Vibration characteristics of pipe organ reed tongues and the effect of ...

Vibration characteristics of pipe organ reed tongues and the effect of the shallot, resonator, and reed curvature G. R. Plitnika) Department of Physics, Frostburg State University, Frostburg, Maryland 21545

共Received 12 April 1999; revised 13 January 2000; accepted 1 February 2000兲 Pipe organ reed pipes sound when a fixed-free curved brass reed mounted on a shallot connected to a resonator is forced to vibrate by an impressed static air pressure. Five sets of experiments were performed in order to investigate the influence of the most important parameters which could affect the tone of a reed pipe. First, the phase difference between the pressure variation in the shallot and the boot, and its relationship to the motion of the reed tongue were analyzed to compare their phases and their spectra. Next, the frequency dependence of the reed on three basic parameters 共reed thickness, its vibrating length, and the imposed static air pressure兲 was investigated in an attempt to determine an empirical equation for the frequency. For each trial, two of the variables were kept constant while the third was altered in order to construct an equation giving frequency as a function of the three variables. Third, experiments were conducted using three different types of shallots: the American 共or English兲 style, the French style, and the German style. The results show that for each shallot, the frequency increases linearly with thickness and linearly with air pressure 共over the normal operating range of the reed兲. For each of the shallots, frequency varies inversely with length when the other variables are held constant. The effect on the reed spectrum of using the three different types of shallot was also investigated, as was the effect of reducing the interior volume of each type. Progressively filling the shallot interior generally decreases the frequency of the vibrating reed. The effect of the resonators on frequency and spectrum was studied because the resonator is an integral part of the resulting tone; virtually every reed stop has some type of resonator. The resonator tends to raise the Q of the impedance peaks and reduce the fundamental frequency. Finally, the influence of the type and degree of curvature on reed vibration was briefly examined; increasing the reed curvature tends to decrease the vibration frequency and increase the sound intensity by creating a richer spectrum. © 2000 Acoustical Society of America. 关S0001-4966共00兲01505-8兴 PACS numbers: 43.75.Np 关WJS兴

INTRODUCTION

During the 19th century, scientific advances in the rapidly growing field of acoustics began to unveil some of the mysteries of sound production in musical instruments. It became possible, with aid of resonators, to dissect a tonal structure and quantify sounds which previously could only be qualitatively assessed. The mood of the time was to assume that science would set all things in order; thus, when the searchlight of science was focused on pipe organs, it was with the intention of transforming organ building from art to science so that empirical rules could be supported by theoretically derived equations. Arguably, organ reed pipes were first subjected to scientific analysis by the German experimental physicist Wilhelm Weber in 1830; this research guided him to an essentially correct theory for the interaction between a vibrating reed and the input impedance of the resonator. The results of Weber’s investigations, as well as an analysis of a vibrating reed tongue, are summarized in Smith’s book Sound in the Organ and the Orchestra 共1911兲. During the 19th century very few experimentalists concerned themselves with organ reed pipes, and the limited theory developed was of scant use to a兲

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organ builders 共see Audsley, 1905, Vol. I, pp. 401–402兲, as even a cursory examination of Tyndall and Helmholtz, two of the 19th century’s greatest investigators of musical acoustics, reveals. In The Science of Sound, Tyndall 共1875兲 devotes a mere two pages to organ reed pipes, and even this is marred by several misleading statements which reveal his lack of understanding of organ building practice 共Tyndall, pp. 234–236兲. Helmholtz 共1954兲 fares only slightly better; On the Sensations of Tone devotes only about four pages to organ reed pipes and most of this is devoted to free, rather than striking, reeds. 共Helmholtz mistakenly believed that striking reeds were an historical curiosity which had passed from general use among organ builders; in fact, the reverse was more nearly true.兲 His major contribution to the science of beating reeds was his discussion of the reed-resonator interaction and the formulation of a general theory for determining the phase angle between the pressure maximum and the vibration for inward 共as well as outward兲 striking reeds 共Helmholtz, pp. 390–394兲. When, in 1888, the Rev. Max Allihn revised and extended Topfer’s 1833 tome on the art of organ building 共Topfer, 1939兲, one entire book 共out of seven兲 was devoted to reed pipes 共Allihn, pp. 277–396兲, and no fewer than six pages to Weber’s theory 共Allihn, pp. 326–331兲. Despite this heroic attempt to understand the functioning of reed pipes,

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the great organ expositor Audsley was compelled to conclude 共by quoting Allihn兲 that despite all the mathematical formulas, ‘‘The subject of reed pipes is a question so intricate and obscure that it creates a sensation as of one trying to find a path through a forest with which one is unacquainted, and in which numberless stray paths lead from the right road’’ 共Audsley, 1919, pp. 453–454兲. Audsley thus dismisses 70 years of organ reed research by wryly commenting ‘‘As we are not prepared ... to enter the forest, with its ‘numberless stray paths’ which lead one away to pitfalls of false conclusions and thickets of perplexity, we shall rest content in directing the ardent student to the pages of the works above quoted from, for the special information they afford’’ 共Audsley, 1905, Vol. II, p. 589兲. The most recent incarnation of Topfer, continuing his preoccupation with scaling rules for organ pipes, was the excellent book by Ellerhorst, first published in 1936 共Ellerhorst, 1966兲. Unfortunately, in this 850-page tome a scant 40 pages is devoted to reed pipes, and half of this is descriptive material. The other half attempts to give a scientific understanding for the scaling of reed tongues and resonators. Although replete with equations and formulas, no real advance in the understanding of the physics of organ reeds is presented. The first real 20th century advance in the scientific understanding of vibrating reeds of all types was promulgated by Bouasse, who devoted a large part of Instruments a Vent 共Bouasse, 1929兲 to a comprehensive review and an expansion of Weber’s seminal work on organ reeds, as well as extending it to include the cane reeds of woodwind instruments. Among other results, he reported that the frequency of a reed pipe increased as the blowing pressure was raised. However, since Bouasse was more interested in orchestral reeds than in organ pipes, most of the subtle, but extremely important parameters unique to organ reeds 共reed curvature, type of shallot, etc.兲 are not even mentioned. Several years later, Bonavia-Hunt 共1933兲 identified two dozen parameters which allegedly effect the power and tonality of reed pipes, but no attempt was made to explain the details of exactly how most of these factors influence speech and tone quality. In 1905, Audsley, after explaining the vibrating mechanism or a reed pipe remarked, ‘‘The operation of the mechanism above described appears to be extremely simple, while the acoustical phenomena which attend it have never been satisfactorily explained’’ 共Audsley, 1905, Vol. I, p. 397兲. Bouasse’s work notwithstanding, Audslsey’s comment remained essentially true for more than four decades, until two researchers, Bonavia-Hunt and Homer, undertook the first truly comprehensive study of organ reed pipes. The results of their experiments, in addition to a wealth of information about the voicing of reed pipes, appeared as The Organ Reed, unfortunately now out of print 共Bonavia-Hunt, 1950兲. The entire second half of this work, subtitled ‘‘The Mechanical Properties of Reed Pipes,’’ summarizes the results of an elaborate set of experiments performed to determine how the reed tongue vibrates, how wind pressure influences the frequency, the nature of the air pulses produced, and the equivalent sound waves emitted 共synthesized by means of a mechanical integrator!兲. Based on experimental results, they 3461

