ttonon
-
Posts
359 -
Joined
-
Last visited
Posts posted by ttonon
-
-
Thus, if one makes a tube of length equal to one quarter (or ¾, or 5/4, etc.) wavelength of a musical tone, with one end closed and the other end open, one has a sound dampener for that wavelength, when a sound source is placed near the opening. With a source of oscillating pressure placed next to the closed end, inside the tube, sound is amplified. The reason for this is really simple, though it will take a bit to explain, as follows.
Take a tube of quarter wavelength, closed at one end, open at the other and apply a source of oscillating pressure to the open end. When a (+) pressure pulse is made at the open end, it moves inside the tube and reflects from the closed end (as a +) and arrives again at the open end in a half period of time (total travel distance is a half wave) - the same time that it takes for the sound source at the open end to go negative. The exiting (+) pulse thus “cancels” the newly generated (-) pulse.
The conclusion here is incorrect, though the logic leading to it is correct. A quarter wave tube, closed at one end and open at the other offers zero impedance while looking into its opening. I hadn't gone through these concepts in many years, and got things confused. The "cancellation" I concluded (which is correct) does not mean that the tube will prevent a reed from speaking, but rather, will result in zero impedance, which is the condition for resonance.
On the other hand, a half wave tube, closed at one end and open at the other will produce infinite impedance, and it is this situation that one likely finds in practice when reeds mounted under such conditions fail to speak. Richard, can you verify that non-speakable reeds are mounted with half wave tubes?
In a new thread, I hope I make the situation with quarter wave and half wave tubes more clear.
Best regards,
Tom
-
-
In discussions here, Helmholtz resonators, quarter wave tubes, and half wave tubes have been mentioned as possible models that might simplify free reed resonant chamber construction. Practical considerations – primarily space and dimensional requirements – limit the applicability of each of these models, and it appears that no single model can be applied to all reeds spanning the useful pitch range of bass, baritone, and tenor instruments.
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.
Helmholtz resonators:
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.
Best regards,
Tom
www.bluesbox.biz
-
Apart from Helmholtz resonator and quarter wave models, there are other ways to view amplification of sound by free reed cavities. I just posted comments in this regard in this same forum under "Big Reed Performance," and include that reference here because of its implications on cavity design.
Best regards,
Tom
www.bluesbox.biz
-
I don't rule out some form of resonant factor, prefering to keep an open mind.
There’s another physical effect involving acoustic waves that may help shed light on the existence, or non existence, of so-called “resonance.” When a plane wave strikes a wall, the wave rebounds, reproducing the same waveform, only now, the reproduced waveform is moving in the opposite direction as the original wave - in the same medium as the original. Let’s say the original wave is moving from right to left. It strikes a wall, and the new wave moves from left to right. Consistent with the linear formulation of musical acoustics, any such wave modifies the pressure of the medium through which it passes by (algebraically) adding its “delta p” to that of the medium. Thus, when the right moving wave passes into the medium which is also carrying the left moving wave, delta p’s are algebraically added. The result is that, at fixed positions in the medium, the amplitude of the pressure fluctuations is twice the amplitude of the original wave. This makes sense simply because where we once had one wave, we now have two, producing twice the amplitude as the single wave.
Thus, whenever a cavity is constructed about a sound source, such as a free reed, the pressure fluctuations, because of multiple rebounds from the cavity walls, within the cavity will be significantly larger than otherwise. Notice that this pressure amplification occurs whether or not the cavity is “tuned for resonance.”
This effect can be easily observed by playing your concertina into a closed corner of a room and noticing how the sound is much more “resonant” than when the instrument is played out in the open, away from walls. This effect is, I believe, largely responsible for the “cassoto” (tone chamber) effect in many accordions. Have you ever wondered why a single, fixed geometry like a tone chamber can affect the sound of reed with pitch spanning that of the entire instrument? With the concertina played in a corner, or indeed, very near a single wall, the rebounding pressure waves add to the waves emanating directly from the instrument. Moving away from the wall greatly diminishes the effect because now, the waves are no longer plane, or one-dimensional, and are dispersing into 3-D space. Of course there’ll be an optimum range of frequencies for which one can clearly hear the effect, determined by how close one can get one’s ear to the wall and by how much dispersion of the wave has occurred before striking one’s ear.
