This is a more detailed critique of the paper by Brown called
"Mathematics, Physics, and A Hard Day's Night", which is referenced in
the article that is linked to in the earlier thread called "FYI: AHDN
opening chord solved?"
As I mentioned in a post to that thread, I believe that the analysis and
conclusions in that paper are incorrect.
Here is a link to the paper in question:
http://www.mscs.dal.ca/~brown/n-oct04-harddayjib.pdf
Some of this critique is based on the details of the mathematics. Maybe
a mathematically inclined person would be willing to comment on this
analysis (rags?). However most of the critique, and the conclusion that
I reach (namely, that Brown's conclusion is incorrect) is based on
arguments involving interpretation rather than mathematics per se.
The critique is broken up into three parts: (1) assumptions; (2)
mathematics, and (3) interpretation.
(1) Assumptions:
Brown's paper essentially begins with three "straw-man" suggestions,
which he states are derived from examination of musical transcriptions.
However, what is astounding to me is that none of these suggestions
incor****ates George Harrison's own comments as to what he played in the
song! According to Wikipedia (and many other sources), George stated
that he played an Fadd9:
"It is F with a G on top..."
http://en.wikipedia.org/wiki/Hard_Day's_Night_(song)#Opening_chord
Now let's review Brown's "straw men", which he refers to as "Versions".
He properly concentrates on what George is playing with his 12-string
Rickenbacker guitar, since this is the instrument that dominates the
chord. The Versions are as follows:
Version 1 consists simply of the notes that are played on a guitar with
the entire third fret barred, i.e., (3 3 3 3 3 3) where the numbers
represent the frets involved in the chord starting with the
lowest-pitched (E) string on the guitar and progressing upwards in
pitch. The corresponding notes are G, C, F, Bb, D, and G, respectively.
One obvious problem here is that there is obviously no Bb in the chord,
and I don't believe that there's a credible source that suggests
otherwise. Try it on piano or guitar! The Bb note lends a minor
tonality to the chord that simply isn't present.
Version 2 is a reasonable approximation to the actual chord; it is
basically a G7sus4 chord, played on the guitar as (3 5 3 5 3 3) with
corresponding notes G, D, F, C, D, and G. This chord is a reasonable
substitute for a single, unaccompanied guitar; however, it is missing
the A note that appears in the chord mentioned by George.
Version 3 is a poor-man's approximation to Version 2. Unlike in the
other versions, Brown also suggests what Paul and John are playing in
this version. He has George playing a Gsus4 chord, played as (3 5 5 5 3
3) and containing the notes G, D, G, C, D, and C, Paul playing a D note
in the bass, and John playing a simpler form of the Gsus4 (x x 0 0 1 3)
with notes D, G, C, and G (x indicates that no note is played on that
string and 0 indicates an open string). Not only is the A note
described by George missing in this version but the F note that appears
in the other two versions is missing as well.
Thus, only one of the three versions is even part-way credible, namely,
version 2. All knowledgable sources claim that the chord as a whole
consists of five notes: D, F, A, C, and G. The only issue in dispute
really is which notes are attributable to which instruments.
Interestingly, the Wikipedia article summarizes Pedler's analysis which
I believe is the most accurate description available, with both George
and John playing the Fadd9 chord, Paul playing D on bass, and George
Martin playing two D notes and a G note on piano.
It is my contention that if Brown had included George Harrison's
statement as to what he played in the chord as one of his "straw men",
it is more likely that he would have reached the correct conclusions.
However, as I describe in more detail below, there are flaws in both the
mathematical analysis and in his interpretation of his results that may
have prevented that outcome.
(2) Mathematics:
The mathematical analysis is based on the Fourier transform, which
breaks a signal up into its various frequency components. Musical notes
have definite frequencies, so in principle Fourier analysis can answer
the question as to which musical notes are played. There is a
complication, however: musical instruments produce so-called harmonics,
which have frequencies that are integer multiples of the notes that are
played. These harmonics are what gives each instrument its
characteristic tone. Brown's analysis partially takes into account the
presence of harmonics, but as I describe in the "interpretation" section
below, there is additional information available on the harmonic
characteristics of various instruments that he doesn't account for and
that may have steered him in the correct direction.
