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Hearing in breifly explain place theory of hearing, physical sound intensity, logarithms, fundamental tracking and aural harmonics.
Typology: Lecture notes
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Human ear is, probably, the mostremarkable organ. It has a verysophysticated construction, issensitive to the sound of frequencyfrom 20 Hz to 20 KHz, that is, in therange of three decades in frequency,and of intensity from
−
W/m
2
to 1
W/m
2 , that is, twelve decades in the
intensity! Ears of some animals areeven better.
Human ear (schematic)Human ear (realistic)
It was proven experimentally that thesound of a particular frequency createswaves in the cocklear liquid that haveantinodes at particular well-definedplaces of the basilar membrane alongits length. That is, each particularfrequency excites the hair cells at aparticular distance from the base of thebasilar membrane. The brain knowsfrom where the nerve signals arecoming and decodes the positionsalong the basilar membrane into thesensation of frequency. It was shownthat equal distances between thesensitive points for any two soundscorrespond to equal ratios of thefrequencies of the two sounds. That is,the frequency coding is logarithmicalthat is the explanation of a largefrequency range the ear is sensitive to.
Here the frequencies double each time, and this resultsto the same shift down along the vertical axis
The response of the ear to the sound intensity
is even more remarkable than its frequency
response, so that we can hear both very quiet and very loud sounds. The intensity response ofthe ear is logarithmic
, too. Below the sound-intensity level
(SIL) in decibel
is shown on the left and
the physical sound intensity in W/m
2
is shown on the right. The curves are the sould intensity
levels that are percepted by an average person with normal hearing as equally loud.
0
20
40
60
80
100
(^210) - log
10 ) x (
x
2
10
log
100
log
, 1
10
log
, 0
1
log:
Examples
log
then
,
10
Let
2
10
10
10
10
10
=
=
=
=
=
=
y
x
y
x
)
log(
)
log(
1
log
)
log(
)
log(
)
log(
)
log(
)
log(
1
a
a
a
a
a
b
a
ab
=
=
=
−
α
α
Properties: Particular values:
0
2000
4000
6000
8000
10000
1.00E+0088.00E+0076.00E+0074.00E+0072.00E+0070.00E+
x
y
=
x
2
Standard plot of a quadratic function. Drawback: The region of small
x
is not well represented
1
10
100
1000
10000
0.1 0. 10 1 1000100 10000010000 1000000
1E8 1E
x
y
=
x
2
Double logarithmic (log-log) plot of the same function. All ranges of
x
, small and large, are
equally well represented. The ear was constructed to hear both very quiet and very loudsounds of very small and very large frequencies. This is why the ear hears logarithmically.
The response of the ear is nonlinear
, so that loud sounds get distorted in the ear. This phenomenon
is similar to the „clipping“ in electric circuits. As a result, the signal remains periodic but it is nolonger pure sinusoidal. Its Fourier spectrum contains harmonics of the main tone. Since theseharmonics are produced by the ear, they are called aural harmonics
In the case of two incoming signals with frequencies
f
1
and
f
, nonlinearities in the ear result in the 2
appearance in the Fourier spectrum of the signal (in the ear!) of many combinational tones withfrequencies
2
1
2
1
where
m
and
n
are natural numbers. The combinational tones are weaker than the main tones but
they can be made apparent by adding a weak tone with the frequency close to one of thecombinational tones to produce beats.Our brain does an additional job of (mis)interpreting the incoming sound wave. In the case of twosounds with frequencies that can be different harmonics of one fundamental, our brain adds thisnonexistent fundamental to our perception of the sound. For instance,
f
=500 Hz and 1
f
=700 Hz, we 2
also hear the „fundamental“ 100 Hz. The same happens for
f
=600 Hz and 1
f
=700 Hz, of course. 2
But for
f
=550 Hz and 1
f
=700 Hz, we do not hear this „fundamental“. 2
200
400
600
800
1000
1200
1400
20 15 10 5
Fourier spectrum of the signal consisting of the two pure tones
f
=500 Hz and 1
f
=700 Hz distorted 2
by a nonlinearity in the ear (or in the speakers!)
Main tones
Difference
tone
Sumtone
Second
aural harmonic
Second
aural harmonic
