Note: In the original hardcopy publication, Equations 5 and 7 contained errors, which carried through to the Table of that publication. These errors, however, are not large enough to alter the main conclusions made in that publication. In this HTML publication, these errors have been corrected, and there are corresponding differences between the Table and Text here and the Table and Text of the original hardcopy publication. I wish to thank Johann Pascher for first pointing out these errors to me – Tom Tonon
Background
Resonance occurs in acoustic systems, and reed cavities in concertinas and other free-reed instruments are no exception. Investigations of reed cavity resonance have been carried out in Europe and in the United States.1 It is my intention here to focus on the practical details of reed cavity resonance, and I hope this article can assist towards a more thorough and broadly accessible discussion. I present here qualitative and quantitative aspects of resonant cavity design, including suggestions on how resonance can possibly enhance as well as detract from reed performance. The approach here incorporates simple acoustic models, and is based both on suggestions from the work of others and on the limited experiments I have done myself. At the end of this article I present tabulated examples of resonant cavity geometries calculated from these models, as applied to the musical range of the concertina family of instruments.
Fundamentals
An important function of the reed cavity is to provide secure and airtight mounting of the reed in order that a uniform stream of air can be directed through the reed and that tongue vibration can proceed without interference.2 Practical free-reed instruments cannot exist without cavities, and very few people have heard the sound of a free reed without an associated cavity. Builders are aware that cavity shape can influence reed performance, and is this feature of the cavity that concerns us here.
The vibrating reed tongue and the air within and about the cavity are acoustically coupled together. In some designs, the effect of the cavity on the musical tone is small, or negligible; in other designs, the effect of the cavity can significantly modify the musical tone; and in still other designs, the acoustic effect of the cavity can prevent the reed from speaking properly.
Coupled Vibrations
Mechanical systems possessing mass and elasticity experience natural modes of vibration called resonance,3 and two or more systems can participate in coupled vibrations. The reed tongue is one such system, and the air within and about the cavity comprises another. The combined tongue/cavity system is coupled together by the air pressure/velocity behavior at and near the vent (slot) through which the tongue passes, since both the vibrating tongue and the vibrating air mass within and about the cavity influence this region. The tongue vibration occurs at a frequency very slightly lower than its natural frequency—vibrating as a bar with one end fixed and the other free—inducing air and pressure oscillations about the vent, and thus causing air inside the cavity to vibrate.4 Because of the coupling mechanism, this influence can amplify or diminish the fundamental of the musical tone, and the precise nature of this influence depends in general upon the particular resonant mode of the cavity and the position of the vibrating tongue with respect to the cavity. The degree of any influence to be expected is difficult to determine without experiment, though there are at least two parameters that must be considered in this determination. One is the frequency match between the resonant mode of the cavity and the vibrational frequency of the tongue, and another is the size of the cavity, or, better stated, the power output of the mass of air that is induced to resonant vibration in comparison to the vibrational power output of the reed itself. Bigger cavities will have larger potential influence.
As an illustration, consider a wider scope of reed instruments. With woodwinds—the clarinet, for instance—the body of the instrument takes on the role of the cavity, which is relatively very large, and the air mass in and about this geometry resonates as the primary source of vibrational energy, with the reed vibration itself contributing very little. The beating reed of the clarinet functions as a pressure-controlling device, is placed at a pressure antinode, and vibrates at a frequency equal to that of the cavity-mode vibration, which is well below the reed’s own natural frequency. Reed vibration and air vibration are strongly coupled, and air vibration dominates, with the soft, pliable reed simply tagging along. At the other extreme are the tone chambers (cassottos) placed in some accordions. Here, the reed/tongue system is weakly coupled to the air geometry associated with the tone chamber, and resonant vibrations of air in and about the tone chamber have little effect back on tongue vibration, which provides the predominant amount of acoustical power. Tone chamber vibrations, however, do significantly alter the sound of the musical tone (volume and timbre) for those reeds that present partials with frequencies that match the resonant frequencies of the tone chamber (a phenomenon discussed in more detail below).5
An intermediate example is the beating reed used in organ pipes,6 which is again a device closely coupled to air vibration and placed near a pressure antinode. Here, the tongue is not nearly as compliant as the beating reed in a woodwind instrument, but it is also not as stiff as that in free reeds, and the resonant frequency of the combined system is a compromise between the tongue and air column systems. A final example is the Asian free reed, which normally functions with a pipe resonator, to which it is closely coupled. This reed is placed near the closed end of the resonator (bawu,with the other end open), approximately a quarter length away from the closed end (khaen), or at the open end of an open pipe resonator (sheng). Such varied placement indicates that the coupling between this reed and its resonator is complicated. For the khaen and sheng, the resonator frequencies are closely matched and the playing frequency is typically slightly above the resonant frequencies of both the reed and the resonator. For the bawu, the playing frequency is near the pipe-resonant frequency, which is considerably above the resonant frequency of the reed.7
With the Western free-reed system, tongue and air vibration coupling is also strong, but a key feature of the mechanism for vibration here is different from those of both the beating reed and the Asian free reed. With the Western free reed, tongue vibration is self-excited and does not require a vibrating air mass (resonator) in order to transfer a steady air pressure difference into oscillatory motion. The mechanism for self-excitation takes place in the neighborhood of the vent and occurs at the frequency at which the tongue vibrates: that is, the fundamental of the musical tone. The tongue can thus be made to vibrate without any cavity, and it can be made to vibrate at frequencies far from any resonant frequency of the cavity. Far from cavity resonance, air vibration in the cavity is small and will thus have relatively little effect on air motion in the critical vent region. As tongue vibration frequency and cavity mode resonant frequency become closer, however, cavity air vibration can become large enough to influence the self-excitation mechanism. Whether this influence assists or interferes with tongue vibration and the resulting musical tone depends upon the resonant mode of the cavity and how the reed is mounted in relation to the cavity. In what follows, I apply the Helmholtz, quarter-wave, and full-wave models as a way to understand the resonant modes possible with the reed cavity and how they might influence reed performance.
