We can work it out – The Beatles’ A Hard Day’s Night Chord

By Dr Kevin Houston

More than 50 years after their split, The Beatles, remain a cultural phenomenon. There’s a recent eight-hour documentary, Get Back, just last month a remastered version of Revolver was released to appropriate acclaim and then there’s the steady stream of books, articles, and videos written about the smallest details in their songs and lives. The bit I want to talk about is arguably one of the smallest details. It’s a single crashing chord lasting no more than 3 seconds that has perplexed musicians since its recording nearly 60 years ago. The John Lennon-penned song A Hard Day’s Night was chosen to open the album accompanying their first film. Their producer wanted a dramatic opening, so Lennon provided a chord played by the members of the band before they launch into the song.

Over the years many musicians have attempted to replicate the sound and discover what notes are being played and by whom. There are many competing theories and one book on the Beatles devotes 40 pages to discussing this single chord.

It’s a quest that smacks of pure maths – searching for knowledge, trying to describe the apparently indescribable, explaining wonder.

So, it’s entirely appropriate that the solution to the problem can be found with mathematics.

What’s really surprising though is that the mathematics has its roots in topics seemingly unconnected to analysing sound. The mathematical tool employed was developed in the early 1800s for studying the flow of heat within objects. This is a common feature of mathematics. Tools developed for one problem regularly find application in completely unrelated areas.

This tool is called the Fourier Transform. Joseph Fourier (1768 -- 1830) was a celebrated French mathematician – celebrated enough to be one of a number of scientists and engineers whose names are displayed on the Eiffel Tower. In his heat flow study, building on the work of others, he showed how to reduce everything to sines and cosines -- a topic taught in school and known to Indian mathematicians around 500CE. This is yet another example of one mathematical tool being used in a seemingly unrelated area -- who would have foreseen that the flow of heat could involve sines and cosines?

In our case, we can use the Fourier Transform to separate the Beatles’ opening chord into its constituent frequencies along with their volumes. Although the actual picture of the frequencies is messy there is enough clarity to find an answer which sounds convincingly close to the original recording when played.

But the Fourier transform can be used for much more. One version of it, the Discrete Fast Fourier Transform, is one of the most useful mathematical tools ever created and forms the backbone of many applications. For example, it transforms photos, video, and audio (in the form of JPEGs, MPEGs and MP3s) so that they can be compressed for easier transmission or storage.

The moral is that mathematicians cannot predict where their work will lead. The Indians of antiquity could never have thought that sines and cosines would be used to study the flow of heat. In turn Fourier could never have thought his heat flow work could be used to analyse the compositions of future musical megastars or create JPEGs.

The same will no doubt be true of mathematicians working today. Some will have particular applications in mind while some research an area purely because of its intrinsic interest and may even be unable to see any applications. But in both cases, history tells us we’ll see surprising applications that no one, not even the creators, predicted.

But none of this happens by accident. The mathematical sciences need proper funding. The government promised £300 million in extra funding for maths back in 2020. £176 million of that is yet to be delivered and recent government announcements suggest they will welch on their promise. If we are to unlock the surprising breakthroughs and crucial solutions to today’s policy challenges like climate change, energy supply and the cost of living the Chancellor ought to confirm the full funding at the earliest opportunity. Rishi Sunak recently revealed his favourite band are the Beatles. There’s no doubt he faces many hard days nights to put the economy back on a sound footing. But without funding maths and the science and technology it underpins that project will be doomed to fail.

Dr Kevin Houston is a senior lecturer and mathematician at the University of Leeds.