WIND INSTRUMENTS

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Modal analysis of a complete clarinet M. Facchinettia,c , X. Boutillonb and A. Constantinescuc a Laboratoire d’Hydrodynamique, CNRS - Ecole Polytechnique, 91128 Palaiseau Cedex, France b Laboratoire d’Acoustique Musicale, CNRS - Université Paris 6 - Ministère de la Culture, 11 rue de Lourmel,

75015

Paris, France c Laboratoire de Mécanique des Solides, CNRS - Ecole Polytechnique, 91128 Palaiseau Cedex, France A modal computation of a complete clarinet is presented by the association of finite-element models of the reed and of part of the pipe, and a lumped-element model of the rest of the pipe. This is a continuation of an initial work by Pinard and Laine (unpublished reports of the Ecole Polytechnique) on isolated reeds. The eigenmodes of the complete system are computed and the results lead to a discussion of the following points: flexion and torsion modes of the reed, their coupling to the acoustical field, plane wave hypothesis, equivalent volume approximation in the mouthpiece, and alignment of resonance peaks.

the reed and the beginning of the pipe associated with a lumped elements model for the main part of the pipe (Fig. 1).

k m r

MODELS AND RESULTS FIGURE 1. Lumped elements model for the pipe, FEM for the barrel, the mouthpiece, and the reed of the clarinet.

INTRODUCTION The classical view of a clarinet associates a linear resonator - the pipe - and a nonlinear excitor - the reed. The purpose of our approach is to include the reed in the linear part and to restrict the nonlinear aspects to boudary conditions: the air flow from the mouth to the pipe through the reed-slit and contact forces between the reed and the curved lay of the mouthpiece. The present analysis deals with the linear aspects of the ensemble of the reed coupled to the pipe by means of a modal investigation of the instrument. The simplest reed model - a spring - has been approached experimentally [1, 2, 3] and used in numerical simulations which were successful in describing basic features of the dynamics of a clarinet [4, 5, 6]. The further step in complexity is that of a single oscillator with various possible sophistications [7, 8, 9, 10]. Modeling the reed as a true continuous system is the current state of research. Several examples of modal analysis of isolated clarinet reeds have been presented over the recent years using holographic interferometry [11, 12, 13, 14]. Two examples of finite-element modeling based on measurements of the mechanical properties of cane have been reported [15, 16]. The present model treats the reed as a complex continuous system in association with the aircolumn: fluid and solid finite element models (FEM) for

The reed geometry has been carefully measured. Reed cane is considered as an elastic transversely isotropic, homogeneous material. Since we are concerned with individual modes of the reed, losses are ignored. They would have to be taken into account in the actual dynamics of the instrument. Five parameters are needed to describe the material (in parenthesis, the values used in the computation): density ρ (450 kg=m 3 ), longitudinal and transverse Young’s moduli E L (10 000 MPa) and E T (400 MPa), transverse to longitudinal shear modulus G LT (1 300 MPa), and longitudinal-transverse Poisson coefficient νLT (0.22). The reed is considered rigidly clamped on the section corresponding to the ligature and having a stress-free boundary elsewhere. This model has been implemented using linear Love-Kirchoff plate elements in the Castem finite-element code (www.castem.org:8001). Three of the first modes of a reed are presented in Fig. 2. Acoustical studies of the clarinet have so far represented the mouthpiece of a wind instrument by its equivalent volume. To go beyond this approximation and to compute the 3-D distribution of the pressure in the upper part of the instrument, a coupled fluid-solid model has been used. The air volume inside the mouthpiece and the barrel is modeled with linear tetrahaedric and prismatic finite elements of compressible elastic fluid. The acoustic pressure at points of the open air surfaces is considered to be zero as well as the normal derivative of the acoustic pressure (corresponding to air flow) on the walls of the mouthpiece and the barrel. The boundary condition coupling the reed and the mouthpiece involves the stress in the solid and the velocity of the fluid. The precise for-

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FIGURE 2. Modes at 2417, 4158, and 7020 Hz of an isolated reed.

mulation is given in [17]. Computed modes in the coupled situation match well the modal shapes on real reeds as measured by holographic interferometry. The modal acoustic pressure at an eigenfrequency of 4119 Hz is displayed in Fig. 3. In this mode, the reed undergoes torsion with a characteristic distance smaller than the half the wavelength in air at that frequency (λ  10 cm); the resulting acoustical short-circuit prevents any efficient coupling of the reed to the air in the mouthpiece. This explains the fairly uniform acoustic pressure for this mode, except very near to the reed. VAL − ISO >−2.51E−04 < 2.51E−04

ume. It appears that variations in eigenfrequencies due to the model change are significant with regard to the alignment of resonances, even at low frequencies. In other words, the misalignment of peaks in either model is of the same order of magnitude as the frequency shifts due to the presence of the reed and the prismatic shape of the mouthpiece.

REFERENCES 1. C. J. Nederveen, Acoustical aspects of woodwind instruments 2nd ed., Illinois University Press, Dekalb, 1998.

−2.47E−04 −2.24E−04 −2.00E−04 −1.77E−04 −1.53E−04 −1.30E−04

2. J. Gilbert, Ph.D. thesis, Université du Maine - Le Mans, 1991.

−1.06E−04 −8.24E−05 −5.89E−05 −3.53E−05

3. X. Boutillon and V. Gibiat, J.A.S.A. 100, 1178–89 (1996).

−1.18E−05 1.18E−05 3.53E−05 5.89E−05

4. R. Schumacher, Acustica 48, 73–85 (1981).

8.24E−05 1.06E−04 1.30E−04 1.53E−04 1.77E−04 2.00E−04

5. M. Mcintyre, R. Schumacher, and J. Woodhouse, J.A.S.A. 74, 1325–45 (1983).

2.24E−04 2.47E−04

6. C. Maganza, R. Causse, and F. Laloe, Europhysics Letters 1, 295–302 (1986). FIGURE 3. Computed eigenmode at 4119 Hz in a mixed solidair situation - Acoustic pressure inside the mouthpiece and barrel.

