Then there’s the opening twang of “Hard Day’s Night,” for over forty years one of music’s great mysteries.
How did the Beatles do that? For almost forty-six years, musicians tried to duplicate that opening chord by combining a Rickenbacker twelve-string guitar like George Harrison’s with John Lennon’s six-string and Paul McCartney’s bass. Everyone failed; they couldn’t make the sound.
Then in 2010 a mathematician, of all people, took a shot. Instead of grabbing three guitars, he grabbed a yellow pad and felt pen and performed a calculation called a Fourier transform. From it, he deduced that a fourth instrument was involved, likely played by the Beatles’ composer, George Martin: a piano striking an F note. But that opening note was so surprising and without precedent that it took over forty-five years, and a formula that includes the sequence f(x)e−2ΠixΣ, to explain it.
“Unthinking” by Harry Beckwith.