## A CAUTIONARY TALE ABOUT MATHEMATICS

### ...or how a mathematician got lost in his field

In his much celebrated, widely published and quoted paper "Mathematics, Physics and A Hard Day's Night" Dr Jason I. Brown purportedly unravelled the mystery of the opening chord of The Beatles' song A Hard Day's Night.

Dr Brown used a mathematical process called Discrete Fourier Transform to dissassemble a digital sample into its constituent frequencies in order to ascertain the notes which comprise the chord.

Here is the solution from Dr Brown's paper:

 "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 correspond to the fundamental notes being struck (though there will probably be some of the louder harmonics present, along with possibly some other loud rattles). A threshold was chosen which kept the sound faithful to the original. The table shows the 48 frequencies with amplitude 0.02 or larger." 12-st = George Harrison 12-string guitar 6-st = John Lennon 6-string guitar Name and instrument detail formerly listed in the original paper is integrated into the table for convenience.

This result has been lauded as the definitive solution. Supporters declare the mathematics don't lie. Musicians should be humbled by the exposure of their arty pretensions in the face of cold hard science. Yet several other solutions actually sound more like the original recording than the suggestion above.

How can this be so?

Dr Brown was led to the wrong answer through very bad science.

### Ingoring empirical evidence

From his pool of 48 frequencies Dr Brown happily paired off octave A's and D's based on his knowledge of a 12-string guitar and concluded that George didn't play F3 because there was no corresponding F4 present. He attributes the three F3's to a piano - and claims this as a revelation in his research.
Archival concert footage clearly shows George and John playing identical fingerings of the chord which include F-notes on the D-string (hence F3-F3-F4 and no D's). George called the chord "F with a G on top" and it appears throughout the song with an F-root.
The acoustic guitar is on a separate track and can be completely isolated in the remixed excerpt on the LOVE album.

 The chord played is Fadd9 utilising all six strings availability of this recording postdates Brown's research

More incredible is that having made the revelation of a piano accounting for his 'missing' frequencies and assigning a note to Lennon's 6-string Dr Brown completely ignores a solitary A4. This note is probably the most signficant of all since it creates the cluster of F4-G4-A4 which is the basis for the distinctive 'chime' effect. Its very existence is entirely due to the fact that it is produced by the high-octave of the 3rd-string-pair of the 12-string guitar. This string 'automatically' changes the top note voicing and expands the range of the chords while arpeggios create melodies with close steps instead of wide skips.

48 frequencies is not enough data to represent multiple vibrating strings. The lower harmonic series of each string will produce (12+6+1)x8 = 152 frequencies in the guitars alone; 72 of which appear in the range where notes are likely to be played (up to E5). Dr Brown's result (8+1+1) is 32 frequencies up to E5 before even considering the 14 strings of his five-note piano chord.

### Misunderstanding the nature of harmonics

"We are after the loudest notes, as these correspond to the fundamental notes being struck (though there will probably be some of the louder harmonics present, along with possibly some other loud rattles)."
The fact that the second and third loudest notes in the pool are D7 and C6 should have signified to Dr Brown that these ubiquitous harmonics will undermine any assumptions about the loudest frequencies being a reliable indicator of notes played. The very loudest frequency which overpowers the entire chord was not actually played - D3 is the 2nd-Harmonic of the D2 in the Bass.
The recorded sound has the bass frequencies rolled off below about 90Hz suppressing the D2 as well as F2 in the guitars. D2 and G2 in the piano part recorded on a separate track are filtered out completely. Reducing bass energy was standard practice for pop recordings at the time.
"George Martin 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 unaccounted for, which is taken as a harmonic."
The maths is all very convenient here. Dr Brown looked inside a piano and found double and triple string sets and a handful of notes in the pool fitted nicely. But that piano chord is a nonsensical conclusion musically and the notion that Lennon would play a single note is utterly ridiculous.
While Dr Brown accepts that high frequencies are harmonics of lower notes he does not explain why there are practically no harmonics corresponding to the three F3's in the piano. These should produce F4 C5 F5 A5 C6 Eb6 three times each but the pool has just one C5 and one C6.
It's curious why Dr Brown singled out the C5 as being essential having allocated only ten of the twenty loudest frequencies. Perhaps he realised that John was left with nothing else to do.
"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 definitely missing."
Two B5's in the pool are ignored. If they are presumed to be harmonics, then where are their corresponding fundamentals? Does Dr Brown have any idea which notes could produce B5? Did he re-examine the other 29,327 frequencies to look for B4 or E4 or (yes indeed!) G2 as possible sources?
Why are they not included as part of the piano chord? ...well we know the answer to that is because they didn't fit neatly into his triple-string theory.

