physics of the blues: music, fourier and the wave- particle duality j. murray gibson presented at...
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Physics of the Blues:Music, Fourier and the Wave-
Particle DualityJ. Murray Gibson
Presented at Fermilab
October 15th 2003
Art and Science
• Art and science are intimately connected
• Art is a tool for communication between scientists and laypersons
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
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
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
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 “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
“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
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…..
Swap phasesHelen with Marge’s phases Marge with Helen’s phases
Phases contain most of the information… (especially when no symmetry)
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