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