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concluded: 共1兲 the reed tongue vibrates in a regular and uniform manner with no high-frequency flexing or rippling; 共2兲 the tongue does not strike the shallot when vibrating; 共3兲 the waveform of the vibrating tongue is not simple harmonic; 共4兲 the center of vibration is not the rest position of the reed, but a position which increasingly approaches the shallot face as the static pressure is increased; 共5兲 the force exerted on the tongue by the impinging wind is uniformly variable over each cycle of vibration 共i.e., from maximum when the reed is closed to minimum when fully open兲 and variable over the length of the reed; 共6兲 the curvature of a well-voiced reed tongue is not parabolic; 共7兲 the rate of change of air flow through the shallot is a function of both the instantaneous position of the tongue and the type of shallot; 共8兲 as the tongue swings outward, the increased aperture tends to compensate for the pressure drop in the boot, thus yielding a constant rate of flow when the reed is more than half open; 共9兲 the air in the boot also vibrates, which has a nonnegligible effect on the sound; and 共10兲 the addition of a resonator suppresses the high dissonant overtones and boosts the lower harmonics. An elaborate set of experiments with a 4-ft conical reed, coordinated and reported upon by Jann and Rensch 共1973兲, investigated the effect of varying the following parameters: resonator diameter, shallot shape, shallot diameter, width and shape of shallot opening, cross section of shallot, width and thickness of the reed, length of the boot, toe hole diameter, resonator material, and wind pressure. Voicing differences were excluded, insofar as possible, by using the same reed tongue whenever feasible and by giving different tongues the same curvature. Although a plethora of data was collected, differences resulting from varying one parameter at a time were described on a continuum of vague psychoacoustic terms such as ‘‘dark to bright,’’ ‘‘round to dry,’’ or ‘‘O-like to E-like.’’ Although sets of spectra were included in an attempt to correlate acoustical data to the above vague terms, the spectra, lacking frequency and amplitude scales, were impossible to interpret, rendering them even more useless than the psychoacoustic terms. Perhaps the kindest words which can be written about this paper is that although the concept was excellent, the realization left much to be desired. In an excellent survey article on the acoustics of reed pipes, Lottermoser 共1983兲 discusses the mechanical properties of the reed tongue, the pressure variation in the boot of a vibrating reed, and the mechanism by which oscillation is sustained. He also covers the important concept of the acoustic input impedance of the resonator and how the shallot and tongue assembly interact with the resonator as the tongue length is changed. Unfortunately, this is merely a survey article which does not pretend to advance the scientific understanding of the organ’s lingual pipes. In another survey article concerned mostly with wind systems and organ flue pipes, Angster et al. 共1997兲 briefly examined the attack of each partial for two reed pipes, but the results were considerably less interesting than for flue pipes. Hirschberg et al. 共1990兲 attempted to correlate measurements on an organ reed pipe to a quasistationary flow model based on a two-dimensional Borda tube. Although the theory G. R. Plitnik: Vibration of pipe organ reed tongues

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was necessarily somewhat simplistic, it did give fairly good agreement to the experimental results as well as provide tentative explanations for much of the observed behavior. Of particular interest was their careful documentation of the approximately linear increase of frequency with blowing pressure for a reed without its attached resonator. Although discrete jumps were observed when the resonator was added, over most of the range examined the frequency dependence on pressure was unchanged by the presence of a resonator, thus indicating that the pressure-frequency dependence is not primarily controlled by the acoustics of the resonator. No attempt was made to examine the influence of the important parameters of shallot type, reed thickness, or reed curvature on frequency or tonal quality. Although almost a century has passed since these words were written, Audsley’s statement still rings true. ‘‘It is quite evident that both the dimensions and forms of the tubes or resonators have a strong modifying influence on the sounds produced by the associated reeds, imparting marked varieties of tonal-coloring and distinctive timbres to the resultant tones; but exactly in what manner, or by what peculiar modes of vibration of the air columns in the resonators, the different tones are produced remains a mystery to the most painstaking investigators’’ 共Audsley, 1905, Vol. I, p. 404兲. Perhaps the present state of affairs regarding the scientific understanding of organ reeds and its potential use to organ builders was best summarized by a well-known organ expositor thus: ‘‘We know a great deal about the laws that govern vibrating metal tongues; we are also well-informed about resonance in tubes, etc. But concerning the interaction of all these details, vast numbers of additional factors assert themselves, and the theory becomes hopelessly inadequate for any practical application. ... ‘With an experience coefficient, the formula might have been corrected and put to some use, if it was not already so complicated that it was unsuitable for any practical purpose’ ’’ 共Andersen, 1969兲. The present work is an attempt to rectify this situation somewhat by methodically investigating the effect of varying the most important parameters determining lingual pipe organ tone. Hopefully, after the basic science of reed pipes is understood, practical applications will ensue.

FIG. 1. Structure and components of the reed block assembly.

consists of a curved piece of brass, called the reed, which beats against the shallot at a frequency determined by the thickness of the reed, the length of its vibrating portion, the air pressure introduced into the boot, and to some extent, the reflections back from the resonator. The shallot is a flat-faced hollow tube, closed at the lower end, upon which the reed rests at a length determined by the tuning wire. When the pipe is activated, air rushes into the boot and the positive air pressure forces the reed to temporarily close against the shallot, thus reducing the air flow. The Bernoulli effect of the air flowing through the narrow opening also helps to bring the reed to closure. After the reed has been blown closed, elastic forces, due to its displacement from equilibrium, spring it back open. A returning pulse of air from the resonator may assist in this process, but it is not a necessary condition.1 The opening and closing reed interrupts the air stream in a regular manner, thus producing the vibration which is the sounding frequency of the reed pipe. The puffs of air emerging

I. STRUCTURE AND FUNCTION

The structure of an organ reed 共or lingual兲 pipe is shown in Fig. 1. The main parts are the boot, the block, the shallot, the brass reed 共or tongue兲, the tuning wire, a short socket to hold the resonator, and the resonator 共not shown in this figure兲. Organ reed pipes, like organ flute pipes, create a tone in a vibrating air column. However, reed pipes differ from organ flue pipes in several important ways: 共1兲 a vibrating brass reed modulates the airstream, 共2兲 the energy is applied at an effectively closed end of the tube, 共3兲 all the air passes through the pipe, and 共4兲 although the reed’s vibration is almost independent of the resonator, standing waves in the resonator can exert a non-negligible influence on the vibration frequency of the reed 共Strong and Plitnik, pp. 319–320兲; the loose coupling between reed and resonator allows the pipe to sound over a considerable range of frequencies. The sound-producing portion of the pipe 共see Fig. 2兲 3462


FIG. 2. Detailed view of the shallot and reed. L is the vibrating length of the reed determined by adjusting the tuning wire. G. R. Plitnik: Vibration of pipe organ reed tongues

