56 Chapter 3
favoring it. As we develop our knowledge, we will
probably understand more about when each coding
takes place.
When hearing a complex tone, we are tolerant with
harmonics slightly mistuned.^38 For instance, if three
frequencies of 800, 1000, and 1200 Hz (i.e., the 4th, 5th
and 6th harmonics of a fundamental of 200 Hz) are
combined and presented to a listener, the pitch
perceived is 200 Hz. If all of them are shifted upward
by 30 Hz—i.e., 830, 1030, and 1230 Hz—the funda-
mental, theoretically, is now 10 Hz, and those three
components are now the 83rd, 103rd, and 123rd
harmonics of the fundamental of 10 Hz. However, when
playing this mistuned complex tone, listeners can hear a
clear pitch at 206 Hz, which matches the middle
frequency—i.e., 1030 Hz—the 4th harmonic of the
fundamental. Although the other two frequencies are
slightly mistuned (in opposite directions) as harmonics,
the pitch is very strong.
It is worth mentioning that pitch recognition is an
integrated process between the two ears, in other words,
it is a binaural process. When two harmonics of the
same fundamental are presented to each ear indepen-
dently, the listener will hear the pitch at the fundamental
frequency, not as two pitches, one in each ear.^39
3.9.3 Phase-Locking and the Range of Pitch
Sensation
What is the range of the fundamental frequency that
produces a pitch? Is it the same as the audible range
from 20 Hz to 20 kHz? The lowest key in a piano is
27.5 Hz, not too far from the lowest limit. However, for
the high limit, it is only about 5 kHz. Because the pitch
perception requires temporal coding, the auditory neu-
rons have to fire at a certain phase of each cycle, which
is called phase-locking. Unfortunately, the auditory sys-
tem is not able to phase-lock to frequencies above
5kHz.^40 This is why the highest note on a piccolo,
which is the highest pitch in an orchestra, is 4.5 kHz,
slightly lower than 5 kHz. Notes with fundamentals
higher than 5 kHz are not perceived as having pitch and
cannot be used for musical melodies. One can easily
confirm this statement by transposing a familiar mel-
ody by octaves: when the fundamental is above 5 kHz,
although one can hear something changing, the melody
cannot be recognized any more.
3.9.4 Frequency Difference Limen
The frequency difference limen is another way of say-
ing “the just-noticeable difference in frequency.” It is
the smallest frequency difference that a listener can dis-
criminate. Experiments with pure tones of duration of
500 ms show that, for levels higher than 10 dB above
threshold, between 200 Hz and 5 kHz, the frequency
difference limen is less than 0.5% of the given fre-
quency, corresponding to 9 cents^41 (about of a
semi-tone).3.9.5 Dependence of Pitch on LevelPitch can be affected by level, however, the influence is
not universal across frequency. At frequencies below 1
kHz, the pitch decreases as level increases; whereas at
frequencies above 3 kHz, the pitch increases with
increasing level; and at frequencies between 1 and
3 kHz, varying the level has little effect on pitch. This is
known as Stevens rule. Terhardt et al. summarized sev-
eral studies of level dependence of pure tones and came
up with the following equation for an average listener.^42(3-8)where,
f is the frequency in Hz of a pure tone at a sound pres-
sure level of L in dB,
p is the frequency of a pure tone at 60 dB SPL that
matches the pitch of the given tone f.3.9.6 Perfect PitchSome people, especially some musicians, develop per-
fect pitch, also known as absolute pitch. They can iden-
tify the pitch of a musical tone without help from an
external reference like a tuner—i.e., they have estab-
lished an absolute scale of pitch in their heads. Some of
them describe the feeling of certain note analogous to a
certain color. Some believe that one can establish a sen-
sation of perfect pitch if he or she has a lot of experi-
ence listening to music on certain keys (which is
normally due to musical training) before the age of 4. It
is fair to state that having perfect pitch is not a require-
ment for a fine musician. Other than the advantage of
tuning musical instruments or singing without a tuning
device, there is no evidence that a person with perfect
pitch would sing more accurately in tune. With the help
of a tuner or accompaniment, a good musician without
perfect pitch would do just as well. There is, however, a
disadvantage due to age effect. For senior persons
(especially those above the age of 65), the pitch scales
are often shifted so that they would hear music that is
being played normally to sound out of tune. This might(^1) » 10
100
pf–
f
u---------- 0.02 L 60– f
1000
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