## Archive for November 3rd, 2008

### A Hard Day’s Night: Math finds a missing piano

November 3, 2008

What do you think when you hear the opening chord of “Hard Day’s Night” by the Beatles?

Is it, “I wonder if that opening chord is G, D, F, C, D, G like many web sites suggest or if it is G, D, G, C, D, G (George Harrison’s 12-string), plus D, G, C, G (John Lennon), and D on bass (Paul McCartney), as given in the official Beatles’ score?” And if you don’t know (you don’t know, do you?), do you think, “I wonder if Fourier Transforms could help me figure it out.”

If that’s what you’ve been thinking, you’ll be delighted to learn that the answer is YES! Fourier Transforms are those integrals that allow you to take a function like $f(x)$ and turn it into another function by calculating
$\int_{-\infty}^{\infty} f(k) e^{2 \pi i k x} dk$
and then playing around with it there. As explained in this article on Science Daily,

The process allowed him to decompose the sound into its original frequencies using computer software and parse out which notes were on the record.

The “him” is Dr. Jason Brown, a professor in the Mathematics and Computer Science Department at Dalhousie University in Halifax, Nova Scotia. He looked at the notes, and discovered that neither of the chords mentioned above actually works. What he found, instead, was that there had to be a piano. In particular,

Some music scholars and authors have previously suggested that a piano was included in the sonic layer, and Harrison allegedly offered differing versions of the guitar voicings himself in various interviews. But, how can we be sure that rock and roll’s most famous guitar chord is part piano, and what the most likely guitar voicings were? With a bit more deductive work, I found the presence of the piano did indeed solve the frequency problem, and the voicings I deduced (along with likely positioning on the guitars) are shown in the diagram. The important point is that the piano is there because the math says it is. (From this article on Guitar Player, which is also where the notes on the two chords above came from.)

This story is apparently a few years old, but was recently picked up and is making the rounds. There are more details on Dr. Brown’s web site [with more musical information], plus he has a program he wrote that lets you explore more about math and music. And that’s totally cool.