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from the shallot into the resonator set up a standing wave in the resonator. Cylindrical resonators support mostly odd harmonics, while conical resonators tend to support all harmonics. Although the natural frequencies of the vibrating reed tongue are generally set to agree with a resonant frequency of the resonator, there is considerable latitude. Also, as the tongue is lengthened, it not only vibrates at a lower frequency but with a greater amplitude, thus changing the harmonic structure of the sound. Part of the art of reed voicing is to adjust every note of a set of reed pipes to have a very similar, or only slowly varying over the compass, tone color. When the reed tongue is lengthened, the resonator must be shortened in order to adjust the frequency if the new tone color is that which is desired. If one wishes to change the loudness without modifying the tone color, one must curve the reed differently, a job which requires considerable skill. Reed pipes are very sensitive to a variety of factors which affect the final tonal quality. Some factors under the control of the voicer which effect tonal quality are: the thickness of the reed tongue, whether there is a weight at the vibrating end, the type and position of the shallot opening, whether the shallot is filled with wax or solder, the diameter of the shallot base, and the type of curve on the tongue. A clarinet stop requires a long gradual curve on the reed, while a trumpet uses a somewhat thinner reed with a curve accelerated toward the end. A posthorn 共or state trumpet兲, usually the loudest stop found on large organs, uses a thin reed curved very high, with a bit of extra curve at the very end to give the sound ‘‘crack and power.’’ It may be useful to classify organ reeds into categories because the properties of the resonators, as well as the manner in which the reeds are voiced, are quite different. One can think of reed pipes as falling into two general families termed classic and romantic, each of which may each be divided into two subgroups. The classic reeds include chorus reeds, used for ensemble playing or as a bright solo voice, and the regals. The chorus reeds serve for pipe organs the same purpose contributed by brass instruments to the orchestra. Resonators are usually a single or a double cone and are typically full length or half length, but occasionally doublelength resonators are used in the upper octaves to strengthen the tone. The regal family of reeds, typically found only on larger pipe organs as a supplement to the chorus reeds, are characterized by having fractional length resonators 共1/8 or 1/16 the fundamental wavelength兲 yielding a tone considerably softer than the chorus reeds, lacking fundamental, and having many overtones. The short resonators cannot support wavelengths of the lower harmonics; they serve rather to impose a formant-like envelope to the rich spectrum. The romantic reeds, developed in the late 19th and early 20th centuries when organ transcriptions of orchestral music were in fashion, attempted to imitate the orchestral reed voices. Stops developed during this time 共such as the French horn, orchestral oboe, bassoon, clarinet, and saxophone兲 actually endeavored to imitate the orchestral brass and woodwind instruments. The romantic reeds may also be divided into two categories, the solo reeds 共orchestral imitators兲 and special-effect reeds. The special-effect reeds are to the ro3463


FIG. 3. An assortment of reed resonators 共from Strong and Plitnik, p. 323兲 where a, b, c, and d are classical chorus reeds; e and f are regals; g and h romantic reeds; and i and j special-effect reeds.

mantic organ what the regals are to the baroque organ. Just as the regals may be used to add tonal color to baroque flue stops, the special effect reeds are employed to add unique and interesting timbres to romantic flues by employing fractional length resonators. The two best-known examples are the vox humana 共as voiced for romantic organs兲 and the kinura. Figure 3 presents a diagram of the resonator shapes typical of the various tonal families. Although many shapes and sizes of shallots have been employed in lingual pipes, there are only three major shallot styles typically employed for chorus reeds, the most common reed stops utilized on classical pipe organs. The three different types of shallots, the only types investigated in this paper, termed the American 共or English兲, the French, and the German, are diagramed in Fig. 4. The American shallot is tapered with a triangular opening, the French shallot is narrow with parallel sides and a parallel opening, while the German style is wider with parallel sides and a large triangular opening.

FIG. 4. The faces of the three types of shallots studied. G. R. Plitnik: Vibration of pipe organ reed tongues

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II. MEASUREMENTS

Of the two dozen parameters which could affect the tone of a reed pipe, five areas 共considered to be the most influential兲 were investigated. First, the phase relationships among the pressure variation in the shallot and the boot, and their relationship to the motion of the reed tongue was examined. These pressure variations are due to the motion of the reed, but are found on opposite sides of the air flow path. The pressure variations and the reed vibration were also analyzed to compare their phase and their spectra. Next, experiments were performed to determine an empirical equation for the frequency of the reed as a function of three basic parameters: the reed thickness, the reed length, and the static air pressure in the boot. An attempt was made to find an empirical equation giving frequency as a function of these three variables, i.e., F⫽ f (T,L, P). Third, the effect on the reed spectrum of using the three different types of shallot was investigated, as was the effect of reducing the interior volume of each type of shallot by progressively filling their interiors with wax. Fourth, the effect on the spectra of adding different types of resonators, both full length and fractional length, to the reed block was studied. Finally, the effect of different reed curvatures was briefly examined. Two Radio Shack wide-range response electret condenser microphones 共270-092B兲 were used to obtain the pressure variations within the boot and the shallot, respectively. To obtain readings with a minimal disturbance to the signals, the microphones were placed firmly into one end of a small section of rubber tubing, while a tapered cork was placed in the other end such that the cork was against the microphone face. A piece of hollow wire insulation was then pushed into a small hole drilled through the cork. The resonant frequency of this tube is about 3000 Hz, well above the highest frequency of interest for this study. Reducing the input area to the microphone also helped to prevent overloading or clipping of the signals. One microphone 共mic 1兲 was placed in a small hole drilled in the boot, while the other 共mic 2兲 had the insulation tube extend through a hole in the boot through a second hole in the back of the shallot opposite the reed, as shown in Fig. 5. The signals from each microphone were simultaneously recorded on a Tektronix 2232 100-MHz digital storage oscilloscope, so that phase relationships could be analyzed. To record the effects of the shallot, reed curvature, and resonator, the microphone was carefully positioned at the sound output from the block. The accelerometer measurements were performed on an Endevco model 122 Picomin piezoresistive accelerometer attached to the vibrating end of the reed with plastic cement. The lead wires were fed through the boot’s foot hole to an Endevco model 2775A signal conditioner for processing the signal before it was sent to the oscilloscope. The static pressure in the boot was measured through a small access hole with a water manometer. This device could measure only the static or average pressure 共in inches of water2兲 during reed vibration as the pressure variations detected by the microphone were too small and too rapid. To operate the reed pipe, tubing from an air valve was fed into the foot hole of the boot and the desired operating pressure was set on the manometer. 3464


FIG. 5. The experimental setup. Mic-1 records the signal in the boot, mic-2 recores the signal in the shallot, and the accelerometer the vibration of the reed tongue.

