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What is Physics of Music (Musical Acoustics)

What is Physics of Music (Musical Acoustics). Investigation of the relationships between the perceptual and physical attributes of musical sound Basic correlations: Loudness <-> Intensity Pitch <-> Frequency Timbre (“color”) <-> Spectrum. Examples of more complex questions:

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What is Physics of Music (Musical Acoustics)

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  1. What is Physics of Music(Musical Acoustics) • Investigation of the relationships between the perceptual and physical attributes of musical sound • Basic correlations: • Loudness <-> Intensity • Pitch <-> Frequency • Timbre (“color”) <-> Spectrum

  2. 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?”

  3. 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)

  4. 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 …..

  5. 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

  6. 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

  7. The most difficult math 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

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

  9. 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.

  10. 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 = 2*v/2L Mode 4 f4 = 4*v/2L N N N

  11. 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 …)

  12. 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

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

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

  15. Fourier synthesis

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

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

  18. Beats and the Critical Region • When two coherent sound waves superimpose, they will go in phase and out of phase at a rate corresponding to the difference of the two original frequencies, producing “beats” with a frequency f(beats) = |f1 – f2| • A simple DEMO varying the beat frequency demonstrates the existence of a “critical region” where your brain is no longer able to count the beats, yet the frequency difference is not yet large enough for you to perceive two independent sounds. Two sounds with such frequency difference produce a rough, unpleasant sensation.

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

  20. Consonance and Dissonance • Combination of the two above ingredients cannot but remind you of the Pythagoras’ recognition that tones with frequencies in ratio of small integers are consonant. • Example: musical “fifth”: an interval with the frequency ratio of the fundamentals of the two tones 3:2 From the well known mathematical theorem: 3 times 2 = 2 times 3 • we conclude something quite non-trivial: every 3rd harmonic of the bottom tone will coincide with every second harmonic of the upper tone. Even when the “fifth” is slightly mistuned, this will results in slow beats, not the unpleasant roughness. And the other harmonics (5th, 7th etc) will be so far from each other that they will be “out of the critical region”, and therefore they will not produce any roughness either.

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

  22. the smaller the integers involved are, the more justified is the above reasoning. • Therefore, the “unison” (frequency ratio 1:1) is the most consonant (and also quite boring) interval, followed by the fifth (3:2), fourth (4:3), Major 3rd (5:4) and minor 3rd (6:5). • That just about does it for the consonant intervals (the Major and minor 6ths are just complements of the minor and Major 3rds). • The “theoretical” frequency ratios for the dissonant intervals (such as 16:9) should be taken with a (large) grain of salt.

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

  24. Intensity -> Sound Intensity Level -> Loudness Level -> Loudness W/m2 -> dB -> phons -> sones approx.: L = 2 (LL-40)/10 Sound intensity as perceived by the humans ear/brain => loudness

  25. Addition of sounds: • “coherent” sounds: add amplitudes, square the result to get the intensity • “incoherent sounds”: [ ] within critical band: add intensities [ ] outside critical band: add loudnesses [ ] way outside: perceive two (or more) independent sounds

  26. Heisenberg Uncertainty Principle

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