Loudness Scales

Our perception of loudness is closely related to the amplitude of air pressure oscillations associated with sound waves. Increasing the amplitude of the oscillations makes a sound appear louder, but it doesn't do this in a linear way. To show this, we listened to a recording of a sound that was reduced in loudness in ten "equal" steps. The corresponding amplitudes of the pressure oscillations (in some arbitrary units; remember the units of pressure are N/m2) follow a sequence

512 ; 256 ; 128 ; 64 ; 32 ; 16 ; 8 ; 4 ; 2 ; 1

This produces a sequences of sounds that we perceived to have equal step decreases in loudness. It's clear that the sequence above does not have an equal step decrease in the amplitude of the sound waves. At the beginning of the sequence, the amplitude is reduced by 256 units and at the end of the sequence, it is reduced by 1 unit. Instead of equal steps, the next amplitude in the sequence corresponds to dividing the preceding amplitude by a factor of two.

The disparity between step size and sound wave power is even more pronounced. The sound wave power is proportional to the square of the amplitude of the pressure oscillations. The sound sequence you heard corresponds to a sound wave power sequence of

262,000 ; 65,000 ; 16,400 ; 4,096 ; 1,024 ; 256 ; 64 ; 16 ; 4 ; 1

obtained by multiplying each amplitude in the first sequence by itself. The first step corresponds to ~200,000 units and the last step to 3 units, whereas we perceived that the loudness decrease was in equal steps throughout the sequence. Each element in the above sequence is obtained by dividing the preceding sound wave power by a factor of four.

We can make the above sequence have a constant, additive step size by taking the logarithm of each number. You can do this on your scientific calculator by entering a number and then hitting the LOG (not the LN) key:

5.4 ; 4.8 ; 4.2 ; 3.6 ; 3.0 ; 2.4 ; 1.8 ; 1.2 ; 0.6 ; 0.0

Now, it's clear that the sequence follows what we heard. There is an equal stepsize (-0.6) between each number of the sequence. This means that perceived loudness is related to the logarithm of the amplitude of pressure oscillations!

Logarithms

I'm assuming that not everyone has had experience dealing with logarithms. We'll have a brief review of their properties before we use them for sound. As already stated, your scientific calculator can take the logarithm of a number by using the LOG key (be careful to not use the LN key for most of the calculations involving sound; this key gives the "natural logarithm" which is different from logarithms we'll use in this course). But what does this really mean?

Logarithms of numbers are analogous to representing numbers in scientific notation. In scientific notation, we represent a number as the product of two numbers

500 = 5 x 102
5,000,000 = 5 x 106

The logarithm of each of these two numbers is analogous in the sense that it equals the sum of two numbers

log(500) = log(5 x 102) = 2.699 = 0.699 + 2
log(5,000,000) = log(5 x 106) = 6.699 = 0.699 + 6

The two terms in the sum are as follows: log(5) = 0.699 and the second term is the power of ten used in writing the number in scientific notation.

This also works for positive numbers less than 1:

0.05 = 5 x 10-2
log(0.05) = log(5 x 10-2) = -1.301 = -2 + 0.699
0.000005 = 5 x 10-6
log(0.000005) = log(5 x 10-6) = -5.301 = -6 + 0.699

Keep in mind, you can't take the logarithm of a negative number. If you try to do this with your calculator, you'll get some sort of error message.

We can summarize the properties of logarithms with several identities. Let's consider two arbitrary positive numbers, A and B

log(A x B) = log(A) + log(B)
log(A / B) = log(A) - log(B)

And finally, since A3 = A x A x A, log(A3) = log(A x A x A) = log(A) + log(A) + log(A) = 3 log(A). This can be generalized for raising the number A to an arbitrary power, n:

log(An) = n log(A)

Getting back to the subject of sound, in qualitative terms it should be clear how we relate constant steps in loudness to the amplitude of pressure oscillations (the physics description of sound waves): we take the logarithm of the amplitudes! We'll talk more about this on Wednesday.

Return to P105 Course Schedule

Last updated: 17 Oct 1999
Comments: bland@indiana.edu