Introduction to Acoustics

In a variation of this procedure, the 1000 Hz sinusoid
is fixed in level, and the test sound is adjusted to
give a loudness match, again with alternating presentation.
If this is repeated for various different frequencies
of a sinusoidal test sound, an equal-loudness contour
is generated [13.52, 53]. For example, if the
1000 Hz sinusoid is fixed in level at 40 dB SPL, then
the 40 phon equal-loudness contour is generated. Figure
13.10 shows equal-loudness contours as published
in a recent standard [13.53]. The figure shows equalloudness
contours for binaural listening for loudness
levels from 10–100 phons, and it also includes the absolute
threshold (MAF) curve. The listening conditions
were similar to those for determining the MAF curve;
the sound came from a frontal direction in a free field.
The equal-loudness contours are of similar shape to the
MAF curve, but tend to become flatter at high loudness
levels. Note that the MAF curve in Fig. 13.10 differs
somewhat from that in Fig. 13.1, as the two curves are
based on different standards.
Note that the subjective loudness of a sound is not
directly proportional to its loudness level in phons. For
example, a sound with a loudness level of 80 phons
sounds much more than twice as loud as a sound with
a loudness level of 40 phons. This is discussed in more
detail in the next section.
13.3.2 The Scaling of Loudness
Several methods have been developed that attempt to
measure directly the relationship between the physical
magnitude of sound and perceived loudness [13.54].
In one, called magnitude estimation, sounds with various
different levels are presented, and the subject is
asked to assign a number to each one according to its
perceived loudness. In a second method, called magnitude
production, the subject is asked to adjust the
level of a sound until it has a loudness corresponding
to a specified number.
On the basis of results from these two methods,
Stevens suggested that loudness, L, was a power function
of physical intensity, I:
L = kI 0.3 x , (13.3)
where k is a constant depending on the subject and the
units used. In other words, the loudness of a given sound
is proportional to its intensity raised to the power 0.3.
Note that this implies that loudness is not linearly related
to intensity; rather, it is a compressive function
of intensity. An approximation to this equation is that
the loudness doubles when the intensity is increased
Loudness (sones) (log scale)
100
50
20
10
5
2
1
0.5
0.2
0.1
0.05
0.02
0.01
0.005
Psychoacoustics 13.3 Loudness 469
0.002
0.001
0 20 40 60 80 100
Loudness level (phons)
Fig. 13.11 The relationship between loudness in sones and
loudness level in phons for a 1000 Hz sinusoid. The curve
is based on the loudness model of Moore et al.[13.3]
by a factor of 10, or, equivalently, when the level is increased
by 10 dB. In practice, this relationship only holds
for sound levels above about 40 dB SPL.Forlowerlevels
than this, the loudness changes with intensity more
rapidly than predicted by the power-law equation.
The unit of loudness is the sone. One sone is defined
arbitrarily as the loudness of a 1000 Hz sinusoid at
40 dB SPL, presented binaurally from a frontal direction
in a free field. Fig. 13.11 shows the relationship between
loudness in sones and the physical level of a 1000 Hz sinusoid,
presented binaurally from a frontal direction in
a free-field; the level of the 1000 Hz tone is equal to its
loudness level in phons. This figure is based on predictions
of a loudness model [13.3], but it is consistent with
empirical data obtained using scaling methods [13.55].
Since the loudness in sones is plotted on a logarithmic
scale, and the decibel scale is itself logarithmic, the curve
showninFig.13.11 approximates a straight line for
levels above 40 dB SPL. The slope corresponds to a doubling
of loudness for each 10 dB increase in sound level.
13.3.3 Neural Coding
and Modeling of Loudness
The mechanisms underlying the perception of loudness
are not fully understood. A common assumption is that
Part D 13.3