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

Physics of the Blues: Music, Fourier and the Wave-Particle Duality. J. Murray Gibson Presented at Fermilab October 15 th 2003. The Advanced Photon Source. Art and Science . Art and science are intimately connected Art is a tool for communication between scientists and laypersons.

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

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

  2. The Advanced Photon Source

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

  4. The Poetry of Mathematics

  5. Music is excellent example…

  6. 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?

  7. 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..

  8. 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?)

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

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

  11. How to make a scale using notes with overlapping harmonics Bflat7/4 G3/2 E5/4 1 2 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….

  12. The pentatonic scale * * * * * C D E G A 9/8 5/4 3/2 27/16 1 Common to many civilizations (independent experiments?)

  13. 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)

  14. Natural Scale Ratios

  15. Diatonic Scale C D E F G A B C “Tonic” is C here Doh, Re, Mi, Fa, So, La, Ti, Doh….

  16. 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)

  17. 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

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

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

  20. Equal temperament scale Note Frequency (Hz) Difference from Just Scale (Hz) Step (semitone) = 2^1/12 Pianoforte needsmultiplestrings to hidebeats!

  21. The Well-Tempered Clavier 1 2 3 4 6 5

  22. Mostly Mozart From his Sonata in A Major

  23. D dim c.f. D min

  24. Minor and Major

  25. 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”

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

  27. 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

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

  29. Beethoven’s “Moonlight” Sonata in C# Minor 1 5 F# dim 9 13 F# (or C) dim

  30. “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

  31. Crushed notes and the blues

  32. Not quite ready for the blues

  33. Four-tone chords • Minimum for Jazz and Contemporary Music And more: 9th, 11th s and 13th s (5,6 and 7note chords)

  34. 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!)

  35. 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

  36. 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

  37. Ants Pant! Phase-enhanced imaging Westneat, Lee et. al..

  38. 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…..

  39. Fun with phases… Helen Gibson Margaret Gibson

  40. Fourier Transforms Helen Marge Amp Phase

  41. Swap phases Helen with Marge’s phases Marge with Helen’s phases Phases contain most of the information… (especially when no symmetry)

  42. Sound Examples Beethoven Clapton Clapton with Beethoven’s phases Beethoven with Clapton’s Phases

  43. 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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