Presumably, the goal of resonant design is to produce forceful (quick starting), possibly amplified, and pleasant sounding timbre. Much of these effects are, however, subjective, and although the goal itself can be a contentious issue, we assume here that the desired affect will be achieved by tuning resonance to the fundamental and/or lower partials (overtones). With quarter wave tubes, tuning to a given frequency will also result in tuning to odd multiples of that frequency, and with half wave tubes, all multiples will be tuned for. With the simple Helmholtz resonator, only the design pitch will be affected (more on this below).
As a slight digression, I believe the ideal sound spectrum of an isolated free reed is composed of overtones that are perfect harmonics (whole number ratios to the fundamental), simply because there is no excited overtone of the vibrating cantilever. (If such an overtone were excited, noise would result.) In reality, there is thus very little “stretching” that occurs with other (real) sound sources. Any stretching that does occur will be due to nonlinearities in resonant air columns mounted about the reed.
Thus, in using these models, the maker must first decide what overtone to design for, and this decision will often be linked to space requirements within the instrument. For instance, the fundamental of a low note may require an unduly large geometry, but the geometry that resonates with the first overtone would be considerably smaller. Thus, for certain reeds, one model designed for the fundamental may be practical, and for other reeds, a different model designed for the first overtone may be more appropriate. The hope is that such a patchwork application of models and designs can provide chambers that span the entire range of a given instrument. When model calculations result in a promising design, a prototype should be built, and hopefully, minor tweaking can achieve a final result.
For the desired effects, it’s not sufficient to provide a reed with only a properly tuned resonator. The resonator (including tubes) must also be large enough. The air associated with the resonator must have sufficient mass so that the energy of its vibration can influence the overall sound. For instance, a quarter wave tube is, in principle, a fine resonator, though its diameter must be sufficiently large so that it can influence the overall sound level.
For the simple resonator, in which all significant dimensions are much less than the wavelength of concern, the resonant frequency is given by
f = (c/2/pi)*sqrt(A/V/(t + k*d)),
where c is the speed of sound, pi = 3.14, A the area of the aperture (pad hole), t the thickness of the reed plate (that is pierced by the aperture), d the diameter of the aperture, k a number in the approximate range 0.8 to 1.2, with the higher values chosen if the pad remains within a distance d of the hole upon opening (pads that remain close to the hole will decrease the resonator pitch.), and V the volume of the cavity = L*w*h, where w is cavity width, and h is cavity height. As I mentioned in a previous post, the term k*d is an “end correction” due to the mass of air vibrating in the vicinity of the opening.
From a practical point of view, L and w will often be related to the reed plate, assuming that one side of the cavity will be comprised of the reed plate itself. The quantities d and t can be somewhat independently chosen, or varied, and thus, the above expression will serve to calculate h, for given f.
It’s expected that this model will apply to some range of reed pitches, and in applying this model, the designer must always check the basic assumption associated with the Simple Helmholtz Resonator; i.e., that x/lamda << 1, where x is every characteristic length comprising the resonator.
At the lower frequency end of this range, designing for the fundamental may result in theoretical dimensions (values for x) that are too large to be accommodated inside the instrument. One then moves the design frequency to twice the fundamental (first overtone), which will result in a ½ scale resonator, as compared to the first design. If these dimensions are also too large, one must then go to the second overtone, and so on. Important here is that, always, x/lamba must be << 1 for the above equation to be valid. One can of course stretch this criterion, realizing simply that resonance can still be achieved, though the above expression for resonant frequency becomes less accurate. Furthermore, there is a question on how desirable it would be to amplify an overtone of the musical tone. Would this tend to shift its presence out of its proper octave, as perceived by the listener?
If it turns out that moving to higher design frequencies results in a violation of x/lambda <<1, without a solution to be found, then the simple resonator model becomes inapplicable, and more complicated calculations are required, should one still desire a Helmholtz geometry design.
For these latter calculations, one might consult “The Physics of Musical Instruments,” Fletcher and Rossing, Springer, pgs. 227 – 232. The calculations suggested in the reference allow resonator dimensions to be comparable to the wavelength in the “long” direction, but still require the transverse dimensions of the resonator to be much less than the wavelength. With these more complicated calculations involving wavelength effects, there then arises the possibility of overtones emanating from the Helmholtz geometry, which, in truth, is no longer functioning as the classical resonator.
I proceeded with the above approach and have come up with some interesting results, which are presented in table form here. But before presenting those results, I briefly describe the quarter wave and half wave models, which can perhaps be useful for the very highest concertina pitches.
Quarter wave and half wave tubes:
A tube the length one-quarter wavelength, with one end open and the other end closed, is a compact resonator. In theory, when the tip of the reed is placed near either the open end of the tube or inside the tube, near the closed end, the fundamental and odd number overtones will be in resonance with the air in the tube. With the reed plate hole placed at the open end of the tube, we have a candidate for a resonant cavity. In calculating dimensions, one should allow for an “end correction,” similar to that mentioned in reference to Helmholtz geometries. Thus
L + t + kd = lambda/4,
where L is the cavity length, t the concertina reed mounting plate, d the diameter of the air hole, and k is a number from between about 0.4 and 0.8, depending how close the lever pad remains to the hole.
A tube of length one-half wavelength, with both ends open can also perhaps serve as a useful model. Perhaps the most practical way to incorporate half wave tubes would be to make the cavity length equal to one whole wavelength and mount the reed along one side, with the free tip of the reed approximately centered from the two open ends. With this arrangement, there must be two air holes and two key pads that simultaneously close and open these holes. Additional lever pads may unduly complicate instrument design, but I present this concept here more or less for completeness. Thus,
2*(L + t + kd) = lambda
For both quarter wave and half wave tubes, the model is applicable for length to tube diameters much larger than unity.