We have a choice here: whether or not to consider this effect “resonance.” I don’t think this is an easy question to answer, and this difficulty, I believe, may often cause confusion (and arguments) on the existence of resonance in free reed cavities. Perhaps the best way to look at it is that resonance is more complex an issue than most people recognize. It’s much easier to define “resonance” as that phenomenon which occurs with more familiar geometries, such as the Helmholtz resonator and quarter wave tubes, with clear analogies to mechanical spring-mass-damper systems. Confusion can then occur when other mechanisms associated with other geometries produce similar audible effects.
I think this effect is present in most all free reed cavities, and that its magnitude is influenced by the ratio of oscillatory wavelength to certain cavity dimensions. I’d be pleasantly surprised if the effect can be made as large as that produced by a properly tuned geometry, though I’d also expect that the effect is significant with many cavities. Because of this, I can understand very well what David has experienced with the deleterious effects of leaks and poorly mounted reeds in so-called "non-resonant" cavities, all which would tend to allow sound pressure to escape in dissipative ways (through small openings fraught with frictional effects). I also commend him on his open mindedness, which I take as recognition of complexities that are not yet fully understood.
Best regards,
Tom
www.bluesbox.biz
-
Goran now:NO *Significant chamber resonance* ....I agree that of course there is physically resonance present related to high overtones but since there is no regular progression of the measures of chambers corresponding to the overtone progression of frequences it will be purely accidental and with the traditional construction impossible to practise.
Hi Goran,
I apologize for misunderstanding your point. I think you mean that there is no theoretical basis that is or has been used to design chambers with resonance, and on this, I think you're probablly right. Also, as you wrote originally, my calculations seem to support your view.
Best regards,
Tom
-
The above conclusions in my previous post concerning the limited usefulness in treating the chamber geometry as a Helmholtz resonator does not rule out the possibility that the chamber can still resonate with the reed motion, resulting in an enhancement of fundamental and/or overtones. If resonance does occur, it may be because of physical interactions other than those described by the simple Helmholtz geometry. Some of the recent posts in this thread have touched upon these other mechanisms.
If I recall you might have experimented with chamber size in developing your pitch bender, and if so are probably one of the few who has heard a note sounded and then listened as the chamber volume was changed. If so, what difference does it make..?I know from my experience that the sound of a free reed can be greatly influenced by the presence of a resonator. This influence is most often an increase in loudness with a more “hollow” or “mellow” tone. The asian free reeds are striking examples of such effects.
Some time ago, I began to experiment with reed tone in mock-up chambers...using a piston type of slide, I could vary chamber volume as I sounded a reed, a little like the way a harmonica's tone can be varied by mouth position, hand placing, etc. I confess that I didn't get too far with the experiment.Bravo, I appreciate your experimental energy. For this to work, and considering the dimensions of interest, I’d guess that you need a very tightly fitting piston. If there were even small air leaks, the acoustic coupling between the reed and the air mass in the resonator would be destroyed. This point is taken up in reference to Richard’s post, below.
First, many thanks to Tom for the substantial survey of what I have said dozens of times in the simple form that there is NO theoretical reason to assume that chamber resonance is significant for concertina tone.However, all I can truthfully say is that I cannot discern any real difference between my radial pan and the Jeffries type. I have not used any acoustical devices to test my observations, by the way!!My guess is that neither your cavities nor the Jeffries cavities are of dimensions that exploit acoustic coupling with their associated reed.
My experience is that the air travel path is far the more accoustically influential than is chamber volume or shape.Goran, I appreciate your comment, though I myself cannot say there is NO theoretical basis for chamber resonance. It may be true that few instruments exploit chamber resonance, and if so, perhaps only by happenstance, but, as I said above, there can, at least theoretically, be resonance that is not explained by the simple Helmholtz resonator. You may be arguing that any theoretical basis cannot be put completely into practice. On that, you are perhaps right, though, considering the full pitch range of the concertina, there will be some pitches whose resonant geometries have realistic dimensions, whether or not this resonance can be easily explained. There's more on this below.
Take the universally accepted standard that the free end of the reed is located at the opposite end of the chamber as the outlet pad is and wonder why? The tone and response of the reed changes considerably to less robust and slower acting when you move the outlet hole closer to the free end of the reed - and when directly over it - will sometimes render the reed inable to speak at all.
Richard, you do have a point with “air travel path,” though, for completeness – and for acoustic meaning – the length of this “air travel path” must be related to the wavelength of oscillation, as follows. As with many physical phenomenon, quantities have meaning only in relation to other quantities of the same dimension. So-called non-dimensional groups are of great value in physics and engineering.
