Tubes and What Goes on in them When They Play a Note

Tubes and what goes on in them when they play a note

At the LBPS Collogue, pipemakers Jon Swayne, Sean Jones and Mike York gave a memorable talk and demonstration of bagpipe acoustics. Jon Swayne provided this transcript

THIS IS a fairly lighthearted look at the physics of sound in tubes, a subject with which, as pipers, we are very much concerned. I should say that none of us has any formal training in musical acoustics, so what we have to say is the result of our own experience and reading, and we sincerely hope we are not attempting to teach any grandmothers to suck eggs.

As an instrument which provides its own accompaniment, you could argue that a bagpipe is perhaps required to be better in tune with itself than any other. The tuning of a piano, for example, is compromised in order that it can play equally out of tune in any key, by a system called equal temperament. A bagpipe on the other hand (and perhaps I should qualify that by specifying a bagpipe set in the context of modern Western European folk music) is usually optimised to play in its home key, determined by the pitch of the drone. As we know, if the drone is not correctly tuned to the chanter, and the chanter scale is not correctly constructed in relation to the drone, then much of the point of the instrument is lost. So in order to arrive at a better understanding of the implications of tuning, of being in tune or not, we are going to look at the mechanisms of sound in tubes. And we are going to do so under four main headings. Resonance, Harmonics and Timbre, Intervals, and Scales and Temperaments. To a certain extent we are going to overlap in the way which we discuss these separate subjects.

Resonance

Music depends for its existence on the phenomenon of resonance. Without it there wouldn't be any music. It can be defined as the tendency of a system to oscillate particularly strongly at a specific frequency or groups of frequencies, a system being, for example, air in a tube, a stretched string, a stretched membrane.

It's worth pointing out that we are not concerned with the resonance of the material of which the tube is made, but with that of the air contained within the tube. For example, a lead or indeed wooden organ pipe constitutes a superb resonator for the air contained within it, but as a tubular bell it would be a complete failure. But have you ever wondered what resonance is, and what it is that brings about oscillation at one specific frequency? I can't say that I had, until I came across a striking explanation in a book about musical acoustics by Ian Johnston with the rather neat title of Measured Tones.

He starts with an analogy for a vibrating system which may be familiar, that of a mass on the end of a spring, and introduces the concept of impedance which can loosely be described as the resistance of a system to oscillation — how much force you have to put into the system to get it to get it to oscillate at a given frequency. He explains that whereas the impedance of a mass increases with frequency, that of a spring decreases. So if the frequency is either very high or very low the total impedance will be large, because one component will be large even though the other is small.

However there is one frequency somewhere in between where the two parts of the impedance have the same value and cancel each other out. At that frequency a small force can produce a large response, which is what is meant by resonance.

Oscillation in an air column can be initiated in various ways, some are familiar such as the lip reed on a trumpet, the so-called air reed on a flute and the cane or plastic reed in a bagpipe. An initial pulse sent from the reed end of the tube travels down it and is reflected from the open end; if energy continues to be fed into the system by air pressure acting on the reed, a standing wave is set up at the resonant frequency of the tube.

An example of an extremely unusual method of excitation is that of Rijke's Tube. It is also an example of how small a force can be needed to cause excitation when a body is resonant.

Here Swayne and company held a practical demonstration using a piece of aluminium scaffold pole about 1.2m long held vertically, with a “birds nest” of thin iron wire placed about 1/4 of the way from the lower end. A lighted gas jet is directed in the lower end. After a period of heating, the jet is withdrawn. The tube “sings ” at a frequency close to that appropriate to a wavelength twice that of the tube, until the wire cools. (Petrus Leonardus Rijke. 1812-1899. Professor of Physics at Leiden University)

See http//hyperphysics.phy-astr.gsu.edu/hbase/waves/rijkev.html#cl for practical demonstration and explanation).

Over the page are some theoretical diagrams of the way resonances behave in open and closed tubes.

It's important to realise that the wavy lines represent graphs of air velocity and not the waves themselves, which are longitudinal. Where the line approaches the tube wall, the air is moving at maximum velocity. At the centre, the air is still and pressure is at a maximum. So at top left (representing a tube open at both ends, like a flute or whistle) you can visualise the air sloshing in and out of the tube at both ends in opposite directions; meanwhile the air at the centre (a velocity node and pressure antinode), varies between high and low pressure but does not move. Such a tube can contain half a wavelength.