III. RESULTS A. Phase relations

Phase differences between the accelerometer and the two microphones 共one inside the shallot and one in the boot outside the shallot兲 were first investigated. For any one trial, either an accelerometer reading and the output from one microphone, or the output from both microphones were recorded simultaneously on the dual trace oscilloscope. Only qualitative measurements of phase were recorded. The frequency of the air vibration measured in the boot was identical with that measured in the shallot, but with the expected phase shift of 90 deg 共this was also confirmed by creating a Lissajous figure with the two signals兲. Accelerometer measurements also yielded the same frequency, and the acceleration showed no phase shift relative to the air vibration in the boot recorded by microphone mic-1 for lower static pressures. However, at the higher operating frequencies caused by increasing the driving pressures, the acceleration lagged the boot pressure by approximately 90 deg 共see Fig. 6兲. Fourier analysis of the waveforms from mic-1, mic-2, and the accelerometer verified that there were no substantive differences in fundamental frequency. Therefore, all further signal analyses were made using the input of microphone mic-1 only. B. Effect of reed thickness, pressure, and reed length

To investigate the frequency dependence of each of these three variables, two variables were held constant while the third was varied. The resulting waveform was recorded so that frequency and spectrum could be recorded. To study the relationship between reed thickness and frequency, a set of eight identically voiced reeds having the same length and width, but varying in thickness, was employed. Only the American shallot was used in tests of dependence of freG. R. Plitnik: Vibration of pipe organ reed tongues

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FIG. 6. 共a兲 Pressure waveform in boot. 共b兲 Acceleration of reed tip.

quency on reed thickness; for each trial, the tuning wire was set at a fixed position and the pressure was adjusted to identical values. After the trials were complete, the thickness was measured at several points by a micrometer and averaged. Next, the dependence of frequency on pressure and length for each of the three different types of shallots was examined. To determine the pressure variation, one particular reed was used and the tuning wire was set at a standard position for an average operating pressure 共which yielded a sound of good quality兲. The pressure was then varied from the lowest pressure at which the reed would sound to the maximum pressure 共when the pressure is too high, the reed is permanently blown against the shallot face and no vibration ensues兲. To study the frequency dependence on length, the tuning wire was used to change the length of a particular reed at a standard operating pressure. The length was changed from a position where the tuning wire is against the wooden wedge 共which sounds approximately one octave below the operating frequency兲 to the shortest length which would still support vibration. Figure 7 presents the results of determining the frequency dependence on reed thickness. The graph combines results of six different trials; for each trial the length of vi3465


brating reed and the boot pressure were held constant. Data were collected at two different lengths and at three different pressures for each length. The lower group of data is for L ⫽2.96 cm and the upper for L⫽2.11 cm. The three different points in each group correspond to three different pressures

FIG. 7. Frequency versus reed thickness for three different pressures 共3, 4, 5 in.兲 with two lengths of reed. The lower set of data points is for L ⫽2.96 cm and the upper for L⫽2.11 cm. G. R. Plitnik: Vibration of pipe organ reed tongues

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FIG. 8. Frequency versus pressure for the American shallot. Length #1 ⫽3.50 cm 共lowest curve兲, length #2⫽2.60 cm 共middle curve兲, and length #3⫽1.98 cm 共top curve兲.

used, P⫽3, 4, and 5 in., respectively. Since the three different pressures did not change the frequency very much, the three points corresponding to each thickness were averaged to obtain one representative frequency. These data were then fit by least-squares analysis to a linear curve which is superimposed on the data of Fig. 7. The linear graph fit the average data quite well, indicating that the frequency is directly proportional to thickness, as is also true for a freely vibrating cantilevered beam. In determining the frequency dependence of the reed on varying static pressure, three separate types of shallot, the American, the German, and the French, were used. For each shallot, the same reed was employed, and measurements were made with the tuning wire set at the same three lengths, which were 3.50, 2.60, and 1.98 cm, respectively. For each length, the pressure was varied from the lowest possible pressure at which the reed sounded 共about 2 in. of water pressure兲 to the maximum achievable 共about 10 in.兲. This is shown for the American-style shallot in Fig. 8 for the three lengths of reed. In order to fit a linear curve to this data, it was necessary to delete the lowest and highest pressure points. This is not unreasonable because these points are far removed from the normal operating range of this reed. The area of confinement, 3 to 7 in. of water pressure, is the normal operating range of this organ reed; outside this range, it is not even certain that the reed is functioning in the same mode of operation. Thus, linear fits were used over the normal operating range of the reed, where it appears that the sounding frequency is directly proportional to the static pressure. Figure 9 shows that similar results are obtained for the French and German shallots, both reeds having been set to the same length of 2.60 cm. For a freely vibrating cantilevered beam, the fundamental frequency of oscillation is inversely proportional to the inverse length squared. The data from this experiment indicate that this is not the case for the driven organ reed, as can be seen in Fig. 10 where frequency versus inverse length is plotted for each of the three shallot types. A linear curve fits the data nicely for these cases 共and for a range of other pressures兲, indicating that the frequency of an organ reed is inversely proportional to the vibrating length for each shallot over the range of lengths and pressures tested. Combining the above relationships and assuming that for organ reeds the frequency F is given by F⫽C * 共 T * P 兲 /L, 3466


共1兲

FIG. 9. Frequency versus pressure for the French shallot 共upper curve兲 and the German shallot 共lower curve兲. L⫽2.60 cm in both cases.

where C is a dimensional constant, T is the thickness of the reed, and L its length, and using data derived from Figs. 7–10, C may be determined by any one of the following relationships, written as an empirical equation of the form: C⫽ 共 F/T 兲 / 共 P/L 兲 ⫽ 共 F/ P 兲 / 共 T/L 兲 ⫽ 共 F/ 共 1/L 兲兲 / 共 T * P 兲 .

共2兲

Note that the numerator of each ratio is simply one of the slopes from the linear curve of best fit and that the denominator is a product or quotient of the appropriate constant parameters for the trial from which the corresponding slope was obtained. In order to reduce the possible number of variables 共such as shallot shape and design兲, C was calculated using only trials involving the American shallot. Unfortunately, the attempt to obtain a simple empirical equation for the operating frequency as a function of static pressure, thickness of the reed, and vibrating length of the reed was not entirely successful. Using Eq. 共2兲, the values obtained for the ‘‘constant’’ C varied by as much as a factor of 10. Although shallot variations were avoided, there must be other important variables inadvertently ignored, such as the exact degree of curvature of the reed or the elastic properties of the brass used. An exact equation could not be obtained, but the attempt has led to a number of other interesting conclusions about the vibrating reed. First, the frequency dependence on static pressure is independent of, or at least not strongly affected by, the vibrating length of reed chosen to perform the measurements. This may be easily seen by considering the slopes of the three lines fitting the F vs P

FIG. 10. Frequency versus inverse length; American, French, and German shallots. G. R. Plitnik: Vibration of pipe organ reed tongues

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data of Fig. 8. The numerical values of the slopes are 12.6, 11.7, and 11.5 Hz/in., respectively. While the pressure dependence of frequency appears relatively unaffected by length, it does seem to be strongly influenced by shallot type. The slopes of the F vs P plots for the French and German shallots were 9.03 and 4.64 Hz/in., respectively. This indicates that the type of shallot 共which influences the flow dynamics of the air兲 has a substantial effect on the manner in which pressure influences frequency. Next, it was shown that the operating frequency of a reed is linearly proportional to the inverse vibrating length for all types of shallots and within the normal operating pressures of the reed. This property has long been known to reed voicers as the rule of thumb that the pitch produced by a reed pipe when the tuning wire is completely extended 共against the wedge兲 is approximately one octave below the desired pitch. To bring the pipe to its correct pitch the tuning wire must be pushed about halfway down the exposed length of reed, thus halving the length of reed doubles the frequency. An examination of the slopes of F versus 1/L indicates that the nature of this dependence is not strongly affected by the shallot type. Slopes for the three types of shallot varied only by a factor of about 1.3. The dependence of frequency on thickness was the relationship having the greatest degree of uncertainly, due to the fact that in order to vary thickness different reed tongues had to be inserted. Although the professional reed voicer who curved the reeds had been instructed to voice these to give as identical a sound as possible, achieved by varying the curvature 共i.e., thinner reeds had to be curved more so they would operate correctly at the chosen pressure兲, different curvatures introduce another possible parameter 共which will be considered later兲. C. Effect of shallot’s interior geometry on spectrum

