This chapter started with a discussion of how to classify wind instruments. Now we have looked into some details of the physics behind the four traditional families, it is interesting to end the chapter by revisiting the question of classification. Of the four types of instrument we originally described, only the air-jet instruments really look like a clear-cut category with no “blurring at the edges”. The other three families — the woodwind reeds, the brass and the free-reed instruments — are interrelated, as we can now recognise from the fact that we were able to learn something about all three using closely related models. Then we looked at the human voice, and found that it has some things in common with all three, without fitting neatly into any of the categories. As we will see shortly, reed organ pipes also fall between categories — and have some things in common with the voice.
Reed woodwinds, the brass family, the free-reed instruments and the voice all feature some kind of “reed”: a mechanical component which vibrates in response to air flow and plays a central role in making the sound. For the reed woodwinds, this is a piece of natural cane or plastic (or two pieces, in the case of the double-reed instruments). For the free-reed instruments it is a piece of metal. For the brass family, the player’s lips provide the “reed”, and for the voice the vocal folds play a similar role.
Apart from the material of the “reed”, there is another important way in which these instruments differ. In the reed woodwinds, the pitch of a note is governed almost entirely by the resonance properties of the tube as set by the combination of open and closed tone holes — unless the player produces an inadvertent squeak, at a frequency based on a resonance of the reed. For a free-reed instrument like the harmonica, it is the opposite way round: a note is normally governed by the fundamental resonance of the reed, but the player can sometimes modify the pitch by adjusting the behaviour of their vocal tract.
The voice is rather like the harmonica is this regard: a sung pitch is governed by the behaviour of the vocal folds. The overtone content of the sound is strongly influenced by resonances of the vocal tract, but the fundamental frequency is rarely influenced by them. For the brass family, things are a bit more subtle. A played note is usually associated with a tube resonance, but the player also has to adjust their “reed resonance” to lie in the vicinity of that note.
Reed organ pipes, which we haven’t talked about before, display a different variation on this theme. Figure 1 shows a schematic diagram of a typical reed pipe. A tube called the “shallot” is contained within a chamber called the “boot”. When the pipe is sounded, air is allowed to enter through the bottom of the boot. The shallot has a slot in it, and a flexible brass reed called the “tongue” is fixed above that slot. The tongue is slightly curved, so that its resting position leaves a gap over the slot. But, similar to a clarinet reed, when it vibrates it can close down against the edges of the slot. The shallot tube passes through a block, and into a resonator. These resonators can have various shapes (more about that in a moment), but the commonest is a conical shape as shown in the figure. Images of typical blocks, shallots and tongues can be seen in Fig. 2, and images of some reed pipe resonators can be seen in Fig. 3.
You might imagine that the reed pipe works in a similar way to a saxophone: they both feature a mouthpiece with a “beating reed”, placed at the narrow end of a conical tube. However, the figures give an immediate clue that something is different. A stiff wire, the tuning wire, passes through the block and presses against the tongue. This wire can be slid up and down, changing the effective vibrating length of the tongue and hence changing the reed resonance frequency. If this wire is used to adjust the tuning of the pipe, the implication is that the pipe must sound at a frequency strongly linked to the reed resonance: not at all like a saxophone, more like a concertina or harmonium.
But neither the concertina nor the harmonium have resonators attached to each note. Furthermore, the reed pipe resonator is equipped with a means to adjust its effective length, so there is a second means of “tuning”. A conical resonator usually has a small slot with a piece of deformable metal that can be used to adjust the resonance frequencies: you can see one in the left-hand image of Fig. 3. A cylindrical resonator may have a slide to adjust the length, or a movable cap at the end of the pipe that can modify the end correction: you can see one in the central image of Fig. 3.
So is the sounding note of the pipe governed by the reed resonance or the resonator? Presumably, the answer involves both in some coupled combination, but we will find that the predominant influence is from the reed. We will return to this question a bit later, when we develop a model of a reed pipe. But first, we should think about the wider question of what determines whether the note of a wind instrument is governed by the reed resonance or by resonance frequencies of its tube or resonator.
I can advance a hypothesis: a critical parameter seems to be the damping of the reed, relative to the damping of the resonances of the tube or other resonator. If the reed damping is low compared to the tube, or at most comparable with it, that seems to be the condition that favours vibration near the reed resonance frequency. But if the reed has significantly higher damping than the tube then the feedback interaction between the flow and the instrument favours a tube resonance. I will show some direct evidence in support of this hypothesis shortly, but first we can show how this idea applies to the various instruments we have looked at.
Figure 4 shows a diagrammatic representation. The three rows represent the three types of “reed” we have met: “clarinet-type” reeds, fixed above a rigid mouthpiece and oriented so that the static effect of increasing the blowing pressure is to close the reed; “harmonica-type” reeds, placed above a slot which is slightly larger than the reed so that it can pass right through; and the kind of “reed” represented by vocal folds or by a brass-player’s lips. On the horizontal axis is the damping of the reed relative to that of the tube or resonator. The background colour gradient indicates, qualitatively, this damping ratio: red when the reed damping is high relative to the tube, blue when the reed damping is comparable or lower. But, of course, reed damping is continuously variable so that a reed could be designed to fall anywhere along this horizontal scale.
The diagram is annotated to indicate the various types of instrument we have discussed. Reed woodwinds like the clarinet or saxophone fall in the top right corner: the resonances of the tube have Q-factors of the order of 30, while the reed is in contact with the player’s lips so it has a rather low Q, perhaps around 3 (recall that Q-factor measures the inverse of damping: low damping means high Q, and conversely). The reed organ pipe has a reed of the same general kind but it is made of metal so it will have quite a high Q-factor, perhaps in the 100s. The resonator will have similar damping to the clarinet, significantly higher than that of the reed, so the instrument is placed towards the right-hand side of the diagram.