At top right, a tube closed at one end is shown. A small pipe chanter or a drone behaves as such, the reed behaving effectively as a closed end. Next to the closed end, there must be a velocity node (no displacement) and a pressure antinode, so this tube supports a quarter wavelength.

Fig. 1

Note what happens when both tubes resonate in their second mode; this occurs when the tube is “overblown” or as the behaviour of the harmonic of a complex tone. The second mode of an open tube has twice the frequency of the first mode, and the tube supports a whole wavelength. Because the closed end remains a velocity node, the second mode of a closed tube has three times the frequency of the first, and supports a three-quarter wavelength.

This pattern continues for higher modes and shows that even-numbered modes are not possible in closed tubes, which explains, for example, why the clarinet overblows at the twelfth, and accounts for the ‘hollow' quality of the sound from closed cylinder bores.

In order better to visualise the formation of nodes and antinodes within a tube at resonance, we prepare a demonstration known as Kundt's Tube. This is an experimental apparatus devised by the 19th century German physicist August Adolf Eduard Eberhard Kundt (1839-1894) for the purpose of displaying and determining the position of nodes and antinodes at resonance within a tube closed at both ends.

Here they demonstrated a perspex tube about 1.5m long and 75mm in diameter containing foamed polystyrene granules. The tube was closed at one end, and driven by a loudspeaker at the other. A variable sine wave signal generator was connected via an amplifier to the loudspeaker. The signal generator was adjusted until the frequency entering the tube reached a point of resonance, at which the polystyrene granules clumped together illustrating the presence of a standing wave. See for example http.//www. doflick. com/View Video. aspx?vId=203

Slinky

The children's spring toy was manipulated by Sean to behave as an analogue of a longitudinal sound wave in a tube.

Struck resonance of plastic tube

They used a 2 metre length of plastic waste pipe to demonstrate the relationship between length and frequency (pitch). Striking the end of it elicited a short percussive note.

The pitch can be calculated from the equation Frequency = velocity of sound ÷ wavelength. The velocity of sound is usually specified at a particular temperature and altitude for reasons which we can look at later. At 15 deg C at sea level it is 340m/s. At room temperature, about 20 deg C it is 343 m/s, a fact which is significant for tuning.

We know that an open tube can support a 1/2 wave, so the actual wavelength must be 4m.

343÷4 = 85.75 Hz (cycles per second).

When it is needed to make more accurate calculations it is necessary to take into account a phenomenon known as “end correction” This follows from the fact that the air moving in and out of the end of the tube does not stop at the precise end of the tube, but moves out a little way into the open atmosphere. The distance is usually taken to be 0.6 x the radius of the tube. For a tube open at both ends the correction most be performed twice.

So in this case, 343÷((2 x 2)+ ((0.6 x (.038/2)) x 2)) = 85.26Hz.

Comparing the struck note with a tone from the tuner, gives a pitch somewhere between E and F

Digression on temperature compensation

As we said above, the speed of sound depends on the prevailing pressure and density of the air. The density in turn depends on the temperature, and decreases as temperature rises. The British Standard for tuning woodwinds specifies A=440Hz at 20 deg C.

A bagpipe tuned thus will play at A=447Hz at 30 deg C; in other words, about 30 cents sharp, or more than a quarter of a semitone. If you are playing outdoors in winter, you might easily encounter a temperature of 10 deg C, when your bagpipe would be playing 30 cents flatter than at 20 deg. This has obvious consequences for playing with fixed pitch instruments, as no doubt most of us have experienced.

The lesson for the instrument maker is , rather than keeping his workshop at a constant 20 deg C, he may adjust his tuner by reference to a temperature compensation chart in order to take the prevailing temperature into account.

Overleaf is a temperature chart which I use in my workshop.

lkin

Because wind instruments involve blowing down a tube, it can be tempting to think that the flow of air down the tube away from the performer is somehow necessary to the process. Struck resonance goes some way to demonstrate that this is not so. But, on the contrary, you can still have an oscillating system if you have air flowing towards the performer.

The Nolkin is organologically classified as a sucked trumpet, and is a traditional instrument of the Mapuche Indians in south-central Chile. It consists of the central stem of a naturally hollow plant, forming a tube about 1-1.5m in length, with a slightly tapering bore of between 4 and 5mm in diameter.