The interference described above can completely prevent the tongue from vibrating: the reed becomes choked. Choking is predicted, then, under certain conditions when tongue vibration frequency is in some neighborhood of cavity mode frequency. Builders occasionally encounter choking in the higher-pitched reeds, and later in this article, I provide calculations illustrating why such reeds are likely candidates for choking, based upon resonant effects. As a remedy, builders sometimes provide a ‘vent hole’ in the cavity, or file off a corner of the tongue, in order to allow the reed to sound properly. These gaps function by allowing air leakage, destroying the offending resonant mode of the cavity, and perhaps also by incorporating a larger degree of damping in the cavity mode vibration. Three other methods are sometimes used by builders to prevent choking: eliminate leather valves on both reeds mounted in the cavity, reduce cavity height to an absolute minimum, and mount the reed with the reed tip near the air opening to the cavity.8 The first of these provides the same function as the ‘vent hole’ described earlier, the second of these changes the resonant geometry, when the cavity is functioning as a Helmholtz resonator, and the third is useful when the cavity is functioning as a quarter-wave tube. I discuss these resonant geometries more fully in the following sections.
An interesting experiment is to take a reed block out of an accordion and attempt to sound the reeds by blowing or sucking through the air passages while making a tight seal between your lips and the block opening. For many reeds, weak, or even no sound results, suggesting an offending resonant mode of the large cavity created by the addition of your pulmonary system to that of the reed block. Some of these reeds, however, can be made to play well simply by holding your nostrils closed while blowing or sucking. Thus, closing your nostrils changes the resonant geometry to one that contains no offending mode. Another way to sometimes allow voicing is simply to suck in rather than blow out, or visa versa, which both engages a different reed—which may have only a slightly different pitch or a pitch one or two semitones different—and also causes a different airflow direction through the reed block. Changing reeds can change the resonance relation, and a different mean flow direction can make a slight change in the cavity resonant frequency.9 Finally, allowing some gap between your lips and the reed block, or breathing also a little through your nose can also restore clear voicing, since your pulmonary system is then partially de-coupled from the rest of the system. Such demonstrations reveal strong coupling between tongue and cavity vibration and suggest that cavity resonance can have a major effect on self-excited tongue vibration.
Fundamental, Overtones, and Partials
The vibrating reed tongue periodically chops the air stream that forces its motion, resulting in complex pressure pulses whose waveform contains many partials (the fundamental, defined by the vibration frequency of the tongue, and overtones).10 These partials have frequencies very close to whole number ratios of each other, and are thus called harmonic. Departure from harmonicity could accompany excitation of additional vibrational modes of the tongue, though such excitation is very small, occurring only at very high blowing pressures.11 In any event, acoustic waves produced by select partials can interact with resonant modes of the cavity. As a result, these partials can be strengthened or weakened, just like the fundamental, as explained above, but there is a diminishing consequence to reed performance as resonance moves up to higher and higher partials. For higher partials, then, the tongue/cavity system is weakly coupled to tongue vibration, and for these partials, one might expect the cavity to function like the weakly coupled tone chambers described earlier. One would not expect a noticeable effect on musical tone if a cavity mode resonates with, for instance, the 10th partial of a musical tone; however, for the lower partials, say less than the 4th, an attentive listener might notice a difference in tone, and even volume.
There does not appear to be an extended effort concerning resonance exploitation on the part of squeezebox builders. Perhaps this is because the last 150 years of development have taught builders that any benefits to be gained are small compared to the required effort. This seeming lack of interest is also understandable because of the danger of destructive interference, which, as I show below, becomes possible in the neighborhood of resonance with certain cavity modes and for certain reed mounting positions. There is also the danger of uneven reed performance within the musical range of the instrument. Such dangers, however, should be reduced both if the techniques for resonance exploitation are well understood and if the builder emphasizes the reinforcement of higher partials of the musical tone, staying away from the fundamental in cases where it is known that fundamental resonance will be destructive. I will explore conditions under which builders can expect such destructive interference, suggest remedies, and, where convenient, show how one might exploit cavity resonance as a way to alter, and perhaps improve, musical tone.
