What Is Sound ?
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Sound is a complex phenomena involving physics and perception. Perhaps the simplest way to put it is that sound involves at least three things:
Sinewave applet
- something moves
- something transmits the results of that movement;
— and though this is philosophically debatable:
- something (or someone) hears the results of that movement.
All things that make sound move, and in some very metaphysical sense, all things that move (if they don’t move too slow, or too fast), make sound.
As things move, they "push and pull" at the surrounding air (or the water, or whatever medium they occupy), causing pressure variations (compressions and rarefactions). Those pressure variations, or soundwaves are what we hear as sound.
Sound is often represented visually by figures like the following:
In this case, one of the authors spoke the immortal line, "I love you, Elmo!" Try to figure out which part of the image corresponds to each word in the spoken phrase. In this image we can see the almost sample-by-sample movement of the waveform (we'll learn later what samples are). You can see, however, that sound is pretty much a symmetrical type of affair (compression and rarefaction): what goes up usually comes down. This is more or less a direct result of Newton's law of equal and opposite actions.
Figure x Most sound waveforms end up looking pretty much alike. It's hard to tell a lot from this sort of time-domain plot of a soundwave.
Figure x This view shows a very minute portion of the soundwave file seen in the previous figure, zoomed in many times.
The above graphs/pictures/charts of sounds are often called functions, and we'll take a bit of space below to really clarify what it is we mean by that. The concept of function is the simplest glue between mathematical and musical ideas.
Sound as a function
We bet most of you probably have a favorite song. Something that reminds you of a favorite place or person. But how about a favorite function? No, not something like a black-tie affair or a tailgate party.
Our favorite function: Tailgate party with Tuxedo-ed gentleman. No, we don't know who these guys are... There is no permission for this.
Figure .x
We mean a favorite mathematical function! In fact, maybe songs and functions aren't so different! Music, or more generally, sound, can be described as a function. And it's this discovery that is at the heart of things like compact disc players, cellular phones, and even radio broadcasts.
Mathematical functions are like machines that take in numbers as raw material and from this input, produce another number, which we'll call the output.
There are lots of different kinds of functions. Sometimes, our functions operate by some easily specified rule, like squaring. When a number is input into the squaring function, the output is the number squared, so the input of 2 produces an output of 4, the input 3 produces an output of 9, and so on. For shorthand, we'll call this function s (for squaring, Sally, silly-example...) and we'll write:
s(x) = x2
s(2) = 22 = 4
s(3) = 32 = 9
The last expression is really just an abbreviation which says for any number given as input to s, then the number squared is the output. If the input is x, then the output is x2.
Sometimes, the input/output relation may be easy to describe, but often the actual cause and effect may be more complicated. For example, we can make a function that works as follows:
There is a thermometer on the wall, and starting at 8 AM, for any number t of minutes that have elapsed since 8 AM, our function gives an output of the temperature at that time. So, for an input of 5, the output of our temperature function is the room temperature at 5 minutes after 8AM. The input 10 gives the output of the room temperature at 10 minutes after 8 AM, and so on.
Once again, for shorthand we can abbreviate this and call the function f (for fantastic-frivolous-fun filled-example, or Fred) and then:
- f(5) = room temperature at 5 minutes after 8AM
- f(10) = room temperature at 10 minutes after 8AM
- f(t) = room temperature at t minutes after 8AM
You can see how this temperature function is a little like our previous sound amplitude graphs. The easiest way to understand the temperature function is according to its graph, a picture which helps us visualize a function. The two axes are the input and output. If an input is some number x units from 0, and the output is then f(x) units (which could be a positive or negative number) then we place a mark at f(x) units above x.
Now let's say
- f(0) = 30
- f(5) = 35
- f(10) = 38
here is what happens when we graph the three temperatures that we wrote down (we'll leave x-axis in real time, but to be more precise we should have probably written 0, 5, and 10 there!). We'll join these marks together by straight lines.
So, how do we get a function out of sound or music?
A Kindergarten Example
Think about an entire kindergarten class piled on top of a trampoline in your neighbor's backyard (all right... we know this is dangerous!). The kids are jumping up and down like maniacs and the surface of the trampoline is moving up and down in a way that is seemingly impossible to describe. But now, we'll try to describe it, at least in part.
Suppose that before the kids jump on the trampoline we paint a bright, flourescent yellow dot on some place on the trampoline and we ask (threaten, beg!) the kids not to jump on that dot so that we can watch how it moves up and down. Before the class starts bouncing around, the surface of the trampoline is at rest. The class climbs on. We take a stopwatch out of our pocket and yell "GO!" and press "START." As the kids go crazy our job is to measure at each possible instant how far the yellow dot has moved from its rest position. If it is above the initial position we measure it as positive (so a displacement of 3 cms up is recorded as +3). If the displacement is below the rest position we measure that as negative (so a displacement of 3 cms down is recorded as -3).