An oxhorn is fixed to the wider end to act as a bell. The blowing end is cut to a V shape.

Fig. 3

Here Mike demonstrated a laboratory version, consisting of a length of plastic home- brew tubing with a polythene funnel acting as a bell.

Four-metre drone

You would probably agree that reeds are fairly mysterious objects. They spend their time out of sight in the darkness performing a vital task. What do they look like when they are working? We decided to build an outsize drone and reed and, so to speak, bring things out into the open.

Jon, Sean and Mike assembled a drone reed of plumwood about 250mm long and 35mm diameter to which was tied a tongue made of 2mm tufnol, 150mm long and 30mm wide.

This was inserted into 4 metre waste pipe drone, 38mm bore, via a transparent stock consisting of a plastic drinks bottle. Air was supplied via a length of hosepipe from a standard bagpipe bag and bellows.

A guess at the pitch of the drone might produce an answer around 215Hz, based on the formula we have already used, (speed of sound÷wavelength, 343÷(4 x 4), 1/4 wavelength in a tube closed at one end). The presence of the reed will lower the actual pitch somewhat.

To determine the actual pitch of the note:

We played a recording of the working drone, speeded up four times to bring it to an easily recognisable and singable pitch. Comparison with the tuner found the note so produced to between Eb and E, which at the octave concerned is about
We looked at the waveform of a single cycle of the tone, and from the timescale ruler we could read the period of the waveform to be about 50ms. 1000-50 = 20 Hz.

Another way of determining this note pitch: if A=440 Hz, transposing down four octaves gives 44016 = 27.5 Hz. The difference in cents between 27.5 and 20 is given by 3 log (fl/f2) = 551 cents = 4th + 1/4 tone. So going down 500 cents from A takes you to E, and another 51 cents to 49 cents above E flat.

Helmholz resonator

Hermann Ludwig Ferdinand von Helmholtz (1821-1894), author of On the Sensations of Tone, available in the translation by Alexander Ellis.

Before the availability of modern electronic measuring apparatus, many ingenious devices were employed to carry out research on the physics of sound. A Helmholtz resonator consists of an enclosed mass of air with a single opening consisting of a short tube; it resonates to only one pitch, determined by the elasticity of the enclosed air and the mass of the air in the tube.

One of the things it was used for was to identify harmonics within a complex waveform, such as a musical tone. Thirty years ago Fourier analysis or spectrum analysis required very expensive dedicated apparatus.

Nowadays, such a task is relatively trivial for an ordinary computer [such as the laptop used for this talk; it is running a shareware tuner program which is has an spectrum analysis function].

On the right is a more sophisticated use of an array of Helmholtz resonators.

Believe it or not, it is a spectrum analyser carrying an array of resonators connected to gas jets, viewed in a rotating mirror, which vary their height according to the strength of the relevant harmonic.

The human mouth can act as variable Helmholtz resonator, and indeed we can use it to find drone harmonics.

They demonstrated by varying the mouth cavity next to the bell of a bass drone, suggesting that those interested should try the exercise in the privacy of their own homes!

A more accurate variable resonator can be made from an empty adhesive canister (of the type designed to fit a trigger operated gun). It already has a piston, to which you screw a wooden dowel for adjusting the position of the piston.

They demonstrated the resonator picking up the pitch of a tuning fork. It can also pick out the harmonics of a drone.

The resonator was fitted with a miniature tie-clip microphone to amplify the selected harmonic. The mouth of the resonator was held close to the bell of a bass drone in C, two octaves below middle C on the piano. Odd- numbered harmonics up to the 9th and beyond can be heard.

Sean then talked about harmonics, putting them in the context of what they do and how they control the tone of instruments, and explained the next demonstration. “You know when children blow across a blade of grass stretched between their thumbs to make a sound. The grass is in resonance and is working as a torsion reed, that is, one that is twisting. If you get a blade of grass as big as a highway and stretch it over a river and then blow a wind across it at exactly the right velocity, it'll go into resonance just as the grass did. This happened of course at Tacoma and resulted in the bridge becoming unstable and breaking up.

Jon then explained another intriguing demonstration of the power of resonance. “We have four Helmholtz resonators here, constructed from small Canada Dry cans. As you can see they are glued to the ends of the arms of a symmetrical cross pivoted at its centre, so as to form a kind of turbine. If we fire the right frequency at it, it should start revolving.”

