Part 1

Hoots, Hertz and Harmonics Part 2
Hoots, Hertz and Harmonics Part 3

How does my flute work? This is a question that I think every flautist (or flutist if you live over the water) should ask him or herself. ‘Why?’ you ask, ‘all I need to do is blow in the right way - whatever that is - put my fingers in the right place at the right time and, hey presto, I’m a flute player’. This is true up to a point but, as an amateur flute player and a professional physicist, I find that understanding what I am trying to do is an enormous help when it comes to actually doing it. So, in these short articles (don’t be frightened because they’re about most people’s unfavourite subject: physics) I am going to try to explain in as simple a way as possible various aspects of the science of the flute. It has to be said that trying to understand completely and precisely why any musical instrument behaves as it does is probably harder than trying to solve some of the really big problems that physicists tackle like understanding the Universe or probing deep into the atomic nucleus. Even so, we can still understand a lot about our instruments using quite simple ideas and, with any luck, use that understanding not only to become better players but better teachers. Those of us who teach (in any subject) always find that students learn more easily and rapidly when they understand why they are doing something rather than if they are just told ‘do this!’

It is useful to start by answering this question. Sound consists of pulsations in the air which are generated by a vibrating object and which travel through the air at a speed of about 330 metres per second (or 760 miles per hour if you don’t like going metric!). When these pulsations meet the ear they set a small diaphragm inside the ear into vibration and the brain does very clever things with the resulting electrical signals from the ear mechanism. If the pulsations are particularly simple - see below - our brain tells us that we are listening to a tuning fork or something similar. If the pulsations are very complicated then we sit back and enjoy Bruckner’s seventh or whatever.

It is helpful to go a little further than this before we get on to flutes so that I can use two important ideas - frequency and wavelength - later on. Sound travels as a wave and I am sure that everybody has watched waves on the sea or ripples on the surface of a pond when you throw a stone in. The water surface forms a wavy pattern which moves along the surface but, if you watch a leaf on the pond or a seagull floating on the sea you will notice that the water itself just moves up and down and does not move along with the wave.

If you measure the distance between the wave crests - typically a few metres at the seaside or a few centimetres on the pond - you are measuring the wavelength of the wave - see figure 1. If you stand at the water’s edge and measure how many wave crests arrive per second you are measuring the frequency of the wave. This would be one every few seconds for typical seaside waves and several per second on the pond.

The frequency and wavelength are related to one another. If the wavelength is small, the frequency is large and if the wavelength is large, the frequency is small. We can go further and do some simple mathematics. Suppose some pond ripples have a wavelength of one centimetre and that ten wave crests pass the stem of a small plant standing in the pond every second. A little thought shows that ten centimetres length of ‘wave’ passes that stem every second and the speed of the wave is therefore ten centimetres per second. This is summarised by a very simple formula:

Wave speed = frequency times wavelength

and I am going to use this from time to time when talking about the flute.

Just a few more words about sound waves. They do not go up and down like water waves (or waves on a violin string) and this makes it a little difficult to describe them. However, I shall try.

In the picture (figure 2) I have shown a totally out of scale picture of what the air might look like as a sound wave goes through it. You can think of the dots as molecules if you like or just particles of air. In one place the air is slightly compressed and the molecules/dots are closer together than normal. A little further on the air is slightly rarefied - less dense than normal - and the molecules are further apart. A little further on still it is compressed again. This pattern of rarefaction and compression moves just like the sea waves in the direction of the arrow and you can think of the compressions and rarefactions as the pulsations which I talked about earlier. Just like the water waves the air particles do not move along with the wave but simply move backwards and forwards by a tiny amount. When a loudspeaker diaphragm moves backwards and forwards it makes the air in front of it move backwards and forwards and that small movement is propagated away at the speed of sound forming a sound wave.

When we show a picture of what shape the wave is, as in figure 1 for example, we cannot simply draw the shape that we see as we do for a water wave. One thing we can do is to plot a graph of how the air pressure or density varies at different points in the wave. On the sea we talk about wave crests, these are the top of the wave, with sound the crest is at a point where the pressure is a little higher than normal, i.e. the molecules are packed a little more tightly together than they usually are. This makes visualising sound waves a bit tricky but you soon get used to it.

