Fundamentals of Musical Acoustics

What is sound? air pressure time

Time something else (air pressure) time ? t = 0

These variations in air pressure over time can be decomposed into sine waves with amplitude and frequency

Peak KE Peak PE Simple Harmonic Motion (Makes sine waves) t

y x  cos  sin  1 Sinusoids y = sin  x = cos 

1     a t a = sin(t) a = cos(t)

A A -A Amplitude a = A sin(t)

y x  A cos  A sin  A y = A sin  x = A cos 

Frequency How frequently does the sin/cos complete a whole cycle?

C B A D FH G E    t   phase    90  radians degrees a t ACGBDEFACGBDEFHHA

Frequency 1 cycle 22 1 cycle / sec. 2  / sec. f cycles / sec.. 2  / sec. f

Radian Frequency  = 2  f f =  /2 

y x  A y = A cos (2  ft) x = A cos (  ft)  ft

a t a = A sin(2  ft) a = A cos(2  ft)

a t Period T T = period f = 1 T fraction of a second in which one cycle repeats

Phase  a = A sin(2  ft +  )

f = 1 T  2  = 1 period T t A -A phase

Two Domains of Description Time Frequency a t a f

(Amplitude)A a f freq (Phase)    a = A sin (2  f +  ) Frequency Domain Representation freq f

Frequency Domain What can I hear? Fletcher & Munson Frequency (Hz) IL (dB)

a f F2F3F4F5F Modes of Vibration N = node N NN NNN

Joseph Fourier ( ) f(t) = A sin(2  nFt +  ) nn  n=0 ∞ periodic function f(t) = f(t+T) T period Integral multiple harmonics A1A1 A2A2 A3A3 A4A4 A5A5 A6A6 A7A7 A8A8 A9A9 A 10 A 12 F 2F 3F 4F 5F 6F 7F 8F 9F 10F 11F 12F A 11

Joseph Fourier ( ) Harmonic Series Modes of Vibration Fundamental Summation F2F3F4F5F6F7F8F9F10F11F12F 2F 3F 4F 5F 6F F

Superposition of Harmonics

Ongoing Superposition / Interference constructive destructive 180° out of phase Identical Frequencies Different Frequencies

a f F2F3F4F5F6F harmonic overtone harmonic a inharmonic partials f

a f f0f0 f f0f0 behind    Resonance

Frequency (Hz) Time Amplitude Input: Amplitude Output:

a f f0f ff ff f0f0 Q = a f f0f0 High Q Low Q

Frequency (kHz) Input admittance level (10 dB/div) FIG Input admittance (driving point mobility) of a Guarneri violin driven on the bass bar side (Alonso Moral and Jansson, 1982b)

Time Domain Real world Sounds

a t a f formants cutoff attacksustaindecay envelopes temporal envelope spectral envelope

Ongoing Spectral Change Pressure Wave Amplitude

3-D graphs a f t low high soft loud start stop

Time a t a f Frequency