The tone started at around the pitch of middle C on the piano. The turbine started to move. When the tone generator changed to a different frequency, the turbine stopped.

When original frequency was revisited, the turbine starts again. Sean explained that there was an analogy to the way a jet engine works.

At resonance, pulses of air are moving in and out of the neck of the resonators. The outward moving air produces a more focused transfer of energy than the inward, so there is a net reaction force away from the resonators, causing them to move in the opposite direction.

For the simplest illustration of resonance we can think of a guitar string, where motion takes place in the middle of the string and the ends are fixed, unlike a wind instrument where at the ends the air is moving in and out - - Sean manipulated a slinky to illustrate the point.

The guitar string image allows us to look at the phenomenon of harmonics. Very few systems have simply one mode of vibration going on at any one time. Guitar strings don't just do that. The waveform is quite a different shape; they have inherent in what is going, harmonics - like this one of course (slinky illustrated the two halves moving in opposite directions).

If you put the finger in the middle of the string as it's played, you fix the middle of the string, you stop it moving and you get a harmonic. You get this motion when the guitar is playing the second harmonic. It's also what you get when the guitar is playing any note at all. When it plays this one (whole spring moving, max in the centre), it's the fundamental. If you pluck a guitar string in the middle, you stimulate mainly the fundamental, you get few harmonics, and a smoother sound, because you get a lot of this one, and very little of this one. If you pluck the guitar string at this end, of course, you get a shape which isn't a nice smooth one. You actually get a wave which looks like this, which is a combination of this (the fundamental), this (the second harmonic) and the third harmonic. So these are all combined together in this complicated guitar string motion which looks like that. And that is absolutely fundamental to the tone we hear. If you pluck the string in the middle you get a softer tone; you get lots of the first harmonic. If you pluck near one end you stimulate more higher harmonics as well, giving a harder brighter sound.

Another Helmholtz resonator is an ocarina (Sean blows one). The air inside is being pressed in and pulled out, squashed and expanded. It's not a tube and it doesn't support harmonics, so you get a clear, soft tone, similar to a flute which has lots of first harmonic, and weaker higher ones.

Sean also projected from a software programme to demonstrate Fourier Synthesis, named after Jean Baptiste Joseph Fourier (1768-1830), whose work establishes (among other things) that a musical tone is composed of a series of pure tones (sine waves) such that each is a whole number multiple of the fundamental.

Fig 4 shows a sine wave. When we only have one sine wave we have this fundamental ocarina type of sound. Then we can add a second harmonic

(octave above the fundamental) and the tone changes, then a third (a fifth above the octave) and the tone changes again. What's interesting to me is that as the level of each harmonic increases, you hear that pitch as an octave or a fifth, but soon you stop hearing the individual harmonics, and the sound merges into a single complex tone. In the same way as when Jon was using the resonator to pick out the harmonics from the drone, what you normally hear is the overall tone. The ear collects all those harmonics and creates the overall sound, the quality of which is governed by the strength of the individual harmonics.

The demonstration tone changed until it was what they termed a sawtooth, which contained every harmonic, with its harder, brighter sound.

Then Sean produced another tube, which can generate these harmonics - a yellow corrugated plastic tube which he whirled round his head, commentating that it was actually very difficult to get it to play the fundamental.

Sean blew gently as a way of estimating what the fundamental would be, then whirled the tube so as to sound the second and then third harmonics at the octave and twelfth respectively.

It's actually the corrugations in the tube causing turbulence in the airflow, and kicking off the sounding of the harmonics.

Fig. 5

The real relevance of all of this for us is one of tone quality In terms of chanters, it's the difference between a small pipe chanter and border or highland chanter, because they contain different recipes of harmonics, different dynamics of each harmonic.

Now let's look at a square wave (see right).

What we have here is a strong fundamental, then we are missing the second harmonic; we've got a lot of the third, none of the fourth, the fifth but none of the sixth. We've only got the odd

numbered harmonics and that gives that hollow nasal sound which is characteristic of the small pipe and the clarinet, because the clarinet and small pipe don't have the even harmonics in their And that is what distinguishes a small pipe and clarinet, as compared with a border pipe or an oboe. In short, the sound quality or timbre is governed by the number and strength of the harmonics.