Sample calculations and results
Sample calculations are presented in the table provided. The reed plate widths and lengths and pitches are taken from actual reeds. I don’t have English-style concertina reeds before me, so I’m for the moment restricted to the measurements given, which are for accordion reeds.
speed of sound 1100 fps = 13200 ips
width 0.687 in
Note -freq -rp ln -wvlth -x -w / 4 -w / 2 ---d-- ---t-- ---L-- ---h-- ------Solution
A2 220.00 2.12 60.00 6.00 15.00 30.00 0.30 0.38 2.12 6.56
A2 880.00 2.12 15.00 1.50 3.75 7.50 0.30 0.38 2.12 0.41 -----Hlmhtz, 2nd overtone
A4 440.00 1.60 30.00 3.00 7.50 15.00 0.30 0.38 1.60 2.17
A4 880.00 1.60 15.00 1.50 3.75 7.50 0.30 0.38 1.60 0.54 -----Hlmhtz, 1st overtone
E5 659.00 1.50 20.03 2.00 5.01 10.02 0.25 0.38 2.00 0.58 -----Hlmhtz, fundamental
A5 880.00 1.31 15.00 1.50 3.75 7.50 0.50 0.25 1.50 1.45
A5 880.00 1.31 15.00 1.50 3.75 7.50 0.30 0.25 1.50 0.71
A5 880.00 1.31 15.00 1.50 3.75 7.50 0.50 0.38 1.50 1.24
A5 880.00 1.31 15.00 1.50 3.75 7.50 0.30 0.38 1.50 0.58 -----Hlmhtz, fundamental
F6 1397.00 1.12 9.45 0.94 2.36 4.72 0.30 0.38 1.12 0.31 -----Hlmhtz, fundamental
D7 2349.00 0.88 5.62 0.56 1.40 2.81 0.30 0.38 0.88 0.14 -----Hlmhtz, fundamental
D7 2349.00 0.88 5.62 ----- 1.40 2.81 0.25 0.25 0.95 ----- -----quarter wave tube
A7 3520.00 0.81 3.75 0.38 0.94 1.88 0.30 0.38 0.81 0.07 -----Hlmhtz, fundamental
A7 3520.00 0.81 3.75 ---- 0.94 1.88 0.25 0.25 0.49 ----- -----quarter wave tube
A7 3520.00 0.81 3.75 ---- 0.94 1.88 0.25 0.25 2.85 ----- -----2 half wave tubes
The table doesn't come out too well, but I think the columns are lined up reasonably well. If you can look carefully, the top column headings lists the musical note, its frequency, the reed plate length, the wavelength, the maximum size for Helmholtz geometry dimensions, x, for the simple Helmholtz model to apply, the quarter wavelength, half wavelength, aperture diameter, aperture (or concertina reed mounting plate) thickness, cavity length, cavity height, and the kind of solution arrived at.
It may be instructive to first look at the calculations for note A5. Here the diameter and aperture thickness were modified in order to obtain a reasonable cavity height, which came out to 0.58 inch. Notice the large affect of hole diameter. Is the final hole diameter (3/8 inch) reasonable? Would this allow sufficient sound to escape? Of course, with resonance, much more sound would come out of a 3/8 inch hole than a 1/2 inch hole. Moving upward, note E5 also appears suitable to the Helmholtz model, for the fundamental, though note A4 requires a first overtone target and note A2 requires a second overtone target. Proceeding downward, note F6 appears suitable to the Helmholtz model, for the fundamental, though higher pitches produce conflicts with this model. For note D7, the cavity length is becoming significant in comparison to the wavelength (larger than x). Note however, that this doesn’t mean there won’t be resonance, but only that the calculation becomes less accurate, and perhaps more experimenting would be required. A quarter wave model is also applied to note D7, producing a reasonable value for cavity length. Note however that the cavity width (0.69) is significant with respect to the cavity length, tending to reduce the accuracy of one-dimensional flow inside the tube. Situations like this can be improved by putting a partition down the center of the cavity, thereby halving the effective width of the tube. With note A7, neither the Helmholtz or quarter wave models apply. The reed plate length is larger than x, and the quarter wave cavity length is shorter than the reed plate length. Of course, alternate reed mountings could correct this issue, and a double half wave tube also appears workable, though with additional cautionary notes with regard to one-dimensional flow.
From the above calculation, I think one can see that reasonably sized geometries can be designed for resonant cavities in concertinas, using different models for the entire pitch range of the instrument. This is a much more optimistic outlook from my previous posts. Furthermore, the simple Helmholtz model can apparently be applied over most of the pitch range of the instrument - provided of course that higher overtone targets produce aesthetically pleasing results. In light of this, I'm curious whether other builders have tried to incorporate these concepts, and if not, why not, for the results seem workable. If we look at the primary result of the above table; namely, the values for h, the height of the cavity, some of these height are perhaps larger than what you find in existing instruments, though, I'd guess they could be accommodated because of the rewards offered by tuned cavities. In building the table, I required that all reasonable values for h be a maximum of about 0.6 inch. How does this sit with actual concertina builders? Richard?
In conclusion, there appears to be at least theoretical grounds for successfully incorporating resonant cavities into concertinas. Of course, the builder would have to build models and verify these results, and it's expected that these theoretical results can only be starting points for these experiments. My hopes are, however, that these results are not far off, and also that resonant cavities can be as rewarding as I'm assuming.
Edited by ttonon, 19 September 2003 - 10:39 AM.