Text: Reeds and Resonators in Organ Pipes By Johan Liljencrants [ Continuing the topic which began in 000403 MMD, "Free Reeds [ and Beating Reeds", by Prof. Johan. -- Robbie The reason you use reed pipes at all is that they give louder sound and have higher and stronger harmonics as compared to flue pipes. But they are more difficult to fabricate and handle. In particular, with a change in temperature they are detuned by a smaller amount than flue pipes -- the pipe frequency is partly controlled by the temperature indifferent reed, not only by the resonator column. The inward beating reed (or sometimes the shallot, as with clarinets) is curved such that the passage is open at rest. Increasing blowing pressure and/or mechanical force (clarinetist's lips, organ tuning crutch) shortens the free length of the reed and frequency rises. The difficult thing in voicing a reed is to perfect this curvature and to have the reed vibrate 'cleanly'. The frequency is quite a lot higher than the natural one for the reed (as determined only from its mass and elasticity) because it does not oscillate in a sinusoidal fashion like a free reed. It rather goes in a sequence of half-periods -- when trying to enter the 'negative' half-period it bangs into the shallot and immediately bounces back to start another 'positive' half-period. The flow through the reed resembles a full-wave rectified sine wave. Because of its rather sharp minima it is very rich in harmonics. It is this bouncing against the shallot (or the other reed in double reed instruments) that makes the beating reeds behave very unlike the free reeds and makes them much more difficult to analyze. When you have the proper bend in the reed and there is no resonator connected it may be able to work over a big frequency range just by varying the blowing pressure. An important partial mechanism in maintaining the oscillation is the 'Bernoulli force' -- in the closing phase, when the airflow passes between reed and shallot, it gets a high linear speed creating a local underpressure that increases closing speed. When the reed opens again this force is smaller because the air has not yet acquired such speed because of its inertia. When you hook on the resonator, the acoustic pressure in this will give an additional force on the reed. But this force is oscillating with the natural frequency of the resonator, not the one of the reed. When the force is big enough it tends to synchronize the reed oscillation to the resonator frequency. So when you blow with increasing pressure the frequency will no more rise evenly; in a graph of frequency vs. pressure it will have plateaus at the resonance frequencies. Mostly you want tight control of frequency so you want to maximize the feedback force: the resonator should determine the frequency rather than the reed and the pressure. The magnitude of the feedback force depends upon the reed area exposed toward the resonator and the resonator acoustical impedance at the point of connection. High impedance means high pressure in relation to the flow. A reed is a periodic open/shut valve that delivers pulses of high pressure P (the blowing pressure) and little flow Q into the resonator. It is a _high_ impedance device, impedance Z = P/Q. As a contrary, a flue mechanism converts the blowing pressure into a jet of high velocity but a very low pressure, almost zero, same magnitude as the audible sound pressure. The flue generator is a _low_ impedance device. To have any good resonator action you must connect the generator at a point where there is a reasonable impedance match to the resonator. A high impedance source (reed) connects to where there is a pressure maximum and flow minimum, essentially a _closed_ end of the resonator. When you measure the resonator impedance here as function of frequency then this corresponds to an impedance _maximum_. Example instruments are all the brasses, the clarinet (cylindrical resonator), bassoon and oboe and saxophone (conical resonator), and the speech vocal tract. To be efficient a low impedance source should be connected at a pressure minimum and flow maximum, that is, to an _open_ end of the resonator. When you measure the resonator impedance as function of frequency then this corresponds to an impedance _minimum_. Examples are the flute, recorder, ocarina, organ flue pipe. The pipe cutup is an open end. The common pipe resonator is a tube of uniform cross section. Its resonances are determined by its length L, the speed of sound c, and the 'boundary conditions' -- it may be open both ends or only one end. (Closed both ends would not give any external sound; instead, this case leads toward the different discipline of room acoustics). When open _both_ ends the fundamental resonance frequency f1 = c/2L and the higher resonances come at 2, 3, 4, 5, .. etc. times this frequency. This is an example often illustrated in physics textbooks as sinusoidal patterns of standing waves; at the resonances there is always an integer number of half-periods over the length of the tube. At all those frequencies there are sharp impedance minima at both ends. Always low pressure (because of the openness), but at resonance there are flow maxima. This suits the flue generator, but never the high impedance reed generator. When open at only _one_ end there is always a flow node at the closed end. The possible resonance standing wave patterns -- zero flow at the closed end, maximum flow at the open -- now come out with an odd number of quarter periods along the tube. The fundamental resonance frequency f1 = c/4L and the higher