The shallot’s interior geometry was investigated because, although not widely known outside the world of organ reed voicers, it has a profound effect upon the reed’s tone 共and spectrum兲. Changing the interior structure by filling in the shallot with solder or hard wax is a trade secret which has long been used by reed voicers to brighten the tone of a rank of pipes. Because changes in the interior geometry change the magnitudes of the forces acting on the reed, both fundamental frequency and the spectrum change. One reed length on each of the three shallots was chosen for this investigation. Each shallot was investigated by measuring the fundamental frequency and the spectrum as it was progressively filled with wax. The static boot pressure was maintained at 4 in. and for each measurement the length of the reed was adjusted to exactly the same set of lengths as progressively larger portions of wax were added to the interior of the shallot. To insure that the thickness, curvature, and shape were identical, only one reed was used for all measurements in this investigation; its length was varied over five different values, the shortest of which was between 1.5 to 2.5 cm from the reed tip 共depending on the shallot type兲. Reed-length adjustments were made consistent by measuring the longest reed length possible inside the boot, and then measuring its corresponding value on the tuning 3467


FIG. 11. Wax fills #1 through #8 for shallots. Fill #1 is the least wax added; the amount of wax increases by a constant mass until fill #8 共the greatest amount of wax兲 is achieved.

wire outside the boot. This enabled the exact determination of the vibrating length of reed inside the boot by measuring the protruding portion of the tuning wire with a Vernier caliper. For this set of measurements the sound output from the socket was picked up by a microphone and input to the digital storage oscilloscope. The stored data was then Fourier transformed to produce a spectrum. The spectrum was used to compare the effects of changes to the interiors of each of the three types of shallot. After obtaining spectra for five different reed lengths for each unfilled shallot, the shallots were filled with approximately 0.065 grams of wax 共having a density of 0.878 g/cc兲. After a small ball of wax was inserted into each shallot, the shallots were heated until the wax melted. The shallots were cooled at a slight tilt of 10° so that the wax was concentrated at the base end of the shallot 共where the face opening is largest; see Fig. 11兲. The same five lengths of reed were used to obtain spectra. Then, another 0.065 g of wax was added to each shallot and the process was repeated. This was done for a total of eight fills, ending with about 0.52 g of wax in each shallot. Table I shows the volume changes, due to the addition of the wax, for the American, French, and German shallots, respectively. Table II presents the fundamental frequencies for five different reed lengths for the unfilled shallot and fills #1 through #8 for each type of shallot. The lengths in a region where the pipe would typically be tuned are 2.86 cm for the German shallot, 2.92 cm for the French shallot, and 2.39 cm for the American shallot. 共These lengths, marked with stars in Table II, are those used for the following discussion.兲 Homer Lewis, a reed voicer with over 50 years of experience, has stated that, for most types of shallot, ‘‘If a small amount of wax is added to the shallot, the pitch will go sharp, but if a larger amount of wax is added, so as to restrict the air flow, then the pitch will go flat’’ 共Lewis, 1996a兲. The present experiments partly confirm this intuition, at least for the American shallot; for the French and German shallots the frequency decreases for all fills. The German shallot exhibits a sudden frequency drop for fill #1; the frequency then increases somewhat to fill #4, after which there is a general decline with successive fills. Table II shows that this trend is typical of all the reed lengths studied for the German shallot. G. R. Plitnik: Vibration of pipe organ reed tongues

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TABLE I. Shallot volume changes. American

French

German

Fill

Wax 共gm兲

V wax 共cc兲

V air 共cc兲

% Wax

V air 共cc兲

% Wax

V air 共cc兲

% Wax

Unfill #1 #2 #3 #4 #5 #6 #7 #8

0 0.065 0.130 0.195 0.260 0.325 0.390 0.455 0.520

0 0.074 0.148 0.222 0.296 0.370 0.444 0.518 0.592

1.58 1.51 1.43 1.36 1.28 1.21 1.14 1.07 0.988

0 4.7 9.4 14.1 18.7 23.4 28.1 32.8 37.5

1.22 1.15 1.07 0.998 0.924 0.850 0.776 0.702 0.628

0 6.1 12.1 18.2 24.3 30.3 36.4 42.5 48.5

1.36 1.29 1.21 1.14 1.06 0.990 0.916 0.842 0.768

0 5.4 10.9 16.3 21.8 27.2 32.6 38.1 43.5

Figure 12 presents the spectral changes with decreasing shallot volume; by plotting the amplitudes of the first five harmonics for the unfilled case, fill #3, fill #5, and fill #8. This figure clearly demonstrates that higher harmonics increase in amplitude relative to the fundamental as the shallot is filled, thus verifying Mr. Lewis’ statement that wax is added to the shallots in order to obtain a brighter tone. As the French shallot was progressively filled with wax it exhibited the same general trend as the German shallot, except that the frequency dropped with the first fill and remained relatively constant for the next several fills, followed by a frequency decrease. Due to the open characteristic of the shallot, successive fills after the first do not seem to affect the fundamental frequency appreciably. Figure 13 shows that the effect on the spectrum of filling the shallot is, again, to increase the upper harmonics relative to the fundamental. By the eighth fill, however, the shallot passageway is so constricted that all the harmonics are somewhat suppressed, giving a spectral envelope similar to that of the unfilled shallot. The American shallot behaves somewhat differently from the German and French shallots as wax is added. The TABLE II. Comparison of results for the filled shallots. Fills consist of approximately 0.065 g of wax added each time to the unfilled shallot. 共* indicates lengths discussed.兲 Fill number—fundamental frequency 共Hz兲 Reed length 共cm兲

Unfilled

#1

#2

#3

#4

#5

#6

#7

#8

American shallot L⫽3.19 L⫽2.79 * L⫽2.39 L⫽1.99 L⫽1.59

219 339 419 558 757

180 258 339 457 696

200 284 399 578 817

179 269 419 588 787

179 309 458 678 897

169 280 409 608 787

180 279 423 607 849

169 269 419 618 877

179 289 449 653 877

French shallot L⫽3.52 L⫽3.22 * L⫽2.92 L⫽2.62 L⫽2.32

200 229 299 359 419

159 190 259 319 397

159 189 260 329 389

150 189 260 418 518

155 170 240 318 428

159 190 269 349 468

159 180 250 329 468

159 169 249 349 488

129 179 229 329 489

German shallot L⫽3.76 L⫽3.46 L⫽3.16 * L⫽2.86 L⫽2.56

189 199 259 299 359

150 169 210 275 336

159 170 229 289 357

139 160 200 289 403

150 160 195 290 406

159 160 189 279 379

150 159 179 269 369

149 159 219 269 398

149 150 179 219 349

frequency first drops, then progressively increases to a value higher than the original frequency, then drops again and levels off at a frequency approximately that of the original unfilled case. This confirms Lewis’ 共1996a兲 prediction, based on his extensive experience with American shallots, that a moderate amount of wax causes the frequency to go sharp 共fill #4兲, while larger amounts will cause a frequency decrease. Figure 14 shows that the amplitude of the fourth harmonic increases relative to the fundamental up through fill #3; further increasing the wax in the shallot progressively lowers the relative amplitude of the fourth harmonic until fill #8, where the relative amplitude of harmonic four is about the same as for the unfilled case. The observed trends exhibited by these experiments can perhaps be explained by considering the traction forces on the reed. It is hypothesized that as the air flows through the shallot, the ‘‘pocket’’ formed at its base induces viscous losses due to the formation of eddy currents, reducing the energy available to the higher harmonics. As this pocket is progressively filled with wax, the flow is guided more smoothly through the channel, there are fewer viscous losses, and more energy is distributed to the higher harmonics, resulting in a brighter tone. The greater the boot pressure relative to the shallot pressure, the more rapidly the air will travel through the shallot opening, which increases this effect and causes the reed to vibrate more rapidly. With the first addition of wax the boot pressure, when the reed is at its maximum extension above the shallot, decreases because of the increased air flow through the shallot. The lower pressure