7. B. Gazengel, J. Gilbert, and N. Amir, Acta Acustica 3, 445– 72 (1995).

In order to simulate the modal behavior of the complete clarinet, the FEM of the top of the pipe is associated with lumped elements representing the rest of the pipe. Simulations such as presented in Fig. 3 show that the acoustical field consists essentially of plane waves. The rest of the pipe can therefore be adequately represented by its acoustic input impedance. The lumped-element oscillators are coupled to the finite elements by means of a rigid plate with negligible mass. We used measurements provided by Vincent Gibiat. Each impedance peak is associated with an oscillator represented in its generic form in Fig. 1. The eigenmodes and eigenfrequencies have been computed for several fingerings of the instrument. We have then revisited the classical question of the harmonicity of the eigenfrequencies. The traditional model of the mouthpiece is that of a cylinder of equivalent vol-

9. S. Stewart and W. Strong, J.A.S.A. 68, 109–20 (1980).

8. E. Ducasse, Journal de Physique 51, 837–40 (1990). 10. S. Sommerfeldt and W. Strong, J.A.S.A. 83, 1908–18 (1988). 11. P. Hoekje and G. Roberts, J.A.S.A. 99, 2462 (A) (1996). 12. I. Lindevald and J. Gower, J.A.S.A. 102, 3085 (A) (1997). 13. F. Pinard and B. Laine, “Etude préliminaire d’une anche de clarinette,” Ecole Polytechnique (1997). 14. B. Richardson, personal communication. (1999). 15. D. Casadonte, J.A.S.A. 94, 1807 (A) (1993). 16. B. Laine and F. Pinard, “Etude numérique des modes propres d’une anche de clarinette,” Ecole Polytechnique (1998). 17. M. Facchinetti, X. Boutillon, and A. Constantinescu (2001), submitted to the J.A.S.A.

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The Sounding Pitches of Brass Instruments D. M. Campbella , J. Gilbertb and A. Myersa a Department

of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, U.K. Email: [email protected] b Laboratoire d’Acoustique de l’Université du Maine, UMR CNRS 6613, 72085 Le Mans, France Brass instruments generate sound through an aeroelastic coupling between the resonances of the player’s lips and the instrument’s air column. The air column can be treated as an acoustic waveguide, but both non-planar modes and non-linear sound propagation may be of significance. Some success has been achieved in modelling the lips as a mechanical oscillator with one degree of freedom, but stroboscopic observation has revealed that lip motion is relatively complex, and attention is now directed towards more sophisticated models. Linear stability analysis has shown that a model coupling two mechanical lip modes and one acoustic mode can reproduce the essential features of the "lipping" technique. One application of brass instrument modelling is the prediction of the sounding pitches of historic brass instruments based on non-invasive input impedance or input impulse response measurements.

INTRODUCTION The long-term goal of current research in the acoustics of lip-reed excited instruments is to find a physical model of the complete system including the player’s windway and lips, the instrument resonator and the radiated sound field. The generation of sound in a brass instrument occurs through aeroelastic coupling between the mechanical resonances of the lips and the acoustic resonances of the instrument’s air column [1]. The model should correctly predict the threshold mouth pressure at which destabilisation of the coupled system occurs and the playing frequency just above threshold. It should also be able to reproduce the large amplitude behaviour characteristic of fortissimo playing in brass instruments, for which the internal acoustic pressure can be greater than 180 dB. In addition, features of brass playing technique such as the player’s ability to pull the frequency of the sounded note above or below the frequency of an acoustic resonance ("lipping") should be demonstrated by the model. One approach to this formidable task is to attempt to develop separate models of the different subsystems, and then to incorporate these sub-models in a global model of the coupled system. In recent years much of the effort has concentrated on modelling the lips and the acoustic resonator, since the coupled system formed by these two elements essentially determines the playing pitches of the instrument [2, 3, 4]. The present paper describes some current work in this vigorous and expanding research field.

AIR COLUMN RESONANCES Most models of the resonating air column of the instrument treat it as an acoustic waveguide, terminated at the upstream end by the lips and at the downstream end by the radiation impedance of the open bell. The coupling

to the lips is described in terms of the input impedance (in the frequency domain) or the input impulse response (in the time domain). Experimental methods are available for measuring these properties of real instruments [5]. The assumption is often made that in a wind instrument only the lowest frequency plane wave mode can exist. This is not strictly true; in the vicinity of a bore discontinuity or a side hole evanescent higher modes are present, and in the flaring bell of a trumpet or trombone higher modes can propagate. These higher modes can affect the reflection coefficient for the plane wave mode [6], and therefore influence the coupling with the lips. The characterisation of the air column in terms of its input impedance is based on the assumption of linear sound wave propagation in the air column. It is now well established that this linearity breaks down in very loud playing on the trombone, leading to the formation of shock waves [7]. The resultant transfer of energy from low to high frequency acoustic modes dramatically alters the timbre of the sound, and by affecting the interaction with the lips could in principle also modify the sounding pitch.

MODELLING THE LIPS In the nineteenth century Helmholtz modelled the brass player’s lip as a mechanical oscillator with a single degree of freedom [8]. The human lip is a complex mass of tissue and muscle, and it seems unlikely that such a highly simplified mechanical model could adequately represent its vibratory behaviour. Indeed, stroboscopic observation has revealed that the motion of a brass-player’s lip is often far from one-dimensional [9, 10]. Nevertheless, the Helmholtz "swinging door" or "outward-striking" model describes many of the properties of real lips, and physical models of trumpets and

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trombones using single degree of freedom lip models have generated remarkably realistic sounds [11, 12]. It has also been shown that important aspects of the vibratory behaviour of human lips can be experimentally reproduced using artificial lips made from water-filled rubber tubes [13]. The artificial mouth can obtain notes from a trumpet or trombone which are difficult to distinguish from notes played by a human performer [14]. A tentative conclusion which may be drawn from these observations is that, while the complexities of the human lips must influence the more subtle aspects of brass instrument playing, the essential features of the non-linear interaction with the air column can be modelled using a much simplified lip model. An advantage of the artificial lips is that their mechanical response can be readily measured. Such measurements on embouchures capable of sounding a trombone have shown that the lips typically display several mechanical resonances in the frequency range of the sounding pitches [4]. Some of these show the classic Helmholtz "outward striking" character, while others in contrast have the "inward striking" behaviour usually associated with woodwind reeds. These experimental observations are useful in guiding the choice of lip model and in supplying realistic values of model parameters.

