The classic Beatles song, “A Hard Day’s Night”, opens with a complex ringing chord that has had songbooks (and musicians) arguing among themselves for decades. Complicating the answer is that even Paul McCartney can’t exactly remember what was done.
Full disclosure: I relate to the Beatles much the way I relate to Mozart: I recognize their musical genius but much of their most popular music does not ‘move’ me either intellectually or emotionally. But I love a good musical puzzle as much as can be.
In principle, given modern computer technology, the problem of transcribing a piece of music should be simple: digitize the audio, carry out a Fourier analysis, and convert the resulting frequencies to note names. Right?
Well… Feed in unaccompanied flute and this will work fine. (As anybody who’s owned an analog synth knows, a triangle wave is a pretty decent starting point for a flute sounds — and while a triangle does have some harmonics, the fundamental is very strong and there are only odd harmonics so you can tell apart the fundamentals pretty easily from the rest in the Fourier spectrum.) Feed in a Hammond organ with just a single drawbar open: ditto. Feed in a more complex sound but with restricted harmony (e.g., a violin playing only single notes), no problem. Feed in a complex chord played by multiple instruments on top of each other, and things get hairier. Have some of the multiple instruments not be quite in tune, or let some be in equal temperament and others in just intonation, and things gets even worse.
An applied mathematician at Dalhousie University did a Fourier analysis on the opening chord some time ago and turned that into a paper. Does this sound like an academic with too much time on his hands, “partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada,” no less? Well, to me it sounds like a good “torture test” for the robustness of a musical transcription code. And where it comes to science popularization, this definitely hits the spot with the musically minded: only yesterday I saw another popular article about the now a decade old analysis being linked on Instapundit.
Just retaining all frequencies with relative amplitudes above 0.02 still gave him 48 frequencies, from which he squeezed a solution that looks good in theory but just doesn’t sound “quite right”.
A musical transcription site run by somebody with the delightful pseudonym “Waynus of Uranus” points out a fly in the ointment that people who grew up with digital recording wouldn’t even have thought of. Back in the day, loud bass tones meant pushing against the limitations of vinyl singles and lo-fi audio equipment alike, so the deep end of the bass (about 80 Hz and lower) was routinely rolled off with an equalizer or a highpass filter during mixing or mastering. What this means, for example: if Paul were to strike an open D string on his bass guitar (or an A string at the fifth fret) his fundamental would be below the filter cutoff, and the Fourier spectrum would instead have the second harmonic much stronger — leading to claims like “Paul played a D3 and a soft D2 at the same time”. I know bass players like Geddy Lee or Rush or Steve Harris of Iron Maiden play lots of double-stops, but this really is a progressive rock or metal thing to do, not a pop thing.
Applied mathematician Kevin Houston takes it from there and digs further in a very geekish way. While the original record was mono, it turns out there is a stereo mix made for the movie—and in the early days of stereo, it was not unusual for recording engineers to just put some instruments all left and others all right, with the vocal in the center. (This is, pretty much, how I used to jam along with Deep Purple records: Jon Lord’s organ and Ritchie Blackmore’s guitar were usually at opposite end of the stereo image, so you could single out their parts by listening to one stereo channel at a time.) In the stereo
In the stereo mix of AHDN, Paul (bass) and George (12-string guitar) are off to one side, and John (acoustic guitar) off to the other, together with producer George Martin on piano. Better still: after subtracting the left channel from the right (i.e., “phase-inverting”), it becomes clear that the acoustic is playing an Fadd9 chord. (That means: an F major chord with an added ninth, a.k.a. a “Steely Dan chord“. It differs from a major ninth chord F9 in that the seventh is omitted.)
To cut a very long story short (some mathematicians can get quite verbose ;)), this is the solution (which relies on a good dose of Occam’s razor/the Law of Parsimony as well):
- Paul just plays a low D2, but because of EQing off the deep end, the D3 overtone/second harmonic comes through louder than the fundamental, hence the acoustic illusion that the bass note played is D3
- John plays F2 A2 F3 A3 C4 G4 (in standard tuning, frets 1-0-3-2-1-3)
- George plays the same chord, but on a 12-string in standard tuning—where the bottom four “courses” have the second string one octave higher. Hence aside from the slight tuning discrepancy with John, he adds F4 A4 as new pitches
- Finally, George Martin on the piano, with the sustain pedal down, plays D2 G2 D3 G3 C4, which one could call a Gsus4/D chord. Sympathetic resonance from the undamped piano strings adds the wash of low-level extra pitches that befuddles the Fourier analysis.
Not only does this not require attributing instrumental acrobatics to the Beatles that are out of character for them, but actually playing those notes on the respective instruments does produce a sound quite like the record. (Listen at 7:17 in the video below.)
Kevin and his collaborators could not readily find an electric 12-string, so they simulated that by layering two six-string electric chords: once fretted 1-0-3-2-1-3, the second time 13-12-15-14-1-2 with an extra hand. “Fake Nashville Tuning“, if you like.)
If this isn’t the solution, it sounds much closer than anything else I’ve heard. Enjoy the above video!