Music and Computers, 1.2 Amplitude

—> To Continue with Chapter 1

Amplitude and Loudness

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 .2 doesn't really help much 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 both of them together (in a way described by something called the Fletcher-Munson curves). But to describe that complex pyschoacoustic, or cognitive aggregate called loudness, we need to first 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 ones are much higher level, and often incorporate related effects from several acoustic phenomena. We often give the chart below to our students to help them sort out this nasty little terminological jungle!

Figure .x Acoustic phenomena resulting from pressure variations, and their psychophysical, or psychoacoustic (perceptual) counterparts.
This chart gives some sense of the way that 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 (especially when you're talking to sensitive computer musicians like us) — they're not the same thing.

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, then pluck it again, this time harder, what is the difference in the sounds you hear? On all of the guitars we’ve ever plucked it’s like this: we 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 verify this by drawing a series of sinewaves, 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 sinewaves 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, one is psychophysical), but for the moment, let's not make that distinction too carefully.

Figure .x Two sinewaves with the same frequency, different amplitude.

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

Figure .x We can draw an envelope over a soundfile, which is something like the average, or smoothed amplitude of the soundwave.
This is roughly equivalent to what we will 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 .x
This figure shows two sinewaves starting at different phase points. All that means is that they 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 a lot more about phase and frequency in later sections and chapters.

But this simple picture shows us something interesting about amplitude. 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 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 soundwaves going positive and negative all over the place. It's a wonder we hear anything at all! (what? what'd you say? I was rarefacting momentarily...).

Intensity

Soundfile x
This soundfile, sometimes called a chirp in acoustic discussions, sweeps a sinewave over the frequency range from 0 Hz to 20,000 Hz.

You can think of this as a function that begins life as a little old sine function: f(x) = sin(x), and then over time, morphs into a really fast oscillating sine function, like f(x) = sin(20,000x), and as it morphs, we'll continually listen to it. Just so you know, this is not named for a bird chirp, but in fact, for a radar chirp, a special function used in some radar work (like finding aliens).

The amplitude of the sinewave does not change, but the perceived loudness changes as it moves through areas of greater sensitivity in the Fletcher-Munson world. In other words, how loud we hear something is mostly a result of amplitude, but also a result of frequency. So much for that independence we've been assuming.

This simple example shows how complicated our perception of even these simple phenomena are (the sinewave is thought to be the kind of simplest sound). We'll talk about Fletcher-Munson curves in more detail later on.

What part of this (really annoying) sound seems loudest to you? We want to know how to really annoy you!

We measure amplitude in volts, pressure, or even just sample numbers: it doesn’t really matter. As we've seen, it's a rather simple concept. We just kind of 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 (6th graders on the trampoline) we say it's got a big amplitude. If in general, things didn't move very much (we drop a bunch of mice on the trampoline, but we promise, no animals, other than the authors, were harmed in the making of this function), we say that that the sound function had a small(er) amplitude.

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

The medium we’re usually interested in is air (at sea level, at 72 degrees F, with the aroma of freshly brewed coffee and baked bread), and we measure intensity in 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 meter ² (or wm²).

As is the case with the perception of frequency as pitch, our perception of intensity as loudness is logarithmic. Whoa — hold on there Batman! What the heck that does that mean? Logarithmic perception means that it takes more and more of a change in the simple quantity (amplitude) to produce the same perceived change in loudness.

Bubba & You

Think of it this way. Let's say you work for McDonald's 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 McDonald's corporation hits it big with their new McBroccoli sandwhich, 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. Bubba thinks this is great!

That means the first year you only get $8.56, and Bubba gets $10.00. He gets a dollar, and you get 56¢. But 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, but his remained the same. The equation for your salary for any given month is:

newsalary = oldsalary + (.07 * oldsalary)

— and Bubba's is:

newsalary = oldsalary + 1

You're getting an increase by 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 year 1/10 — at this rate he'll approach a zero percent raise if he works for McDonald's for an infinite period of time!). After years of working, here's what the salary raises look like as a function:

You pass him after about a year and a half, and it just gets worse after that!

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

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