LIP-RESONATOR COUPLING The simplest model of the interaction between the lips and the resonator is one in which a single mechanical lip mode is coupled to a single acoustic mode of the air column. It has been shown that this model is incapable of displaying the lipping technique [4]. On the other hand, linear stability analysis of a model which simultaneously couples two mechanical lip modes and one acoustic mode has shown that it can exhibit behaviour equivalent to lipping. A model in which two lip modes with constant parameters are coupled to a single mode of a cylindrical resonator of variable length predicts a threshold playing frequency which changes continuously from below to above the acoustic resonance frequency as the tube length is increased, in qualitative agreement with experiments using the artificial lips [15]. A potential application of brass instrument modelling is in the evaluation of the playing properties of historic instruments [16]. Museum curators are increasingly unwilling to allow early instruments to be blown by human players because of the destructive effect of warm, moist breath. A measured input impulse response could be coupled to a realistic lip model to allow the instrument to be virtually played, giving valuable information to makers of reproductions. It will be necessary to take into account

non-linear energy loss mechanisms at side holes and bore discontinuities if the model is to be capable of predicting the sounding pitch and timbre at realistic playing levels.

ACKNOWLEDGMENTS The authors are grateful to many colleagues at Edinburgh and Le Mans for helpful discussion and collaboration. Financial support from EPSRC, CNRS and the Royal Society is acknowledged. ————————————————

REFERENCES 1. S. J. Elliott and J. M. Bowsher, J. Sound Vib. 83, 181-217 (1982). 2. S. Adachi and M. Sato, J. Acoust. Soc. Am. 99, 1200-1209 (1996). 3. F. C. Chen and G. Weinreich, J. Acoust. Soc. Am. 99, 12271233 (1996). 4. J. S. Cullen, J. Gilbert and D. M. Campbell, Acustica 86, 704-724 (2000). 5. D. B. Sharp, Acoustic Pulse Reflectometry for the Measurement of Musical Wind Instruments, PhD Thesis, University of Edinburgh (1996) [http://acoustics.open.ac.uk/publications.htm]. 6. J. A. Kemp, N. Amir and D. M. Campbell, Proceedings of 5ième Congrès Francais d’Acoustique, Lausanne September 2000, 314-317. 7. A. Hirschberg, J. Gilbert, R. Msallam and A. P. J. Wijnands, J. Acoust. Soc. Am. 99, 1754-1758 (1996). 8. H. J. F. Helmholtz, On the Sensation of Tone 1877, trans. A. J. Ellis, repr. Dover 1954. 9. D. C. Copley and W. J. Strong, J. Acoust. Soc. Am. 99, 1219-1226 (1996). 10. R. D. Ayers J. Acoust. Soc. Am. 108, 2567 (2000). 11. R. Msallam, S. Dequidt, R. Caussé and S. Tassart, Acustica 86, 725-736 (2000). 12. C. Vergez and X. Rodet, Acustica 86, 147-162 (2000). 13. J. Gilbert, S. Ponthus and J.-F. Petiot, J. Acoust. Soc. Am. 104, 1627-1632 (1998). 14. J. Gilbert and J.-F. Petiot, Proceedings of International Symposium on Musical Acoustics, Edinburgh August 1997: Proc. Inst. Acoustics 19(5), 391-400 (1997). 15. M. A. Neal, O. Richards, D. M. Campbell and J. Gilbert, Proceedings of the International Symposium on Musical Acoustics, Perugia, September 2001 (in press). 16. A. Myers, Characterization and Taxonomy of Historic Brass Musical Instruments from an Acoustical Standpoint, PhD thesis, University of Edinburgh (1998).

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Brass Player's 3-D Lip Motion and 2-D Lip Wave Shigeru Yoshikawa and Yoko Muto* Department of Acoustical Design, Kyushu Institute of Design, Fukuoka, 815-8540 Japan *Rion Inc., 3-20-41 Higashi-Motomachi, Kokubunji, 185-8533 Japan

Abstract: A slow-motion picture of the upper lip motion was taken by the stroboscope from the frontal and side directions. The experimental data were obtained when five (two advanced, two medium, and one beginning) players blew a common natural horn equipped with a transparent mouthpiece for the French horn. Three-dimensional understanding of the upper lip motion is acquired by combining the frontal- and side-view analyses. Basically 3-D lip opening motion tends to be twodimensionalized by reducing needless lateral motion in accordance with the improvement of the playing skill. The 2-D trajectory of the wave crest on the upper lip surface illustrates the lip-wave propagation. These 3-D lip motion and 2-D lip-wave propagation are superposed each other. The lateral wave propagation (with the speed of about 2 to 4 m/s) creates the outward lip opening in the lowest F2 (78 Hz) tone. Non-advanced players cannot generate a distinct lip wave. The vertical wave propagation governs the upward lip opening in higher-mode tones. The application of the Rayleigh-type surface-wave assumption and the averaged vertical wave velocity (about 1.8 m/s) to the estimation of lip tissue elasticity approximates the shear modulus as about 4000 N/m2. The generation and propagation of the lip wave was first defined quantitatively by the present stroboscopic visualization.

