Our sense organs 45

waves at different frequencies, exerting the same
pressure, are not perceived subjectively as being
of equal volume. A sound at a level of 20 dB,
having a frequency of 63 Hz, would have to be
made about 30 times stronger in order to sound
as loud as a 1,000 Hz signal at the same dB level.
From the formula already given, that means it
would have to be increased by 20 x log(30) =
29.5 dB.
By connecting the points of equal loudness at different
frequencies on a dB-Hz diagram, curves
known as isophones result. By definition, the
measured sound pressure in dB at a frequency of
1,000 Hz is the loudness, expressed using a unit
known as the phon. So for example, if one wants
to find the 50 phon isophone, a test subject listens
to a 1,000 Hz signal having a sound pressure
of 50 dB. At all other frequencies the person
adjusts a control indicating dB values, until it
sounds just as loud as the 1,000 Hz signal. In this
way one can plot the dB values corresponding to
each of the frequencies to obtain the 50 phon
curve. Only at a frequency of 1,000 Hz is the phon
scale numerically equal to the decibel scale.
The pressure at which a sound becomes audible
is called the threshold of audibility. This corresponds
to the 4 phon isophone. If a sound is so
loud that it causes pain, the threshold of pain is
reached. Its isophone is 130 phon. If our ears
were purely mechanical sound detectors, then all
isophones would have been horizontal lines.
We are able to distinguish very clearly between the
loudness of two sounds. At low sound intensities
at a given frequency, a difference of 1 dB is sufficient.
At louder levels this difference is even less.
12 orders of magnitude without switching:
The ear has the amazing ability of detecting a
range of sound pressure extending over 120 dB.
Keeping in mind that 6 dB represents a doubling
of sound level, this means that the human ear
can handle intensities ranging over 20 powers of
2 (120/6 = 20; 2 20 = 1,048,576 = approximately
one million). In the case of sound energy, doubling
occurs every 3 dB because of the physical
relationships involved. The human ear thus has
the unique ability of detecting differences in sound
energy over a very wide range. The relevant factor
is 40 powers of 2 (120/3) which is equal to
12 powers of ten (2 40 = 1024 4 = 1.099 x 10 12 ).
Expressed differently: The range between the
pain threshold and a barely audible sound
encompasses an energy ratio of one million million
to one. This is an astonishing feat, since it is
accomplished with just one range of measurement.
No known technical measuring apparatus
can do this without switching from one range to
another. If, for example, we want to measure
voltages in the range from 1 volt to 10,000 volts
(4 powers of ten), it can only be done with a single
instrument by switching the measuring
range.
Sound energy I
Pain threshold
Music
Threshold
Speech
Range of normal human hearing
The threshold of audibility is a curve, meaning that the
ear is more sensitive to some frequencies than to others.
The optimal range is between 1 kHz and 5 kHz, where
sound pressures as low as 2 x 10 -5 N/m 2 can be detected.
This is equivalent to an intensity I (sound energy) of 10 -16
W/cm 2 . The intensity-frequency ranges for speech and
music are shaded. The maximum range of hearing lies at
about 2 kHz. At this frequency, the range of sound energy
we are able to detect spans an almost unimaginable
13 orders of magnitude (powers of ten).
of
audibility
Frequency f in kHz
Furthermore, the human ear is an optimally constructed
measuring system whose sensitivity
reaches the limits of physical possibility. Sound
waves are pressure waves having very small
23
Sound pressure p