Digital Sound & Music: Concepts, Applications, & Science, Chapter 4, last updated 6/25/2013
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The discussion above is given to explain why it makes sense to use the logarithm of the
ratio of to express the loudness of sounds, as shown in Equation 4.4. Using the logarithm of
the ratio, we don’t have to use such widely-ranging numbers to represent sound amplitudes, and
we “stretch out” the distance between the values corresponding to low amplitude sounds,
providing better resolution in this area.
The values in column 4 of Table 4.1, measuring sound loudness in decibels, come from
the following equation for decibels-sound-pressure-level, abbreviated dBSPL.
( *
Equation 4.4 Definition of dBSPL, also called
In this definition, is the air pressure amplitude at the threshold of hearing, and is the air
pressure amplitude of the sound being measured.
Notice that in Equation 4.4, we use as synonymous with dBSPL. This is
because microphones measure sound as air pressure amplitudes, turn the measurements into
voltages levels, and convey the voltage values to an audio interface for digitization. Thus,
voltages are just another way of capturing air pressure amplitude.
Notice also that because the dimensions are the same in the numerator and denominator
of , the dimensions cancel in the ratio. This is always true for decibels. Because they are
derived from a ratio, decibels are dimensionless units. Decibels aren’t volts or watts or pascals
or newtons; they’re just the logarithm of a ratio.
Hypothetically, the decibel can be used to measure anything, but it’s most appropriate for
physical phenomena that have a wide range of levels where the values grow exponentially
relative to our perception of them. Power, intensity, and air pressure amplitude are three
physical phenomena related to sound that can be measured with decibels. The important thing in
any usage of the term decibels is that you know the reference point the level that is in the
denominator of the ratio. Different usages of the term decibel sometimes add different letters to
the dB abbreviation to clarify the context, as in dBPWL (decibels-power-level), dBSIL (decibels-
sound-intensity-level), and dBFS (decibels-full-scale), all of which are explained below.
Comparing the columns in Table 4.1, we now can see the advantages of decibels over air
pressure amplitudes. If we had to graph loudness using Pa as our units, the scale would be so
large that the first ten sound levels (from silence all the way up to subways) would not be
distinguishable from 0 on the graph. With decibels, loudness levels that are easily
distinguishable by the ear can be seen as such on the decibel scale.
Decibels are also more intuitively understandable than air pressure amplitudes as a way
of talking about loudness changes. As you work with sound amplitudes measured in decibels,
you’ll become familiar with some easy-to-remember relationships summarized in Table 4.2. In
an acoustically-insulated lab environment with virtually no background noise, a 1 dB change
yields the smallest perceptible difference in loudness. However, in average real-world listening
conditions, most people can’t notice a loudness change less than 3 dB. A 10 dB change results in
about a doubling of perceived loudness. It doesn’t matter if you’re going from 60 to 70 dBSPL
or from 80 to 90 dBSPL. The increase still sounds approximately like a doubling of loudness. In
contrast, going from 60 to 70 dBSPL is an increase of 43.24 mPa, while going from 80 to 90
dBSPL is an increase of 432.5 mPa. Here you can see that saying that you “turned up the
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