Then there was a demonstration using the Fourier program of the consequence of progressively increasing the amplitude of a sine wave while clipping it at a low level. The waveform on the screen became more and more square until it reached the shape of the square wave in the previous demonstration. Meanwhile the quantity of the harmonics could be seen to increase, and the sound changed from its initial ocarina-like quality to the hollow sound typical of a clarinet.

Sean projected diagrams of waveforms in tubes, similar to those Jon talked about (see Fig. 1)

Referring to the diagram of the closed tube, Sean said it was an example of what they demonstrated with the Kundt's tube, which is a harmonic going on in the Kundt's tube, with no movement in the middle. These complex images of what is happening in tubes are rather hard to follow, but they are important because they help us to understand the difference between small pipes and border pipes.

Looking at the diagram of the open tube, this shows what happens in a flute. A flute appears to be closed at one end, but actually the mouth hole causes it to be open. And there are a lot of flutes in the world which are more obviously open at both ends, such as the South American kena, the Japanese shakuhachi, the ney and the kaval. The fundamental created between the open ends is the dominant sound we hear. The higher harmonics of these instruments are weak and this is why flutes, recorders and the like produce a sweet clear tone - rather like the guitar string plucked in the middle.

Reed instruments behave differently from flutes; for one thing they generate much stronger high harmonics giving them brighter sounds. If the bore is cylindrical, reeds actually act as closed ends to tubes. Even though air is passing through them, they reflect the standing waves inside the tube as though the end were closed. As a result, the harmonics we see in the flute won't fit inside. What does fit is a standing wave that has a pressure antinode at the reed end and a pressure node at the open end. It's hard to get around the technical terms here. The upshot is that the full set of harmonics won't fit inside the tube because the reflections at an open end have a different effect from reflections at a closed end. The odd harmonics are missing and we hear a square wave type of sound that we investigated earlier. As we have also seen, a cylinder bore sounds one octave lower than a flute or conically bored reed instrument of the same length.

Border chanters, and other conically bored reed instruments are different again. The conical bore makes the standing wave reflect at the reed end as if it were open. They actually have the full set of harmonics as though they were open at both ends. The reed, the amplifying effect of the conical bore, plus the fact that they play an octave higher than cylindrically bored reed instruments of the same length, combine to give border and other conical chanters their bright sound, rich in high harmonics.

Mike then took over, talking about what happens when two or more pipes are sounded together, as with a bagpipe. As we have seen, he explained, a musical note is composed of a set of separate pure tones or harmonics, and it is the interaction of these sets of harmonics when we combine two musical notes which causes pleasing, in-tune intervals or displeasing out-of-tune nastiness.

If we first look at what happens when we sound two pure tones together, we will be better placed to appreciate what happens when we sound two complex tones or musical notes together. Taking two sine tones of similar but not identical frequency and sounding them together, we hear a phenomenon (familiar to all of us) known as beating. The greater the difference between the two frequencies the faster the beating becomes and the more unpleasant the sound becomes.

They then carried out a sine wave demonstration, using two generators, one set to 260 Hz, the other starting at 260Hz and gradually increasing to 290Hz, causing a strong beating, very slow at first but gradually increasing to a rate of 30Hz. They used a visual analogy to explain why this happens.

This series of vertical lines represents a sound wave, each line being a pulsation of sound energy.

And this series represents a sound wave of a slightly different frequency.

The first has has 150 pulsations whereas the second has 155 within the same space.

When we combine them we see this ...

...an interference pattern, which is the visual equivalent of beating. The darker areas represent times when the combined pulsations are bunching up, and causing an increase in sound level. The areas in between should be lighter than they show here, since if they were sound waves they would cancel out, but this is just a limitation of this visual analogy.

It is worth noting that there are five “beats” here and that this is also the difference between the number of pulsations. It follows that if we combine two sine tones one of 260 Hz (cycles per second) and one of 261 Hz, we get 1 beat per second, while with 260 Hz and 262 Hz we should get 2 beats per second

So a unison is found when two tones have exactly the same frequency. Hence they do not beat.

And when two musical notes are in unison, all their harmonics are also identical and they do not beat either. The white notes here represent the fundamentals and the black notes the next 5 harmonics.

