1. 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 A coherent(?) synthesis: Music, Science and Human Affairs, With Exuberance and Humility Physics of Music => Physics AND Music

2. PHYS 207 enrollment 2010

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

10. … 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

23. 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.

25. Examples of more complex questions: “ What is the role of imperfections in creating the perception of perfection?” “ What is the role of sound in Music?”

27. 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 …..

28. 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

30. The most difficult math we will use in PHYS207 Mode 1: L= λ /2 Mode 2: L=2 λ /2 Mode 3: L=3 λ /2 Mode 3: L=4 λ /2 Mode n: L=n λ /2 i.e. 1/ λ =n/2L n=1,2,3,… 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) = f n = n f(1) where the fundamental frequency f (1) = f 1 = v/2L f = f(1) f = 2 f(1) f = 3 f(1) f = 4 f(1) λ L

31. Vibration modes of a system with 2 and 3 transverse degrees of freedom.

32. 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.

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

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

39. Heisenberg Uncertainty Principle

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

42. Fourier synthesis

43. waveform spectrum Instrument 1: f1 = 440 Hz Instrument 2: f1=440 Hz f1=1175 Hz

44. waveform spectrum Instrument 3: f1=58 Hz f1 = 196 Hz f1 = 440 Hz

46. Beats between tones of slightly different frequencies f1 and f2 f1 f2 f1 and f2

48. Spectra of two tones in musical “fifth” (frequency ratio 3:2) a) Note C b) Note G frequency intensity intensity

50. The ear, as see by A physician A physicist An electrical engineer/c.sci.

52. Intensity -> Sound Intensity Level -> Loudness Level -> Loudness W/m 2 -> dB -> phons -> sones Sound intensity as perceived by the humans ear/brain => loudness approx.: L = 2 (LL-40)/10