Consider a simple tube, which provides a very useful model for acoustic phenomenon (and particularly those involved with musical instruments) when the wavelength of oscillation is comparable to, or less than, the air masses defined by the geometry in question. In the ideal case, the length of this tube and its “end conditions” are the only meaningful acoustic parameters. Thus, if one makes a tube of length equal to one quarter (or ¾, or 5/4, etc.) wavelength of a musical tone, with one end closed and the other end open, one has a sound dampener for that wavelength, when a sound source is placed near the opening. With a source of oscillating pressure placed next to the closed end, inside the tube, sound is amplified. The reason for this is really simple, though it will take a bit to explain, as follows.
Take a tube of quarter wavelength, closed at one end, open at the other and apply a source of oscillating pressure to the open end. When a (+) pressure pulse is made at the open end, it moves inside the tube and reflects from the closed end (as a +) and arrives again at the open end in a half period of time (total travel distance is a half wave) - the same time that it takes for the sound source at the open end to go negative. The exiting (+) pulse thus “cancels” the newly generated (-) pulse.
With the oscillating pressure source placed near the closed end inside the tube, the sound from the source will be amplified, as follows: A (+) pulse is made at the wall (or hits the wall and bounces back as a +) and travels towards the open end. When it arrives at the open end, it bounces off the opened end (returns) as a negative pressure pulse. That’s important: the boundary condition provided by the open end requires that a positive pulse be bounced back as a negative pulse. (This is mathematically and physically necessary in order to maintain the open end at constant pressure; i.e., open to infinity.) So, the (-) pulse now comes back a half period later just when the source is also (-). Sound is thus enhanced. (I should mention that not all of the energy is bounced back from the open end. Some of the energy escapes the tube and is heard as the musical tone.)
What about three-quarter wavelengths and five-quarter wavelengths, etc.? One can see the reasoning here by picturing a sine wave. At the origin, the signal is zero. Every three-multiple of a quarter wave away, the signal is a maximum. One can see graphically that the result in these cases is the same as for the quarter-wavelength tube with one end open, one end closed.
There are similar lines of reasoning for a tube opened at both ends, but this is not the case here.
Richard, I suppose that drilling a hole in the side of the tube will, as you say, release pressure, and destroy the acoustic coupling between the reed and the tube, when the reed is mounted near the opening, thus allowing the reed to speak. It would be interesting if you could measure the length of the cavity in those instances when you notice stifling and compare it to the wavelength of oscillation. My guess is that the presence of leather valves would greatly enhance the stifling influence of the tube, working just the opposite as drilling a hole.
(Digression, mainly for completeness: Is the reed a pressure source of sound, or a velocity source? The answer to this question is important in understanding how cavities affect reed sound. Without thinking much on it, my guess is, as I assumed above, that the reed is a pressure source, because of the fact that a free reed can be blown and be made to sound without any cavity around it. It is thus not like an edge tone, which is, in reality, a flow-control valve.)
A relevant question here is, can one come up with a quarter-wave design approach to produce resonant cavities in concertinas? Let’s explore this question and pay particular attention to the pitch range of the instrument. There are always complicating issues when attempting to apply idealized models to reality. In the simple tube, the air is assumed to be an ideal gas with no friction, but the most important assumption relevant to us is that the air motion within the tube is considered one-dimensional, and equivalently, the tube diameter (or cross section) is much smaller than the wavelength of oscillation. These assumptions are okay for quarter wave tubes with lengths at least several times larger than the cross section, but in present concertinas, this may not be the case. There are also other issues; e.g., the reed is mounted on the side of the tube, and so the pressure pulses made by the reed occur over some length of the tube, on one side of the cross-section, with perhaps much of it removed from the closed end. I also agree with Richard that the size of the opening hole and the proximity of the pad may have a significant effect.
Middle “C” (262 hz) has a quarter wavelength of about one inch. A three-quarter wavelength is about 3.2 inches. So if someone could make a reed with a plate of between one inch and 3.2 inches and mount it in a cavity with cross section less than about one quarter inch, there may be a place to start experimenting. There are similar results for lower pitches and bass instruments.
At 1,000 hz, the wavelength is 1.1", with quarter wavelength 0.3", three-quarter wavelength 0.8", and five-quarter wavelength 1.4". Here again we can perhaps find accommodation in a realistic instrument, but the tube cross section should be less than about an eighth of an inch. Things are becoming very tight.