For those interested in such exploitation of resonance, note that, for resonant design to be optimum, air leaks through cavity walls should be eliminated. Often in practice, two reeds share the same cavity, and in such cases, both reeds should be valved, so that the non-speaking reed does not provide a leak of acoustic energy. With English-system instruments, both reeds are of the same pitch, whereas in other, bi-sonorous, designs (the Anglo, for example), the two pitches differ, usually by one or two half tones. Since resonant cavity design depends upon the pitch of the musical tone, the question arises as to which pitch to use with bi-sonorous cavities. As a starting point, one might simply design for the average of the two pitches, at least for initial investigations, with the possibility for subsequent tweaking. More elaborate treatment of these cases would require the construction of separate cavities by means of partitions,12 which is beyond the scope of this article.
The Wavelength of Oscillation
An important parameter in every acoustic phenomenon is the wavelength of oscillation, 
, defined by (Equation 1)
where c is the speed of sound (1130 ft/sec for air at room temperature) in the wave medium and v the frequency of oscillatory motion. The wavelength is our characteristic length, and all dimensions of the cavity must be compared to this length in order to draw valid conclusions concerning their acoustic relevance. In our case, the frequency, v, will be that of the partial of interest.
Comparing the various cavity dimensions to the pertinent wavelength of oscillation allows us to predict what kind of resonant modes are possible for a given cavity geometry. Practically speaking and in simple terms, a given geometry will behave at resonance in one of two ways: as a Helmholtz resonator or as a quarter-wave tube. With special instrument construction not normally utilized, full-wave resonance can also be produced, as discussed below.
Helmholtz Resonator13
Typical cavities consist of a volume of air connected to a necked-down region where vibratory air motion can be concentrated. Such geometry resembles that of the classic Helmholtz resonator. When is much larger than all cavity dimensions, we can expect the cavity geometry to behave according to this model, in which case, all pressure fluctuations within the cavity will be spatially uniform. Figure 1 depicts this geometry, situated so that the reeds are placed behind, out of view, and the necked-down region produced by the air hole in the concertina Action Board is identified with the Aperture of the figure. In operation, pressure oscillations in the cavity impart oscillations in the air that travels through this aperture, and this air motion has a non-zero time average that corresponds to the net airflow in or out of the bellows. The response of this geometry will increase as some partial of the musical tone approaches its resonant frequency. In effect, this construction is a mechanical system, equivalent to the more familiar spring/mass system, with the compressible air in the cavity corresponding to the spring, and the vibrating air in the vicinity of the aperture corresponding to the mass.
Most all of us are familiar with how easy it is to excite a Helmholtz resonator; we can simply blow across the mouth of a soda bottle. One might thus question whether a Helmholtz resonator can be excited by a reed placed in the wall of the resonator, as in Figure 1, and not, for instance, near the outside of the aperture. The free reed, being a flow-controlling device, introduces mass into the Volume of the resonator in periodic fashion, resulting in cavity air pressure oscillations. Theoretically then, the Volume is excited in the very way it functions as part of the resonator, and at resonance, the coupled system should behave very differently from its behaviour far away from resonance. Although it is difficult to determine solely on theoretical grounds just what this resonance behaviour will be, it is, as shown below, a simple matter to calculate the resonant frequency of this geometry.
With this geometry, my own limited experimentation has shown that, when the cavity experiences Helmholtz resonance with the fundamental of reed-tongue vibration, interference occurs, and the reed-tongue vibration is seriously hampered, even choked. This interference occurs even for Helmholtz resonant frequencies somewhat below the fundamental and suggests that resonant cavity air vibration feedback to the critical vent region upsets the self-excitation mechanism, at least for those cavities large enough to supply sufficient energy. In other words, at resonance, the reed-tongue vibration is not ‘stiff’ enough to completely dominate cavity resonance. Hence the practice of some builders to provide air leaks in the cavity, or incorporate very small cavity volumes as a way to change and/or reduce the resonance response. Alternatively, one can make other adjustments to resonator geometry, by utilizing the expressions given here for the calculation of cavity resonant frequency. In addition, my own experiments show the following: for Helmholtz resonant frequencies a little larger than the fundamental, interference does not occur, and I have even observed volume amplification. Similar behaviour occurs during Helmholtz resonance with the second partial (first overtone), though with reduced intensity and less interference, with the absence of choking. The resonance effect drops off rapidly for even higher partials. A general effect on tone seems to be a reduction in sound contribution from the partials with frequencies well above the cavity resonant frequency. These observations appear to be insensitive to where exactly the reed tip is located in the cavity wall.
Fig. 1. Helmholtz model.
As noted above, the acoustic wave associated with in the calculation of the resonant frequency need not arise from only the frequency of tongue vibration (fundamental). Higher partials of the pressure waveform produced by the vibrating tongue should also be considered, since such partials may still result in wavelengths that are significantly larger than all cavity dimensions, which validates the Helmholtz model. The same cavity, of course, will cease to function as a Helmholtz resonator for frequencies so high that the associated wavelengths are comparable to some resonator dimension. In these cases, the cavity can perhaps function as a quarter-wave tube (see below). The Helmholtz resonator represents an extreme end of the range of resonant geometries and has only one resonance mode. By definition, overtones do not exist in its operation, simply because such overtones imply that some cavity dimension is comparable to the wavelength associated with such overtones.