FIG. 12. Harmonic amplitude changes for different fills, German shallot. 3468


G. R. Plitnik: Vibration of pipe organ reed tongues

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TABLE III. Comparison of results for the resonators. American shallot Reed length

FIG. 13. Harmonic amplitude changes for different fills, French shallot.

in the boot means there is less force on the reed, causing the contact point of the reed on the shallot to move toward the tuning wire, thus lowering the vibration frequency. As the wax progressively fills the shallot base, reduced viscous losses allow a smoother and more rapid air flow which reduces the shallot pressure, thus effectively increasing the force on the reed which results in progressively higher frequencies of vibration. As more wax is added, the effect continues until some maximum point is reached 共before the shallot is half filled with wax兲. Further filling of the shallot constricts the air flow, which causes the frequency to decrease, with an attendant reduction in the intensity of the sound. The larger base of the American shallot 共0.3 cm larger where the wax is added兲 suggests that these effects will be less for the American than for the French or German shallots. D. Effect of the resonator

To investigate the effect of the resonator on reed vibration, seven different resonators were successively attached to the socket. The shallot chosen was the American; the reed was the same one used previously. The reed pipe was tuned to middle C 共263 Hz兲 and the output from each resonator was recorded at the open end. The same procedure was also used to record the waveform from each resonator after the shallot had been filled twice with wax 共only amounts of wax corre-

FIG. 14. Harmonic amplitude changes for different fills, American shallot. 3469


Fundamental frequency 共Hz兲 L⫽2.64 cm L⫽2.79 cm L⫽2.66 cm Unfilled Fill #2—0.13 g #8—0.52 g

Apparatus 共middle C—263 Hz兲

259

259

269

Resonator #1—trompette L⫽30.5 cm

259

239

249

Resonator #2—oboe L⫽49.5 cm

239

259

249

Resonator #4—brass trompette L⫽41.5 cm

258

250

249

Resonator #5—orchestral oboe L⫽50.0 cm

219

230

229

Resonator #7—holz regal L⫽18.0 cm

258

259

259

Resonator #8—vox humana L⫽21.5 cm

258

269

259

Resonator #11—fagot L⫽46.3 cm

239

249

249

sponding to fill #2 and fill #8 were applied兲. The resonators used were an assortment of typical resonators for middle C found on pipe organ reeds. The results of this experiment are summarized in two tables. Table III lists the fundamental frequency for three different shallot fills 共unfilled, the #2 typical fill, and the extreme fill #8兲 of the American shallot without resonator 共row 1兲 as well as the respective fundamentals when seven different resonators are attached. Table IV tabulates the relative log amplitudes of the first seven harmonics for each fill combined with each resonator. As the resonators were varied, the shallot fills and reed lengths remain unchanged. Adding a resonator to the boot and socket has two effects on the resulting spectrum. First, the signal-to-noise ratio is substantially increased, yielding well-defined harmonics with relatively high Q as compared to the spectra obtained when no resonator is attached. Second, the fundamental frequency is decreased by the resonator. This effect, which can be easily understood as adding an effective acoustic mass to the system, is well-known to reed voicers 共Lewis, 1996b兲. An examination of the spectra show that the fourth harmonic 共about 1000 Hz兲 is almost always the largest in amplitude. This can, perhaps, be explained by the resonance of the boot. It has long been recognized by reed voicers 共Lewis, 1996b兲 that boot resonance can, especially for short-length resonators, cause problems with tonal stability. If the boot is considered as a closed–closed tube, its resonant frequency would be about 900 Hz, in the same general region of the fourth harmonic of middle C. Each resonator affects the spectrum differently, depending on the resonator’s length, shape, and the nature of the terminal end 共open, partially capped, or closed with a small opening兲. In order to reinforce the reed vibration, the resonator must be either a ‘‘full length’’ 共i.e., equal to one-half the desired wavelength for conical type resonators, or oneG. R. Plitnik: Vibration of pipe organ reed tongues

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TABLE IV. Log amplitudes of harmonics for various fills and resonators. Har.#

boot

res #1

res #2

res #4

1 2 3 4 5 6 7

6.40 4.40 7.20 7.45 5.90 6.00 5.25

2.20 4.95 4.15 6.65 4.70 3.20 1.60

6.80 6.20 6.90 9.20 7.40 4.85 4.00

1 2 3 4 5 6 7

6.40 4.80 5.90 7.20 5.90 6.30 4.80

4.45 5.15 4.60 6.30 5.30 3.45 2.80

5.10 5.20 6.45 7.30 6.00 5.85 3.70

Fill #2 3.30 4.00 5.60 6.30 4.90 4.10 3.75

1 2 3 4 5 6 7

5.15 4.30 6.15 7.10 5.90 5.90 5.60

2.05 4.45 4.40 5.60 4.85 3.20 2.70

4.35 4.05 4.80 6.20 5.00 4.65 3.70

Fill #8 2.50 3.40 5.20 5.90 4.90 3.70 3.33

res #5

res #7

res #8

res #11

6.20 6.55 6.30 7.00 6.50 4.90 4.90

6.80 6.60 6.00 8.20 6.00 4.85 4.30

3.50 4.05 5.30 6.50 5.25 3.70 ¯

4.35 4.15 4.60 6.55 6.55 3.20 3.20

5.30 5.35 5.25 5.90 6.10 3.35 3.80

6.10 5.90 5.40 7.85 7.00 6.25 3.20

5.90 5.55 6.85 7.80 6.70 6.15 4.70

3.00 4.05 4.85 6.60 6.30 4.50 3.20

5.70 6.10 5.55 6.95 6.90 5.35 4.80

5.90 7.25 6.05 7.85 8.00 6.30 4.10

5.60 5.10 6.30 7.00 5.40 5.55 4.10

Unfilled shallot 2.70 3.40 3.90 3.10 5.00 3.70 5.90 5.10 4.70 4.10 3.40 2.60 3.40 1.80