So, follow the bouncing dot! It rises, and then falls a bit, sometimes a lot, sometimes a little, again and again. If we chart this bouncing dot on a moving piece of paper, we get the the kind of function (of pressure, or deformation, or perturbation) that we've been talking about.
Let's return to the idea of writing down list of numbers corresponding to a set of times. We're going to turn that into the graph of a mathematical function! We're going to call that function F. (For "Fabulously Far-out First Function", or Fernando.)
We mark off on the horizontal line (the x-axis) the equispaced numbers, 1, 2, 3, and so on. Then, we also mark off on the vertical axis (the y-axis) numbers 1, 2, 3, etc. going up and -1, -2, -3 etc. going down. The numbers on the x-axis stand for time, and on the y-axis is the displacement. If at time N we recorded a displacement of 4 say, we put a dot at 4 units above N. We say that "F(N) = 4". If we recorded a displacement of -2, we put a dot at the position 2 units below N, and we say "F(N) = -2". Each one of the values F(N) is called a sample of the function F.
We'll learn later (in 2.1, when we talk about sampling a waveform) that this process of "every now and then" recording the value of a displacement in time is referred to as sampling, and it's fundamental to computer music, and the storage of digital data. It's actually pretty simple. We regularly inspect some continous movement, and record its position. Just like watching a marathon on television, you don't really need to see the whole thing from start to finish — checking in every minute or so gives you a good sense of how the race was run.
But, suppose you could take a measurement at absolutely every instant in time — i.e., take these measurements continuously. That would give you a lot of numbers (in fact infinitely many, because who's to say how small a moment in time can be?). Then we would have numbers above and below every point, and get a picture something like the ones above, which appears to be continuous!
Actually, calling these axes x and y is not so instructive. It is better to call the y-axis "Amplitude" and the x-axis "Time." The graphs below let you play with the notion of pressures in time a bit.
Press the moving sound wave icon to hear the speaking sound file seen to the right.
Press the moving sound wave icon to hear the bird song sound file seen to the right.
Press the moving sound wave icon to hear the gamelan sound file seen to the right.
When you hear something that (ahem) sounds like music to your ears, this is in fact the end result of a very complicated sequence of events in your brain that was initiated by vibrations of your eardrum. The vibrations are caused by "billions and billions" of air molecules crashing against the eardrum. Together they act a bit like waves crashing against a big rubber seawall (or those whacky kids on the trampoline).
These waves are in turn the result of things like speaking, plucking a guitar string, sighing, hitting a key of the piano, the wind blowing leaves, or Bill Clinton blowing into a saxophone. Each of these actions causes the air molecules near the instrument or sound source to be disturbed, like dropping many pebbles into a pond all at once. The resulting waves are sent merrilly on their way towards you, the listener, and your eagerly awaiting eardrum. The corresponding function takes as input the number representing the time elapsed since the instrument was played (the sound was initiated), and returns a number that measures how far, and in what direction your eardrum has moved at that instant. But what is your eardrum actually measuring? That's what we'll talk about below.
Amplitude, Pressure
In the graphs of sound waves at the beginning of this section, time was represented on the x-axis, amplitude on the y-axis. So as a function, time is the input, amplitude (or pressure) is the output. Just like the temperature example.
As we'll point out again and again in this chapter, one way to think about sound is as a sequence of time varying amplitudes, or pressures. Or more succintly, as a function of time. The amplitude (y-) axis of the pictures of sound above represents the amount of air compression (above zero) or rarefation (below zero) caused by a moving object, like vocal chords. Note that zero is the "rest" position, or pressure equilibrium (silence). Looking at the changes in amplitude over time gives a good idea of the amplitude shape or envelope of the soundwave.
Actually, this amplitude shape might correspond closely to a number of things, including:
- the actual vibration of the object
- the changes in pressure of the air, or water, or some other medium
and even, and perhaps most importantly,
- the deformation (in or out) of the eardrum.
We might also, when discussing electronic sound production, use this graph, or picture, to represent voltage (which is just an electronic measure of pressure), plotted against some other quantity, like time. This picture of a sound wave, as amplitudes in time, is a nice visual metaphor for the idea of sound as a continuous sequence of pressure variations. When we talk about computers, this just becomes a picture of a list of numbers plotted against some variable (again, time). We'll see, in Chapter 2, how these numbers are stored and manipulated. Any of these concepts might make good labels for the y-axis in a graph such as the ones above.
Frequency: a preview
Amplitude is just one mathematical, or acoustical characteristic of sound, just as loudness is only one of the perceptual characteristics of sounds. But sounds aren't just loud and soft...
People often describe musical sounds as being "high" or "low". A bird tweeting may sound "high" to us, or tuba may sound "low" to our ears. This seems to be one of our fundamental verbal means of describing sounds.
