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

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Presentation on theme: "Physics of the Blues: Music, Fourier and the Wave- Particle Duality J. Murray Gibson Presented at Fermilab October 15 th 2003."— Presentation transcript:

1 Physics of the Blues: Music, Fourier and the Wave- Particle Duality J. Murray Gibson Presented at Fermilab October 15 th 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 CDEGAFB 1 step = semitone 2 steps = whole tone C D E F G A

11 How to make a scale using notes with overlapping harmonics Pythagoras came up with this…. Concept of intervals – two notes sounded simultaneously which sound good together Left brain meets the right brain… C G 3/2 E 5/4 B  flat 7/4

12 The pentatonic scale CDEGA 1 5/43/29/827/16 * **** Common to many civilizations (independent experiments?)

13 Intervals Unison (“first”) Second Third Fourth Fifth Sixth Seventh Octave (“eighth”) Major, minor, perfect, diminished.. Two notes played simultaneously 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)

14 Natural Scale Ratios Interval Ratio to Fundamental in Just Scale Frequency of Upper Note based on C (Hz) (C-C) Unison Minor Second25/24 = (C-D) Major Second9/8 = Minor Third6/5 = (C-E) Major Third5/4 = (C-F) Fourth4/3 = Diminished Fifth45/32 = (C-G) Fifth3/2 = Minor Sixth8/5 = (C-A) Major Sixth5/3 = Minor Seventh9/5 = (C- B) Major Seventh15/8 = (C-C’) Octave

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

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 overlap minimum 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 (Middle C) C C # 4 /D b D4D D # 4 /E b E4E F4F F # 4 /G b G4G G # 4 /A b (Concert A) A A # 4 /B b B4B C5C Frequency (Hz) Step (semitone) = 2^1/12 Pianoforte needs multiple strings to hide beats! Difference from Just Scale (Hz)Note

21 The Well-Tempered Clavier

22 Mostly Mozart From his Sonata in A Major

23 D dim c.f. D min

24 Minor and Major

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26 The “Dominant 7 th ” The major triad PLUS the minor 7 th 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 Hz, –B flat is Hz –B is 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”

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

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29 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

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

31 Beethoven’s “Moonlight” Sonata in C # Minor F # dim F # (or C) dim

32 “Blue” notes Middle C = Hz E flat = Hz Blue note = perfect harmony = 5/4 middle C = Hz – slightly flatter than E E = Hz Can be played on wind instruments, or bent on a guitar or violin. “Crushed” on a piano 12 Bar Blues - C F 7 C C F 7 F 7 C C G 7 F 7 C C

33 Crushed notes and the blues

34 Not quite ready for the blues

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

36 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!)

37 Using Music to Explain Physics Quantum Mechanics –general teaching Imaging and Phase –phase retrieval is important in lensless imaging, e.g. 4 th generation x-ray lasers

38 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  melody Particle-nature  percussive aspect TT 2  /f

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

40 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…..

41 Fun with phases… Helen GibsonMargaret Gibson

42 Fourier Transforms Helen Marge Amp Phase

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

44 Sound Examples Clapton Beethoven Clapton with Beethoven’s phases Beethoven with Clapton’s Phases

45 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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