Part 2

Hoots, Hertz and Harmonics Part 1
Hoots, Hertz and Harmonics Part 3

In this part we are going to look at some more aspects of the science of the flute using some of the ideas which I introduced in part one. We’ll start with a small experiment.

One thing that is true of most instruments is that the shorter something is - a violin string, the tube of a wind instrument etc. - the higher the pitch is. So we ought to be able to do a simple calculation to find out what frequency or pitch is going to be produced with a tube whose length we know. Now I explained in part one how a brief pulse of sound in the embouchure hole of a flute travels down the tube at the speed of sound (330 metres per second or 760 mph in more homely units !). It then ‘bounces off’ the far end and travels back up again where it does another bounce and so on. If we hold, say, a microphone near the bottom end it will detect a little ‘blip’of sound every time the pulse arrives there and if we want to know the frequency or the pitch of the sound wave all we have to do is work out how many blips per second it detects. Now the time for a return trip is equal to twice the length of the tube divided by the speed of the pulse just like a car journey from, say, Leeds to London and straight back again. If we know the journey time we know how many return journeys there are every second and this is just what we want - the frequency. A bit more maths then tells us that we don’t have to calculate the time but can find the frequency straight away by dividing the speed by twice the length of the tube.

Let’s try it and see how it works out. Measure the distance from the embouchure hole of your flute to the end of the foot joint. It will be close to 60 cm or 0.6 metres (unless you have a B foot). The speed of sound is 330 metres per second so we can say

frequency of sound = 330 ¸ (2 x 0.6) = 275 Hz

Now the frequency of low C is 262 Hz so we are pretty close. The reason why it has not come out exactly right is because, as always, the situation is actually a bit more complicated and, as far as the sound wave is concerned, the effective length of the tube is always longer than the measured length. In other words the sound pulse I was talking about earlier is not reflected until has emerged a short distance from the end of the tube.

What about other notes ? When we open keys starting from the lower end all we are doing is changing the effective length of the tube. The sound pulse is reflected from wherever it meets an opening to the outside world. So all we have to do is measure from the embouchure to the first open hole and do the same calculation as before. Again the answer is not quite right because the effective length depends how big the hole is. The smaller the hole the longer the effective length and the lower the pitch which is why you can do glissandos on an old flute without keys, or a clarinet by making the whole steadily smaller. You can do it on an open hole flute as well but it is a bit trickier to bring off.

I have talked about a sound pulse travelling up and down the flute tube. This is OK but we usually like to think in terms of waves. Now the bouncing sound pulse rapidly turns into a bouncing sound wave. Unfortunately physicists are strange people and describe this wave as a stationary wave! This does not mean that the wave is not moving but that its wave pattern doesn’t move. What does this mean ? Think of a violin string. When it is plucked in the centre it simply vibrates from side to side as shown in figure 5 yet there is actually a wave travelling up and down the string just like the wave travels up and down the flute tube. The wave pattern is always the same and it is obvious when you look at it that the string at each end does not move and the middle moves the most. The ends are called nodes and the middle, which is moving the most, is called an antinode. The same kind of pattern exists inside a flute tube although, of course we can’t actually see the nodes and antinodes. We can draw pictures of them however and they are quite useful in helping us understand how our instrument works.

We can find out where the nodes and antinodes are by thinking what is happening at each end of the flute when it is sounding, say, low C. Each end is open to the air so the air is free to vibrate as much as it likes and these point must be antinodes (like the middle of the the violin string). In the middle of the tube there is a node (like the ends of the violin string, which are fixed) and the air is not vibrating at all. This is not very obvious so we’ll have a look at how the air in the tube is vibrating.

The top picture in figure 6 shows how the air is vibrating when the flute is sounding low C. The arrows indicate (with a huge exaggeration) how much the air is vibrating at a particular point. The air at each end is pumping in and out in opposite directions so the air in the centre can’t be moving at all and there is a node there. It is helpful to draw the pattern of nodes and antinodes in another way, rather as they look on a vibrating string, and the lower picture in figure 6 shows this. Now it is the vertical arrows that show how much vibration there is at a particular point in the tube.

The useful thing about these pictures is that we can find the wavelength of the sound from them and this gives us the frequency and pitch. How is this? In part one I explained that the wavelength of a sound wave is equal to the speed of the wave divided by its frequency. If you look back at the calculation we did earlier you will see that this means that the wavelength of the sound produced by a flute playing low C is close to twice the length of the tube which is also twice the distance from one antinode to the next. All we have to do therefore is measure the distance between nodes, or antinodes, double it and, hey presto, that is the wavelength.

