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D. Rocchesso: Sound Processing
Fechner's idea
Weber's law
Physics
Psychophysics
Physical Sound
Perceived Sound
Intensity I, I
Loudness L, L
Frequency f, f
Pitch p, p
Duration d, d
Apparent Duration ~
d, ~
d
Table C.2: Physics vs. Psychophysics
the symbol preceding the physical or psychophysical variable name, in the
latter case being a mnemonic for the internal noise variance.
The construction of psychophysical scales relies on the Fechner's idea
that:
The value of the perceived quantity is obtained by counting the
JNDs, and the result of such counting is the same whether we count
physical or sensed JNDs. There is a "zero level" for sensation, i.e.,
the scale of sensations is a ratio scale (all four arithmetic operations
are allowed).
For instance, for loudness:
L · N
JN D
= L N
JN D
=
L
L
.
(11)
If the JND is not constant:
N
JN D
=
L
0
dL
L(L)
.
(12)
From the Fechner's idea we have
N
JN D
=
dL
L(L)
=
dI
I(I)
.
(13)
Fechner's psychophysics is based on two assumptions (exemplified for loudness):
1. L is constant;
2. I is proportional to I, or
I
I
= k, with k constant (Weber's law).
Based on the two assumptions, the Fechner's law is derived as
L = L · N
JN D
= L
dI
kI
= ~
k log(I) ,
(14)
for a certain value of the constant ~
k.
For the loudness of pure tones neither the assumption 1 nor 2 are valid.
Therefore, the Fechner's law (14) does not hold
. However, the Fechner's paradigm
is the basis of new developments that provide models matching the experimental
results quite closely. More details can be found in [42, 4].
1
Gustav Theodor Fechner (1801-1887) is considered the father of psychophysics.
2
Weber's and Fechner's laws are taken for granted quite often in human-computer interac-
tion.