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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 waveswith
amplitude and frequency
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Peak KE
Peak PE
Simple Harmonic Motion(Makes sine waves)
t
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y
x
cos
sin 1
Sinusoidsy = sin x = cos
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1
-1
a
t
a = sin(t)
a = cos(t)
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A
A
-A
Amplitude
a = A sin(t)
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y
xA cos
A sin A
y = A sin x = A cos
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Frequency
How frequently does the sin/cos complete a whole cycle?
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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
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Frequency
1 cycle 2
1 cycle / sec. 2/ sec.
f cycles / sec. .2/ sec.f
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Radian Frequency
= 2f
f = /2
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y
x
A
y = A cos (2ft)x = A cos (ft)
ft
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a
t
a = A sin(2ft)
a = A cos(2ft)
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a
t
PeriodT
T = period
f =1T
fraction of a second in which one cycle repeats
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Phase
a = A sin(2ft + )
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a = A sin(2ft + )
f =1T
2 = 1 period
T
t
A
-A
phase
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Two Domains of Description
Time
Frequency
a
t
a
f
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(Amplitude)A
a
f
freq(Phase)
a = A sin (2f + )
Frequency Domain Representation
freq
f
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Frequency DomainWhat can I hear?
Fletcher & Munson
Frequency (Hz)
IL (dB)
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a
fF 2F 3F 4F 5F
Modes of Vibration
N = node
N
N N
N N N
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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
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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
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Superposition of Harmonics
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Ongoing Superposition / Interference
constructive
destructive180° out of phase
Identical Frequencies Different Frequencies
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a
fF 2F 3F 4F 5F 6F
1 2 3 4 5 61 2 3 4 5
harmonicovertone
harmonic
a
inharmonicpartials
f
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a
ff0
ff0
behind
Resonance
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Resonance
Frequency (Hz)
Time
Amplitude
Amplitude
Input: Amplitude
Output:
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a
ff0
.7071
1.0
fff0Q =
a
ff0
High Q
Low Q
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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)
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Time Domain
Real world Sounds
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a
t
a
f
formants
cutoff
attack sustain decay
envelopes
temporal envelope
spectral envelope
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Ongoing Spectral ChangePressure Wave
Am
plit
ude
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3-D graphsa
f
t
low
high
soft
loud
startstop
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Time
a
t
a
fFrequency