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PHYS207: Physics of Music

PHYS207: Physics of Music. Vladimir (Vladi) Chaloupka Professor of Physics. PHYS 207 enrollment 2010. PHYS207: Physics of Music. Vladimir (Vladi) Chaloupka Professor of Physics Adjunct Professor, School of Music Affiliate, Virginia Merrill Bloedel Hearing Research Center

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PHYS207: Physics of Music

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  1. PHYS207: Physics of Music Vladimir (Vladi) Chaloupka Professor of Physics

  2. PHYS 207 enrollment 2010

  3. PHYS207: Physics of Music Vladimir (Vladi) Chaloupka Professor of Physics Adjunct Professor, School of Music Affiliate, Virginia Merrill Bloedel Hearing Research Center Affiliate faculty, DXARTS Adjunct Professor, Henry M. Jackson School of International Studies and BY FAR the most modest professor on campus (BY FAR !) www.phys.washington.edu/users/vladi/PHYS207 A coherent(?) synthesis: Music, Science and Human Affairs, With Exuberance and Humility Physics of Music => Physics AND Music

  4. PHYS 207 enrollment 2010 NOTE: similar enrollment in PHYS216/SIS216: Science and Society (5 credits NW or I&S; www.phys.washington.edu/users/vladi/phys216)

  5. Einstein as Scientist, Musician and Prophet • Einstein as scientist: In 2005 we celebrated the Centenary of Einstein’s Annus Mirabilis • Einstein as musician: from a review: “Einstein plays excellently. However, his world-wide fame is undeserved. There are many violinists who are just as good.” • Einstein as prophet: “Nuclear weapons changed everything except our way of thinking.”

  6. Preview of the Physics of Music Syllabus and Schedule • And the square feet per person: 536 square feet vs. (536 feet) squared = 250,000 square feet! => An important goal of the course: improve (or install) your BS detector

  7. Music and Science, with Exuberance and Humility Pythagoras’ integers Kepler’s Harmonia Mundi Superstring Theory: all elementary particles as “modes of vibration” of the same string (ergo: “Princeton String quartet”) Laser Interferometer Space Antenna: “listening to the gravitational Symphony of the Universe” Music as an example of emergent complexity: parts of Art of Fugue “sound like parts of the Mandelbrot set” Goedel Escher Bach Exuberance and Humility: Two Pipe Organs

  8. LISA: Laser Interferometer Space Antenna Will detect the change of the distance as small as 10-11 m !!!

  9. LISA: listening to the gravitational symphony of the Universe

  10. LISA orchestra, soloists and the first 0.000 000 000 000 1 seconds

  11. The Andromeda Galaxy: 2 million light years away. The most distant object visible by naked eye (you have to know where to look, and find a really dark place, but the experience is very much worth it!) Note: for details on when and how to see Andromeda, see http://www.physics.ucla.edu/ ~huffman/m31.html

  12. J.S.Bach as Amadeus The central Theme of Amadeus (play/movie) applied to Bach The Bach genetic phenomenon Bach myths: BACH = 14 JSBACH = 41 even (from a doctoral Thesis [sic]): “the Unfinished fugue breaks off at bar 239 because 2+3+9 = 14” !

  13. Number of (male) Bach’s doing music at any particular year

  14. … finally I realized that to me, Goedel and Escher and Bach were only shadows cast in different directions by some central solid essence. Douglas Hofstadter

  15. Goedel Escher Bach Hofstadter A musico-logical fugue in English Goedel Undecidability Theorem: “In every sufficiently powerful formal system, there are propositions which are true, but not provable within the system” (i.e. “Truth if more than Provability”) Relief provided by fanciful Dialogues

  16. Hofstadter’s GEB Dialogues(in the spirit of Lewis Carroll) ….. Meaning and Form in Mathematics Sonata for Unaccompanied Achilles Figure and Ground Chromatic Fantasy, and Feud Brains and Thoughts English French German Suite Minds and Thoughts …..

  17. Mandelbrot Set Tour (optional) 1) z(0) = 0 2) z(n+1) = z(n)^2 + c and back to 2) 3) if z(n) finite then c belongs to the set Amazingly, this simplest of algorithms results into an object of infinite complexity (and arresting beauty). One cannot but recall Dirac’s claim that the Quantum Electrodynamics explains “most of Physics and all of Chemistry” … Also: the varied copies of Mandelbrot “body” are reminiscent of various versions of Art of Fugue theme, and the filaments are like the secondary motifs …

  18. Exuberance and Humility in Music and Science Left: The pipe organ at the St. Marks Cathedral in Seattle Above: the 1743(Bach was just composing the Art of Fugue then!) instrument at the College of William and Mary in Williamsburg.