But what are we really saying?. It turns out that there's a fundamental characteristic of these graphs of pressure in time that is less obvious to the eye, but very obvious to the ear. Namely, if there is (or is not) a repeating pattern, and if so, how fast it repeats.That's frequency!
Soundfile.x Tuba sounds
Soundfile.x Bird Sounds
When we say that the tuba sounds are low and thebird sounds are high, what we are really talking about is some result of the frequency of these particular sounds — how fast some pattern in their picture repeats. In terms of waveforms, like what you saw and heard above, we can, for the moment, somewhat fuzzily state that the rate at which the air pressure fluctuates (moves in and out) is the frequency of the sound wave. We'll learn a lot more about frequency, and its related cognitive phenomenon, called pitch, in a later section.
How our Ears Work
Mathematical functions and kids jumping on a trampoline are one thing (actually, two things), but what's the relationship to sound and music? (Other than the songs and yells of our kindergarten class!). Just moving a little eardrum in and out can't be the whole story! Well, it isn't. The ear is a complex mechanism that tries to make sense out of these arbitrary functions of pressure in time, and sends them to the brain. The brain, well, that's another story entirely.
We've already used the physical analogy of the trampoline as our eardrum and the kids as air molecules set in motion by a guitar string, harmonica, washboard, car horn, etc. But to talk about it more fully, we need to discuss how sounds interact, via the eardrum, with the rest of our auditory system (include the brain). Our eardrums, like microphones and speakers, are in a sense transducers — they turn one form of information or energy (as you can see from the above, the difference is not at all clear!) into another.
When soundwaves reach our ears they vibrate our eardrums, transferring the sound energy through the middle to the inner ear, where the real magic of human hearing takes place in a snail-shaped organ called the cochlea. The cochlea is filled with fluid and is bisected by an elastic, hair cell-covered partition called the basilar membrane. When sound energy reaches the cochlea, it produces fluid waves that form a series of peaks in the basilar membrane, the position and size of which depend on the frequency content of the sound.
Different sections of the basilar membrane resonate (form peaks) at different frequencies: high frequencies cause peaks towards the front of the cochlea, while low frequencies cause peaks towards the back. These peaks match up with and excite certain hair cells, which send nerve impulses to the brain via the auditory nerve. The brain interprets these signals as "sound," but as an interesting thought experiment, imagine extra-terrestrials who might "see" soundwaves (and maybe "hear" light). In short, the cochlea transforms sounds from their physical, time domain (amplitude v. time) form, to the frequency domain (amplitude v. frequency) form which our brains understand. Pretty impressive stuff for a bunch of goo and some hairs!
There is no permission for this.
Figure .x A picture of the inner ear. This diagram shows how soundwaves that enter through the auditory canal are transformed into peaks, according to their frequencies, on the basilar membrane. In other words, the basilar membrane serves as a time-to-frequency converter, in order to prepare sonic information for its eventual cognition by the higher functions of the brain.
Wow — who'd have thought it was this complicated. But keep in mind — and this is a kind of hint of a lot of cool things that are coming up — that soundwave pressure picture is just raw data, there's no frequency, timbral, or any other kind of information in the picture itself. It needs a lot of processing, organization, consideration, and tender loving care to attain any sort of meaning to us higher species.
We've made it sound pretty simple, but actually there's a lot of controversy in current auditory cognition research about the specifics of this remarkable organ, and how it works. As we understand more and more about it, musicians and scientists have an increasing sense of understanding how we perceive sound, and some think, how we perceive music. It's an exciting field of research, and an active one!
How Do We Describe Sound?
Sound can be, and is, described in many ways. We have a lot of different words for sounds, different ways of speaking about them. We can call a sound "groovy," "dark," a "burpy kind of thing," "bright," "intense," "low and rumbly," etc. In fact, our colloquial language for talking about sound, from a scientific viewpoint, is pretty imprecise. Part of what we're trying to do in this webbook and in computer music in general is to try and formulate more formal ways of describing sonic phenomena. That doesn't mean that there's anything wrong with our usual ways of talking about sounds, it actually works pretty well as the list below shows:
A Trumpet Sound
A loud sound that gets softer
A noise the makes my eyes bleed! Whooshing
Chirping
A Phat Groove
But to manipulate digital signals with a computer it is useful to have access to a different sort of description, So instead, we need to ask (and answer!) the following kinds of questions:..
- How loud is it?
- What is its pitch?
- What is its spectrum?
- What frequencies are present
- How loud are the frequencies?
- How does the sound change over time?
- Where is the sound coming from?
- What's a good guess as to the characteristics of the physical object that made the sound?
— and many others. Even some of these questions ("Pitch?" — we're not even exactly sure what we mean by that) can be broken down into lots and lots of smaller questions. Taken together, the answers to these questions and others help describe the various characteristics and features that for many years have been referred to collectively as the timbre (or "color") of a sound. We'll get to that in a few sections, but before we talk about timbre, let's start with something simpler: amplitude and loudness (section 1.2).