Fundamentals of Psychoacoustics
167
critical band
Just Noticeable Difference
JND
to the power law
L[sones] =
1
15.849
I
I
0
0.3
.
(9)
Roughly speaking, an increment by 9 phons is needed to double the perceived
subjective loudness in sones. This holds for tones at the same frequency or within
the same critical band. In a physiological perspective, the critical band can be
defined as the band of frequencies whose positions along the basilar membrane
stay within the area excited by a single pure tone (see figure 2 and section C.4).
We can say that the intensities of uncorrelated signals effectively sum:
I = I
1
+ I
2
; p
2
= p
2
1
+ p
2
2
p =
p
2
1
+ p
2
2
.
(10)
For uncorrelated pure tones within a critical band, if the law represented by
the straight line in figure 4 does apply, if we double the intensity we have 3 phons
of increment. Therefore, 3 doublings (×8) are needed to have an increase by 9
phons. This is the increase that roughly corresponds to a doubling in loudness.
For example, 8 violins playing the same note at the same loudness level are
needed to effectively double the perceived loudness.
If two sounds are far apart in frequency, their intensities sum much more
effectively. In this case, using two sources at different frequencies also doubles
the loudness.
20
40
60
80
100
1
0.01
0.1
10
100
Loudness Level (phons)
Loudness (sones)
Figure 4: Sones vs. phons
C.2.1
Psychophysics
In psychophysics, the Just Noticeable Difference (JND) of a physical quantity
is the minimal difference of that quantity that can be noticed in two stimuli,
or by modulation of a single stimulus. Sincy our perception is driven by neural
firings statistically distributed in time, the appropriate way to measure JNDs
is by subjective experimentation and statistical analysis. The random nature of
perception is indeed the cause of JNDs, because the accuracy of our internal
representations is limited by the intrinsic noise of these random processes.
The relation between physics and psychophysics is represented in table C.2.1
by means of three important acoustic quantities. The JNDs are represented by