Fundamentals of Musical Acoustics

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Fundamentals of Musical Acoustics
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What is sound?
air pressure
time
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Time
something else (air pressure)
time
? t 0
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These variations in air pressure over time can
be decomposed into sine waves with amplitude and
frequency
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Simple Harmonic Motion
(Makes sine waves)
t
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Sinusoids
y sin q x cos q
y
1
sin q
q
x
cos q
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a sin(t)
a cos(t)
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Amplitude
A
A
-A
a A sin(t)
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y
A
A sin q
q
x
A cos q
y A sin q x A cos q
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Frequency
How frequently does the sin/cos complete a whole
cycle?
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C
B
D
A
E
F
H
G
radians
p
3p
3p/2
5p/2
phase
2p
p/2
t
f
180
90
270
90
180
360/0
degrees
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Frequency
1 cycle
2p
1 cycle / sec.
2p / sec.
.
f cycles / sec.
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Radian Frequency
w 2pf
f w/2p
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y
q 2pft
A
q
x
y A cos (2pft) x A cos (2pft)
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a
a A sin(2pft)
t
a A cos(2pft)
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Period
a
T
t
T period
fraction of a second in which one cycle repeats
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Phase
f
a A sin(2pft f)
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a A sin(2pft f)
phase
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Two Domains of Description
a
t
Time
a
f
Frequency
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Frequency Domain Representation
a
(Amplitude)
A
freq
f
p
(Phase)
f
freq
f
-p
a A sin (2pf f)
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Frequency Domain
What can I hear?
IL (dB)
Frequency (Hz)
Fletcher Munson
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Modes of Vibration
a
N
N
N
f
F
2F
3F
4F
5F
N
N
N
N node
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Joseph Fourier (1768 - 1830)
periodic function
f(t) f(tT)
T
period
Integral multiple harmonics
A2
A1
A3
A6
A7
A5
A4
A8
A9
A10
A12
A11
F 2F 3F 4F 5F 6F 7F 8F 9F 10F 11F
12F
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Joseph Fourier (1768 - 1830)
Harmonic Series
F
2F
3F
4F
5F
6F
7F
8F
9F
10F
11F
12F
Fundamental
F
2F
Modes of Vibration
3F
4F
5F
6F
Summation
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Superposition of Harmonics
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Ongoing Superposition / Interference
constructive
Identical Frequencies
Different Frequencies
180 out of phase
destructive
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a
partials
inharmonic
f
a
1
4
overtone
2
3
5
1
4
2
3
5
6
harmonic
harmonic
f
F
2F
3F
4F
5F
6F
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Resonance
a
f
f0
p
f0
f
f
behind
-p
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Resonance
Input
Amplitude
Output
Amplitude
Frequency (Hz)
Amplitude
Time
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a
High Q
Low Q
f
f0
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Input admittance level (10 dB/div)
Frequency (kHz)
FIG. 10.14 Input admittance (driving point
mobility) of a Guarneri violin driven on the bass
bar side (Alonso Moral and Jansson, 1982b)
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Time Domain
Real world Sounds
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a
temporal envelope
t
attack
sustain
decay
envelopes
formants
a
spectral envelope
cutoff
f
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Ongoing Spectral Change
Pressure Wave
Amplitude
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a
3-D graphs
loud
soft
low
t
stop
high
start
f
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a
t
Time