### Misunderstanding the perception of pitch

"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."
While laypersons have some concept of how the harmonics affect the timbre and distinguish for example brass instruments from strings, what is not so familiar is how that combination of harmonics relates to pitch.
Paul H. Erlich in MusicTheory@yahoogroups.com explains:
Pitch is a perception formed from many partials and corresponds to the best-fit fundamental for a harmonic series that would include those partials. Even if the fundamental is physically absent, and all the other partials are there, you hear only one pitch, and it corresponds with the frequency of the missing fundamental.
This has serious consequences for the pool of notes when:
"A threshold was chosen which kept the sound faithful to the original."
Dr Brown should have found six F's and six A's but they are not in his pool because their amplitude is below the arbitrary threshold - having been filtered in the recording and mixing process. His selective data pool consists largely of harmonics - 33 frequencies from C5 to G7 plus some of the lower D's F's and G's - hence most of the notes which he is allocating among the instruments are not the notes which were actually played.

Effective analysis requires examining the harmonics as members of a related series rather than simply discarding lots of numbers to make things easier. In Dr Brown's ensemble of fourteen notes it turns out that seven are actually harmonics - often emanating from instruments other than what he specifies:

Bass  D3
Piano  D5  G5  E6
12-string Guitar  D3  D4
6-string Guitar  C5

### Misunderstanding tuning and Equal Temperament

"we see that some of the instruments could have been better tuned as not all of the numbers are close to their nearest integer"

Dr Brown's data includes an E6 calculated to be 30.5 semitones above A-220 falling midway between Eb6 and E6. It's not an out of tune instrument, it is the 13th-Harmonic of the "definitely missing" G2. Instrument sounds do indeed contain "harmonics, whose frequencies are multiples of the fundamental tone’s frequency" however the musical scale to which western music is tuned is based on intervals which are not integer multiples of any fundamental frequency. Thus the harmonics are not precisely in tune with the scale tones - which makes most of the frequencies fairly easily distinguished as either harmonics or fundamentals and ultimately identified as played notes.

As it happens the Bass Guitar is tuned sharp - a fact that can be used to confirm that it plays D2 not D3 - and the lowest F-notes in the guitars are pushed a little sharp because they are being fretted with the thumb over the top of the neck. The lowest notes of the piano D2 and G2 will appear to be flat because piano tuners stretch the octaves to make their harmonics more in tune with the higher notes. (harmonics sound sharp due to the inharmonicity of thick strings - being physically unable to vibrate into the nodes)

### Disregarding the principles of music

Dr Brown establishes that the guitars are playing the notes A D G C. It's a stack of fourths with no particular tonality but a very distinctive suspended sound. Presumably that is what The Beatles wanted. He then asserts that a piano contributes unrelated notes F and E.
• Why would more notes be added which not only drastically alter the character of the chord but also severely compromise its reproduction in live performance?
• How would George Martin have decided musically what notes to add to this chord... and come up with an F and an E?
• What musical value is there in the chord proposed by Dr Brown?

### Failing to verify results

"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)."
Did Dr Brown ever play this? The 'chime' is not a result of open-strings playing wide intervals. It is the cluster of F4-G4-A4 as heard even more clearly in the coda where the notes (all fretted) are played separately but allowed to ring into each other.

"I actually had to run down to a piano store and stick my head in a few grand pianos..." [on radio]
If Dr Brown had gone to his music store and played his piano chord instead of counting the strings he might have realised that nobody (even himself) recognised it.
The ensemble does contain the notes suggested by the "musical forensics" (except for that high-E) but the haphazard allocation of notes to instruments doesn't sound much like the original recording.

### Misquoting references

"George Martin makes a point of saying "it shouldn't be expected that people are necessarily doing what they appear to be doing on records" and likens recording to filmmaking, where all sorts of effects are carried out in the background in order to create illusions."
Whatever the context of Martin's remark in his long and varied career there is nothing unorthodox about the composition, performance or recording of this chord. The entire session was three hours including five complete takes as well as overdubs of vocals, acoustic guitar, percussion, guitar solo and piano. The group were very busy filming during the week that the song was written including the day of the recording. One can hardly expect that they devoted much time at all to concocting one chord.

### Making unsubstantiated claims

"We see that sometimes mathematics can unravel the best mysteries."
Hmm... yeah, right.