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Music_ Fourier and the Wave-Particle Duality

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					Physics of the Blues: Music, Fourier and the WaveParticle Duality
J. Murray Gibson Presented at Fermilab October 15th 2003

The Advanced Photon Source

Art and Science
• Art and science are intimately connected

• Art is a tool for communication between scientists and laypersons

The Poetry of Mathematics

  0
2

Music is excellent example…

Outline:
• What determines the frequency of notes on a musical scale? • What is harmony and why would fourier care? • Where did the blues come from? (We' re talking the "physics of the blues", and not "the blues of physics" - that's another colloquium). • Rules (axioms) and ambiguity fuel creativity • Music can explain physical phenomena
– Is there a musical particle? (quantum mechanics) – The importance of phase in imaging?

Overtones of a string

Fourier analysis – all shapes of a string are a sum of harmonics

f ( x )   cn cos(nx / L )
n

Harmonic content describes difference between instruments e.g. organ pipes have only odd harmonics..

Spatial Harmonics
• Crystals are spatially periodic structures which exhibit integral harmonics
– X-ray diffraction reveals amplitudes which gives structure inside unit cell

• Unit-cell contents? (or instrument timbre?)

Semiconductor Bandgaps…
• Standing waves in a periodic lattice (Bloch Waves) – the phase affects energy and leads to a bandgap

Familiarity with the Keyboard

A B C

D

E F

G

1 step = semitone 2 steps = whole tone CDEFGA

How to make a scale using notes with overlapping harmonics
E 5/4

G Bflat 3/2 7/4

1 C

2

3

4

5

6 7 8

Concept of intervals – two notes sounded simultaneously which sound good together
Left brain meets the right brain… Pythagoras came up with this….

The pentatonic scale
*
* * * *

C

D

E

G

A

1

9/8 5/4

3/2 27/16

Common to many civilizations (independent experiments?)

Intervals
• • • • • • • • Unison (“first”) Two notes played simultaneously Second Third Major, minor, perfect, diminished.. Fourth Fifth Not all intervals are HARMONIC Sixth (although as time goes by there are more.. Harmony is a learned skill, as Beethoven Seventh discovered when he was booed) Octave (“eighth”)

Natural Scale Ratios
Interval (C-C) Unison Minor Second (C-D) Major Second Ratio to Fundamental in Just Scale 1.0000 25/24 = 1.0417 9/8 = 1.1250 Frequency of Upper Note based on C (Hz) 261.63 272.54 294.33

Minor Third
(C-E) Major Third (C-F) Fourth Diminished Fifth

6/5 = 1.2000
5/4 = 1.2500 4/3 = 1.3333 45/32 = 1.4063

313.96
327.04 348.83 367.93

(C-G) Fifth
Minor Sixth (C-A) Major Sixth Minor Seventh

3/2 = 1.5000
8/5 = 1.6000 5/3 = 1.6667 9/5 = 1.8000

392.45
418.61 436.06 470.93

(C- B) Major Seventh (C-C’) Octave

15/8 = 1.8750 2.0000

490.56 523.26

Diatonic Scale

C

D

E F

G

A B C

“Tonic” is C here
Doh, Re, Mi, Fa, So, La, Ti, Doh….

Simple harmony
• Intervals
– “perfect” fifth – major third – minor third – the harmonic triads – basis of western music until the romantic era
• And the basis of the blues, folk music etc.
The chords are based on harmonic overlap minimum of three notes to a chord (to notes = ambiguity which is widely played e.g. by Bach)

The triads in the key of C
CEG M3 P5 C Major Triad DFA m3 P5 D Minor Triad

EGB
FAG

m3 P5 E Minor Triad
M3 P5 F Major Triad

GBD AC E

M3 P5 G Major Triad m3 P5 A Minor Triad
BDF m3 d5 B Diminished Triad

Three chords and you’re a hit!
• A lot of folk music, blues etc relies on chords C, F and G

Baroque Music

Based only on diatonic chords in one key (D in this case)

Equal temperament scale
Note (Middle C) C4 C#4/Db4 Frequency (Hz) 261.63 277.18 Difference from Just Scale (Hz) 0 4.64

D4 D#4/Eb4 E4
F4 F#4/Gb4

293.66 311.13 329.63
349.23 369.99

-0.67 -2.83 2.59
0.4 2.06 Step (semitone) = 2^1/12

G4 G#4/Ab4 (Concert A) A4
A#4/Bb4

392.00 415.30 440.00
466.16

-0.45 -3.31 3.94
-4.77

Pianoforte needs multiple strings to hide beats!