There is another subtlety involving the use of the Fourier transform in
this analysis, and that is related to the fact that he samples a
relatively small (~ 1 second) time interval within the chord. He also
uses digital sampling. Both of these indicate that he uses a version of
a Discrete Fourier Transform (DFT) to analyze the chord. He doesn't go
into much detail as to how this is done but one can draw some general
conclusions regardless. The main point that I want to make here is that
there is a maximum frequency resolution associated with the finite time
duration of the sample. This is related to the so-called
Nyquist-Shannon theorem:
http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem
which states a relation****p between the time spacing between sampled
points and the maximum frequency observable in the transform. A
corollary relates the duration of the time sample to the frequency
resolution, which winds up being inversely pro****tional to the time
interval sampled. Theoretically the resolution limit is 1/(2T) (where T
is the duration of the sample, 1 second in this case), but there are
other technicalities that conspire to make the practical resolution
available larger.
The most im****tant technicality is the need for so-called apodization.
As it turns out, abruptly truncating the sampled signal at its beginning
and end leads to artifacts in the Fourier transform that make accurate
interpretation difficult. Thus, numerical DFT routines use apodization
functions which smooth out the "rough edges" of the sampled signal
before the transform is performed:
http://en.wikipedia.org/wiki/Apodization
One effect of the use of apodization is a decrease in the frequency
resolution achievable. Brown in his paper doesn't mention these
subtleties; however, it is reasonable to assume that the practical
frequency resolution is 1/T instead of 1/(2T); consequently, given his
1-second sample, the practical frequency resolution is 1 Hz.
Why is this im****tant? Simply, it is because in his table of
frequencies extracted from the FFT there are many frequencies listed
that are within 1 Hz of each other (see the table at the top of his p.
3). He claims that these multiple frequencies represent multiple
occurrences of the same note in the chord, and I believe that this
misleads him into an incorrect conclusion as to who played what!
(3) Interpretation:
As an example of the type of error made above, consider the list of
notes that appears near the bottom of the left-hand column on p. 3. In
that list he includes four separate occurrences of the note D3 (which is
the D note almost an octave below middle C, which would correspond to
the open D string on a guitar). However, comparison with the table at
the top of this page indicates that these four D notes correspond to the
following frequencies (in Hz): 145.619, 148.621, 149.372, and 150.123.
However, note that the second and fourth of these are within 1 Hz of the
third, and it is therefore incorrect to conclude that they represent
four distinct notes! It is likely that the "extra" frequencies are
artifacts resulting from either the finite sample or the use of
apodization. The first frequency is reasonably separated from the
others, so at best one can conclude that at least two D3 notes are being
played, and not necessarily four.
Brown uses this erroneous conclusion to infer that George played a D
note in the chord. Not so, according to George! And not necessarily
so, according to a properly conducted analysis! The same error results
in supposed multiple occurrences of the notes F3, C4, and so on. From
this point on, the analysis simply falls apart. His conclusion? That
George played (x 0 0 0 1 x) on his guitar (with notes A, D, G, and C),
that Paul played a D on bass (which is the one thing that everyone
agrees on anyways), and that John simply played a high C note (high E
string, eighth fret) on his guitar! And that none of the guitars played
an F note! The only F note played in the chord is deduced as coming
from George Martin's piano part.
To a trained musician, the conclusion is nonsense, but it is fair to ask
what it is that allows a musician to associate certain notes with a
given instrument and other notes with other instruments. The answer
takes us back to harmonics. Each instrument has its own characteristic
tone, and as I mentioned earlier, this tone is associated with the
presence of certain harmonics. A piano simply does not sound like an
electric 12-string guitar.
However, one can analyze the harmonic structure of notes produced by the
instruments taken one at a time in the same manner as that employed by
Brown in his analysis of the entire chord; in fact, it's much simpler!
This suggests a more intelligent and thorough method of analyzing the
chord; simply put, one would record single notes played by each of the
instruments and Fourier analyze each separately. What one would find is
that each note on each instrument has its own characteristic harmonic
content; for example, the F note played on John's acoustic guitar might
have a second harmonic that has 1/3 the intensity of the fundamental in
its Fourier transform.
One would then try to match up the ratios of fundamentals and harmonics
obtained through analysis of each instrument separately with the results
of Brown's Fourier analysis (properly corrected, of course, for the
frequency-resolution issue described above). Better yet, if one is
capable of analyzing the instruments separately, re-do Brown's Fourier
analysis with a longer time sample and thereby achieve better frequency
resolution.
It sounds like a challenge to me. I've got the Ric 12 guitar, a couple
of acoustic guitars that probably come close to John's Gibson acoustic,
a bass, and a piano. Now all I need is a grant :-)
RichL
--
All follow-ups are directed to the newsgroup rec.music.beatles.moderated.
If your follow-up more properly belongs in the unmoderated newsgroup,
please
change your headers appropriately. -- the moderators
--
|