The resonant frequency, v0, for the Helmholtz geometry is given below (Equation 2):
where also, from Equation 1, 0 = c/v0; 
= 3.14; 
is the square root function; A is the area of the aperture (air hole); t is the length of the Aperture (thickness of the Action Board); d is the diameter of the aperture; V is the net air volume within the cavity, and k is a number in the approximate range 0.43 to 0.80, with the higher values chosen if the fully open pallet remains within approximate distance d of the aperture. Lower values are chosen for k if this distance is about twice d (pallets that remain close to the hole will decrease the resonator pitch). The accuracy of this ‘end correction’ term, kd, decreases as 0 becomes smaller and no longer large compared to the product (2d).14
The calculation of V depends upon the construction of the cavity. For traditional English construction, one reed is situated outside the cavity, often with its leather valve situated inside the cavity, and another reed is situated somewhat inside the cavity. For accordion-reeded instruments, the entire reed is situated outside the cavity, with a slight addition of air space due to the thickness of the cavity wall supporting the reed. In the simple case of an orthogonal cavity, of length L, width W, and height H, we calculate (Equation 3):
where Vadj is the volume adjustment because of how the reed is mounted. With this notation, the volume of any reed part within the cavity proper will contribute negatively to Vadj (reed volume is subtracted). Note that in Figure 1 the Helmholtz geometry is, for the sake of simplicity, assumed to be such an orthogonal structure. Sample calculations using these expressions will be presented below.
Quarter-Wave Tube Resonator
A tube is defined as a cylinder whose transverse dimensions are much less than its length,15 with a quarter-wave tube resonator being such a tube—of length one-quarter wavelength—with one end open and the other end closed. Reed cavities somewhat resemble tubes, and Figure 2 depicts a cavity that functions as a quarter-wave tube. This drawing depicts traditional-style English concertina reeds that are mounted with the free tip of the tongue near the closed tube end.
Fig. 2. Quarter-Wave Model.
From Figure 2, an immediate conclusion is that, with the cavity functioning as a resonant quarter-wave tube using the tongue vibration frequency (fundamental) for the relevant wavelength, there is likely to be serious interference between tongue vibration and cavity air vibration. The explanation is as follows. At resonance, the air within the cavity must vibrate with a velocity node (minimum) at the closed end and a velocity antinode (maximum) at the open end. The self-excited free reed mechanism, however, requires a large velocity oscillation near the freely vibrating tip of the tongue, which is in the vicinity where the tube air vibration requires a minimum. Thus, neither vibrating system satisfies the requirement of the other, and interference with the reed’s self-excitation mechanism is likely. I have experimentally verified such interference, including choking, which is similar to the choking caused by cavities resonating as Helmholtz resonators (as explained above). Even with (effective) tube lengths a bit different from one-quarter wavelength, the reed might speak only weakly.16 The suggestions on how to avoid Helmholtz resonance interference explained above also apply here, but with quarter-wave resonance, an alternative method to provide better voicing would be reorient the reed so that the free tip of the tongue lies near the open end of the cavity. Builders sometimes adopt this practice, which is illustrated in Figure 3, and doing so will likely result in amplification in musical tone, since each vibrating system then satisfies the requirement of the other. I have observed such amplification experimentally, and such amplification is theoretically possible both at the fundamental frequency and at overtones whose frequencies are odd-numbered multiples of the fundamental. With conventional reed placement, and if a higher partial of the musical tone provides the pertinent wavelength with which to measure the length of the tube, choking is less likely, though weak tones are still possible.
Fig. 3. Quarter-Wave Model with Alternate Reed Mounting
The resonant quarter wavelength geometry is given by (Equation 4):
where, from Equation 1, vo = c/o , where vo is the resonant frequency, and the ‘effective tube length’ is given approximately by (Equation 5):
where L is the cavity length, t is the thickness of the Action Board, = 3.14, d is aperture diameter, and as in the Helmholtz model above, k is a number from between about 0.4 and 0.8, depending on how close the pallet remains to the aperture. In Equation 5, it is assumed that o is large compared to the product (2d) and large compared to the difference (Leff – L).17
For some cavity geometries, W, the cavity width, is not very much smaller than L, and in such cases, there may be an occurrence of transverse standing modes, though on simple analysis, they do not appear to require much concern here.18 style> For those interested in exploiting the effect of quarter-wave tubes on musical tone, it may be advantageous to divide the cavity with a lengthwise partition, effectively separating the two reed tongues that share the same cavity and significantly increasing the ratio L/W. Figure 4 illustrates this partition, with the resulting reduction in the size of W. Such a partition may also be more useful for bi-sonorous cavities. As mentioned earlier, a quarter-wave geometry (with reed tongue tip mounted as in Figure 3) that amplifies a partial of one frequency will also amplify partials having frequencies that are odd multiples of this frequency.