3-D IMAGE OF THE UPPER LIP MOTION The visualization such as stroboscopic viewing is essential to know real motion of a vibrating body such as the lips of brass players. Particularly it is very useful to observe real motion from various directions. Such observation makes it possible to acquire proper 3-D image of the movement. Figure 1 shows sequential pictures from frontal (a) and side (b) viewings of the lip motion of an advanced player A when he played a F2 tone in forte on a natural horn (made by the Lawson Brass Instruments Inc.). An acrylic transparent mouthpiece was used for observation. The coordinate system for motion analysis of the lip opening is also indicated in left frames. The x-, y-, and zaxes respectively tend to the outward direction almost parallel to the mouthpiece axis, the lateral direction along the lower lip surface, and the vertical direction. Note that a x-z plane for side-view analysis vertically cuts the upper lip at the center of the lip.

(a)

(b)

Fig. 1. Frontal (a) and side (b) viewings of lip opening movement when an advanced player A played a F2 tone. The x-, y-, and z-axes for motion analysis are also shown. : lip tip wave crest.

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(a)

(b)

9 8 7 6 5 4

-8

-6

-4

-2

0

2

4

6

8 (

Fig. 2. An example of how to calculate the wave velocity propagating on the lip surface (player A; F2). (a): lip-opening shape in y-z plane; (b): lip-wave trajectory in x-z plane. In our experiment five French horn players, who consist of two advanced(A and B), two medium(C and D), and one beginning (E), played notes F2 to C5 with a common natural horn. The individuality of players is well observed. Although the motion of the upper lip in Fig. 1(a) looks 2-D, that of a beginning player E looks 3-D because of appreciable lateral movement toward the center. Certainly, the shape of lips is given by nature. However, this lateral movement, which is due to the unstableness of lip edge points, is appreciable in non-advanced players C, D, and E. We may thus consider that the lateral movement should be needless and avoided because such motion produces an inappropriate modulation of the flow. Frontal-view analysis gives the average lip-opening lateral width and its deviation as well as the lip opening shapes [cf. Fig. 2(a)], while side-view analysis gives 2-D (x-z) trajectories of the upper-lip tip [ in Fig. 1(b)] and the wave crest [ in Fig. 1(b) and cf. Fig 2(b)]. Particularly, when the lip motion for higher modes is examined, we may recognize the x-z traj ectory of wave crest as an effective measure of player's individuality or playing skill.

LIP-TISSUE ELASTICITY ESTIMATED FROM LIP-WAVE PROPAGATION Although the trajectory of wave crest is plotted in x-z plane in Fig. 2(b), this should be transformed to that in y-z plane because the wave really propagates on the lip surface. However, such transformation is not straightforward. Here we will concentrate on deriving the propagation velocity of wave crest from combining lip-wave trajectory in x-z plane with lip-opening shape in y-z plane (see Fig. 2). The wave crest propagates almost along the y-axis from frame #1 to #11 in Fig. 2(b) because the x-axis overlaps on the y-axis in our (slightly slant) side viewing. The crest appears near the lip edge in frame #1 and propagates toward the lip center in frame #11 [cf. Fig. 1(b)]. This propagation distance from the lip edge to the lip center may be estimated as about 5.3 mm from Fig. 2(a). Also, the time required for the wave to propagate between these points is calculated as about 2.6 ms (the period of F2 = 1/87.3 = 11.5 ms; the frame-to-frame time interval = 11.5/45 = 0.26 ms). So, the propagation velocity v y- toward the negative y direction is estimated as 5.3/2.6 = 2.0 m/s. The propagation velocity v z+ in the positive z direction is directly given as 2.7 m/s from the distance 4.1 mm and the interval 1.5 ms between frame #11 and #17. Similarly v z- is estimated as 1.0 m/s, however, it is difficult to estimate vy+. The propagation path is roughly estimated. Until now we have considered the wave propagation without defining the wave type. If it is allowed to suppose the Rayleigh-type surface wave by omitting any discussion on this validity, the surface-wave velocity of isotropic soft tissues can be approximated as cs = 0.955 (G ) 1/2 , where G and are the shear modulus and the density of the lip, respectively. Since is close to be 1.1 × 103 kg/m3, G is evaluated as 1.21 × 103 cs2 . Next problem is to determine which velocity obtained in our measurement should be the most relevant to cs . The averaged value v z = (vz+ + v z- )/2 seems to be the best for cs since this v z can minimize the influences of the blowing pressure and surface tension due to the embouchure. As a result, G is estimated as 4.1 × 103 N/m2 for player A and 3.5 × 103 N/m2 for player B with respect to F2 tone. It was difficult to define an appropriate wave velocity for nonadvanced players C, D, and E. Although exact separation between the lip wave and the lip opening motion was postponed, our new information mentioned above will serve to establish proper lip modelling and simulations in more realistic situations.

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Pipe Impedance and Reed Motion for a free Reed Coupled to a Pipe Resonator James P. Cottingham Physics Department, Coe College, Cedar Rapids, IA 52402, USA, [email protected] The Asian free reed mouth organs employ metal free reeds mounted in resonating pipes. Measurements of the input impedance curves have been made for some representative pipes from these instruments. These complement earlier measurements of reed position and phase using a laser vibrometer system, as well as measurements of sound spectra produced when the pipes are played. In agreement with theory for an outward striking reed coupled to a pipe resonator, the sounding frequency of the reedpipe combination is typically found to be above the natural frequency of the reed and slightly above the frequency of the first impedance peak of the pipe.

INTRODUCTION The Asian free reed mouth organs employ metal free reeds mounted in resonating pipes. Unlike the free reeds found in Western instruments such as the reed organ, accordion, and harmonica, the reeds of these instruments are approximately symmetric, so that the same reed can operate on both vacuum and pressure (inhaling and exhaling). The best known of these are perhaps the sheng of China and the sho of Japan. For a detailed survey of the Asian free reed instruments see the article by Miller [1]. Figure 1 shows typical reeds from a sheng and an American reed organ. The khaen (also spelled kaen, khene, caen) is a free reed mouth organ indigenous to northeastern Thailand and Laos and is perhaps the most important musical instrument of the Lao people of this region. The sheng, sho, and khaen all employ one reed per pipe, thus requiring a separate pipe for each pitch.

The experimental work discussed in this paper involves the khaen and the sheng. Impedance curves have been measured for several pipes from a sheng and for simulated khaen tubes constructed from PVC pipe.

