Physics of the Blues Music, Fourier and the Wave-Particle Duality by qfa20129

VIEWS: 0 PAGES: 45

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


            0
            2
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

       f ( x )   cn cos(nx / L )
                n




       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?)
   Semiconductor Bandgaps…
• Standing waves in
  a periodic lattice
  (Bloch Waves) –
  the phase affects
  energy and leads
  to a bandgap
Familiarity with the Keyboard




             A B C   D   E   F   G




                             1 step = semitone
                             2 steps = whole tone
    CDEFGA
How to make a scale using notes
  with overlapping harmonics
       E        G Bflat
       5/4      3/2 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

Common to many civilizations (independent experiments?)
                        Intervals
•   Unison (“first”)
                             Two notes
•   Second                   played simultaneously

•   Third
•   Fourth           Major, minor, perfect, diminished..

•   Fifth
•   Sixth             Not all intervals are HARMONIC
                      (although as time goes by there are more..
•   Seventh           Harmony is a learned skill, as Beethoven
                      discovered when he was booed)
•   Octave (“eighth”)
                 Natural Scale Ratios
                       Ratio to Fundamental   Frequency of Upper Note
      Interval
                             in Just Scale         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   F   G   A B C


                      “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.

          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
         CEG    M3 P5 C Major Triad


         DFA     m3 P5 D Minor Triad


         EGB     m3 P5 E Minor Triad

          FAG     M3 P5 F Major Triad


          GBD      M3 P5 G Major Triad


          AC E     m3 P5 A Minor Triad


                  BDF    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
    Note         Frequency (Hz)   Difference from Just Scale (Hz)

(Middle C) C4       261.63                 0
   C#4/Db4          277.18                4.64
     D4             293.66               -0.67
   D#4/Eb4          311.13               -2.83          Step
                                                        (semitone) =
     E4             329.63                2.59          2^1/12
     F4             349.23                0.4
   F#4/Gb4          369.99                2.06
     G4             392.00               -0.45           Pianoforte
   G#4/Ab4          415.30               -3.31           needs
(Concert A) A4      440.00                3.94           multiple
                                                         strings to hide
   A#4/Bb4          466.16               -4.77           beats!
     B4             493.88                3.32
     C5             523.25                 0
The Well-Tempered Clavier




     1       2       3




 4               6
         5
       Mostly Mozart




From his Sonata in A Major
D dim
c.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 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
              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)




     1                            2




3                      4                 5




                                   C diminished (Fdominant 7th)
Beethoven’s “Moonlight” Sonata in
            C# Minor
         1




     5


                                  F# dim
     9




     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
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
                                Df DT ~ 2 
                             2/f




                                DT

    Demonstrated by musical notes of varying duration
    (demonstrated with Mathematica or synthesizer)
    Wave-nature  melody
    Particle-nature  percussive aspect
Ants Pant!



           QuickTime™ an d a
         Cinepak decompressor
     are need ed to see this p icture .




                                          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
Helen   Fourier Transforms   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



        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

								
To top