Fig. 4. Quarter-Wave Model with Partition
Full-Wave Tube Resonator
A full-wave tube resonator is a tube of length one wavelength, with either both ends open or both ends closed. Because of the end conditions, such geometry does not normally exist in squeezebox construction; however, from a theoretical point of view, and for those interested in how such geometry might be exploited for its resonance possibilities, Figure 5 illustrates one way in which this could be done. The configuration here incorporates the open-end conditions. Note the partition in Figure 5, which creates a tube of one wavelength from a cavity whose length is closer to one-half wavelength. Note also the placement of the free tip of the reed tongue, which is near the air hole, at the top. With this arrangement, the requirement for maximum air velocity by both tongue and cavity air vibrations is satisfied, and amplification should theoretically occur for the design partial, as well as for partials having frequencies that are whole number multiples of the frequency of the design partial.19
Fig. 5. Full-Wave Model with Partition.
The resonant geometry for the full-wave tube geometry is given by (Equation 6):
where, from Equation 1, vo = c/o, vo is the resonant frequency, and following the approach taken with the quarter-wave geometry, Leff is the ‘effective length’ of the air cavity, expressed by (Equation 7):
where L is the cavity length, t is the thickness of the Action Board, d is aperture diameter, and as in the Helmholtz model above, k is a number from between about 0.4 and 0.8, depending on how close the pallet remains to the aperture. For accuracy, the same restrictions noted in reference to the quarter-wave geometry also apply here.20
As can be seen in Figures 4 and 5, there is a small difference between the partition in a quarter-wave cavity and that of a full-wave cavity. Provided the reed tips are mounted as shown, it is a simple matter to physically change a quarter-wave cavity to a full-wave cavity, though the full-wave cavity must be excited at a frequency four times that of the quarter-wave cavity.
In looking at the Helmholtz, quarter-wave, and full-wave geometries depicted in Figures 1, 2, and 5, one might ask: what’s the difference? The difference is the magnitude of the wavelength, o, which corresponds to the frequency of the partial contained in the pressure waveform produced by the vibrating tongue that is being investigated. In general, the same geometry can behave like a Helmholtz resonator at one wavelength, like a quarter-wave resonator at another, and like a full-wave resonator at still another, provided the quarter-wave tube has one end open and the other closed, and the full-wave tube has both ends open.
Complications
The resonant geometries and corresponding equations for resonant frequency and resonant cavity lengths for the Helmholtz, quarter-wave, and full-wave geometries are only models, and inaccuracies can be expected with comparison to the real world. Some sources of inaccuracy have been pointed out, especially those associated with an estimate of the effective mass (value for k), the assumed comparative sizes among cavity dimensions, and the comparison of these dimensions with the wavelength of oscillation. In practice, such limitations are usually stretched to the limit, and often beyond, in order to utilize such expressions as experimental guidelines. With the quarter-wave and full-wave models, we should also mention that air motion through the reed vent (slot) is not entirely concentrated at the end of the cavity; the vibrating tongue moves through the vent with finite clearance, causing some leakage of acoustic energy from the cavity. In addition, the presence of reed parts inside the tube causes changes in cross section that can influence the resonant frequency. Calculations performed according to suggestions here can be effective illustrators of concepts involved, but should, because of inaccuracies, be considered only as starting points for experimentation.
Notes:
All lengths in inches
Note is nomenclature for piano keyboard
Partial is partial number
v is frequency of corresponding partial (Hz) is wavelength of corresponding partial
Vadj is volume adjustment to orthogonal cavity structure, Equation 3 (cubic inches)
W is orthogonal cavity width in Helmholtz model
L is orthogonal cavity length in Helmholtz model
d is aperture diameter
t is aperture length (Action Board thickness)
H is calculated orthogonal cavity height for Helmholtz resonance, Equation 2 & 3, k = 0.6
Smax = 0.15 o is about maximum size of any component for Helmholtz model to remain accurate
Hfixed is cavity height used for tube calculations in next two columns
L-QW is length of cavity for quarter-wave model, Equations 4 & 5, k = 0.6
L/2-FW is length of cavity for partitioned full-wave model, Equations 6 & 7, k = 0.6
BOLD numbers indicate regions on the musical scale where resonance occurs and/or where reed choking may occur (when Partial = 1)
Sample Calculations
Table 1 presents the results of calculations that illustrate how close reasonable cavity dimensions come to resonant geometries. One can study Table 1 and draw conclusions on where along the musical pitch range there is greater or lesser tendency for cavity resonance to approach the frequencies of various partials of the musical tone. Table 1 also gives an idea of how much cavity geometries need to be adjusted in order to arrive at geometries that will resonate at the frequencies of various partials of the musical tone.