IMPEDANCE CURVES For khaen pipe impedance measurements, the pipes used were constructed from PVC tubing with dimensions similar to actual khaen pipes. Each of the PVC khaen pipe impedance curves was measured following the method of Benade and Ibisi [3]. A piezo buzzer was positioned in place of the reed. A small hole was drilled, just large enough for a probe microphone tip to be inserted so that the probe tip was inside the tube near the position of the reed. With the microphone connected to a spectrum analyzer, spectra were obtained using a swept sine wave. For the sheng, bamboo pipes from a real instrument were used. In this case no hole was drilled, but the flexible probe microphone tube was fed through the existing finger hole, which was otherwise sealed with putty.

Results for the PVC khaen tubes FIGURE 1. Reeds from a sheng (left) and an American reed organ (right) from Gellerman [2].

The sheng employs a free reed at the end of a closed pipe resonator of conical-cylindrical cross section, while in the khaen the free reed is placed at approximately one-fourth the length of an open cylindrical pipe. In both cases the effective length of the pipe is determined by one or two tuning slots cut in the pipe. These slots not only shorten the effective length of the pipe, but also complicate the system of pipe resonance, affecting the tone quality of the played notes by adding end sections several centimeters long .

These measurements appear to be the first input impedance measurements on khaen pipes. Some typical results are shown in Figures 2 and 3. Previous observations and measurements had concluded that such a free reed pipe will sound at a frequency above the reed frequency and close to a resonant frequency of the pipe [4,5]. This resonant frequency is normally the fundamental, but if the length of the tube places the fundamental below the reed frequency, the reed-pipe can sound near the second harmonic of the pipe [4].

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FIGURE 2. Impedance curve (dashed) and sound spectrum (solid) of a 59 cm PVC khaen tube with no end sections.

FIGURE 4. Impedance curve (dashed) and sound spectrum (solid) of a sheng pipe.

impedance peak, with the second harmonic of the sound spectrum apparently reinforced by the strong nearby peak in the impedance curve.

SUMMARY

FIGURE 3. Impedance curve (dashed) and sound spectrum (solid) of a 59 cm PVC khaen tube with end sections.

Another earlier observation [4] confirmed and to some extent explained by these results is the difference in tone quality observed in comparing khaen pipes with and without the end sections. This difference in tone quality seems closely related to the differences observed here in the input impedance curves for the two cases, illustrated by comparing Figures 2 and 3.

The free reeds coupled to pipe resonators in the instruments under consideration seem to behave as "opening" or "outward striking" reeds as discussed by Fletcher [7]. The sounding frequency of the reed-pipe combination is above the natural frequencies of both the reed and the pipe, regardless of the direction of airflow. (For the bawu, a free reed pipe with finger holes, the sounding frequency can be pulled by the pipe resonance nearly an octave above the reed frequency [5].) Previous measurements of reed motion using a laser vibrometer verify that, for these instruments, the free reed tongue makes only slight excursions in the upstream direction, thus approximating the outward striking reed model [8].

REFERENCES 1.

Results for the sheng pipes 2.

The sheng pipes are cylindrical over most of their length, but the lower portion in which the reed is mounted is conical. The frequencies of the resonances identified by the impedance peaks are not harmonic. Calculations have been made of the impedance maxima modeling the sheng pipe as a compound conicalcylindrical horn following Fletcher and Rossing [6]. These give qualitative results similar to the measured impedance curves, but do not accurately predict the frequency ratios of the impedance maxima. The pattern observed for the sheng is that the sounding frequency of the pipe is again above that of the first

3. 4. 5.

6. 7. 8.

Miller, T.E., "Free-Reed Instruments in Asia: A Preliminary Classification," in Music East and West: Essays in Honor of Walter Kaufmann, Pendragon, New York, 1981, pp. 63-99. Gellerman, R.F., The American Reed Organ and the Harmonium, Vestal Press, 1996, pp. 3-4. Benade, A.H. and Ibisi, M.I. , J. Acoust. Soc. Am. 81, 11521167, (1987). Cottingham, J.P. and Fetzer, C.A., "Acoustics of the khaen," Proceedings of the International Symposium on Musical Acoustics, Leavenworth, Washington, 1998, pp. 261-266. Cottingham, J.P, "Acoustics of a symmetric free reed coupled to a pipe resonator," Proceedings of the Seventh International Congress on Sound and Vibration, Garmisch-Partenkirchen, Germany, 4-7 July 2000, pp. 1825-1832. Fletcher, N.H. and Rossing, T.D., The Physics of Musical Instruments, 2nd ed., Springer, New York, 1998, pp.. 216-217. Fletcher, N.H., Acustica 43, 63-72 (1979). Busha, M. and Cottingham, J.P., J. Acoust. Soc. Am. 106, 2288, (1999).

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Kinetic Studies of Air Column Acoustics R. Dean Ayers and Mark T. McLaughlin Department of Physics and Astronomy, California State University Long Beach, Long Beach, CA 90840-3901, USA The linear behavior of an air column can now be illustrated very easily using a personal computer. Concepts that may have been somewhat abstract become more concrete with kinetic images on the computer screen. One basic example is a representation of standing waves with imperfect nodes in terms of several dependent variables, each of which displays a “lurching” or “galloping” behavior. Because of the emphasis that they receive in introductory courses, undamped traveling waves and ideal standing waves can play too prominent a role in a student’s understanding of realistic waves, which are subject to damping in propagation and on reflection. A detailed examination of the intermediate “lurching” waves may help to relegate those idealized special cases to their proper roles as limiting behaviors. The treatment of the example presented here is analytic rather than numerical, with the computer just providing kinetic images of the solutions. The use of scrollbar controls for physical parameters of the wave images encourages informal experimentation. Allowing students to write their own program lines for the calculation of dependent variables may help them to make the transition from using real sinusoids to working with complex exponentials.