At 2,000 hz, wavelength is 0.55", quarter wave is 0.14", and seven-quarter is 0.96". A 2,000 hz reed can probably be made about an inch long, though the cross section of the tube would need to be no larger than about 1/16" in order that the cross section be much smaller than the wavelength. Is this realistic? My concern here is that, even if the cross section can be made so small, would there be enough air mass in the tube to achieve the proper effect? One would have to experiment here.
For frequencies above 2,000 hz, the wavelength is becoming so small that the cross section of any practical tube will be larger than the wavelength. The simple tube assumption thus breaks down and one must devise another model by which to tune the cavities. Incidentally, the "C" three octaves above middle "C" has a pitch of 2,093 hz.
As with the Helmholtz approach, the simple tube approach also presents challenges - because of the small wavelengths involved.
A question arises: Would it be objectionable to have an instrument with some, or even most, of the pitches mounted with resonant cavities and others not? Perhaps the reeds without a resonant cavity would sound very weak, in comparison to the others, as if they belong to another instrument. Is this an all or nothing endeavor?
Perhaps not. If all the non resonant chambers are confined to one end of the pitch range, perhaps the mismatch would not be overly noticeable. Other instruments have similar issues and through the years, makers find ways to minimize tonal nonuniformities. There’s no "perfect" instrument.
How important is tonal quality? I suppose this gets back to Chris’s recent question. How possibly better is the tone and how much weight and expense would people be willing to trade for that better tone?
If I were an instrument maker, I’d perhaps put more effort into finding out how feasible it would be to build an entire instrument with resonant chambers, even bearing in mind the risks. I say this because I'm aware of the extent of tonal enhancement, though I'm sure this issue has been present with the concertina all the way back to Wheatstone’s time. I've seen papers on the web describing Wheatstone's experiments with (large) tubes as resonators for free reeds, though I don't believe these tubes were associated with cavities. Nevertheless, here’s a suggestion. Someone – a maker with extra time and enthusiasm, or an interested experimenter – can make a modified instrument with at least one (but better, a few) resonant cavities. He/she can take it to one of the concertina gatherings, or mail it to selected people to hear what it sounds like. In this way, a consensus can perhaps be obtained on how valuable such a sound is. Looking at the size of the chambers can help one estimate the size of a completely resonant instrument, and so an estimate on weight and expense might then be also estimated.
Best regards,
Tom
-
Recent posts in this forum (Chris Ghent and d.elliot) and other posts elsewhere have queried on methods that might enable the free reed designer to more easily match the chamber geometry to the associated reed, in order to produce "more pleasant" sounding instruments. As d.elliot notes here, this is not a simple task, and there appears no known method by which this can be accomplished. In thinking about this problem, my first impulse would be to apply the simple principle of a Helmholtz resonator to the design, though in carrying out the details, certain difficulties become apparent.
Before stating off hand what these difficulties are, let me outline the approach, which still may prove useful in certain special cases.
If we look upon the reed chamber as a simple Helmholtz resonator, then perhaps we can fit its design into a simple mathematical framework. A Helmholtz resonator is, in its original form, a volume with a single, small aperture, and the resonant frequency of such a geometry is given by f = (c/2/pi)*sqrt(A/L/V), where c is the speed of sound, pi the ratio of the circle's circumference to its diameter, A the cross sectional area of the port, L the "effective length" of the vibrating air mass, and V the resonator volume.
It's good to keep in mind the assumptions behind this simple formula, which is derived assuming a "lumped parameter" system; i.e., all dimensions of the resonator are much smaller than the wavelength of oscillation under question. Necessarily with this assumption, the air within the resonator will vibrate uniformly, defined by one pressure, temperature, etc., and the only significant air motion associated with the vibration will be concentrated in and about the aperture. This lumped parameter assumption is the crux of the problem that often will arise when one follows this simple design approach. (One must always check this assumption with the above formula for resonant frequency.) In order to point this out, we proceed as follows.
With concertina reed chambers, there's a walled volume with a hole controlled by a valve, and a free reed mounted elsewhere in the wall. The approach is to consider the volume, V, of the resonator to be the volume of the chamber, and the area A and length L to be associated with the hole. The idea is that any overtone of the reed that matches the resonant frequency of the resonator will be enhanced. My guess is that the "most mellow" sound of the reed will be achieved when the resonator frequency matches the fundamental pitch of the reed. I don't believe the simple resonator has overtones. This is simply because, if it did, the assumption of lumped parameters would be violated, and other analyses would have to be performed in order to calculate the pitch of these overtones. In such a case, the object would not be functioning as the classic Helmholtz resonator.