resonances come at 3, 5, 7, .. etc. times this frequency. Putting a flue generator at the open end you get the classical stoppered organ pipe which has its dominant harmonics at those frequencies [the odd harmonic series]. Instead, looking at the closed end, at those same frequencies you have impedance maxima -- high pressures and zero flow because of the closure. This is the proper place to connect a high impedance reed generator. Responding to Robert Linnstaedt's question in MMD 2000.04.06-03 there is another screw to the resonator problem. Instead of cylindrical you can make the resonator conical like in the oboe, bassoon, and saxophone. Then the waves inside the resonator are no more the simple plane waves, they are spherical with their center at the tip of the cone. Close analysis calls for the solution to the spherical wave equation which is too complicated to discuss here. Let us leave with the correct comments by John Nolte in MMD 2000.04.11-07 that the resonances come at 2, 3, 4, .. etc. times the fundamental and that the tube has to be some 1.5 times longer than the cylindrical one with the same fundamental resonance. The clarinet and the (straight soprano) saxophone are exceptionally similar except for one crucial thing: the bore that is cylindrical and conical respectively. This conicity then also necessitates the holes and pads in the saxophone to be progressively bigger toward its low end. The unlike distributions of the higher resonances explain their difference in timbre. Another most important consequence appears when you overblow the conical instruments: the sound goes to the octave (fundamental*2), whereas the cylindrical clarinet goes to the octave and a fifth (fundamental*3). When overblowing the player increases blowing and lip pressures to raise the reed frequency, and he normally also opens a special valve [the register key] placed near a pressure node for the overblown note. This has little consequence at the overblown frequency, but it destroys the Q value for the fundamental resonance. It is important that the reed really does have a high impedance in order not to destroy the proper action of the closed end resonator. When the reed is mounted the closed end is only approximately so. The consequence is a somewhat impaired resonance Q and the need to shorten the resonator slightly. The most sophisticated and difficult to analyze resonators are those of the brass instruments. With those the fundamental resonance is never used because it is grossly out of tune with the higher resonances that instead are accurately in the relations 2, 3, 4, 5, etc. It is a great feat in instrument evolution to find out the proper shape of the bell flare in order to reach this accurate intonation. Because the player overblows to any of them, all must be in tune. It is weird to know that there are infinitely many shapes of horns that can fulfill this criterion, still each such shape must be made with high precision. Good examples are the trumpet, the cornet and the fluegelhorn -- three instruments which to a naive observer might appear identical looking at working principle, resonator length, and tonal range. But they have distinctly different flare shapes and this settles their differences in timbre, the various higher resonances get differing sets of Q values. Some band organs feature trumpet-like flared resonators. This is attractive to look at, but perhaps not always necessary from a technical point because with each pipe you never give more than one note. It is then not so important that the higher resonances are precisely tuned so the more unpretentious straight conical horn speaking at its fundamental is often found adequate, and at the same time this saves space. The 'inverted' layout using the boot as a resonator mentioned MMD 2000.04.08-05 by Bill Chapman, and commented on by John Nolte in MMD 2000.04.13-08, is a very interesting variant I have never seen. It appears to me the boot resonator is made to lock the reed fundamental at the correct pitch for the note. The small top resonator then works quite independently of pitch just to shape the spectrum envelope of all the higher harmonics. This has some similarity to speech production where the pitch is primarily controlled by mass and tension in the vocal cords (mechanical rather than acoustical resonator). The vocal tract 'top resonator' has its fundamental resonance in the range 200 to 800 Hz, generally a lot higher than the vocal cord pitch and with no direct connection to it. The Vox Humana designation is just to the point. It would be interesting to know what is the internal height of the boot, and the pitch of one of those pipes. Then it should be easy enough to characterize the type of boot resonance. It is also of special interest to know the size of the foot hole. This is probably important for the Q value of the boot resonance and should give some decoupling from a rather unpredictable influence from the windchest and playing valve. Here I come to think of my trumpet instructor who long ago taught me to tune my mouth cavity with my tongue -- the same kind of thing you do to control the note when whistling. I believe all wind instrument players do this. Once I also saw a research video of the vocal cords taken with a fiber endoscope through the nose of a flutist and a trumpet blower. They did things with their vocal cords to control the upstream airflow (an adjustable foothole!), something the players had probably learned by practice, but were completely unaware of. Johan Liljencrants

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