A harmonica will have metal reeds with low damping, while the associated “resonator” is the player’s vocal tract, with high damping. It therefore lies at the right-hand edge of the diagram. Finally, both brass players’ lips and human vocal folds will have rather high damping, with Q-factors perhaps as low as 1. In the case of a brass instrument, this “reed” is coupled to a tube with much lower damping. But in the case of the vocal folds, it is coupled to the singer’s vocal tract, with quite high damping (Q-factors perhaps around 5). So these two instruments are placed towards the left-hand and right-hand sides of the diagram, respectively.
With this arrangement, all the instruments on the red side of the diagram ordinarily play at a pitch governed primarily by a resonance of the tube. By contrast, all the instruments towards the blue side play at a pitch governed primarily by the reed resonance. This pattern is interesting and suggestive, but you might well feel that I have cheated in some way by making a hypothesis that forced everything into this pattern. We need to do some modelling, to see if the idea stands up to scrutiny.
As a first step, we have a very simple option available. Back in section 11.3 we developed a simple model of a clarinet. We used that model to explore the “pressure-gap diagram”, with axes representing two important control parameters for the player. The top row of Fig. 5 reproduces two of those pressure-gap diagrams, taken from Figs. 32 and 33 of section 11.3. The colour shading shows the playing frequency for cases that produced a note rather than silence: these cases fall within a wedge-shaped region of the plane. The only difference between these two cases is that the Q-factor of the reed has been doubled in the right-hand plot.
As was noted back in section 11.3, the left-hand diagram shows only reddish colours in the wedge, but in the right-hand diagram a small white region has appeared. This corresponds to what a clarinettist would call a squeak: a sound at a rather high frequency, close to the reed resonance (which was 2.8 kHz in these examples). Now look at the diagrams in the bottom row of Fig. 5. These show the result of doubling the Q-factor of the reed two more times, and it is immediately obvious that the white region grows each time the reed damping is reduced. By the lower right-hand diagram, there are only a few pixels left that play near the expected frequency. Everywhere else, the “clarinet” squeaks.
Looking back at Fig. 4, what we have done is to move progressively rightwards along the top row, and the behaviour is as that figure predicted. No designer of any instrument would want a case like the lower left-hand plot in Fig. 4: this “instrument” would be equally likely to “play” or “squeak”, and would not be at all popular with players. This is the reason I annotated Fig. 4 showing all the instruments grouped near the extreme ends of the horizontal axis — intermediate cases would be expected to show this kind of very poor playability behaviour.
By the end of the sequence in Fig. 5 we have created a kind of clarinet-shaped reed organ pipe, which exhibits a large region of the parameter space in which it plays a note close to the reed resonance. However, no organ-builder would choose to use this design, because it involves a pipe which is quite unnecessarily long for such a high note.
The simplest way to think about reed organ pipes, in the light of this discussion, is that the note is governed by the reed resonance, and that the resonator is used to modify the nature of the radiated sound by adjusting the balance of the harmonics. To be effective in that role the resonator presumably needs to be carefully tuned in relation to the reed resonance. From this point of view, resonator tuning will have more to do with voicing the pipe to sound “right”, rather than with tuning the pitch of the note. But of course, if the resonator has a resonance frequency close to that of the reed there may be some interaction affecting the pitch. We will return to that question when we construct a more realistic simulation model for a reed pipe.
This description of how a reed organ pipe functions should sound familiar: what I have just described is a version of the “source-filter model” that we used when discussing the voice. For the voice, the resonances of the vocal tract are used to modify the sound from the vocal folds, especially in the context of producing different vowel sounds. The resonator of a reed pipe can play a similar role.
We need to remember that the organ is very unusual among wind instruments, in that it has a separate pipe for each note. The organ builder can optimise the voicing of each pipe with that one note in mind, whereas a designer of clarinets or saxophones has to make compromises so that the same tube can produce many different notes. To make pipes with different sound qualities, to correspond to different stops on the organ, the organ builder will want to make deliberate and consistent modifications to the harmonic content. Different designs of resonator give an important resource for that purpose, in the case of ranks of reed pipes.
Some of the options for resonators are easy to understand. If we want to reinforce all the harmonics of a note, we need a resonator that has a full harmonic series of resonances. Given that the pipe is acoustically closed by the shallot and tongue, a conical resonator would be the natural choice, and indeed this is the commonest form of reed pipe resonator. Another option would be to emulate the flaring shape of a brass instrument like the trumpet — we have seen earlier that such a bore shape can produce harmonically related overtones. The tallest pipe in the collection on the right-hand side of Fig. 3 is of this type. It is designed to be installed facing forwards rather than upwards like most organ pipes, to project bright, “brassy” sound out to the listeners.
If the organ builder chooses a cylindrical resonator, this will tend to have a series of resonances close to odd-numbered harmonics: 1, 3, 5 etc. The result might be a clarinet-like tone quality. For the converse effect, a conical resonator of half the length will have resonances near harmonics 2, 4, 6 etc. A suitable reduced-length cylindrical resonator could be designed to reinforce harmonics 2, 6, 10 etc. There are many other possibilities: the right-hand picture in Fig. 3 shows a wide variety of other resonator shapes. No doubt most of these shapes have been found empirically, to give distinctive tone quality to particular stops.