Pierwszy akord 'A Hard Days Night' (ang.).pdf

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

Lewisohn quotes the Beatles’ producer, George Mar-

tin, “We knew it would open both the lm and the

soundtrack LP, so we wanted a particularly strong

and eective 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):

• Version 2: G D F C D G (another favorite, and

one that often appears on Internet sites as the

“one” that George Harrison (GH) played).

• Version 3: This one has George, John Lennon

(JL) and Paul McCartney (PM) playing, with

George (on his twelve string electric) playing G

D G C D G, slightly dierent than in version

2, while John plays D G C G and Paul plays D

on his bass (this is the transcription from [1]).

2 Musical Forensics

We need to discuss briey the physics and mathe-

matics of sound (see [5, 2] for more details). Pure

tones have a frequency, which corresponds to its

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

le, 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-

tles). A threshhold was chosen which kept the sound

faithful to the original. The table shows the 48 fre-

quencies with amplitude 0 . 02 or larger.

Is there any way to tell which of these three ver-

sions is the right one? Mathematics will help direct

us to the answer.

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,

so choosing the reference note A 220 Hz, the fre-

quencies were converted to the number of semitones

above or below A 220 (by applying the function

f ( x ) = 12 log 2 ( x/ 220)). Here is the list of semitones

(we see that some of the instruments could have been

better tuned as not all of the numbers are close to

their nearest integer):

how the notes were played we’ll need to make some

deductions.

Some of the notes (especially in the higher range)

must be harmonics, as they are well beyond what

instruments can play. In fact, the range of a guitar

is from E2 to about E6 and the bass guitar from

E1 to about D4. Notes could have arisen on either

George Harrison’s or John Lennon’s guitar or Paul’s

bass. The analysis now shows why the three well

known transcriptions of the opening chord must all

be wrong: each has a low G2 being played, but this

note is denitely missing.

It is well known that for the album A Hard Day’s

Night , George used a 12 string guitar and its sound

can denitely be heard on the solo in A Hard Day’s

Night . Thus it seems safe to assume George used

this guitar on the opening chord as well. The twelve

string guitar has each string doubled, with the bot-

tom four in octaves, so the strings are, from lowest

to highest, E2 E3 A2 A3 D3 D4 G3 G4 B3 B3 E4 E4.

It seems reasonable that notes on strings of roughly

the same thickness struck on one instrument would

be roughly of the same amplitude. Looking back at

the frequencies and their amplitudes, we see that one

D3 is extra loud, with an amplitude of 0.175. This

is taken as a bass note from Paul’s Hofner bass (no

other single frequency is nearly as loud).

Now A2 and A3 can be paired o, both likely

coming from George’s 12 string (a nice open pair of

strings). But even with one of the D3’s accounted

for on Paul’s bass, what about the other three D3’s?

− 11 . 9466, − 7 . 14367, − 6 . 79035, − 6 . 70313,

− 6 . 61635, − 4 . 04686, − 3 . 97239, − 3 . 89825,

− 2 . 07421, − 0 . 124124, 3.02237, 3.07191, 5.34031,

10.0254, 11.9353, 15.0472, 17.0118, 17.0339, 17.056,

22.0254, 22.0584, 22.075, 25.5205, 25.8951, 27.0719,

29.1659, 30.5752, 30.9449, 31.0238, 31.0337, 33.099,

34.699, 35.9056, 41.078, 41.133, 41.1385, 41.1439,

41.1604, 41.1659, 41.1714, 43.0042, 43.0091, 43.7514,

43.8127, 45.708, 46.0626, 46.0667, 46.1244

In musical circles, middle C is written as C4, with

the second number indicating the octave, so A 220

Hz is written as A3. Here are the frequencies above

rounded to the nearest semitones:

A2, D3, D3, D3, D3, F3, F3, F3, G3, A3, C4, C4,

D4, G4, A4, C5, D5, D5, D5, G5, G5, G5, B5, B5,

C6, D6, E6, E6, E6, E6, F#6, G#6, A6, D7, D7,

D7, D7, D7, D7, D7, E7, E7, F7, F7, G7, G7, G7,

Many of the notes appear in the various versions

of “the chord”. But to argue what was played and

Only one can come from George’s 12 string, and even

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 dierent,

but each string on a piano is individually tuned and

is likely to be slightly o 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 eect). George Martin

(GM) played D3 F3 D5 G5 E6 on the piano. The

other notes are fairly high and could be attributed

to harmonics of these notes, except that there is a

loud C5, which could have been played by John high

up on his six string. There is also one extra E6 un-

accounted for, which is taken as a harmonic.

3 The End

The notes played on the piano interweave well with

the notes on the 12 string, starting a bit higher (at

D3) from the lowest note played on the guitar and

ending higher (at E6). The amplitudes show why the

piano is so well hidden; it is mixed perfectly, with

amplitudes almost identical to those of the higher

strings played on Harrison’s guitar.

In All You Need is Ears [4], George Martin makes

a point of saying “it shouldn’t be expected that peo-

ple are necessarily doing what they appear to be

doing on records” and likens recording to lmmak-

ing, where all sorts of eects are carried out in the

background in order to create illusions. We see that

sometimes mathematics can unravel the best mys-

teries.

References

[1] T. Fujita, Y. Hagino, H. Kubo and G. Sato, The

Beatles Complete Scores , Hal Leonard, Milwau-

kee, 1993.

[2] T.W. Gray and J. Glynn, Exploring Mathe-

matics with Mathematica , Addison Wesley, New

York, 1991.

[3] M. Lewisohn, The Complete Beatles Recording

Sessions , Doubleday, Toronto, 1988.

Acknowledgements

[4] G. Martin, All You Need Is Ears , St. Martins’

Press, New York, 1979.

This article was partially supported by a grant

from the Natural Sciences and Engineering Research

Council of Canada.

[5] J.S. Rigden, Physics and the Sound of Music ,

Wiley, New York, 1977, pg. 71.

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