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 waveswith

amplitude and frequency

Peak KE

Peak PE

Simple Harmonic Motion(Makes sine waves)

t

y

x

cos

sin 1

Sinusoidsy = sin x = cos

1

-1

a

t

a = sin(t)

a = cos(t)

A

A

-A

Amplitude

a = A sin(t)

y

xA cos

A sin A

y = A sin x = A cos

Frequency

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

CB

A

D

F HG

E

t

phase

90

radians

degrees

at

A C GB D E F A C GB D E FH H A

Frequency

1 cycle 2

1 cycle / sec. 2/ sec.

f cycles / sec. .2/ sec.f

Radian Frequency

= 2f

f = /2

y

x

A

y = A cos (2ft)x = A cos (ft)

ft

a

t

a = A sin(2ft)

a = A cos(2ft)

a

t

PeriodT

T = period

f =1T

fraction of a second in which one cycle repeats

Phase

a = A sin(2ft + )

a = A sin(2ft + )

f =1T

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 (2f + )

Frequency Domain Representation

freq

f

Frequency DomainWhat can I hear?

Fletcher & Munson

Frequency (Hz)

IL (dB)

a

fF 2F 3F 4F 5F

Modes of Vibration

N = node

N

N N

N N N

Joseph Fourier (1768 - 1830)

f(t) = A sin(2nFt + )n nn=0

∞

periodic functionf(t) = f(t+T)

Tperiod

Integral multiple harmonics

A1

A2

A3

A4

A5

A6A7

A8 A9 A10 A12

F 2F 3F 4F 5F 6F 7F 8F 9F 10F 11F 12F

A11

Joseph Fourier (1768 - 1830)Harmonic Series

Modes of Vibration

Fundamental

Summation

F 2F 3F 4F 5F 6F 7F 8F 9F 10F 11F 12F

2F

3F4F5F6F

F

Superposition of Harmonics

Ongoing Superposition / Interference

constructive

destructive180° out of phase

Identical Frequencies Different Frequencies

a

fF 2F 3F 4F 5F 6F

1 2 3 4 5 61 2 3 4 5

harmonicovertone

harmonic

a

inharmonicpartials

f

a

ff0

ff0

behind

Resonance

Resonance

Frequency (Hz)

Time

Amplitude

Amplitude

Input: Amplitude

Output:

a

ff0

.7071

1.0

fff0Q =

a

ff0

High Q

Low Q

Frequency (kHz)

Input admittancelevel (10 dB/div)

FIG. 10.14Input 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

attack sustain decay

envelopes

temporal envelope

spectral envelope

Ongoing Spectral ChangePressure Wave

Am

plit

ude

3-D graphsa

f

t

low

high

soft

loud

startstop

Time

a

t

a

fFrequency