Physics of the Blues: Music, Fourier and the Wave-Particle Duality J. Murray Gibson Presented at Fermilab October 15th 2003 Not serious science – My hobby - An Amuse-Guele Maybe not original content, but certainly unusual for Physics Colloquium, and I hope fun Outfox Harvey Drucker talking about the new ways of war..

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

Music is excellent example… Of all the seminar halls In all the physics joints In the whole world You had to walk into this one…

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 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?) Ribosome

Semiconductor Bandgaps… Standing waves in a periodic lattice (Bloch Waves) – the phase affects energy and leads to a bandgap “Bloch Waves sent my kids to college” said a solid-state physicist,” and sent them back home” said a student..

Familiarity with the Keyboard C D E F G 1 step = semitone 2 steps = whole tone C D E F G A

How to make a scale using notes with overlapping harmonics Bflat 7/4 1 2 3 4 5 6 7 8 C 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 Demonstrate with Scotland the Brave and Chinese/Native American Melody Harpo 1 9/8 5/4 3/2 27/16 Common to many civilizations (independent experiments?)

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

Natural Scale Ratios Interval Ratio to Fundamental in Just Scale Frequency 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 “Tonic” is C here C D E F G A B C Doh, a deer “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. Demonstrate 2 note and 3 note chords re a piece Melody can also fill in Mostly harmony, but also counterpoint Disonance is also a part of the story (but limited) AMAZING GRACE? 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 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

Three chords and you’re a hit! A lot of folk music, blues etc relies on chords C, F and G Dylan Folk example – blues later e.g. Blowing in the Wind? Gives you Edelweiss – who you cgnna impress

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

Equal temperament scale Note Frequency (Hz) Difference from Just Scale (Hz) (Middle C) C4 261.63 C#4/Db4 277.18 4.64 D4 293.66 -0.67 D#4/Eb4 311.13 -2.83 E4 329.63 2.59 F4 349.23 0.4 F#4/Gb4 369.99 2.06 G4 392.00 -0.45 G#4/Ab4 415.30 -3.31 (Concert A) A4 440.00 3.94 A#4/Bb4 466.16 -4.77 B4 493.88 3.32 C5 523.25 Step (semitone) = 2^1/12 Pianoforte needs multiple strings to hide beats! The Well-Tempered Clavier

The Well-Tempered Clavier 1 2 3 4 6 5

Mostly Mozart From his Sonata in A Major Note erroneous corrections on manuscript preserved for centuries – Must have frustrated Mozart (stupid referees – somethings are universal) From his Sonata in A Major

D dim c.f. D min

Minor and Major Major and Minor on piano Brilliant major, e.g. Beethoven, Tchaikovsky

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 Musical example? And circle and more notes

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 Chopin’s Nocturne in E Flat C diminished

Romantic music.. A flat diminished (c.f. B flat dominant 7th) 2 1 3 4 5 C diminished (Fdominant 7th)

Beethoven’s “Moonlight” Sonata in C# Minor 1 5 F# dim 9 Diminished based on C base only – very clever and foreboding…. 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 The “St Louis Blues”

Crushed notes and the blues

Not quite ready for the blues 1904

Four-tone chords Minimum for Jazz and Contemporary Music The Rich sounds of James Bond – For Your Eyes Only Also classic Motown 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!) Similarity with mathematics

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 p 2p/f DT Demonstrated by musical notes of varying duration (demonstrated with Mathematica or synthesizer) Wave-nature  melody Particle-nature  percussive aspect

Ants Pant! 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 Transforms Helen 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 Beethoven Clapton 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 “My Kind of Town”