We address one detail here; i.e., there's a kind of hole in the wall associated with the reed, and the question arises, should this hole be considered in the determination of A, as used in the above formula. I think not, simply because the reed is a driving force for the resonator behavior and is thus not greatly affected by the resonator. If the pitches of the reed, including its fundamental and overtones, are not matched to the pitch of the resonator, the resonator is off design. With proper design, the independant pitches of both the reed and the resonator will not change when both are coupled into a single system.
A word on L. As mentioned, all air vibration associated with the resonator is confined to, and about, the aperture. Thus, one can visualize a small slug of air vibrating in and out of the aperture. If one were to contain this vibrating air mass into a cylinder with cross section A, its length woul be L. For a very thin walled resonator then, L would be considerably larger than the wall thickness. In fact most people write, L = t + c, where t is the wall thickness, and c is a correction that accounts for the amount of air vibrating immediately inside and outside the hole. The correction c has been calculated fairly accurately for special cases, and in our case, c should be a little less than the diameter (or major dimension) of the hole. Since there's an end correction on both sides of the aperture, one might approximate: L = t + 1.5d, where d is the diameter of the hole. (We are assuming that the valve opening the aperture is moved far enough away from the aperture so that it does not interfere with the vibrating air mass. )
Before getting into numerical calculations, we can easily see what the effects of various chamber dimensions have on tuning the chamber to the pitch of the reed, as long as the lumped parameter assumption is valid. Thus, the resonator pitch decreases with increasing wall thickness and increasing resonator volume, and larger apertures produce higher resonator pitches.
As an example, consider chamber dimensions which are typical of existing concertinas: a chamber 2" long, 1/4" deep, and 3/8" wide, with an aperture 1/4" in diameter and a wall thickness of 1/4". The resonator pitch calculates to 1360 hz, with a corresponding wavelength of 9.7". Changing only the aperture diameter to 1/2" gives a pitch of 2150 hz, with a wavelength of 6.1". Notice two things: Firstly, the pitches of these resonances are rather high, and secondly, the lumped parameter assumptions are breaking down. Herein lies the difficulty of this approach and why producing mellow sounding reed chambers is a difficult chore.
For current constraints on concertina dimensions, we conclude, 1) the reed cavities cannot be made large enough to resonate in a simple way with many of the pitches played by the instrument, and 2) simple lumped parameter methods of design cannot be applied.
Both these issues would, in principle, disappear if larger reed cavities can be made to fit into the instrument. For instance, if we take the following (unrealistic) dimensions: a chamber 4" long, 1" deep, and 3/4" wide, with an aperture diameter 3/8" and an aperture extension of length 1/2", we get a resonant frequency of 391 hz, and a corresponding wavelength of 33.7". The pitch here is close to "G" above middle "C", and the lumped parameter assumption is reasonable.
Of course, with more realistic dimensions, there can be favorable resonances, but in these cases, standing waves exist within the reed cavity, and calculations in these cases become exceedingly more difficult than the ones presented here.
In conclusion, a simple Helmholtz approach to concertina reed cavity construction appears not generally useful for existing instruments. A suggestion that arises from these ashes would perhaps be to find a way to install resonant tubes of small diameter inside the instrument, but such an approach may also require a larger instrument.
Best regards,
Tom
www.bluesbox.biz
Big Reed Performance
in Instrument Construction & Repair
Posted
Gene,
What problems with standing waves are you referring to? When standing waves exist in an air column, this is called resonance. The whole of your article seems to extoll the benefits of resonance, so what is the problem with resonance? Also, do you mean to say that when the walls of the chamber are not parallel, standing waves do not form? In truth, the angle of the walls only affects the kinds of overtones, or the kinds of standing wave modes, that can occur. For instance, in a cylinder (parallel walls), with one end open and the other end closed, only odd harmonics can form standing waves, but for a conical shape, all harmonics can form standing waves. An example of the first of these is the clarinet, and of the second, the saxophone. I believe the walls of the cavity are angled to one another simply in order to allow the reed plate to wedge into place.
Best regards,
Tom
www.bluesbox.biz