Handbook for Sound Engineers

Psychoacoustics 53

(3-3)

where,

k and are constants accounting for individuality of
listeners,

I is the sound intensity,

p is the sound pressure,

D varies with level and frequency.

The unit for loudness is sones. By definition, one
sone is the loudness of a 1 kHz tone at a loudness level
of 40 phons, the only point where phons and SPL meet.
If another sound sounds twice as loud as the 1 kHz tone
at 40 phons, it is classified as 2 sones, etc. The loudness
of pure tones in sones is compared with the SPL in dB
in Fig. 3-15. The figure shows that above 40 dB, the
curve is a straight line, corresponding to an exponent of
about 0.3 for sound intensity and an exponent of 0.6 for
sound pressure as in Eq. 3-3. The exponent is much
greater for levels below 40 dB, and for frequencies
below 200 Hz (which can be confirmed by the fact that
the equal loudness contours are compact for frequencies
below 200 Hz on Fig. 3-13).

One should note that Eq. 3-3 holds for not only pure
tones, but also bandpass signals within an auditory filter
(critical band). The exponent of 0.3 (<1) indicates
compression within the filter. However, for a broad-
band signal that is wider than one critical bandwidth,
Eq. 3-3 holds for each critical band, and the total loud-
ness is simply the sum of loudness in each band (with
no compression across critical bands).

3.8.4 Loudness versus Bandwidth

Due to less compression across critical bands, broad-

band sounds, such as a rocket launching or a jet aircraft

taking off, seem to be much louder than pure tones or

narrow bands of noise of the same sound pressure level.

In fact, as in the example in Section 3.3.3, increasing

the bandwidth does not increase loudness until the criti-

cal bandwidth is exceeded. Beyond that point multiple

critical bands are excited, and the loudness increases

markedly with increase in bandwidth because of less

compression across critical bands. For this reason, the

computation of loudness for a wide band sound must be

based on spectral distribution of energy. Filters no nar-

rower than critical bands are required and octave fil-

ters are commonly used.

3.8.5 Loudness of Impulses

Life is filled with impulse-type sounds: snaps, pops,

crackles, bangs, bumps, and rattles. For impulses or

tone bursts with duration greater than 100 ms, loudness

is independent of pulse width. The effect on loudness

for pulses shorter than 200 ms is shown in Fig. 3-16.

This curve shows how much higher the level of short

pulses of noise and pure tones must be to sound as loud

as continuous noise or pure tones. Pulses longer than

200 ms are perceived to be as loud as continuous noise

or tones of the same level. For the shorter pulses, the

pulse level must be increased to maintain the same

loudness as for the longer pulses. Noise and tonal pulses

are similar in the level of increase required to maintain

the same loudness. Fig. 3-16 indicates that the ear has a

time constant of about 200 ms, confirming the time

window on the order of 100 ms, as discussed in Section

3.7. This means that band levels should be measured

Figure 3-14. Levels with A-, B-, and C-weightings. (Refer-
ence 27.)

Frequency—Hz

Gain—dB

A-weighting

C-weighting

B-weighting

Loudness k I

D

= u

=kcup^2 D

kc

Figure 3-15. Comparison between loudness in sones and

loudness level in phons for a 1 kHz tone. (Plack, Reference

15, p118, data from Hellman, Reference 28.)

Level—dB SPL

Loudness—sones

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