the comments below.

Everybody has a view

So the chord became not just a mystery, but a favourite subject of debate among Beatles fans and musicians.

Rock star Randy Bachman, of The Guess Who and Bachman Turner Overdrive, was pretty sure he'd solved the mystery when he got behind the scenes at Abbey Road Studios with George Martin's son Giles.

Bachman heard what he thought were the original source tapes, reproduced track-by-track from a computer, and pieced together his own version of the chord.

"It was George on a 12-string [electric guitar] ... And it's an F chord, but you put a G on top, and you put a G on the bottom, and you put a C next to that G," he explains. "Paul's playing a D on the bass, and John's rhythm guitar was a Dsus4."

And it was close — but not close enough for Dr Brown.

He has his doubts about those "source tapes". That's because when A Hard Day's Night was recorded, the drums, bass and two guitars were all recorded onto the same track.

"The process of separating the instruments was done through software," he explains. But the software isn't perfect.

And Dr Brown would know. He's run a similar analysis himself.

"I'd known about the controversy since I tried to play the chord on the guitar."

As a teenager, he discovered the Beatles at the same time as he was learning to play guitar.

"I would spend eight to ten hours a day in the summer during high school, teaching myself to play The Beatles' songs."

He'd pore over Beatles songbooks with chord charts to all the songs. But every book seemed to have a different transcription for the opening chord of A Hard Day's Night.

"Every book was transcribing what they thought the Beatles had played, but there was no way of telling what was right and what was wrong."

Music + mathematics

In 2004, celebrations for the album's 40th anniversary reignited Dr Brown's fascination with the mystery. By then, he was a maths professor.

"I was in the habit of just picking up math books to read for interest, and one of the books that I picked up had a chapter on mathematics and sound," he explains.

"I thought of combining the music with the mathematics, and whether there'd be a scientific way to decide how the chord was being played — as opposed to just using your ears."

That crucial chapter introduced Dr Brown to Fourier transforms: a mathematical technique that lets you take a signal, like a musical a chord, and break it down into its component parts.

"I could decompose the opening chord into a bunch of these fundamental notes. I got in the order of 30,000 of them," he says.

That's a lot of frequencies for one chord. But of course, some of them are loud, and some are very faint.

"I realised that the notes that were actually being played in the chord would be amongst the loudest ones," Dr Brown says.

"This allowed me to start making some mathematical deductions from the data that I got."

He matched the loud frequencies to the instruments that must have played them, finding a bunch of notes that he — and everyone else before him — expected to be there. But there were also things missing.

"There were other notes that I thought were in the chord that weren't there — that people had transcribed into the chord because they believed that it had to be in the chord, but actually weren't there."

Share The Beatles with fill-in drummer Jimmie Nicol in Melbourne, 1964.

Ghost notes

These are called ghost notes: notes that your brain expects to hear, based on the context they appear in, but that aren't actually present. For example, most Beatles songbooks transcribe the chord with a G note on the lowest guitar string.

"There are Gs in the chord, there's just no low G on the bottom," Dr Brown says.

"In fact, the bottom note being played is a D on Paul McCartney's bass, and the notes being played by George Harrison ... actually have an A on the bottom."

Alongside all of these deductions, Dr Brown also made allowances for human — or Beatle — error.

"When I was moving the frequencies to the closest notes, some were quite a distance away. It's slightly out of tune — forever! That's part of the beauty of the chord."

But even after all this effort, something was still amiss.

No matter which way Dr Brown arranged his data, it never completely lined up with the instruments that were assumed to have played the chord.

This arresting, infuriating musical moment, deconstructed mathematically, just wasn't playable by three Beatles on three Beatle guitars.

"From everything that I had read before ... I had expected that it was just George Harrison's 12-string. But I ran into a roadblock because it wasn't playable on a guitar.

"[Yet] I still ran into logical problems, even if I allowed the chord to be not just being played by George Harrison, but by John Lennon and Paul McCartney."

Dr Brown's roadblock came in the form of three frequencies for a certain F note that couldn't be attributed to any of the guitars, no matter how many possibilities he entertained.

"I almost gave up looking at the chord, but then I realised that I had an assumption I'd carried with me — that the Beatles had played that opening chord.

"What happens if that assumption wasn't quite true? What happens if there was another instrument?"

Another instrument — with keys

"Later on in the song, you can clearly hear a piano doubling George Harrison's lead guitar solo. So I thought, maybe there is a piano in the song.

"A piano typically has three strings tuned identically corresponding to each note, and a hammer hits them. So perhaps those three frequencies could come from a single note on the piano."

Dr Brown ran down to his local music store to test if his hunch was true.

"Before I got thrown out of the store by the manager, I managed to determine yes, it was possible, and it actually told me a little deduction about the piano used in the Abbey Road Studios to record that chord.

"It told me that it was a mid-sized grand piano, as opposed to a much larger grand piano, because of where where that note sits."

Those missing F frequencies, Dr Brown concluded, came from a single note played on a mid-sized grand piano.

The final piece slid into place: buried deep in the mix of that shimmering opening chord, someone — maybe Ringo, but probably George Martin — had played an F on a piano.

Dr Brown remembers that moment very clearly. The feeling that he'd just solved a Beatles mystery — one that had been buzzing around his subconscious brain for decades.

"It was extraordinarily exciting. The chord was a mystery for such a long time, and people still talk about it," he said.

"I think that maybe one of the legacies of the Beatles' music is the brilliance of what they put into their songs, on so many levels, so that people 40-50 years later will still be analysing them, still trying to figure out what made them so great."