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 Reed Cavity Design and Resonance 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.