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Chapter 1: The Digital Representation of Sound,
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 doesnt 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).
Well learn in Section 1.3 that amplitude and frequency are not independentthey 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), well return to an important way that frequency affects loudness (well 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!
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? Youll hear the same pitch, only louder. That illustrates something interesting about the relationship between pitch and loudnessthey 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 pitchso all of the sine waves must be at the same frequency. Well 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 lets not make that distinction too rigid.
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 others positive (rarefaction, perhaps). This is called phase cancellation, but the complexity with which this phenomenon occurs in the real sound world is obviously very greatlots and lots of sound waves going positive and negative all over the place.
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, well 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 doesnt 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 didnt 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 were 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. Lets say you work at a burger joint flipping burgers, and you make $8.00 an hour. Lets say your supervisor, Bubba, makes $9.00 an hour. Now lets 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)
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 1/10at 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.
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 dBbut theyre 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.
Let’s move on to a discussion of frequency in the next section.
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