Physics of the Blues:Music, Fourier and the Wave-

Particle DualityJ. 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

02

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

)/cos()( Lnxcxfn

n

Harmonic content describes difference between instrumentse.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?)

T. A. Steitz, et al., Yale University50S Ribosomal Subunit

Structural Biology Center 19ID

A portion of a diffraction pattern obtained at the Structural Biology Center 19ID beamline fromcrystals of the 50S ribosomal subunit Ń orthorhombic space group C222 1, a = 212 , b = 301 ,c = 576 Ń showing the spatial resolution of the measurement.(Courtesy of Dr. T. Steitz, Yale University)

Semiconductor Bandgaps…

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

Familiarity with the Keyboard

C D E GA FB

1 step = semitone2 steps = whole tone

C D E F G A

How to make a scale using notes with overlapping harmonics

2 3 4 51 6 7 8

Pythagoras came up with this….

Concept of intervals – two notes sounded simultaneouslywhich sound good together

Left brain meets the right brain…

C

G3/2

E5/4

Bflat7/4

The pentatonic scale

C D E G A

1 5/4 3/29/8 27/16

* * * * *

Common to many civilizations (independent experiments?)

Intervals

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

Major, minor, perfect, diminished..

Two notesplayed simultaneously

Not all intervals are HARMONIC(although as time goes by there are more..Harmony is a learned skill, as Beethovendiscovered when he was booed)

Natural Scale Ratios

IntervalRatio to Fundamental

in Just ScaleFrequency of Upper Note

based on C (Hz)

(C-C) Unison 1.0000 261.63

Minor Second 25/24 = 1.0417 272.54

(C-D) Major Second 9/8 = 1.1250 294.33

Minor Third 6/5 = 1.2000 313.96

(C-E) Major Third 5/4 = 1.2500 327.04

(C-F) Fourth 4/3 = 1.3333 348.83

Diminished Fifth 45/32 = 1.4063 367.93

(C-G) Fifth 3/2 = 1.5000 392.45

Minor Sixth 8/5 = 1.6000 418.61

(C-A) Major Sixth 5/3 = 1.6667 436.06

Minor Seventh 9/5 = 1.8000 470.93

(C- B) Major Seventh 15/8 = 1.8750 490.56

(C-C’) Octave 2.0000 523.26

Diatonic Scale

C D E G AF B C

Doh, Re, Mi, Fa, So, La, Ti, Doh….

“Tonic” is C here

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)

The triads in the key of CC 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

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

(Middle C) C4 261.63 0

C#4/D

b4 277.18 4.64

D4 293.66 -0.67

D#4/E

b4 311.13 -2.83

E4 329.63 2.59

F4 349.23 0.4

F#4/G

b4 369.99 2.06

G4 392.00 -0.45

G#4/A

b4 415.30 -3.31

(Concert A) A4 440.00 3.94

A#4/B

b4 466.16 -4.77

B4 493.88 3.32

C5 523.25 0

Frequency (Hz)

Step (semitone) = 2^1/12

Pianoforteneedsmultiplestrings to hidebeats!

Difference from Just Scale (Hz)Note

The Well-Tempered Clavier

1 2 3

45 6

Mostly Mozart

From his Sonata in A Major

D dimc.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 notin 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)

C diminished (Fdominant 7th)

1 2

3 4 5

Beethoven’s “Moonlight” Sonata in C# Minor

F# dim

F# (or C) dim

1

5

9

13

“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

f T ~ 2

Demonstrated by musical notes of varying duration (demonstrated with Mathematica or synthesizer)Wave-nature melodyParticle-nature percussive aspect

T

2/f

Ants Pant!

QuickTime™ and aCinepak decompressor

are needed to see this picture.

Phase-enhanced imaging

Westneat, Lee et. al..

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

Fourier TransformsHelen Marge

Amp

Phase

Swap phasesHelen 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