quarter the wavelength for cylindrical兲 or a ‘‘half length’’ 共one-quarter the desired wavelength for conical type resonators兲. Many other types and resonators have been employed for regals and special effect reeds; these resonators are typically a relatively small fraction of the wavelength 共1/8 or 1/16 are not uncommon兲. For the full or half-length resonator, its length is adjusted, depending on geometry, to match its first and/or second resonance to the desired frequency. The very short resonators act more like a high-pass filter, reinforcing the higher frequencies and attenuating the lower frequencies; feedback from the resonator has little effect upon the reed for these short resonators 共Strong and Plitnik, 1992, p. 327兲. Most of the resonators tested in this portion of the experiment were designed to be used on a pipe sounding middle C 共263 Hz兲. The trompette resonator 关an inverted cone as in Fig. 3共b兲兴 was tested with the American shallot in three conditions 共unfilled, fill #2, and fill #8兲. As already discussed, filling in the shallot lowered the frequency and increased the upper harmonics of the spectrum. For this configuration, fill #2 seemed to be optimum; it brought the lower and upper partials more in line with the strong fourth harmonic, thus giving a much more uniform spectral envelope and increasing the Q of most harmonics. Fill #8, being somewhat excessive, i.e., more than would typically be used for this stop, caused a deterioration of the uniform spectrum by reducing the upper and lower partials relative to the fourth, although their Q values were sharpened somewhat. The effect on the spectra for resonator #2, a typical oboe 关double cone as in Fig. 3共a兲兴 indicates that, in a similar manner, progressively filling the shallot boosts the upper partials, evens out the spectral envelope, and increases the Q. The effect on resonator #4, a brass trumpet 共a large cone 3470


FIG. 15. Changes in the spectral envelope for the orchestral oboe with three different shallot fills. The solid line is the unfilled case, the heavy line is for fill #2, and the dashed line is for fill #8.

with an exponentially flared bell兲, of filling the shallots is that there are only small changes in the fundamental frequency, but considerable change in the spectral envelopes. The lower harmonics are attenuated and the Q becomes higher as the volume of the shallot decreases. The optimum fill seems to be fill #2; after this the fundamental becomes increasing reduced until it is almost missing for fill #8. The effect of resonator #5, orchestral oboe 关a narrow capped cone as in Fig. 3共h兲兴, is for the filled shallots to lower the vibration frequency, to increase the Qs, attenuate the lower partials, and increase the upper. This resonator probably has more effect on the spectrum because it is quite narrow, the only opening being a small slot near the capped top. Adding the resonator decreased the frequency by about 40 Hz, independent of the amount of shallot fill. Figure 15 shows the spectral envelopes for this resonator with three different shallot fills. It can be seen that fill #2 smoothes the spectrum, raises the amplitude of all higher partials, but with a greater relative increase to the higher frequencies. The overall higher amplitude created with fill #2 is most likely due to the reduced shallot interior volume being a better match to the small resonator employed for this pipe. Typically, a smaller diameter shallot would be used. Figure 15 also shows that there is almost no difference between the spectral envelope of fill #2 and fill #8, probably because all flow constrictions in the shallot produce a fairly good match to the small diameter resonator. The effect of resonator #7, holz regal 共a fractional length wooden resonator of square cross section兲, was that adding the resonator did not lower the vibration frequency, except for fill #8. For this resonator, the sharpest Qs occurred for the unfilled shallot; additional filling did not much affect the spectral envelope, but it did reduce the Q values. Although this resonator maintains a constant cross-sectional area over most of its length, all harmonics are present in the spectrum, indicating that the resonator is not controlling the reed so much as acting as a filter with minimal feedback. This is also G. R. Plitnik: Vibration of pipe organ reed tongues

3470

TABLE V. Analysis of reed curvatures. German shallot—reed length 2.5 cm.

FIG. 16. Changes in the spectral envelope for the fagotto with three different shallot fills. The solid line is the unfilled case, the heavy line is for fill #2, and the dashed line is for fill #8.

supported by the observation that adding the resonator did not particularly affect the vibration frequency for most of the filled shallots. The effect of resonator #8, vox humana 关a fractional length capped cylinder as shown in Fig. 3共i兲兴, is to increase the amplitude of the upper harmonics and to increase the Qs as the shallot is progressively filled. The fundamental frequency does not change when the resonator is added 共except for fill #8兲, indicating that this fractional length resonator acts as a filter which exerts very little control on the vibrating reed. The effect on resonator #11, fagotto 关a narrow double cone as in Fig. 3共a兲兴 is exactly that expected for a full-length, conical-type resonator. Adding the resonator lowers the frequency of the fundamental, while progressively filling the shallot sharpens the Q values. In this case, the optimum fill seems to be fill #2 since it has the most uniform spectral envelope, the best developed harmonics, and harmonics with the greatest amplitude, as can be seen by inspecting Fig. 16. Fill #8 gives a fairly uniform, but somewhat attenuated, spectral envelope. The tone produced with this resonator is quite bright due to the prevalence of strong, well-developed upper harmonics.

Curvature number

Amount of curvature 共mm兲

Frequency of fundamental 共Hz兲

#1 #2 #3 #4 #5

1.75 2.54 2.82 3.11 3.73

389 369 319 319 318

creasing the curvature no longer lowers the frequency. This can be explained by considering that as the overall curvature is increased, the effective vibrating length is also increased, i.e., the point of contact between the reed and the shallot moves closer to the tuning wire when the air enters the boot. Likewise, as the curvature is reduced, the point of contact moves away from the tuning wire, thus increasing the vibration frequency. When a large portion of the reed is in contact with the shallot 共low curvature兲, the pressure in the boot will prevent the section of the reed touching the shallot from vibrating, even though the tuning wire does not prevent it. A summary of the change in spectra as the curvature is increased is shown in the spectral envelope of Fig. 18; changing the curvature seems to have only a small effect on the spectrum; the first two harmonics become progressively more attenuated relative to the upper harmonics which are only slightly affected. This agrees with the general consensus that a greater curvature, as would be used for a thinner reed, tends to produce a tone having greater brightness 共see Williams and Owen, 1988兲

IV. CONCLUSIONS

E. Effect of reed curvature on the spectrum

Although the vibration of organ reeds seems to be a much more complicated phenomenon than one may have been led to assume, all overtones were integer multiples of the fundamental and no nonharmonic partials were observed. Concerning the phase difference on opposite sides of the reed, there are two main conclusions. First, there is an approximately 90° phase shift in pressure on opposite sides of the reed, and second, the acceleration of the reed tongue lags

To determine the effect of reed curvature, a typical reed tongue was fitted to the German shallot. The experiment began with the reed having a low initial curvature. After each successive measurement 共to obtain a spectrum兲 a burnishing tool was used to progressively increase the reed’s curvature. For purposes of this experiment, the reed curvature was defined as the height of the reed tip above the shallot, as measured by a micrometer. The results of this experiment are presented in Table V. A professional reed voicer informed us that if the curvature is increased near the tuning wire the pitch will decrease, while increasing the curvature near the tip will increase the pitch 共Lewis, 1996b兲. For this experiment the reed was curved over its entire length; the results, shown in Fig. 17, confirm the voicer’s prediction. The pitch decreases for the first two increases of curvature, then remains constant; in-

FIG. 17. The effect of reed curvature on fundamental frequency. As the curvature increases 共as measured at the tip兲 the frequency decreases.