In Table 1, Column Note shows the musical note, with nomenclature based on the 88-key piano. As can be seen, the calculations represent the musical range of bass (G1 to C5), baritone (G2 to C6), treble (G3 to C7), and piccolo (G4 to C8) concertinas. Column Partial shows the partial number of the pressure waveform produced by the tongue vibration, with fundamental taken as 1, first overtone as 2, etc. Column vo gives the frequency corresponding to the overtone, and Column Vadj gives an approximate volume adjustment, accounting for departures from the orthogonal volume calculation (Equation 3). Column o gives the wavelength corresponding to frequency vo , and Columns W, L, d, t, H, and Smax are used in the Helmholtz resonator calculation, listing (orthogonal) cavity width (W), length (L), height (H), and aperture diameter (d). Using Equations 2 and 3, we may calculate the value of (H) from the other parameters (W, L, d, and t), whose values were adjusted until a reasonable value for (H) was obtained. Column Smax calculates the maximum size that any of the previous five parameters can assume, without the simple Helmholtz calculation becoming inaccurate, as discussed in the previous section on Helmholtz Resonators. Column Hfixed gives the cavity height used in the tube calculations in the next two columns. Column L-QW gives the cavity length for quarter-wave tube resonance (Equations 4 and 5), and Column L-FW gives the cavity length of a full-wave resonant tube (Equations 6 and 7).
As an example, consider the first line in the calculation for note G1; this shows that the Helmholtz calculation using the fundamental as the design frequency yields a very large value for cavity height, H (166 inches!), when reasonable values for W, L, d, and t were chosen. Note that with this calculation, the magnitude of H is much larger than the value for Smax, indicating that the Helmholtz model does not apply; however, we can still conclude—and the unduly large value for H indicates—that the resonant Helmholtz geometry is very different from the cavity geometry that would exist in the real world (which would have a value for H around 0.5 inches). Thus there is no chance that a cavity for this reed pitch could resonate with the fundamental of the musical tone. The second calculation for note G1 uses the ninth partial (eighth overtone) as the design frequency, and a smaller value for H is obtained, though still perhaps not practical (1.11 inches). The third calculation, for the 11th partial, does show a realistic value for H, assuming moderate adjustments to other cavity dimensions. Bold numbers here and elsewhere in Table 1 indicate areas of susceptible resonance matching between cavity modes and various musical tone partials. Thus, one might conclude on theoretical grounds that some partial higher than about 11 for this reed pitch may be altered by Helmholtz resonance of the cavity, though it is doubtful that alteration of such a high overtone would be noticeable to a listener. Similar comments apply to the fourth line, which calculates the results for an even higher overtone. Note, however, that the value for Smax in this last calculation is less than the value for L, which indicates that the Helmholtz model is becoming less accurate.
For note G2, one concludes similarly that there is no chance that a concertina will be built wherein the cavity provides Helmholtz resonance for the fundamental at reed pitch G2. As with G1, however, the possibility for such resonance increases as we consider higher partials, and in particular, one can expect that some partial starting with the 6th or 7th might experience such resonance.
Thus, higher pitched reeds have cavities that display tendencies to resonate with decreasing ‘partial number’. For note G3, we find that 4th or 5th partials and higher give realistic values for H, and thus a possibility to encounter Helmholtz resonance in a range of overtones that could become noticeable. For note G4, we find that 3rd partials and higher are candidates for resonance. For these notes, my own experimentation suggests that the affected partials may experience reinforcement (interference) if the Helmholtz frequency is a little above (or below) the partial frequency. Note C5 produces a 3rd partial as a candidate for Helmholtz resonance, though the pertinent wavelength is becoming a bit small and the accuracy of this calculation is becoming compromised (see value for Smax). When we get to notes between C5 and C7 and upward, we see a possibility that the fundamental itself may experience Helmholtz resonance with the cavity. With notes higher than about C7, wavelengths are becoming so small that the Helmholtz model may contain serious errors, as shown by comparative values of Smax. Such errors, however, do not mean that the cavity will not resonate, but only that another model must be applied, and we retain the bold format to indicate the possibility of some sort of resonance with the fundamental.21 In some of these cases, the tube models become applicable, as discussed below.
I mentioned earlier my own experimental results that suggest interference when Helmholtz resonance is about equal to or a little lower than reed-tongue vibration frequency (the fundamental). Table 1 shows that such interference can be expected somewhere between notes C5 and C6. In theory, a simple fix for compromised reed performance would be to alter some key cavity or aperture dimension, according to the resonance formulas presented in this article. For resonance with higher partials, as with notes G3 to G8, my experimentation has shown that serious interference with the self- excitation mechanism appears unlikely, though some weakening of tone is possible when the Helmholtz frequency is close to, or somewhat lower than, the second or third partial frequency. For Helmholtz resonance at frequencies in a moderate range that is a little larger than these partial frequencies, I have observed possible enhancement, suggesting passive filtration by the cavity resonance, as explained in the sections Fundamental, Overtones and Partials. Should a builder choose to exploit any possible enhancement at resonance, Table 1 suggests that notes above approximately G4 would be likely candidates, and that precisely tuned Helmholtz resonators must be especially made for notes C6 and above, because of the danger of interference leading to choking.
