So far we have seen how the sound is produced and how its frequency is selected and stabilised by the resonating tube. We also saw that you can play a harmonic series (e.g. C₁ , C₂ , G₂ , C₃ , E₃ etc.) by blowing progressively harder with the same fingering. I’m now going to try to explain the science of this and you might find that it helps you to understand some of the problems of playing in tune, how to overcome those problems and why we use such a seemingly complicated set of fingerings for the third octave

Now if you tried the credit card experiment in Part 1 you probably noticed that the pitch of the feeble whistle note went up when you blew harder. Experiments have been done on this and it has been found that the frequency goes up in a very simple way as the air speed increases. In figure 8 I have plotted a graph which shows this. Now when we blow the edge tone on a flute things are quite different and instead of getting a ‘Swanee whistle’ effect as we increase the air speed we find that only certain notes sound and these are the harmonic series for the fingering that we are using. This is because there is a tube of air ‘listening’ to the edge tone and it can only vibrate at one of its resonant frequencies. The first three of these resonant frequencies are shown on the vertical axis of figure 8 as f, 2f and 3f . The almost horizontal lines opposite these four points show the actual frequency of the notes produced and something very strange seems to be happening. Starting from zero air speed nothing happens at first, which is not surprising if you are a flute player! Then the lowest note appears but its frequency is a bit lower than it should be. It then rises gently to a bit more than f and suddenly jumps almost an octave to a little less than 2f. The pattern is repeated until it jumps to around 3f and then to 4f and so on. The whole pattern follows the general trend of the straight line which would be the edge tone without the tube so it seems to make some sort of sense but what we are interested in as flute players is the fine detail.

What is happening is that, when you first blow a note, you produce an edge tone which, with any luck, is not too far away from one of the ‘right’ frequencies. The feedback effect that I talked about in Part 1 then takes over and the vibration of the air in the tube pulls the frequency much closer, but not necessarily exactly, to that frequency. Why not exactly? If you go back to another experiment that I described in Part 1 you found that the flute would ‘sing’ loudly at you when you held a tuning fork near the embouchure hole and fingered the right note. It also sang, but not so loudly, when you fingered a semitone either side of the right note. We describe this by saying that the resonance is not very sharply defined. The result is that the air in the tube will still resonate even if the edge tone frequency is not quite right. The range over which it shows some resonance effect is typically about a semitone or so.

When the note just starts - at point A in figure 8 - you can look on what is happening as a battle between the edge tone, whose frequency is lower than f, and the air in the tube which would like to vibrate at a frequency exactly equal to f . The result is a compromise - somewhere in-between. As the air gets faster the two frequencies match better but eventually, when the air is too fast, the compromise gives too high a frequency - at point B. Shortly after this the mismatch of frequencies is so bad that the best compromise is with the next harmonic of the air in the tube so the note jumps up an octave - or almost an octave to point C and then the whole thing happens again.

Now is a good time to try this on your flute. Choose a note somewhere around the bottom of the range. Blow it starting from nothing and try to increase the air speed slowly and steadily. Listen carefully to the sound and try not to compensate for out of tune notes as you would do naturally when playing. You should be able to follow the shapes shown in figure 8. It is even better to use an electronic tuner since it will be more sensitive to differences in pitch than your ear. It will also show another effect which is that, even in the middle of the loudness range of a particular note, the harmonic series will not give exact intervals. What I mean by this is that the second note will probably be quite a good octave above the first but the third note will probably be a flat twelfth and the fourth an even flatter double octave. It depends a little on exactly what note you choose to start with. It is also quite instructive to do the same experiment with a recorder if you have one since you can’t unconsciously tinker with your embouchure to correct the pitch of a note.

By this time you might be feeling that all this science is getting a bit baffling and ‘what is the point of it?’. What it shows is that a flute, or indeed any wind instrument, can never really be described as being in tune. There are two problems

(a) Because the resonating tube is very complicated and not a plain cylinder, the natural resonant frequencies are not exactly as simple physics predicts but all go a bit flat.

(b) The complicated interaction between the edge tone (or reed tone in other instruments) and the tube resonance means that the pitch of a note changes quite naturally with the loudness of the note.

The flute maker deals with the first problem by adjusting the bore of the tube. In modern flutes the headjoint tapers noticeably towards the cork (Boehm’s famous parabolic curve). In older instruments (and some piccolos today) the head joint is cylindrical and the body tapers towards the lower end. This tapering helps to keep the second harmonic a good octave above the first. It also stops the third and higher harmonics being dreadfully flat but they are still not very useful musically without some assistance. The second problem has to be sorted out by player. Just as with a recorder a given note is only exactly right at one dynamic but we can do something about it on a flute, the recorder player can’t!

Every beginner’s nightmare, all those complicated cross fingerings which seem to be so random when we first learn them. They might still seem random to some players after years of using them. Most of them are actually very logical and with a little science you could invent most of them from scratch even if you had never played a flute!

First of all why do we have to use them? Because if we played the notes using straight harmonics they would be flat. Can we make any sense of them? Yes we can - most of them anyway.

Let’s have another look at the stationary wave pattern in a flute tube. Figure 9 shows the wave patterns for the first three resonances in a flute tube. I have shown the nodes (no vibration) and the antinodes (maximum vibration).