Now the range of frequencies which our ears recognise as sound extends from about 20 vibrations per second at the bottom to about 20,000 vibrations per second at the top. (As you get older this top limit goes down). We use a special name for frequency - the Hertz - after a nineteenth century German physicist. 1 Hertz is one vibration per second and so on.

As musicians we talk about pitch rather than frequency and, although the two are obviously related, it is not a simple relation. Pitch is arbitrary so we have to use some kind of standard to link them. Nowadays most of us use an A above middle C which has a frequency of 440 Hertz (abbreviated 440 Hz) although there is a regrettable tendency to increase this to a slightly higher figure. It is fascinating that all the other A’s that we use are multiples or sub multiples of 440 Hz. A just above the treble stave has a frequency of 880 Hz, the next one is 1760 Hz and so on. The A at the top of the bass stave is 220 Hz. We can summarise this by saying that every time you double the frequency you get an octave up and if you halve the frequency you get an octave down. What a remarkable link between mathematics and music! The brain clearly recognises a factor of two as something very special.

So now we know the frequencies of all the As. What about the other notes in the scale? If you multiply a frequency by one and a half or 3/2 you get a note which is a fifth higher. One and a half times 440 is 660 Hz and that is the frequency of an E. If you multiply by 4/3 you get a fourth higher, by 5/4 a major third higher and by 6/5 a minor third higher. After this it gets complicated and we are into the tricky world of how a scale is put together and what is meant by equal temperament. However, a useful number to have to hand is that a semitone corresponds to a frequency increase, or decrease, of close to 6%. I shall come back to this later on. There are very good reasons why these ratios of numbers lead to common musical intervals, it is not just an arithmetic accident, but that’s another story.

One more thing. I have to get a bit technical now and talk about sine waves. The shape of a wave (think of water waves for the time being) can vary. It can be very simple as shown in the top part of figure 3 or more complicated as in the bottom picture although they both have the same basic wavelength and frequency.

When we study waves in physics the first ones we meet are the simple ones and these have a very special shape which is described by the sine function which you might have met at school if you did trigonometry, which is all about triangles. Just how flutes and triangles (not the musical kind!) are related is a bit of a mystery unless you understand physics and mathematics so we shan’t pursue that any further. The key thing that we need to know is that the sine wave is a kind of basic wave and when I talk about a simple sound having a particular frequency I am thinking about that kind of wave.

The flute, like any other wind instrument has two essential parts - something to make a noise and something to turn that noise into a nice musical sound. The initial noise produced by the flute is called an edge tone and the tube does the job of turning the edge tone into music. Now the noise mechanism in an oboe, clarinet or trumpet is pretty obvious. Something is buzzing - the reed or the player’s lips - and the tube converts that buzz into a musical sound. The edge tone is much more subtle and really very interesting in itself. Whenever you blow air at some sort of obstacle the air does not always flow round or past it smoothly but often becomes turbulent which is only a word for saying that it no longer flows smoothly. Most of the time, all you hear is a rushing noise which is caused by the air moving about rapidly and randomly so that it generates random sound waves having no definite wavelength or frequency. Sometimes however the air is better behaved and it waggles from side to side in a regular way as it goes past the obstacle. When the air goes on one side it generates a small eddy like a whirlpool, a short time later it generates a new eddy on the other side and so on. This makes a sound wave with a definite frequency and we can often hear the result as a whistle. Figure 4 shows a picture of the motion of a stream of air as it passes over sharp wedge shaped object. Think of the wind ‘whistling over the telephone wires’ or ‘whistling through the crack in the door’.

Hold the edge of a credit card or the blunt side of an ordinary table knife near your lips and try blowing it like a flute. With a bit of practice you can make a steady high pitched whistle - good embouchure practice when you are out and about with no flute and want to keep in trim although you might get some funny looks! This is the edge tone which is the basis of how a flute or a recorder or a flue organ pipe works. Of course the sound is very feeble and you won’t earn much money doing gigs on a credit card. But this is where the tube comes in. If you were to hold a tube of just the right size nearby the air in the tube would resonate with the feeble whistle and make it louder. (This would be very difficult in practice since the whistle has such a high frequency but it works when you replace the credit card with the blowing edge of your flute which is, of course, attached to a tube already).

