Physics of the Blues: Music, Fourier and the WaveParticle Duality

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Physics of the BluesMusic, Fourier and the
Wave-Particle Duality

• J. Murray Gibson
• Presented at Fermilab
• October 15th 2003

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The Advanced Photon Source
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Art and Science

• Art and science are intimately connected
• Art is a tool for communication between
  scientists and laypersons

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The Poetry of Mathematics
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Music is excellent example
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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?

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Overtones of a string
Fourier analysis all shapes of a string are a
sum of harmonics
Harmonic content describes difference between
instruments e.g. organ pipes have only odd
harmonics..
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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?)

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Semiconductor Bandgaps

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

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Familiarity with the Keyboard
C
D
E
G
A
F
B
1 step semitone 2 steps whole tone
C D E F G A
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How to make a scale using notes with overlapping
harmonics
G3/2
E5/4
B?flat7/4
2
1
3
6
7
8
4
5
C
Concept of intervals two notes sounded
simultaneouslywhich sound good together Left
brain meets the right brain
Pythagoras came up with this.
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The pentatonic scale





C
D
E
G
A
5/4
3/2
9/8
27/16
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Common to many civilizations (independent
experiments?)
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Intervals

• Unison (first)
• Second
• Third
• Fourth
• Fifth
• Sixth
• Seventh
• Octave (eighth)

Two notesplayed simultaneously
Major, minor, perfect, diminished..
Not all intervals are HARMONIC(although as time
goes by there are more.. Harmony is a learned
skill, as Beethovendiscovered when he was booed)
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Natural Scale Ratios
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Diatonic Scale
C
D
E
G
A
F
B
C
Tonic is C here
Doh, Re, Mi, Fa, So, La, Ti, Doh.
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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 overlapminimum
of three notes to a chord (to notes ambiguity
which is widely played e.g. by Bach)
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The triads in the key of C
C E G M3 P5 C Major Triad
D F A m3 P5 D Minor Triad
E G B m3 P5 E Minor Triad
F A G M3 P5 F Major Triad
G B D M3 P5 G Major Triad
A C E m3 P5 A Minor Triad
B D F m3 d5 B Diminished Triad
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Three chords and youre a hit!

• A lot of folk music, blues etc relies on chords
  C, F and G

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Baroque Music
Based only on diatonic chords in one key (D in
this case)
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Equal temperament scale
Frequency (Hz)
Difference from Just Scale (Hz)
Note
Step (semitone) 21/12
Pianoforte needsmultiplestrings to hidebeats!
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The Well-Tempered Clavier
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2
3
4
6
5
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Mostly Mozart
From his Sonata in A Major
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D dim c.f. D min
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Minor and Major
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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 notin the diatonic scale for C, but
it is for F Also heading for the blues
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Circle of Fifths

• Allows modulation and harmonic richness
• Needs equal temperament
• The Well Tempered Clavier
• Allows harmonic richness

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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
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Romantic music..
A flat diminished (c.f. B flat dominant 7th)
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1
3
4
5
C diminished (Fdominant 7th)
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Beethovens Moonlight Sonata in C Minor
1
5
F dim
9
13
F (or C) dim
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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

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Crushed notes and the blues
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Not quite ready for the blues
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Four-tone chords

• Minimum for Jazz and Contemporary Music

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

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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

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The Wave-particle Duality

• Can be expressed as fourier uncertainty
  relationship
• Df DT 2 p

2p/f
DT
Demonstrated by musical notes of varying duration
(demonstrated with Mathematica or
synthesizer) Wave-nature ? melody Particle-nature
? percussive aspect
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Ants Pant!
Phase-enhanced imaging
Westneat, Lee et. al..
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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..

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Fun with phases
Helen Gibson
Margaret Gibson
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Fourier Transforms
Helen
Marge
Amp
Phase
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Swap phases
Helen with Marges phases
Marge with Helens phases
Phases contain most of the information
(especially when no symmetry)
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Sound Examples
Beethoven
Clapton
Clapton with Beethovens phases
Beethoven with Claptons Phases
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Conclusion

• Music and physics and mathematics have much in
  common
• Not just acoustics
• Musicians 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