style=”font-family: Verdana; color: red;”>We now examine the results associated with quarter-wave tube resonance. style> Column L-QW was calculated using Equations 4 and 5, and with the realistic values in Columns W, L, d, t, and Hfixed. style> The idea here is to compare the numbers in Column L with the numbers in Column L-QW, and quarter-wave resonance is expected for those partials where these numbers are in reasonable agreement. style> Bold numbers again indicate possible resonance areas. style> As in the case of the Helmholtz calculation, there is a general trend, with resonance possibilities occurring for lower partial numbers as the musical pitch increases. style> Thus, the fifteenth partial of note G1, the fourth partial of note G4, the third partial of note C5, the second partial of note C6, and the first partials (fundamentals) of notes C7 and C8 show such behavior. style> In some of these calculations, there is departure from the restrictions placed on Equations 4 and 5, though the trends illustrated here should be still valid.
In general terms, we thus come to a similar conclusion as with the Helmholtz calculation; namely, that the lower-pitched reeds can provide higher partials with frequencies that can match the quarter-wave resonant mode of their cavities, and that, as the pitch of the reed increases, lower-numbered partials can provide such frequencies, until we arrive at the highest-pitched reeds, where the fundamental itself provides such frequencies.
As mentioned earlier, and as shown in Figure 2, concertina cavity designs often situate the free tip of the reed tongue away from the air hole. From Table 1, for notes in the octave about C7, this arrangement invites the possibility of reed choking. As also noted previously, a simple remedy, among others, might be to mount the reed tip at the aperture end of the cavity, which might then result in tone enhancement. (See the section Quarter-Wave Tube Resonator for further explanation of such effects.)
Column L-FW does not normally apply to existing concertinas, since its calculation assumes a partition, with reed orientation shown as in Figure 5. I include it for completeness, and it may be of import to those interested in understanding and exploiting resonance phenomenon. Table 1 shows that only the highest-pitched reeds are expected to show susceptibility for fundamental resonance with full-wave tubes, indicating applicability somewhere in the vicinity of note C8.
One can see that the occurrence of resonant behavior in Table 1 is dependent upon the assumed dimensions of the cavities, and that real concertinas will have other cavity dimensions and other occurrences. It is important to note, however, that the general trend concluded here for both Helmholtz and tube models should be valid for real instruments.
Summary and Conclusions
Reed cavity resonance exhibits a full range of influence on concertina reed performance. In some cases, reed cavities have very little effect, while in other cases, there can be significant effect on the timbre and volume of the musical tone. Finally, resonance effects can cause serious interference with reed tongue vibration and musical tone, particularly for Helmholtz and quarter-wave resonance with high-pitched reeds. The Helmholtz resonator and quarter-wave tube models can explain much of the resonant behavior of reed cavities, and I have presented methods to calculate their resonant geometries. The sample calculations illustrate the range of influence of resonance on bass, baritone, treble, and piccolo instruments. For the lowest range of these instruments, only the higher partials of the musical tone appear open to influence. As one moves up the pitch range of this family of instruments, lower-numbered, more noticeable partials become susceptible to influence from cavity resonance, and for the highest pitches, I suggest resonant interference and reed choking is a danger in some cavity designs. Helmholtz resonance appears to be the more commonly experienced type of resonance, though quarter-wave resonance makes a significant appearance, in a less regular fashion.
In this article, I have presented possible mechanisms for interference and choking, remedies to prevent such behavior, and suggestions on how one might attempt exploitation of resonant effects for improved musical tone. These discussions and suggestions are based, in part, on my limited experimentation on these issues, which I cannot say is universally conclusive. Free-reed operation is a complicated affair, and I hope the discussion here can encourage participation by others.
NOTES
1. See for instance, Gerhard Richter: Akustische Probleme bei Akkordeons und Mundharmonikas, Teil 1: Allgemeine Grundlagen (Kamen, Germany: Karthause-Schmulling, 1985). More recent—and in English—is a series of papers by James P. Cottingham, abstracts of which are published in The Journal of the Acoustical Society of America: ‘Acoustics of American Reed Organs’, 99 (1996), 2461; with Casey A. Fetzer, ‘Modeling Free Reed Behavior using Calculated Reed Admittance’, 102 (1997), 3084; ‘Theoretical and Experimental Investigation of the Air-Driven Free Reed’, 103 (1998), 2835.
2. Care must be taken so that no air jets or concentrations of air streams interfere with the self-excitation mechanism of tongue vibration.
3. Damping (friction) is also present in any real system, though it is not required for resonance. In this article, we neglect the small effect damping has on cavity resonant frequencies.
4. There is a mean (time-average) airflow through the cavity, upon which is superimposed an air vibration with oscillating pressure and velocity. The magnitude of these oscillations depends upon spatial position within and about the cavity.
5. Any concertina player can observe the tone chamber effect by playing the instrument close inside the corner of a room. Sound reflection off the walls in this case produces a tone with timbre noticeably different from that of the tone played out in the open.
6. Free reeds were used in organ pipes, though they were, to my knowledge, discontinued during the 1920’s because their start (attack) transient was regarded as too slow.