This experiment gives a clue as to exactly what you are doing when you play your flute. You are blowing a jet of air, often in the shape of a thin ribbon, at the blowing edge so as to generate an edge tone. It is important to realise that you are not blowing across the hole or into it but are trying to aim the middle of the jet at the edge itself. In practice you might vary the precise direction a little one way or the other to produce particular effects and I shall talk about this in a later article.

I used the word ‘resonate’ just now. What do I mean? It has a special meaning in physics which I shall try to explain since most musical instruments use resonance in some form or other. Suppose you have something which will vibrate by itself at a definite frequency if you hit it or pluck it, a violin string say. Then, if you bring another object in contact with it which is being made to vibrate at the same frequency, the first object will vibrate very strongly in sympathy with the second, this is resonance.

1. Find a medicine bottle or any bottle which holds about 200 ml and has a neck about half an inch across. You might have to hunt around a bit. Blow it like a flute. Gradually pour in water until it blows a sound about a semitone flatter than an A440 tuning fork. Now strike the fork and hold it near the neck of the bottle. The sound gets much louder and you have resonance! It is also fun to blow an empty 2 litre soft drink bottle, it makes a most impressive noise which worries my cat!

2. Pull the head of your flute out about 15 mm or half an inch, finger A, strike the fork and hold it near (NOT ON) the embouchure hole. Again the sound will be much louder and you have resonance again! Try fingering B flat and G sharp while the tuning fork is still sounding. The resonance will still be there but will be very much weaker. I shall return to this later on.

So when we blow our flute it looks as though we first generate an edge tone and the air in the tube resonates and produces a loud sound. But things are not quite so simple unfortunately and it is a little more complicated than that. The edge tone that you blew in the first experiment was very high pitched, i.e. it had a high frequency - probably a few thousand Hz - which was much higher than a typical flute frequency, so how can it be resonance? This is where the clever bit comes in (not from you but from physics). When my students do experiments on edge tones under laboratory conditions the tone is quite loud and has a very well defined and stable pitch. When you blow on your flute the edge tone is not very well defined and is more of a random rushing noise. Somewhere in this noise is a very weak sound whose frequency is about the same as the resonance frequency of the flute tube. This tiny sound is amplified by the tube.

We can describe the way that this works in another way. When you first blow you send a little pressure wave or impulse of sound down the tube at the speed of sound. When it reaches the far end it is reflected back up the tube but not as an impulse with a high pressure, as it started out, but one of low pressure. When that arrives at the embouchure hole the lower pressure sucks the air jet into the tube and sends another, stronger, high pressure impulse back down. This is again reflected in the same way and the process is repeated. Every time the impulse goes up and down it gets a little more powerful and it happens periodically at a frequency which is determined by the speed of sound and the length of the tube. Eventually the air jet is oscillating in and out of the hole strongly at the resonance frequency of the tube of air. A powerful sound wave is now travelling up and down the tube and sound energy is radiated from the openings in the tube to give an audible musical note. The energy for this is provided by the player who keeps the edge tone going. (Just how a sound wave is reflected from the open end of the tube in such a strange way is very difficult to explain without mathematics so I’m afraid we must leave that one there!)

What have we learnt so far? I hope that you have learnt some painless physics, and understand enough about waves and resonance to have some idea of the very complex process which enables you to play music on your instrument. In some ways it amazes me that flutes and recorders work at all. If you compare what we are doing with, say, a trombone player there is a world of difference. A good trombone player will be able control the tension in his or her lips so that they vibrate at the correct frequency to start with and all the tube has to do is resonate. Of course there is still a similar feedback process at work where the sound wave travelling up and down the tube encourages the lips to vibrate correctly otherwise beginners would get nowhere at all. We flute players, however, have no control whatsoever over the edge tone frequency and rely on physics almost entirely to make things work. A final point, physics gives us a very good excuse when we cannot articulate rapidly and loudly in the bottom octave!