Mathematics, Physics And A Hard Day/'s Night

Mathematics, Physics and A Hard Day’s Night
Jason I. Brown, Dalhousie University
Abstract
• Version 1: G C F Bb D G (a favorite by its
ease of play; just play a barre chord at the
third fret).
In this article we shall use mathematics
and the physics of sound to unravel one of
the mysteries of rock ’n’ roll – how did the
Beatles play the opening chord of A Hard
Day’s Night? The song may never sound the
same to you again.
1
Introduction
It was forty years ago that the Beatles ushered in a
new era in pop music with the opening of A Hard
Day’s Night. The importance of the opening chord
was clearly apparent to the Beatles. In The Com-
plete Beatles Recording Sessions [3], author Mark
• Version 2: G D F C D G (another favorite, and
Lewisohn quotes the Beatles’ producer, George Mar-
one that often appears on Internet sites as the
tin, “We knew it would open both the ﬁlm and the
“one” that George Harrison (GH) played).
soundtrack LP, so we wanted a particularly strong
and eﬀective beginning. The strident guitar chord
was the perfect launch.” This seemed to close the
discussion about the origin of the chord but should
it have?
Many a guitarist (whether professional or ama-
teur) has tried to reproduce the chord, but the voic-
ing of the chord has remained a subject of much dis-
cussion over the past 40 years. If you browse musi-
cal transcriptions for the chord, you will come across
three common ones (note that guitar parts are scored
as usual an octave up from where they sound):
1

• Version 3: This one has George, John Lennon
2
Musical Forensics
(JL) and Paul McCartney (PM) playing, with
George (on his twelve string electric) playing G
We need to discuss brieﬂy the physics and mathe-
D G C D G, slightly diﬀerent than in version
matics of sound (see [5, 2] for more details). Pure
2, while John plays D G C G and Paul plays D
tones have a frequency, which corresponds to its
on his bass (this is the transcription from [1]).
pitch, and an amplitude, which corresponds roughly
(but not exactly) to the loudness of the tone, and are
modelled mathematically by sine and cosine func-
tions. All sounds are made up of pure tones, which
add together to give you complex tones and chords
(that is, the functions of the latter are linear com-
binations of the functions for the pure tones).
A
single note sounded on an instrument is made up
of a fundamental (main) pure tone plus other tones,
called harmonics, whose frequencies are multiples of
the fundamental tone’s frequency. All of the ampli-
tudes are added together in this “mix” that we hear.
When a sound is digitized for a CD, the amplitude is
sampled 44,100 times every second. What is hidden
in the process of recording music are the individ-
ual frequencies, and how they were played. Fourier
Transforms can be used to dissassemble the sampled
amplitudes into the original frequencies.
After the song A Hard Day’s Night was opened
in a sound editing program on a computer, a seg-
ment of approximately one second was selected in
the middle of the chord. The sound was saved as a
ﬁle, and using some Mathematica subroutines from
[2, chapter 14] a Fourier Transform was run on the
list of data. There were 29,375 frequencies present,
which included not only the notes being struck, but
also harmonics, as well as any other frequencies that
might have arisen during the recording.
We are after the loudest notes, as these corre-
spond to the fundamental notes being struck (though
there will probably be some of the louder harmon-
ics present, along with possibly some other loud rat-
Is there any way to tell which of these three ver-
tles). A threshhold was chosen which kept the sound
sions is the right one? Mathematics will help direct
faithful to the original. The table shows the 48 fre-
us to the answer.
quencies with amplitude 0.02 or larger.
2