“LURCHING” WAVE BEHAVIOR Two undamped sinusoidal plane waves of the same wavelength λ propagate in opposite directions in a uniform pipe. All acoustic pressures are normalized to the amplitude of the stronger wave, which travels to the right (+ x direction). The amplitude of the weaker wave is R, the pressure reflection coefficient at the origin, and here R is limited to positive values. Thus at time t = 0 both pressure waves have crests at x = 0. At each point in the region x < 0, where the waves overlap, the total acoustic pressure varies sinusoidally in time. The amplitude of that temporal variation is Ap (x) = [1 + R2 + 2Rcos(2kx)] 1/2 ,

(1)

p crest(x) = (1 − R2 ) /[1 + R2 − 2Rcos(2kx)] 1/2 . (2) The velocity of the total wave varies, whether one focuses on a crest or on a point of contact with the upper envelope curve (temporal maximum, O). For either type of feature the average velocity over a half period is v = λ / T to the right. Within that interval the variation can be considerable, as shown in Fig. 2. Crests travel rapidly through nodal regions and slowly near antinodes, while temporal maxima show the opposite behavior. The two normalized velocities have the same time dependence except for a shift by T/4: vx, feature / v = (1 − R2 ) / [1 + R2 ± 2Rcos(2ω t)],

(3)

with + for a crest, − for a contact point, and ω = 2π/ T. 1

3

Relative Velocity

Relative Acoustic Pressure

where the propagation number k = 2π / λ. Imperfect nodes and antinodes are visible in the amplitude envelope, which is the pair of dotted curves ± A p (x) in Fig. 1. (The value R = 0.5 results from dividing the area of the pipe’s cross section by three for x > 0.)

Each frame in a moving picture of the total disturbance is a sinusoidal function of x. A few of these are shown at an interval of ∆t = T / 12 from −T / 2 to 0, with T = the period. The height of a crest in the total wave ( V) must vary as it shifts in order to squeeze through each node. The curve traced by a crest is

0

-1

-0.5

-0.25

0

x/ λ FIGURE 1. Amplitude envelope (dotted) and lurching wave plots (solid) for R = 0.5: V = crest; O = contact point.

1.5

0 -0.5

-0.25

0

t/T FIGURE 2. Normalized feature velocities: solid = crest; dotted = contact (temporal maximum).

SESSIONS

The behavior of the normalized particle velocity is similar to that of the acoustic pressure. In this example its reflection coefficient at x = 0 is just −R, so we can adapt the results obtained for pressure. The amplitude envelope for x < 0 gets shifted by λ / 4, placing a node at x= 0, and the signs for the two velocities in Eq. (3) are exchanged. The antinodes and nodes of the two amplitude envelopes are the only locations where acoustic pressure and particle velocity are in phase with each other, as in a single traveling wave. At those points the lurching wave can be coupled without reflection to an incident wave in an upstream pipe of appropriate diameter. For the odd integer multiples of λ / 4 between a node and an antinode, the diameter of the lurching wave segment must be the geometric mean of the diameters on either side of it to avoid reflection. For the integer multiples of λ / 2 between two antinodes or two nodes, the diameters upstream and downstream from the lurching wave segment must be equal.

ENERGY AND INTENSITY The potential and kinetic energy densities also show lurching behavior. Their normalized values

(a)

The software was developed in MATLAB, version 6. Scrollbar controls on the screen for R and other parameters make this a hands-on tool. Students can check their solutions to analytic problems by writing program lines for calculating the dependent variables. Those who are just learning to use complex exponentials can verify that they give the same results as real sinusoids. This software is also useful for studying waves in higher dimensions, or those with damping in propagation. Input impedance, pressure reflection coefficient, and effective length, all treated as functions of axial position as well as frequency, can also be examined in non-uniform air columns.

1

ACKNOWLEDGMENTS

0 -0.5

This work has been supported by the Paul S. Veneklasen Research Foundation and the CSULB Scholarly and Creative Activities Committee. We are grateful to Nader Inan for his assistance.

-0.25

0

(b) Rel. Tot. Energy Density

IMPLEMENTATION AND USES

2

1

0 -0.5

-0.25

0

x/λ FIGURE 3. Lurching behaviors of energy densities and their envelopes: (a) potential energy density, (b) total energy density; R = 0.5 and ∆t = T/12 for −T/2 < t < 0.

Relative Acoustic Intensity

Rel. P.E. Density

2

are just the squares of the normalized acoustic pressure and particle velocity, respectively. The potential energy density and its envelope are shown in Fig. 3(a). The kinetic energy density has the same envelope shape, but shifted by λ / 4. The total energy density and its envelope are shown in Fig. 3(b). Finally, Fig. 4 shows the envelope and instantaneous plots for the acoustic intensity, pvx,particle, the power flux per unit area. The horizontal lines at 0 and 1 are the upper limits for the two envelope curves, independent of the value of R.

1

0

-0.5

-0.25

0

x/λ FIGURE 4. Acoustic intensity and envelope.

SESSIONS

Experimental Results on the Influence of Channel Geometry on Edgetone Oscillations C.Ségoufin, B. Fabre, L. Delacombe Laboratoire d’Acoustique Musicale, UPMC, 4 place Jussieu, case 61, 75252 Paris cedex 05, France Experimental investigations of edgetone oscillations are usually carried on a flue channel geometry insuring a fully developed flow at its exit. In this paper, the influence of channel length and chamfers on sound production is investigated experimentally by studying aero-acoustical sources at the labium. The dipolar source induced by the jet movement on the labium is always found to be dominant, but with characteristics depending upon the flue channel configuration. Surprisingly, adding chamfers at the end of the long channel induces a strong decrease in the source strength.

