Chapter 1: The Digital Representation of Sound,
Part One: Sound and Timbre

Section 1.2: Amplitude and Loudness

Xtra bit 1.2
Another sonic universe

In the previous section we talked briefly about how a function of amplitude in time could be thought of as a kind of sampling of a sound. (Remember, a sample is essentially a measurement of the amplitude of a sound at a point in time.)

But knowing that at 200.056 milliseconds the amplitude of a sound is 0.2 doesn’t really help us understand much about the sound in most cases. What we need is some way of measuring some form of average amplitude of a number of samples (we sometimes call this a frame). We need a way of understanding how these amplitudes, which are a physical measurement (like frequency), correspond to our perception of loudness, which is a psychophysical (anything that we perceive about the physical world is called "psychophysical") or, more precisely, psychoacoustic or cognitive measure (like pitch).

We’ll learn in Section 1.3 that amplitude and frequency are not independent—they both contribute to our perception of loudness; that is, we use them together (in a way described by something called the Fletcher-Munson curves, Section 1.3). But to describe that complex psychoacoustic or cognitive aggregate called loudness, we first need to understand something about amplitude and another related quantity called intensity. Then, at the end of our discussion on frequency (Section 1.3), we’ll return to an important way that frequency affects loudness (we’ll give you a little preview of this in this section as well).

In fact, it’s very important to realize that certain terms refer to physical or acoustic measures and others refer to cognitive ones. The cognitive measures are much higher level and often incorporate related effects from several acoustic phenomena. See Figure 1.8 to sort out this little terminological jungle!

Figure 1.8  Acoustic phenomena resulting from pressure variations and their psychophysical, or psychoacoustic (perceptual), counterparts.

This chart gives some sense of how the terminology for sound varies depending on whether we talk about direct physical measures (frequency, amplitude) or cognitive ones (pitch, loudness).

It’s true that pitch is largely a result of frequency, but be careful: they’re not the same thing.

Applet 1.2
Changing amplitudes

This applet lets you experiment with changing amplitudes of a signal so that you can experiment with different sound levels.

Amplitude and Pitch Independence

If you gently pluck a string on a guitar and then pluck it again, this time harder, what is the difference in the sounds you hear? You’ll hear the same pitch, only louder. That illustrates something interesting about the relationship between pitch and loudness—they are generally independent of one another.

You can have two of the same frequencies at one loudness, or two different loudnesses at one frequency. You can visualize this by drawing a series of sine waves, each with the same period but different amplitude. Pure tones with the same period will generally be heard to have the same pitch—so all of the sine waves must be at the same frequency. We’ll see that pure tones correspond to variations in that old favorite function, the sine function. Remember that amplitude is not loudness (one is physical, the other is psychophysical), but for the moment let’s not make that distinction too rigid.

Soundfile 1.11
Two sine waves

Figure 1.9  Two sine waves with the same frequency, different amplitude.

Soundfile 1.12
A sine wave and a triangular wave

Figure 1.10  Two sounds may have the same frequency but different waveforms (resulting in a different sense of timbre).

Figure 1.11  We can draw an envelope over a sound file, which is the average, or smoothed, amplitude of the sound wave.

This is roughly equivalent to what we perceive as the changes in loudness of the sound. (If we just take one average, we might call that the loudness of the whole sound.)

Figure 1.12  Two sine waves starting at different phase points. These sine waves have different starting points—like the trampoline we talked about in the previous section. It can start flexed, either up or down. Remember that these starting points are very close together in time (tiny fractions of a second). We’ll talk more about phase and frequency later.

This simple picture directly above shows us something interesting about amplitude. The point where the colors overlap is where the waveforms will combine, either summing to a combined signal at a higher level or summing to a combined signal at a lower level. That is, when one waveform goes negative (compression, perhaps), it will counteract the other’s positive (rarefaction, perhaps). This is called phase cancellation, but the complexity with which this phenomenon occurs in the real sound world is obviously very great—lots and lots of sound waves going positive and negative all over the place.

Intensity

Soundfile 1.13
Chirp

You can think of Soundfile 1.13 as a function that begins life as a sine function, f(x) = sin(x), and then over time morphs into a really fast oscillating sine function, like f(x) = sin(20,000x). As it morphs, we’ll continually listen to it.

Soundfile 1.3, sometimes called a chirp in acoustic discussions, sweeps a sine wave over the frequency range from 0 Hz to 20,000 Hz. (Just so you know, this sonic phenomenon is not named for a bird chirp, but in fact for a radar chirp, which is a special function used in some radar work.)

