Nearly all musical instruments are made up of two basic elements: a generator, which gets the vibration going, and a resonator, which amplifies the vibration and modifies it to create the sound of the instrument.
On the flute, the generator is the mouth hole edge against which the player’s breath is directed. When the breath meets the edge, it does not, as might be expected, divide into two separate air streams. Instead, the air stream rapidly fluctuates between going all into the hole and going all away from the hole. This sets up a rapid vibration at the head of the tube.
The remainder of the flute tube is the resonator—or more accurately, the container for the resonator, the actual resonator being the air within the tube. (The mechanisms on the outside of the flute are for the sole purpose of opening and closing the holes, and they have nothing to do with creating the sound.) Acoustically speaking, this tube is considered to be open at both ends, since the mouth hole acts as if it were an open end. So if we close all the note holes, the resonator-tube can be seen like this:
Because the tube walls constrict air inside, that air acts like a stiff spring, fairly independent of the air surrounding it. When the air stream at the mouth hole begins fluctuating in and out of the tube, this air-spring receives a rapid succession of tiny pushes and begins vibrating.
It does not, however, vibrate at the same rate as the vibration at the mouthhole. The pushes given by the vibration at the mouth hole are strong enough to start the air-spring moving, but not strong enough to control the rhythm of the air-spring’s vibrations. Instead, the air-stream uses the energy imparted to it by these pushes to start vibrating in its own natural rhythm. This natural rhythm is determined by the length of the air-spring. When this vibration is set up, the movement of the air in the tube becomes a series of contractions and expansions that looks something like this:
Because of the constricted nature of the air-spring, it retains a portion of the energy imparted to it and thereby grows in strength. It soon overpowers the weak fluctuations at the mouth hole and makes their timing conform to its own rhythm. When this happens, the pushes given by the mouth hole fluctuations will occur simultaneously with each contraction of the air-spring. This is something like a person pushing a swing. It makes the vibration build to a point at which it can vibrate the air around it, and a note is heard.
This note can be altered very slightly by breath and lip adjustments, but to change it completely (short of changing octaves), the length of the air-spring itself must be changed. This is done by opening a hole in the side of the tube. The hole removes the constriction of the air at that point—it’s almost as if the tube were cut off there. Now the air-spring only goes as far as that open hole, like so:
If another hole is opened closer to the mouth hole, the air-spring will end there instead. The vibrating portion of the tube will always be (at least on the first octave) between the mouth hole and the first open hole beneath it.
The shorter the air-spring, the faster its natural rhythm and the higher the note it will produce. To go up the first octave of the flute, then, the flutist opens one hole at a time from the bottom, shortening the air-spring a little with each hole opened.
To get an idea of the dimensions we’re talking about, on a modern flute with all holes closed, the air-spring will contract and expand 262 times per second; with all the holes opened, it will contract and expand a little more than twice that. The total amount of expansion or contraction on a given note (exaggerated in the illustrations) is an average of one fiftieth of an inch, with an individual molecule moving in either direction only half that distance!
While the air in the flute tube is moving in the pattern described above—called the fundamental vibration—it’s also moving in a number of harmonic vibrations. These are additional natural patterns of vibration for the air-spring.
As an example, lets close up the holes in the tube again and take a look at the first harmonic. In this pattern of vibration, the air-spring acts as if it were divided into two equal sections. Just as the entire air-spring does in the fundamental vibration, each half alternately contracts and expands, but in opposite timing to each other, so that one is contracting while the other is expanding. It looks something like this:
Each section, being half the length of the original air-spring, vibrates at a rate twice as fast as the fundamental vibration.
The air in the tube is also moving in a second harmonic vibration, in which the air-spring is divided into three sections; and so on, up to the sixth harmonic—and even higher for some fingerings. All these ways of vibrating are occurring in the flute tube at the same time. And if we open holes in the tube, all these vibrational patterns will then be occurring between the mouth hole and the first open hole beneath it.
How is it possible for all these vibrations to be happening simultaneously? To understand this it might help to think of these vibrations not so much as actual movements of the air, but as the movement of forces that act on the air. If an individual air molecule inside the tube were pushed by two opposing vibrational forces at the same time, it would move in the direction of the stronger; if they were equal, it wouldn’t move at all. If the two forces were going in the same direction, it would move in that direction an extra amount.
You may have seen something like this if you’ve ever played with a long rope attached at one end. You can send a vibration down the rope, then another one shortly after. The first vibration will bounce back at the other end, and when the two vibrations meet, they will pass through each other.
Let’s return to the first harmonic. We said before that the two equal sections were each vibrating at a rate double that of the fundamental. This produces a separate harmonic note, one octave higher than the fundamental note. Each additional harmonic vibration also creates its own harmonic note, and these are all going on at the same time as the fundamental.
These notes, however, are so closely related that they blend together, and we hear them as one note, rather than separately. Yet they make a big difference in the tone. On any instument, tone is determined by the relative strength of the harmonics. The nature of the vibrations of all instrument sounds is identical—for instance, the fundamental vibration of the note C is the same (except in volume) whether it is played by a flute, a violin, or a tuning fork. What creates the differences in sounds is the combination of the vibrations. This is something like creating many different recipes from a given set of ingredients, by varying selection and amounts. The flute sound has fewer types of harmonic vibration than almost any other instrument, and this is the main factor in the production of its distinctive tone.
Harmonics are also important in the production of the higher octaves.
When the ear hears a fundamental and various harmonics all together, it interprets the lowest note—in this case the fundamental—as “the” note, and hears the rest as tone. To produce the flute’s second octave, then, we have to take out the fundamental, so that the first harmonic—which is one octave higher—will be the lowest note heard.
This is done through lip and breath adjustment. By a combination of speeding up your breath and pushing your lips closer to the mouth hole edge, you cut in half the “travel time” of the air stream. At this point the fluctuations of the air stream “double-time” and hook up with the first harmonic. The fundamental, left without support, drops out, leaving the first harmonic as the lowest note, and causing the second octave to be heard. This is why the second octave can be produced using the same fingerings as the first.
If the travel time of the air stream is cut down still further, the air stream fluctuations will hook into the third and then the fourth harmonics, dropping the previous, slower harmonic at each step. Both the third and the fourth harmonics are used in the production of the third octave.
To make the notes of the third octave easier to play, venting is also used. (On the second octave, venting is used on the D and the D-sharp.) In venting, you open a hole somewhere along the vibrating portion of the tube, thereby removing the constriction at that point and introducing a “weak spot” into the air-spring. This forcibly divides the air-spring, guaranteeing that the lower harmonic will have to drop out. (In the higher octaves, more than a single note hole has to be opened to completely cut off the air-spring at that distance from the mouth hole.) Venting also improves the tuning on some notes.
This is only a brief, partial account of what is known about the complexities of sound production in the flute. What’s more, scientists still have far to go in unraveling the mysteries of this simple tube with holes.