INTRODUCTION Edgetone oscillations have been experimentally and theoretically investigated by several authors, see for instance Holger and Crighton [1,2]. All studies have been carried out on a geometry with a long channel insuring a fully developed flow before the flue exit. Powell [ 3], studied aero-acoustical sources at the labium of the edgetone in terms of Lighthill’s analogy [4]. Using ad hoc assumptions, he shows that the force term is dominant and he relates radiated sound to force induced at the labium by the jet movement. He confirmed it experimentally using a setup where the jet is issuing from a long flue channel. Previous experimental observations carried on a complete recorder flute configuration [5] have shown that small geometrical modifications of the flue channel geometry could have great effects on the jet stability. Therefore these parameters, known as crucial by recorder makers, are suspected to influence sound production. In this paper we discuss in an edgetone configuration the effect of the channel geometry on the sound source, through measurements of the jet force on the labium as well as measurements of the radiated sound.

EXPERIMENTAL SETUP The jet is issuing from a straight flue channel made of a removable block embedded in a fix volume, the foot. Three blocks are used providing three different channelgeometries: a long straight channel of length L=18mm, for which a Poiseuille flow is expected in the flow at the flue exit, a long channel with chamfers at its exit (where edges are cut at ~45° on 1mm length) and a short channel for which a nearly square velocity profile is expected. The height h and

the width H of the two channels are kept constant: h=1mm and H=20mm. The foot is supplied with a mixture of N 2O2 and CO2 from a 50Bar bottle of compressed gas via a regulator. The mean foot pressure is measured using a manometer DIGITRON 2020P. The labium consists in a metal plate on which strain gauges are glued, allowing force measurements [6]. The metal plate is clamped on one end in a heavy steel jaw. The edge of the plate has 90° cut (±45° from the channel axis) and is placed at a distance W=9mm from the flue exit. In the case of the channel with chamfers at the end, the distance between jet separation points and the labium is increased of 0.7mm. The radiated sound is measured using a B&K 1/4” microphone connected to an FFT analyzer, the microphone is placed at a distance x=23cm above the labium. Measurements are carried on the 1 st 2nd and 3rd oscillating modes of the edgetone. Force measurements presented in this paper are limited to the 1 st mode. The frequency of the 1st mode ranges from 400Hz to almost 2kHz. The Reynolds number varies within a range 300-2500. Undesirable acoustical reflections are limited as much as possible by placing glass wool blocks in the directions of maximal radiation of the edgetone.

RESULTS Force measurements are presented Figure 1 as a function of the mean jet speed Uj . The force is made dimensionless using the total jet force: Fa =

(

)

1 h /2 2 ρ 0H ∫ −h / 2 Us (y) dy 2

SESSIONS

(1)

where ρ0 is the air density, Us(y) the flow velocity profile at the flue exit which is considered a function

F (N) 50e-4

F/Fa

10e-4

1

5e-4 0.5

1e-4 7 0.1 5

10

15

30 Uj (m/s)

chamfers. oflong the channelchannel length [5]. An with example of comparison between the measured force and the force deduced from radiated sound measurement using Powell’s approximation is shown Figure 2, in the case of the long channel without chamfers. The two curves fit very well with an evolution pattern that is common to the three configurations. There is however a slight oscillation of the force deduced from radiation that could be related room resonances, although we tried to limit it to the maximum.

DISCUSSION Results obtained for our long channel confirm Powell's data and validates our experimental setup. The driving pressure range has been extended and sound production exhibits a sharp decline. We suspect this is due to turbulence appearing before the jet reaches the labium. This can be checked by flow visualization. The same good fit between measured and deduced force was observed in the three configurations, which means that the dipolar source induced by the jet movement on the labium is always dominant. The configurations can be more carefully compared using Figure 1. The dimensionless force exhibits a similar aspect in all the cases, being approximately constant at the beginning and declining quickly at a given transition velocity. The force exerted by the jet issuing from the short channel is higher compared to the long channel, but when made dimensionless with a reference force taking into account the velocity

20 30 Uj (m/s)

Figure 2: Comparison of the force measured at the labium ( ) and the force deducted from radiated sound measurements (--) as a function of the mean jet speed Uj. Long channel, no chamfers.

20

Figure 1: Force measured at the labium as a function of the mean jet speed Uj for a flue exit/labium distance of 9mm. (-) short channel, (--) long channel, no chamfers, (-.)

10

profile change at the flue exit, the two collapse. When using a short channel, the edgetone oscillation threshold is lower. This behavior is also present on frequency measurements and is the consequence of a higher propagation velocity on the jet when the velocity profile is sharp [5]. The transition velocity is also lowered. Ségoufin&al. [5] have shown that when the channel is shortened, the jet flow is more complex and exhibits numerous developed vortical structures. Maybe this results in turbulence appearing at a lower jet velocity. It has to be noted that despite the presence of turbulence before the labium, the force term induced by the jet movement is still dominant. Adding chamfers to the flue exit of the long channel reduces dramatically the sound production. This effect cannot be explained in terms of velocity profile modification as it is assumed that chamfers don’t modify it. We don't have any explanation for this effect at the time being, and further investigations would involve numerical simulation.

REFERENCES 1. 2. 3. 4.

A.W. Nolle, J. Acoust. Soc. Am., 103, 3690-3705, 1998 D. Crighton, J. Fluid. Mech. , 234, 361-391, 1992 A. Powell, J. Acoust. Soc. Am., 33, 395-409, 1961 M. J. Lighthill, Proc. Roy. Soy. Lond., A 211, 564-587, 1952 & A 222, 1-32, 1954. 5. C. Ségoufin, B. Fabre, M.P. Verge, A. Hirschberg and A. P. J. Wijnands, Acta Acustica, 86, 649-661, 2000

SESSIONS

Tone holes and cross fingering in wood wind instruments. John Smith and Joe Wolfe School of Physics, University of New South Wales, Sydney NSW 2052, Australia, [email protected] Opening successive tone holes in woodwind instruments shortens the standing wave in the bore. However, the standing wave propagates past the first open hole, so its frequency can be affected by closing other tone holes further downstream. This is called cross fingering and, in some instruments, it is used to produce the 'sharps and flats' missing from their natural scales. The extent of propagation of the standing wave depends on frequency, so that different modes of the bore are affected to different degrees, giving different timbres to cross fingered and simply fingered notes. We measure the frequency dependence of the transmission of waves in the bores of flutes in both the regions where tone holes are closed and open, and of the radiation from open holes of different sizes. We use these results to explain the different effects of cross fingering in modern and traditional instruments, and the differences in timbre between cross fingered and normally fingered notes.