Let’s try an experiment to investigate nodes and antinodes. For this you will require

(a) a ruler

(b) a piece of chalk.

(c) a friendly violinist (a violist or cellist or even a double bass player will do - the longer the instrument
the longer the ruler you will require)

Measure the length of an open string and make a light mark on the finger board with the piece of chalk exactly half way along (ask permission first!!). Now ask the player to hold a finger very lightly on a string exactly over the mark and bow the string. You will hear a note an octave higher than the open string. Now make a mark exactly one third of the way from the peg box end (violinists call it the nut) and ask the player to hold a finger lightly at that point. This time the twelfth is produced. The next one to try is one quarter of the way up the string, this will give a double octave. Composers often ask string players to produce such notes, which are called harmonics, and players are sometimes very puzzled as to know what exactly what to do to produce a given note. All they have to do is read PAN and all will be revealed!! Why PAN ? Because flute players also play harmonics without realising it!

Lets see how it all works. Now when the player holds a finger lightly on the string it stops the string vibrating at that point. The point must therefore be a node. The next node down is at the nut so the wavelength of the wave on the string is equal to twice the distance from the finger to the nut. We must though put the finger in a place which divides the length of the string by a whole number, 1,2,3,4.....Why should this be? Because the other end of the string is also a node and all the nodes must be equally spaced. Draw a picture and think about it if that doesn’t seem very obvious.

The really interesting thing is what frequencies are produced when these harmonics are played. We need a little more arithmetic for this. If the player fingers at the half way point the wavelength is halved which means that the frequency doubles. This gives a note an octave higher as you heard if you did the experiment. At the one third point the frequency is three times that of the open string. This is one and a half times the frequency of the octave note and, as you might recall from part one, that means a note a perfect fifth above the octave or a twelfth above the open string note. After that we get a double octave, a double octave and a major third and so on so it all works out nicely so far.

‘You’ve stopped talking about flutes’ I hear you say. ‘Have I picked up the Strad by mistake?’ No, you haven’t because we’re returning to flutes now.

We can produce exactly the same effect with a flute except that we cannot ‘hold a finger on lightly’ to produce a node somewhere in the tube. We have to do something different. One way is simply to blow harder and generate a higher frequency edge tone. The other way is to produce an antinode at an appropriate point along the tube. In practice we normally use a combination of these two ways. Before we look at this let’s summarize what we know about harmonics in a table. We have just done some simple calculations so in the table I have done some more and have labelled the open string note with the number ‘1’, the harmonic produced half way along with the number ‘2’, one third of the way along with ‘3’ and so on. The frequencies are all multiples of some starting frequency which I have called ‘f’. It would be 262 Hz for middle C or 440 Hz for A and so on. The musical pitches start quite arbitrarily, with a C but if, for example, your violinist friend was playing on the E string then the lowest note would be an E. All we need to know really is the interval between the different notes

Every frequency in the table is a whole number times the starting frequency. The intervals are all exact except for number 7 which is a pretty horrible B flat. This series is called the harmonic series and is a remarkable link between mathematics and music and you can use it, for example to understand scales, harmony and, of course, how instruments work.

You can now try another experiment (Which you might have done already).

First finger low C and blow a normal note. Then blow a bit harder, the note will jump up an octave. It will not be a very good octave but never mind. Then blow even harder and you will get a G, then a C then an E and so on. The top two - Bflat (or thereabouts) and C - take a bit of puff but have a go. It is quite fun, and very good practice, to play bugle tunes (see, for example, Trevor Wye’s practice book on Tone Production) using harmonics. It makes you appreciate what a difficult job horn players have, especially those intrepid characters who play older music on valveless horns.

Finally, lets draw some pictures of nodes and antinodes to see what is happening inside the flute. The rules are that there must be an antinode at each end and equally spaced nodes somewhere inside.

Figure 7 shows a few possibilities. The nodes and antinodes are labelled N and A and the top picture shows the starting point (ie the pattern for the lowest frequency). In each picture after this I have drawn in another node and you can see that the wavelength gets shorter each time in the order 1/2 1/3 1/4 and so on. This means that frequencies get larger in the order 2 3 4 and so you can see straight away without doing any tiresome arithmetic why you got a harmonic series when you blew your bugle calls.