B4
C5

493.88
523.25

3.32
0

The Well-Tempered Clavier

1

2

3

4

5

6

Mostly Mozart

From his Sonata in A Major

D dim c.f. D min

Minor and Major

The “Dominant 7th”
• The major triad PLUS the minor 7th interval • E.g. B flat added to C-E-G (in the key of F) • B flat is very close to the harmonic 7/4
– – – – Exact frequency 457.85 Hz, B flat is 466.16 Hz B is 493.88 Hz Desperately wants to resolve to the tonic (F)
B flat is not in the diatonic scale for C, but it is for F Also heading for the “blues”

Circle of Fifths
• Allows modulation and harmonic richness
– Needs equal temperament – “The Well Tempered Clavier” – Allows harmonic richness

Diminished Chords
• A sound which is unusual
– All intervals the same i.e. minor 3rds, 3 semitones (just scale ratio 6/5, equal temp -1%) – The diminished chord has no root
• Ambiguous and intriguing

• An ability of modulate into new keys not limited by circle of fifths
– And add chromatic notes – The Romantic Period was lubricated by diminished chords

C diminished

Romantic music..
A flat diminished (c.f. B flat dominant 7th)

1

2

3

4

5

C diminished (Fdominant 7th)

Beethoven’s “Moonlight” Sonata in C# Minor
1

5

F# dim
9

13

F# (or C) dim

“Blue” notes
• Middle C = 261.83 Hz • E flat = 311.13Hz • Blue note = perfect harmony = 5/4 middle C = 327.29 Hz – slightly flatter than E • E = 329.63 Hz

• Can be played on wind instruments, or bent on a guitar or violin. “Crushed” on a piano • 12 Bar Blues - C F7 C C F7 F7 C C G7 F7 C C

Crushed notes and the blues

Not quite ready for the blues

Four-tone chords
• Minimum for Jazz and Contemporary Music

And more: 9th, 11th s and 13th s (5,6 and 7note chords)

Ambiguities and Axioms
• Sophisticated harmonic rules play on variation and ambiguity • Once people learn them they enjoy the ambiguity and resolution • Every now and then we need new rules to keep us excited (even though we resist!)

Using Music to Explain Physics
• Quantum Mechanics
– general teaching

• Imaging and Phase
– phase retrieval is important in lensless imaging, e.g. 4th generation x-ray lasers

The Wave-particle Duality
• Can be expressed as fourier uncertainty relationship

Df DT ~ 2 
2/f

DT
Demonstrated by musical notes of varying duration (demonstrated with Mathematica or synthesizer) Wave-nature  melody Particle-nature  percussive aspect

Ants Pant!

QuickTime™ an d a Cinepak decompressor are need ed to see this p icture .

Westneat, Lee et. al..

Phase-enhanced imaging

Phase Contrast and Phase Retrieval
• Much interest in reconstructing objects from diffraction patterns
– “lensless” microscopy ios being developed with x-ray and electron scattering

• Warning, for non-periodic objects, phase, not amplitude, is most important…..

Fun with phases…

Helen Gibson

Margaret Gibson

Helen

Fourier Transforms

Marge

Amp

Phase

Swap phases
Helen with Marge’s phases Marge with Helen’s phases

Phases contain most of the information… (especially when no symmetry)

Sound Examples

Clapton

Beethoven

Clapton with Beethoven’s phases

Beethoven with Clapton’s Phases

Conclusion
• Music and physics and mathematics have much in common • Not just acoustics
– Musician’s palette based on physics – Consonance and dissonance
• Both involved in pleasure of music

• Right and left brain connected?
– Is aesthetics based on quantitative analysis?

• Music is great for illustrating physical principles


				
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