  19. Bach’s Kunst der Fuge (Art of the Fugue) as a Triple Art: 1) Art of writing a fugue 2) Art of playing a fugue 3) Art of listening to a fugue. And the main skill a student of the Art of Listening to a Fugue must learn is how to achieve the right balance between the perception of the melodies and the perception of the harmony – you need to cultivate the ambiguity, in order to achieve the fusion of the counterpoint into Music.

  20. Elementary Physics of Music • Vibration of a string: the slowest (fundamental) mode has a “node” at both ends • Faster modes have additional nodes in between • It is not difficult to determine the frequencies of the modes: fn = n*f1 where f1 = v/2L • in general, the frequency spectrum of an arbitrary periodic vibration of the string will consist of equidistant peaks at the above frequencies – this is often called a ”harmonic” spectrum: • harmonic spectrum  periodic sound

  21. The most difficult math/phys reasoning we will use in PHYS207 Mode 1: L=λ/2 f = f(1) Mode 2: L=2 λ/2 f = 2 f(1) Mode 3: L=3 λ/2 f = 3 f(1) f = 4 f(1) Mode 3: L=4 λ/2 λ Mode n: L=nλ/2 i.e. 1/λ =n/2L n=1,2,3,… L Now: wavelength = distance traveled in one period: λ = v T i.e. T = λ/v And frequency is the inverse of period: f=1/T = v/λ = v (n/2L) = n(v/2L) So by a sequence of simple (almost trivial) steps, we have obtained an important and far-reaching result: Frequency of the n-th mode is f(n) = fn = n f(1) where the fundamental frequency f(1) = f1= v/2L

  22. Modes of vibration of a string N N Mode 1 f1 = v/2L Mode 2 f2 = 2*v/2L N • Modes of vibration understood as either standing waves, or as resonances of a system with infinite number of degrees of freedom N N Mode 3 f3 = 3*v/2L Mode 4 f4 = 4*v/2L N N N

  23. Example: Spectra of two tones intensity a) Note C intensity b) Note G frequency 0 f 2f 3f 4f 5f 6f….. octave 5th 4th Major 3rd minor 3rd i.e. the harmonic overtones of a simple tone contains the musically consonant intervals (we will learn about the intervals soon …)

  24. Consequences of these extremely simple considerations are actually far-reaching: • any periodic sound is a mix of several “harmonics”, equidistantly spaced in frequency In the first approximation: • increasing the “amounts” uniformly corresponds to louder sound • changing the frequency of the fundamental corresponds to changing the pitch • using different proportion of fundamental / second/ third / … harmonics means changing the “timbre”, i.e. the “sound color”. • As we will see, these consideration also determine consonance vs. dissonance

  25. Resonance: • Example: a mass on a spring • Great playground for Elementary Physics: • Newton: F = ma Restoring force of a spring: F = -kx • Equation of motion: ma = -kx Energy: Kinetic = ½ mv2 Potential: 1/2kx2 Total = constant (“conserved”) After calculations (not really difficult): resonance at f = (1/2pi) sqrt(k/m)

  26. Resonance corresponds to a peak in the response of the system to a periodic stimulus at a given frequency More complicated systems have more than one resonant frequency; each of them corresponds to a mode of vibration; each mode is characterized by its nodes Practical examples: Child on a swing Car stuck in snow Tacoma Narrows bridge collapse Vibration of the violin string …..

  27. Modes of vibration of a membrane are not harmonic => The sound is not periodic => there is no definite pitch Possible spectrum:

  28. “Standing waves” vs. modes of a system with infinite NDF Vibration modes of a system with 2 and 3 transverse degrees of freedom.

  29. Vibration of a string can be understood as superposition of traveling waves, and/or as modes of vibration of a system with infinite number of degrees of freedom.

  30. Waves: Waves are disturbances propagating in space [ ] Mechanical / electromagnetic / gravitational / quantum /. [ ] Longitudinal / transverse [ ] 1d / 2d / 3d / … NB: consequence for intensity = f(distance) [ ] traveling wave “standing wave” = a mode of vibration = a superposition of traveling waves [ ] reflection off the – fixed end - free end

  31. Heisenberg Uncertainty Principle

  32. V = v(sound) Vs = v(source of sound) Shock wave if Vs > v

  33. Fourier synthesis

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