Freq. (Hz)
Ampl.
Freq. (Hz)
Ampl.
Freq. (Hz)
Ampl.
Freq. (Hz)
Ampl.
110.34
0.0600967
299.494
0.0298296
1050.86
0.0687151
2368.93
0.0221358
145.619
0.025485
392.57
0.0309716
1185.97
0.0372155
2371.19
0.0212846
148.621
0.0264278
438.358
0.0286329
1286.55
0.0231789
2371.94
0.0436633
149.372
0.0656018
524.678
0.0680974
1314.32
0.03819
2372.69
0.036042
150.123
0.175149
587.73
0.020613
1320.33
0.0223535
2637.65
0.0261839
174.142
0.0275547
588.48
0.0310337
1321.08
0.0494908
2638.4
0.0237794
174.893
0.0380282
589.231
0.0231753
1488.47
0.0241328
2754.
0.020001
175.643
0.0407103
785.141
0.0323532
1632.58
0.0205742
2763.76
0.0493617
195.159
0.0405164
786.642
0.0251928
1750.43
0.0234704
3083.52
0.0332062
218.428
0.0448308
787.393
0.0268553
2359.93
0.0366079
3147.32
0.0293723
261.964
0.0302402
960.784
0.0228509
2367.43
0.0267098
3148.07
0.0418507
262.714
0.0234502
981.801
0.02242
2368.18
0.0755327
3158.58
0.0285631
The frequencies need to be converted to notes,
how the notes were played we’ll need to make some
so choosing the reference note A 220 Hz, the fre-
deductions.
quencies were converted to the number of semitones
Some of the notes (especially in the higher range)
above or below A 220 (by applying the function
must be harmonics, as they are well beyond what
f (x) = 12 log2(x/220)). Here is the list of semitones instruments can play. In fact, the range of a guitar
(we see that some of the instruments could have been
is from E2 to about E6 and the bass guitar from
better tuned as not all of the numbers are close to
E1 to about D4. Notes could have arisen on either
their nearest integer):
George Harrison’s or John Lennon’s guitar or Paul’s
bass. The analysis now shows why the three well
−11.9466,
−7.14367,
−6.79035,
−6.70313,
known transcriptions of the opening chord must all
−6.61635,
−4.04686,
−3.97239,
−3.89825,
be wrong: each has a low G2 being played, but this
−2.07421, −0.124124, 3.02237, 3.07191, 5.34031,
note is deﬁnitely missing.
10.0254, 11.9353, 15.0472, 17.0118, 17.0339, 17.056,
It is well known that for the album A Hard Day’s
22.0254, 22.0584, 22.075, 25.5205, 25.8951, 27.0719,
Night, George used a 12 string guitar and its sound
29.1659, 30.5752, 30.9449, 31.0238, 31.0337, 33.099,
can deﬁnitely be heard on the solo in A Hard Day’s
34.699, 35.9056, 41.078, 41.133, 41.1385, 41.1439,
Night. Thus it seems safe to assume George used
41.1604, 41.1659, 41.1714, 43.0042, 43.0091, 43.7514,
this guitar on the opening chord as well. The twelve
43.8127, 45.708, 46.0626, 46.0667, 46.1244
string guitar has each string doubled, with the bot-
tom four in octaves, so the strings are, from lowest
In musical circles, middle C is written as C4, with
to highest, E2 E3 A2 A3 D3 D4 G3 G4 B3 B3 E4 E4.
the second number indicating the octave, so A 220
It seems reasonable that notes on strings of roughly
Hz is written as A3. Here are the frequencies above
the same thickness struck on one instrument would
rounded to the nearest semitones:
be roughly of the same amplitude. Looking back at
A2, D3, D3, D3, D3, F3, F3, F3, G3, A3, C4, C4,
the frequencies and their amplitudes, we see that one
D4, G4, A4, C5, D5, D5, D5, G5, G5, G5, B5, B5,
D3 is extra loud, with an amplitude of 0.175. This
C6, D6, E6, E6, E6, E6, F#6, G#6, A6, D7, D7,
is taken as a bass note from Paul’s Hofner bass (no
D7, D7, D7, D7, D7, E7, E7, F7, F7, G7, G7, G7,
other single frequency is nearly as loud).
G7
Now A2 and A3 can be paired oﬀ, both likely
coming from George’s 12 string (a nice open pair of
Many of the notes appear in the various versions
strings). But even with one of the D3’s accounted
of “the chord”. But to argue what was played and
for on Paul’s bass, what about the other three D3’s?
3

Only one can come from George’s 12 string, and even
accounted for, which is taken as a harmonic.
if John played another one on his six string, there’s
still another to account for. There is no evidence
that any guitar was multitracked, at least on this
opening chord. The two F3s create a much bigger
problem. For no matter how George plays an F3 on
his 12 string, an F4 should be heard as well, and
there is no F4 at all present!
A hidden assumption came to the fore. Beat-
les’ record producer, George Martin, is known to
have doubled on piano George Harrison’s solo on
the track. Could “the chord” be part piano? Pi-
anos have three strings for every note; a hammer
strikes all three at the same time to produce a sound.
That solved the problem of the three F3’s: all could
have come from a piano playing F3. Note that the
frequencies of the three F3s were slightly diﬀerent,
but each string on a piano is individually tuned and
is likely to be slightly oﬀ from one another in the
“triple.”
But what about the three left over D3’s? If all be-
longed to a single piano note, then where would the
single D4 come from? Not from George Harrison’s
guitar (as a D3 or another D4 would be present) and
not from George Martin’s piano (as otherwise there
would be three D4’s present). However, the bottom
ten pitches on a piano are single strings which change
to pairs of strings, and around C3 they change to the
usual triples of strings. But indeed there are some
grand pianos (of medium length) for which the break
occurs right after D3. This implied that two of the
D3s were played on the piano.
What George Harrison played on his 12 string
was nothing like any of the transcriptions: he played
A2 A3 D3 D4 G3 G4 C4 C4, most likely on string
sets 2 through 5 – eight strings with six open strings
in total; (for a great chiming eﬀect). George Martin
(GM) played D3 F3 D5 G5 E6 on the piano. The
other notes are fairly high and could be attributed
3
The End
to harmonics of these notes, except that there is a
loud C5, which could have been played by John high
The notes played on the piano interweave well with
up on his six string. There is also one extra E6 un-
the notes on the 12 string, starting a bit higher (at
4

D3) from the lowest note played on the guitar and
References
ending higher (at E6). The amplitudes show why the
piano is so well hidden; it is mixed perfectly, with
[1] T. Fujita, Y. Hagino, H. Kubo and G. Sato, The
amplitudes almost identical to those of the higher
Beatles Complete Scores, Hal Leonard, Milwau-
strings played on Harrison’s guitar.
kee, 1993.
In All You Need is Ears [4], George Martin makes
a point of saying “it shouldn’t be expected that peo-
[2] T.W. Gray and J. Glynn, Exploring Mathe-
ple are necessarily doing what they appear to be
matics with Mathematica, Addison Wesley, New
doing on records” and likens recording to ﬁlmmak-
York, 1991.
ing, where all sorts of eﬀects are carried out in the
background in order to create illusions. We see that
[3] M. Lewisohn, The Complete Beatles Recording
sometimes mathematics can unravel the best mys-
Sessions, Doubleday, Toronto, 1988.
teries.
[4] G. Martin, All You Need Is Ears, St. Martins’
Acknowledgements
Press, New York, 1979.
This article was partially supported by a grant
from the Natural Sciences and Engineering Research
[5] J.S. Rigden, Physics and the Sound of Music,
Council of Canada.
Wiley, New York, 1977, pg. 71.
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