What happens when the two notes are different? Here we have 6 different intervals. The notes forming the intervals are represented by the white notes and the harmonics are represented by the black notes. In each case the harmonics are written out only far enough in each series to reach a unison. In reality they continue way up into the stratospheric ledger lines. As you can see the more dissonant the intervals become, the further you must continue up the harmonic series to achieve a unison.

If the intervals are mistuned, the unisons between the harmonics will also be mistuned and they will beat. Since the lower harmonics are usually louder than those higher up the series, the beating will be most noticeable with the more consonant interval. For example, look at the perfect 5th. The 3rd harmonic of the A will beat with the 2nd harmonic of the E. But with the major second we've got to go all the way up to the relatively faint 8th and 9th harmonics to find a unison which will beat when the interval is mistuned. On top of that all these close non-unison harmonics further down the series will be beating like mad anyway regardless of the tuning, and it this mass of beating which makes this interval sound dissonant.

In fact it's fair to say that the characters of the different intervals are defined by the specific combinations of unisons and clashes between the harmonics. Take the perfect 4th for example, an interval we all consider to be fairly consonant, but it has a slight edge compared with the 5th, caused by the major second between these the third harmonic of the A and the second harmonic of the D, (which are relatively low in the series and hence quite prominent).

The major third has a few more non-unison relationships in the lower harmonics, and perhaps it is this specific combination of sweet and sour which make it the the perfect harmony.

These intervals where we find pairs of harmonics in unison are called just intervals. As bagpipers we require the notes of our chanter to form just intervals with our drones in order to eliminate beating and achieve that very pleasing sound we all know and love, caused by the harmonics of drones and chanter reinforcing and complimenting one another.

Or at least this is the aim of most modern instruments. However there are numerous traditional instruments whose scales contain non-just intervals. Sometimes these “mistuning” have come about because of technical compromises, as is the case with the fourth degree of Bulgarian gaida which is flat as result of being voiced through a tiny hole, which if enlarged to give a just fourth would fail to perform its other function of raising each note by a semitone when opened. The cabrette of the Auvergne is another example; the leading note is always very flat. But if either of these instruments were altered to give just intervals, the instrument and its repertoire would cease to sound idiomatic.

Here is a just scale, by which I mean that each degree of the scale forms a just interval with the tonic. The frequencies of the notes and their ratio to the frequency of the tonic are as shown. Notice how the consonant intervals are simple ratios octave 2.1, 5th 3:2 - whereas the less consonant ones have more distant relationships, major 2nd 9:8, major 7th 15:8.

In fact when you are making an instrument it's actually quite hard to hear if you've got the 2nds and 7ths correct against the bass drone, but if you introduce a baritone or alto drone tuned to the 5th (in this case E) it becomes much easier because you are back in the territory of consonance and small number ratios. The second (B) creates a perfect 5th with the E drone and the G and G# form minor and major thirds with it respectively.

When there is no drone at the 5th it is possible to listen for the 3rd harmonic of the bass drone, which in this case is an E and acts as kind of virtual alto drone against which these notes may be tuned.

Some people ask for a bagpipe whose drones will also tune up to the 5 finger note (in this case B) to enable them to play minor tunes without having to resort to cross fingering or keywork. Sadly the B minor scale which a perfectly just A scale yields is far from perfect.

Look at the relationship between the B and the F# above. This should be a perfect 5th in the ratio 3:2 but we get the ratio 733:495 or rather 2.9616:2.

This means the F# will be very flat against the B drone. In practise it is sometimes possible to use an alternative fingering or a wide vibrato to overcome this mistuning, but the situation is far from satisfactory. Perhaps it would be better to play without drones if you wanted to play in this key.

The chart overleaf shows the two scales one against the other.

Most fixed pitch instruments such as pianos and accordions are tuned using a system called equal temperament which arrives at a compromise between perfect just intervals and an ability to play in all 12 keys without wild mistunings as in the above example.

The area where the compromise is biggest is with the thirds sixths and 7ths which are 14,16 and 18 cents out respectively - around a sixth or fifth of a semitone out. This is why these notes (C#, F# and G) will sound out of tune when you play them with a piano or accordion, especially one which is dry-tuned. As I mentioned before you can often mask the mistuning to an extent with vibrato, alternative fingering and altered bag pressure.

