Handbook for Sound Engineers

Psychoacoustics 55

fundamental frequency. For example, when a violin is

playing a tone of concert A (440 Hz), the spectrum

includes not only the frequency of 440 Hz but also the

frequencies of 880 (2 × 440) Hz, 1320 (3 × 440) Hz,

and 1760 (4 × 440) Hz, etc.

To perceive a pitch, the sound must be able to match

with a pure tone, i.e., a listener must be able to adjust

the frequency of a pure tone to produce an identical

pitch as the given sound. An opposite example is as

follows. When one hits a small drum, it might sound

higher than a bigger drum. However, normally one

cannot match the sound that a drum produces with a

pure tone. The exception, of course, would be a tympani

or a steel drum. Therefore, the sound that most drums

make does not result in the perception of pitch. Another

attribute of pitch is that, if a sound has pitch, one can

use it to make a melody. One could use a frequency

generator to produce a pure tone at a frequency of 10

kHz, and one could match it with another tone by

listening to the beats. However, it would not be

perceived as a tone, and it could not be used as part of a

melody; therefore it would not be thought of as having

pitch.^34 This will be discussed further in Section 3.9.3.

3.9.1 The Unit of Pitch

The unit of mel is proposed as a measure of the subjec-
tive quantity of pitch.^35 It is always referenced to a pure
tone at 1 kHz at 40 dB above a listener’s threshold,
which is defined as 1000 mels. If another sound pro-
duces a pitch that sounds two times as high as this refer-
ence, it is considered to be 2000 mels, etc. Fig. 3-18
shows the relationship between pitch in mels and fre-
quency in Hz. The frequency axis in Fig. 3-18 is in log-
arithmic scale. However, the curve is not a straight line,
indicating that our pitch perception is not an ideal loga-
rithmic scale with respect to frequency in Hz. This rela-
tionship is probably more important for melodic
intervals (when notes are played sequentially) than for
chords (when notes are played simultaneously). In a
chord, in order to produce a clean harmony, the notes
have to coincide with the harmonics of the root note;
otherwise, beats will occur, sounding out of tune. In the
music and audio industry, it is much more convenient to
use frequency in Hz or the unit of cent based on the
objective quantity of frequency.
Because our hearing is approximately a logarithmic
scale on frequency—e.g., doubling frequency trans-
poses a musical note to an octave higher—musical
intervals between two tones can be described objec-
tively in the unit of cent as defined by Eq. 3-7.

(3-7)

where,

f 1 and f 2 are the fundamental frequencies of the two

tones.

Thus, a semi-tone on a piano (equal temperament) is

100 cents, and an interval of an octave is 1200 cents.

Using the unit of cent, one can easily describe the

differences among various temperaments (e.g., equal

temperament, Pythagorean scale, Just-tuning, etc.).

3.9.2 Perception of Pure and Complex Tones

How does our brain perceive pitch? The basilar mem-

brane in the inner ear functions as a frequency analyzer:

pure tones at various frequencies will excite specific

locations on the basilar membrane. This would seem to

suggest that the location of the maximum excitation on

the basilar membrane determines the pitch. Actually, the

process is much more complicated: besides the place

coding, there is also temporal coding, which accounts

for the time interval between two adjacent neural

spikes. The temporal coding is necessary for perceiving

the pitch of complex tones, the virtual pitch with miss-

ing fundamentals, etc.36,37 The theories based on place

coding and temporal coding have been proposed to

explain the origin of perception of pure and complex

pitches. For either the place theory or the temporal the-

ory, there is experimental evidence supporting and dis-

Figure 3-18. The relationship between frequency, a purely

physical parameter, and pitch, a subjective reaction to the

physical stimulus. (After Stevens et al., Reference 35.)

3k

2.5k

2k

1.5k

1k

500

0

10 20 50 100 200 500 1k 2k 5k 10k

Subjective pitch–mels

Frequency–Hz

Musical Interval in cents 1200log 2

f 2

f 1

= ©¹§·--- -

1200

log 102

--------------- log 10

f 2

f 1

= u ©¹§·--- -