The amplitude of the sine wave in Soundfile 1.13 does not change, but the perceived loudness changes as it moves through areas of greater sensitivity in the Fletcher-Munson curve (Section 1.3). In other words, how loud we perceive something to be is mostly a result of amplitude, but it is also a result of frequency. This simple example shows how complicated our perception of even these simple phenomena is.

We measure amplitude in volts, pressure, or even just sample numbers: it doesn’t really matter. We just graph a function of something moving, and then if we want, we can say what its average displacement was. If it was generally a big displacement (for example, sixth graders on the trampoline), we say it has a big amplitude. If, on the other hand, things didn’t move very much (say we drop a bunch of mice on the trampoline), we say that the sound function had a small amplitude.

In the real world, things vibrate and send their vibrations through some medium (usually gas) to our eardrums. When we become interested in how amplitude actually affects a medium, we speak of the intensity of a sound in the medium. This is a little more specific than discussing amplitude, which is more a purely relative term.

The medium we’re usually interested in is air (at sea level, 72°F), and we measure intensity as the amount of energy in a given air unit, the cubic meter. In this case, energy (or work done) is measured in watts, and we can then measure intensity in watts per meter² (or wm²).

As is the case with the perception of frequency as pitch, our perception of intensity as loudness is logarithmic. But what the heck does that mean? Logarithmic perception means that it takes more of a change in the amplitude to produce the same perceived change in loudness.

Bubba and You

Think of it this way. Let’s say you work at a burger joint flipping burgers, and you make $8.00 an hour. Let’s say your supervisor, Bubba, makes $9.00 an hour. Now let’s say the burger corporation hits it big with their new broccoli sandwich, and they decide to put every employee on a monthly raise schedule.

They decide to give Bubba a dollar a month raise, and you a 7% raise each month. Bubba thinks this is great!

That means the first month you only get $8.56, and Bubba gets $10.00. The next month, Bubba gets $11.00, and you get $9.15. This means that you now got a 59¢ raise, or that your raise went up while his remained the same. The equation for your salary for any given month is:

new salary = old salary + (0.07 X old salary)

Bubba’s is:

new salary = old salary + 1

You’re getting an increase by a fixed ratio of your salary, which itself is increasing, while Bubba’s raise/salary ratio is actually decreasing. (The first month he got 1/9, the next month 1/10—at this rate he’ll approach a zero percent raise if he works at the burger place long enough.) Figure 1.13 shows what the salary raises look like as functions.

Figure 1.13  Bubba & U.

This fundamental difference between ratiometric change and fixed arithmetic change is very important. We tend to perceive most changes not in terms of absolute quantities (in this case, $1.00), but in terms of relative quantities (percentage of a quantity).

Changes in both amplitude and frequency are also perceived in terms of ratios. In the case of amplitude, we have a standard measure, called the decibel (dB), which can describe how loud a sound is perceived. As a convention, silence, which is 0 dB, is set to 10^-12 wm². This is not really silence in the absolute sense, because things are still vibrating, but it is more or less what you would hear in a very quiet recording studio with nobody making any sound. There is still air movement and other sound-producing activity. (There are rooms called anechoic chambers, used to study sound, that try to get things much quieter than 0 dB—but they’re very unusual places to be.) Any change of 10 dB corresponds roughly to a doubling of perceived loudness. So, for example, going from 10 dB to 20 dB or from 12 dB to 22 dB is perceived as a doubling of perceived sound pressure level.

Xtra bit 1.3
More on decibels

Table 1.1  Average decibel levels for environmental sounds. 120 dB is often called the threshold of pain. Good name for a rock band, huh?

Note that even brief exposure to very loud (90 dB or greater) sounds or constant exposure to medium-level (60 dB to 80 dB) sounds can cause hearing loss. Be careful with your ears—invest in some earplugs!

"Audience participation: During tests in the Royal Festival Hall, a note played mezzo-forte on the horn measured approximately 65 decibels of sound. A single uncovered cough gave the same reading. A handkerchief placed over the mouth when coughing assists in obtaining a pianissimo."

—Concert program from the Royal Festival Hall, South Bank Centre, London

Applet 1.3
Amplitude versus decibel level

Two faders, one that changes amplitude (linear, acoustical), one that changes decibel level (exponential, psychoacoustical). Which one would you rather use?

Let’s move on to a discussion of frequency in the next section.