In conclusion, Jon said that musical practice in the west has placed enormous emphasis in teaching and in performance on striving for a maximum degree of in-tune-ness. Particularly perhaps we can see, or rather hear, this in the performances of choirs and string quartets.

But the concept of what is or is not in tune is frequently misunderstood, and this is often because of the existence of the system of tuning known as equal temperament.

As a bagpipe maker I have sometimes had customers say something like: “I have tuned my drone to my tuner; the keynote of the chanter is in tune with my tuner, but according to the tuner, some of the notes of the scale seem to be out of tune”.

I hope we have said enough to show what is wrong with such a statement. It's quite common, even in the musical press, to come across a reference to equal temperament being the scale or tuning system now used in western music. As a generalisation this of course is rubbish, and it can't be emphasised too strongly that equal temperament was devised only for instruments having fixed intonation, and especially keyboards.

We have already seen that a justly tuned scale of A major only works as such in that key, that the notes of a scale in B minor using the notes of a just scaled tuned in A major are not in perfect tuning with a drone tuned to B. In the same way, it is impossible to tune all semi- tones of a piano so that it can be played in tune in all keys. Equal temperament was devised so that the relative tuning between notes of the scale is the same in all keys, in order that one can play in any key without sounding too out of tune. As the name of the system implied, this is done by making all semi-tones equal. Since frequency doubles at the octave, the multiplier for calculating the frequency of one semi-tone from another is 12^th root of 2 or 2¹¹² = 1.059 Thus, for example, A flat = 440 = 1.059 = 415.5 Hz

Here is another way to look at it. If you were to take note A = 22.5 Hz and rise through 12 perfect fifths, one for each semi-tone, you get the following result:

A 2929.29

D 1946.2

G 1297.46

C 864.98

F 576.65

A# 384.43

D# 256.29

G# 170.86

C# 113.91

F# 75.94

B 50.63

E 33.75

A 22.5

On the other hand, 7 octaves up from A 22.5Hz (22.5 x 27) = 2880 Hz. Where you might expect the two notes to have the same frequency, in fact there is a difference of 23.46 cents (known as the comma of Pythagoras) between 2929.29 and 2880Hz. The logic of equal temperament is that this discrepancy of roughly 24 cents is averaged out over the 12 fifths by reducing each one by 2 cents. This tuning results in the fifths and fourths being slightly mistuned by 2 cents each, which is not so easy to hear, but the thirds and sixths are out by about 14 cents which is very noticeable, particularly as we have already pointed out, when paired with a just tuned third.

But I'll say again that all this is only necessary for an instrument having fixed intonation. A choir, a string quartet or any combination of instruments having flexible tuning, will normally aim for maximum consonance as the context of the music demands, no matter what modulations (key changes) occur as the music progresses.

In short, equal temperament is only relevant to bagpipes in so far as it may be necessary to make some adjustments when playing with instruments having fixed intonation.

In conclusion, we've been looking at some of the ways sound behaves in tubes, how complex tones are constructed from simpler ones, how bagpipe scales are made from the structure of the tones themselves and how and why this is important for tuning.

Jon, Sean and Mike then ended their talk with a fairly spectacular demonstration of a Rubens Tube (Heinrich Rubens 1865 - 1922) constructed by Sean. This consisted of a 2 metre aluminium tube perforated at 2 cm intervals. A loudspeaker fed from a signal generator was connected to one end, and a butane gas supply to the other. The gas issuing from the perforations was lit, and the signal generator adjusted to find a resonance at which a standing wave formed in the tube, creating higher and lower pressure points along the tube. Where there was more pressure the flames were higher and vice versa.

See, for example, http.//www.youtube.com/watch?v=HpovwbPGEoo

References and further reading

Arthur H Benade. Fundamentals of Musical Acoustics. OUP

Murray Cambell and Clive Created: The Musician's Guide to Acoustics. JM Dent. Llewellyn S Lloyd & Hugh Boyle: Intervals Scales & Temperaments. MacDonald & Jane's.

Ian Johnston. Measured Tones. The Interplay of Physics and Music. Adam Hilger. Jens Schneider- The Nolkin: a Chilean Sucked Trumpet. Galpin Society Journal XXLVI. Fig. 3 reproduced with the kind permission of the Galpin Society.

Figs. 4, 5 & 6 are due to Paul Falstad at http://www.falstad.com/mathphysics.html