7. See Cottingham, ‘Acoustics of a Symmetric Free Reed Coupled to a Pipe Resonator’, abstract in The Journal of the Acoustical Society of America, 107 (2000), 2896. An important difference between Asian and Western free reeds is that the Asian reed operates always as an “opening” reed, whereas the Western free reed operates as a “closing” reed and sometimes as both a “closing” reed and an “opening” reed. For an explanation of this terminology and the implications, see Neville H. Fletcher and Thomas D. Rossing, The Physics of Musical Instruments, 2nd ed. (New York: Springer, 1999), 401, 413.
8. Aldo Mencasini, owner, Bell-Duovox Accordion, West Nyack, New York, private communication.
9. The presence of a mean flow through the reed vent has a small (second order), but sometimes significant, effect on reed performance.
10. Some people are erroneously under the impression that the partials in the musical tone of squeezeboxes are caused by higher resonance modes in the reed tongue itself, in the same way that a vibrating guitar string produces its partials.
11. See Cottingham, C. Joseph Lilly, and Christopher H. Reed, ‘The Motion of Air-Driven Free Reeds’, The Journal of the Acoustical Society of America, 105 (1999), 940.
12. By partition, I mean a lengthwise divider down the middle of the cavity, with only the vicinity of the port hole allowing acoustical communication between the resulting half-size cavities.
13. Named after Hermann Helmholtz (1821-1894), whose Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (Brunswick, 1863; English translation by A. J. Ellis, 1875/reprinted 1956, as On the Sensations of Tone) is one of the classic studies of musical acoustics. As Allan W. Atlas has noted, Ellis himself played the concertina; see ‘Who Bought Concertinas in the Winter of 1851? A Glimpse at the Sales Accounts of Wheatstone & Co.’, Nineteenth-Century British Music Studies, 1, ed. Bennett Zon (Aldershot: Ashgate, 1999), 63-64.
14. The calculation of k is a complex exercise, though it has been done for cases that approximate its application here. The “end correction” kd is necessary to account for the inertia of the air mass that vibrates in the region immediately outside the air hole, and the extent of this mass depends upon the size of the hole with respect to the wavelength and the proximity of the pallet to the hole. For instruments where the pallet remains closer to the hole than stated, even larger values of k should be used. Such an effect is used in some marimbas, as a way of tuning the associated quarter-wave tube resonator. Note also that radiation loss from the cavity should be small when the stated restrictions are satisfied and is thus neglected in these calculations.
15. The main reason for this requirement is to ensure that the inherent assumptions of one- dimensional flow inside the tube are adhered to. For more rigorous conformity between this quarter-wave model and practice, one can place partitions down the center of the cavity, effectively doubling the ratio L/W, as discussed elsewhere in the text. Since the transverse dimensions are assumed much less than a quarter wavelength, waveform variations in the transverse direction are negligible, and the cross-sectional shape of the tube is not important, unless higher-order effects, such as wall friction, are included.
16. Since the clarinet is basically a quarter-wave tube with one end open and the other closed, and with the reed placed at the closed end, one might ask why the free reed behaves differently. The answer lies in the nature of the two reeds, as explained previously. The clarinet reed is a pressure-controlling device, whereas the free reed is a flow-controlling device. With the tube in resonance, the boundary condition at the wall requires a velocity node and a pressure maximum. Such a condition is compatible with the clarinet’s beating reed and incompatible with the free reed.
18. The appearance of transverse standing waves should not invalidate the expressions given here; they can only add additional modes of vibration. Because of the geometries involved and the wall (zero velocity) boundary conditions in the transverse direction, only those transverse modes that support partials with frequencies having whole number ratios of the pertinent frequency are likely, and these would most often result in frequencies too high to be of interest. The resonant frequencies of mixed longitudinal/transverse modes, however, require a more complicated analysis.
19. One might be curious about why we skipped half-wave tube resonators, which would be shorter and perhaps more practical than full-wave tubes. The reason is that half-wave tubes require both ends open, and oscillations at one end are 180 degrees out of phase with those at the other end. Thus an arrangement with the vibrating reed tip at both ends, as in Figure 5, could not work, since this arrangement excites the two ends of the tube air with the same phase. Of course, one might separate both ends of the (half-wave and full-wave) tubes and excite only one end, but this would require two pallets connected to the same key, and this arrangement is not considered practical.
20. When the aperture area becomes too small in comparison to the tube cross section, WH, the end conditions are no longer simply ‘open’, but in this case, the tube may still function as a full-wave resonator, since even ‘closed’ conditions allow resonance (see also note 14).
21. For those interested in calculating the ‘Helmholtz’ resonant frequency in these cases, see Fletcher and Rossing, The Physics of Musical Instruments, 227–32, where a discussion is given for cases in which resonator dimensions are comparable to the wavelength in the long direction, L, but still require the transverse dimensions, W and H, to be much less than the wavelength. With these more complicated calculations involving wavelength effects, in which the resonator ceases to behave as the simple resonator, overtones occur, introducing additional possibilities for resonance.
