Dominic Pendler dedicated a chapter of his book "The Songwriting Secrets of the Beatles" to this chord, and demonstrates quite convincingly that there was a piano involved. It's a bit disappointing that the Dalhousie professor doesn't point to the existing literature before claiming to have discovered something novel.
Where does this article (the PDF cited by nickb, not the link, which isn't working for me anyway) claim to be doing something novel?
It's a cute demonstration of basic physics, more-or-less at the freshman level (although lots of high school students could follow it). It doesn't claim to be anything else.
UPDATE: Finally got the blog link to load. While the original paper is modest in its claims, the blog hypes them up and then fails to cite the original paper. This blogger does not earn a cookie.
The paper is a little bit hand-wavy about the mathematics of how CDs are sampled, how the Fourier transform works, and how frequencies are converted to semitones. But it does have references which mostly alleviates my desire for rigor :)
Once the author has a list of pitches though, the process of assigning pitches to instruments is very well laid-out. The key to locating the piano was that piano strings are hit in triplets, and each string in a triplet is tuned ever so slightly differently.
Signal processing was one of my favorite courses in college.
It's nice to have an algorithmic way to find the pitches in a chord, but honestly anybody with a good ear can pick that out. Try sitting down with a Bill Evans recording and figuring out the different chords you hear - something jazz students do all the time.
The fact that it's not reproducible on a standard-tuned guitar might have stopped guitar players from figuring it out on the fretboard though.
But what they've debated is the exact voicing and instrumentation (which instrument played which part), NOT the notes of the chord. All this "debate" seems to just be Beatles fanboys looking for a lot more cleverness and complexity in Beatles music than really exists. As a huge fan of music and music theory, and no huge fan of the Beatles, I can assure you that countless abundantly more eluding, complex, and novel chords exist in other music.
"see if he could apply a mathematical calculation known as Fourier transform to solve the Beatles’ riddle."
I feel less dumb today knowing clearly exactly how that would work. Although I would have thought it would in practice be feeding it into an audio analysis app and picking the peaks and mapping them to notes on various tunings. Not really mathematics though, but cool none the less.