3471



3471

FIG. 18. Changes in spectral envelope due to different reed curvatures.

the boot pressure by approximately 90° when the system is vibrating at the higher frequencies induced by the higher boot pressures. Fourier analyses of the signals indicated that each signal was composed of one strong fundamental and a number of higher partials. A consideration of the accelerometer signals from different vibrating lengths of reed also indicates that when the length is shorter 共higher frequency兲 the vibration is considerably simpler 共i.e., less energy is distributed to the overtones兲 than for the lower frequencies obtained when the vibrating length of reed is longer. It is hypothesized that this results from the more intense beating of the reed against the shallot, which ‘‘clips’’ the waveform to produce more overtones of greater amplitude. Although the attempt to obtain a simple empirical equation for the operating frequency as a function of static pressure, thickness of the reed, and vibrating length of the reed was not entirely successful, the attempt to find it has revealed several interesting, if not unexpected, relationships. First, the frequency dependence on static pressure is essentially independent of the vibrating length of reed, but strongly influenced by shallot type. Next, it was shown that the operating frequency of a reed is linearly proportional to the inverse vibrating length for all types of shallots and within the normal operating pressures of the reed. Then, it was shown that the frequency of a sounding reed is linearly proportional to the inverse vibrating length for all types of shallots operating within the normal pressure range of the reed. Furthermore, this dependence is not strongly affected by the shallot type. Finally, it was established that frequency is linearly dependent on reed thickness, even though the varying tongue curvature necessitated by this experiment lends a degree of uncertainly to the conclusion. Concerning the filled shallots, there are three main conclusions: 共1兲 filling the shallot with wax causes the frequency to decrease, 共2兲 as more wax is added the frequency increases to values approaching the ‘‘unfilled’’ case, and 共3兲 wax tends to brighten the tone by raising the amplitudes of the higher harmonics relative to the fundamental. Several conclusions can be drawn from the investigation of the resonators. First, the addition of a resonator almost always lowers the fundamental frequency, increases the spectral amplitudes, raises their Q value, and increases the signal to noise ratio of the spectrum. Second, the amplitude of the fourth harmonic of the spectrum is generally the larg3472


est in magnitude, perhaps due to the resonance of the boot reinforcing its amplitude. Third, all the frequencies of vibration decrease, relative to the resonatorless case, when fill #8 共the maximum amount of wax兲 is applied to the American shallot. This may be attributed to the increased effect of the returning air pulses ‘‘focused’’ by the decreasing crosssectional area of the resonator where it enters the block. Fourth, the longer the resonator, the larger the frequency drop as compared to the waveform produced by the reed with no resonator in place. Finally, the fractional length resonators 共holz regal and vox humana兲 have little effect on the reed frequency, but the holz regal had a profound effect on increasing the Q of the harmonics as well as the signal-tonoise ratio. The preliminary conclusions drawn from the analysis of changes to the reed curvature were that as curvature increases, the frequency decreases, up to a point; beyond this there is no further change. Changes to the spectrum are minimal for reeds curved over their entire length, but according to a professional reed voicer 共Clipp, 1998兲, additional curvature applied near the tip of a reed tongue brightens the tone considerably. The all-important aspect of reed curvature is now being investigated in considerably more detail; the results will be reported in a future paper. ACKNOWLEDGMENTS

The experimental work of my students, Ron Knox and Jeff Ritchie, constitute the foundation upon which this work was constructed. A debt of gratitude is owed to the master reed voicers of Trivo, Inc. 共Hagerstown, MD兲, Homer Lewis and Joe Clipp, for their willingness and eagerness in unselfishly donating their time to discuss reed voicing and to provide reed parts as needed. Finally, the important contributions of Dr. Chandra Thamire, Dr. Paul LaChance, and Dr. William J. Strong who read through several proofs of this paper and offered invaluable suggestions, are gratefully acknowledged. 1

The single known exception is the Zajic regal, developed by the late Adolph Zajic who was a master reed voicer at the M. P. Moller Pipe Organ Co. of Hagerstown, MD. This regal had absolutely no resonator; the opening for sound egress was the top of the block. 2 Pressure is measured in inches of water as this has always been, and continues to be, the standard in the American pipe organ industry. Allihn, M. 共1888兲. Die Theorie und Praxis des Orgelbaues 共B. F. Voight, Weimar兲. Andersen, P. G. 共1969兲. Organ Building and Design 共Oxford University Press, New York兲, pp. 55–56. Angster, J., Angster, J., and Miklos, A. 共1997兲. ‘‘Akustische Messungen und Untersuchungen an Orgelpfeifen,’’ Acta Organologica, Band 25, Kassel 共Merseburger兲. Audsley, G. A. 共1905兲. The Art of Organ-Building, Vols. I and II 共Dover, New York兲, 1965 reprint of the 1905 edition. Audsley, G. A. 共1919兲. The Organ of the Twentieth Century 共Dover, New York兲, 1970 reprint of 1919 edition, pp. 453–454. Bonavia-Hunt, N. A. 共1933兲. Modern Studies in Organ Tone 共Office of Musical Opinion, London兲, p. 91. Bonavia-Hunt, N., and Homer, H. W. 共1950兲. The Organ Reed 共J. Fisher, Glen Rock, NJ兲. Bouasse, H. 共1929兲. Instruments a Vent 共Librairie Delagrave, Paris兲, Vol. I, pp 68–70, 103–117. Clipp, Joe. 共1998兲. Personal interview, Trivo Organ Reed Co., Hagerstown, MD, 28 June 1998. G. R. Plitnik: Vibration of pipe organ reed tongues

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Ellerhorst, W. 共1966兲. Handbuch der Orgelkunde 共Frits Knuf, Hilversum兲, pp. 282–323. Helmholtz, H. 共1954兲. On the Sensations of Tone 共Dover, New York兲, reprint of the 1885 edition. Hirschberg, A., van de Laar, R., Marrou-Maurieres, J., Wijnands, A., Dane, H., Kruijswijk, S., and Houtsma, A. 共1990兲. ‘‘A quasi-stationary model of air flow in the reed channel of single-reed woodwind instruments,’’ Acustica 70, 146–154. Jann, G., and Rensch, R. 共1973兲. ‘‘Experiments with Measurements in Reed Pipes,’’ ISO Information #9 共Feb. 1973兲, pp. 633–646. Lewis, Homer 共1996a兲. Personal interview, Trivo Organ Reed Co., Hagerstown, MD, 9 April 1996. Lewis, Homer 共1996b兲. Personal interview, Trivo Organ Reed Co., Hagerstown, MD, 23 April 1996.

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Lottermoser, W. 共1983兲. ‘‘Akustik der Zungenpfeifen,’’ in Die akustischen Grundlagen der Orgel, Vol. I of Orgeln, Kirchen and Akustik 共Bochinsky Frankfurt am Main兲. Smith, Hermann 共1911兲. The Making of Sound in the Organ and in the Orchestra 共Charles Scribner’s Sons, New York兲, pp. 230–245, 249–250. Strong, W., and Plitnik, G. 共1992兲. Music, Speech, Audio 共Soundprint, Provo, UT兲. Topfer, J. G. 共1833兲. Die Orgelbaukunst nach Einer Neuen Theorie 共Hoffman, Weimar兲共edited and revised by Paul Smets, Rheingold, Mainz, 1939兲. Tyndall, J. 共1875兲. The Science of Sound 共Citadel, New York兲, 1964 reprint of the 1875 edition. Williams, P., and Owen, B. 共1988兲. The Organ 共W. W. Norton, New York兲, p. 36.


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