Now, if we open a hole anywhere in the tube, that point must become an antinode . So if we open a hole halfway along and blow the instrument the wave pattern must look like the second rather than the first diagram and the note will be an octave higher than before we opened the hole. This what is happening when we play a D or E flat and lift our left hand first finger. The small hole under the C key is in about the right place for an antinode to form. If you play low C and open the thumb key you again get the octave for the same reason. These are not really cross fingerings and the extra hole is opened to make the note easier to produce and lift the pitch a little into the bargain. They show the principle though.

If we open a hole about one third of the way along the tube then the wave pattern would be like in the third diagram and we would get a twelfth above the fundamental note. This is what happens when we blow a D above the stave. The starting point is G overblown to the third harmonic (= D). The C key is near an antinode for that note and so opening it gives us the D. At the same time it raises the frequency closer to the correct value which you can show using your electronic tuner if you have one. Unfortunately it’s not easy to explain why the frequency goes up but all we have to remember is that opening a hole to make an antinode will raise the pitch of the note a little above the pitch of the note which is produced with the hole closed.

The next thing to work out is what key should be opened for any other note. It has to be where there is an antinode for the higher note, but we don’t need a calculator and a ruler for this. All we have to know is what other fingering will produce approximately the same high note when overblown.

Let’s have look at F. High F can be played either by fingering F and overblowing to the fourth harmonic two octaves higher or by fingering Bflat and overblowing to the third harmonic a twelfth higher. (Can you find or work out another possibility? There is one!). There must therefore be an antinode (at the open end of the tube remember) very near the left hand middle finger key so this is the one to leave open when fingering high F. For high G we open the thumb key since we can also play a (flat) high G by overblowing C.

High E works in the same way but is, as we all know, a bit of a problem and noticeably harder to blow than F. It can be played by overblowing an A so the antinode must be very close to the upper of the two G keys and only the hole under the left hand third finger should be open. In a conventional closed G sharp flute the hole below it must also be open at the same time. The sound wave does not know quite what is going on and where the antinode should be so, although we get the correct note, the acoustic conditions are not right and the note does not sound readily. An open G# flute or a split E solves the problem (see Daniel Pailthorpe’s admirable article in praise of open G# in December 1997 Pan).

F# is also a problem for the same reason. There are two holes open where there ought to be one. A split F# mechanism has been produced but I have never seen one and I would guess that the added complexity of the mechanism is not good news for reliability. The conventional (and poor) G# works in the same way. It is interesting that adding right hand 2 and 3 improves the note enormously but is not quite so obvious why.

At this stage things get complicated. The wavelengths of the notes are getting shorter and shorter and the positions of the nodes and antinodes less and less predictable. High A is reasonably sensible. It can be played (with some difficulty) by fingering A or by fingering F (remember the harmonic sequence F F C F A C...). So we finger F, lift the left hand third finger to give an antinode somewhere near there and then open the C key to give yet another antinode higher up to help matters along. The hole under the C key is not in exactly the right place for this but since it has to perform a number of different jobs it is made very small and its position is a compromise. A very small hole will act as an ‘antinode assisting’ hole for a large range of notes. Have a look at the octave keys on the reed instruments or think how you produce octaves on a recorder by partially opening the thumb hole. After A the fingerings get less predictable and more hit and miss.

We often see a note, particularly in modern works, with a little ‘o’ written over it. The composer wants a harmonic and I find that players (string players as well) are often very puzzled as to what to do. Now you know all about the harmonic series it’s easy. Just finger the right lower note and blow harder than usual. You will probably have to adjust the pitch a bit as well. They sound a bit weird but that is the whole idea of asking for them. Usually, fingering a twelfth or two octaves below will do the trick. A few examples are: A just above the stave - finger low D; High E - finger A; C above the stave - finger F or low C. You can have a lot of fun working others out and trying different sounds.

Many of you may already have experimented with whistle tones. If you have never done it, finger a top octave note, F is a good one to start with, and blow very, very gently. With a bit of practice you will produce a very quiet top F which is quite different from simply playing the same note normally but pppp. What you are doing is actually being very clever and generating the correct edge tone frequency in the first place. The tube then simply goes into resonance without all the complications that I described in Part 1 where the tube resonance ‘pulled’ the edge tone to the right value. You therefore have to blow with just the right air speed and, since you are putting very little power in, the note is never very loud. With a bit of practice you can play most, if not all, if the top register in this way. It is excellent practice for embouchure stability and breath control and well worth persevering with. It’s also very popular with the neighbours!

You might be puzzled that there is a very big difference between the air speed that you need for a whistle tone F and the same note played as a normal harmonic. I've been telling you that you need just the right speed for a particular note and now I seem to be contradicting myself! As is often the case physics has a complication up its sleeve and the complication in this case is that experiments have shown that there are four different ways in which an edge tone can form and our little experiments on the flute involve at least two of them.

So, that’s it for the time being. You can relax now. Next time (if you, the long suffering reader, and Hannah, our admirable editor, can put up with yet another instalment) I shall talk about the knotty problems of intonation. Just what does ‘being in tune’ mean, why do we have problems keeping in tune and what does science tell us about those problems ?