INTRODUCTION

MATERIALS AND METHODS

On modern woodwind instruments such as the flute, oboe, clarinet and saxophone, a chromatic scale may be played over at least one octave simply by opening tone holes, one at a time, starting at the downstream end. Mechanisms that couple the keys allow the player to cover more tone holes than s/he has fingers. On the recorder, and on the ancestors of the modern flute, oboe and clarinet, opening successive finger holes produces a diatonic scale. The missing notes in the chromatic scale are achieved by cross fingering. In both old and new instruments, a second register is obtained by overblowing, using nearly the same fingerings. Third and higher registers involve cross fingering. Opening a tone hole provides a low inertance shunt from the bore to the external radiation field. If the hole has a large diameter and if the frequency is sufficiently low, the shunt approximates an acoustic short circuit. So, for instruments with large holes (flute and saxophone) cross fingering has relatively little effect in the lowest register. Various authors have modelled tone holes with a range of complications[1-5]. Although the standing wave is attenuated by the tone hole, it still extends past it along the bore, giving an extra length or end effect that varies with frequency. The end effect decreases as the size of the tone hole increases, and depends (to a lesser degree) on the height of its chimney, the extent of covering by keys, and the degree of undercutting (i.e. chamfering of the junction between the tone hole and the bore). In this study we measure the frequency dependence of the propagation of the standing wave beyond the first open tone hole for simple and cross fingerings.

The instruments studied are similar but not identical to those studied previously [6]. An acoustic current was synthesized having frequency components from 0.2 to 3 kHz with 2 Hz increments, and input to the embouchure hole via a short pipe whose impedance approximates the radiation impedance that normally loads the instrument at this point when it is played. To measure the frequency dependence of propagation of the standing wave, a probe microphone was inserted at the embouchure, and also at the open or closed tone holes The technique for the measurement of transfer functions is described elsewhere [6], except that a probe microphone (external diameter 1.0 mm) was used for all measurements.

RESULTS AND DISCUSSION The first plot shows the input impedance spectrum of a reproduction of a 19th century flute (chosen because of its relatively small tone holes) for the fingering used to play E4 or E5. At low frequencies, the spectrum is not very different from that of a shorter version of the bore, as if it were terminated a little beyond the first open hole. The flute, being played with the embouchure open to the radiation field, operates at minima in impedance. For most fingerings, the first few impedance minima are harmonically spaced. (See [6-9] for detailed discussions.) At higher frequencies, the minima are weaker (due to wall losses) and, in the example shown here, they also occur at frequencies lower than harmonic multiples of the fundamental, due to the greater propagation of the wave beyond the first open tone hole.

SESSIONS

acoustic pressure increases with length past the first closed hole. This increases the effective length of the bore with frequency, and so the first minimum is flattened, and the second a little more, etc. This allows the production of flattened notes, but it also produces inharmonic minima, which do not support the high harmonics in a strongly non-linear vibration rŽgime, and hence contribute to the darker timbre of cross fingered notes. Musically useful discussion requires much greater detail. Discussions of fingering effects and a much larger set of results are given on our web site [9].

|Z| __ MΩ 10 5 2 1 .5 .2 .1 .05 f/kHz

FIGURE 1. The acoustic impedance spectrum, in MPa.s.m−3, for a simple fingering that plays E4/5. E

6 7 8

|Z| __ MΩ 10 5 2 1 .5

6

.2 .1

7

.05 f/kHz 8

FIGURE 3. The impedance for a cross fingering.

ACKNOWLEDGMENTS

f/kHz

FIGURE 2. The ratio of the pressure at the open tone holes to the pressure at the embouchure. Figure 2 shows the frequency dependence of propagation of the standing wave into the region of open holes, expressed as the ratio of the acoustic pressure at the 6th, 7th and 8th tone holes to that at the embouchure for the same acoustic current at the embouchure. The sharp peaks coincide with the minima in Z: other frequencies do not generate a strong standing wave for this fingering. At the first open hole (hole 6), the first several peaks (those corresponding to the harmonic minima in Z) have similar magnitudes, but they become weaker at higher frequencies. Further down the bore (holes 7 & 8), the penetration increases with frequency, up to the cut off frequency for the array of open tone holes (for this instrument around 2 kHz), above which there is little difference with position, and the standing wave propagates freely [8]. In modern instruments with larger tone holes, (data not shown), the penetration increases less rapidly with frequency and the cut off frequency is higher. When cross fingerings are used, the analogous curves are more complicated. The wave propagates strongly down the bore, and for most frequencies, the

We thank John Tann and Terry McGee. This work was supported by the Australian Research Council.

REFERENCES 1.

J.W. Coltmann, J.Acoust. Soc. Amer., 65, pp. 499-506 (1979).

2.

W.J. Strong, N.H. Fletcher and R.K. Silk, J. Acoust. Soc. Amer., 77, pp. 2166-2172, (1985)

3. D.H. Keefe, J. Acoust. Soc. Amer., 72, pp. 676-687, 1982. 4.

C J. Nederveen, J.K.M. Jansen and R.R. van Hassel, Acustica 84, pp. 957-966, (1998).

5.

V. Dubos, J. Kergomard, A. Khettabi, J-P. Dalmont, D.H. Keefe, C.J. Nederveen, Acustica 85, pp. 153-169 (1999).

6.

J. Wolfe, J. Smith, J. Tann, and N.H. Fletcher, J. Sound & Vibration, 243, 127-144 (2001).

7.

N.H. Fletcher and T.D. Rossing, The Physics of Musical Instruments. New York, Springer-Verlag, 1998

8.

A.H. Benade J. Acoust. Soc. Amer. 32, 1591-1608 (1960).

9.

J. Wolfe, J. Smith and J. Tann. Flute www.phys.unsw.edu.au/music/flute

acoustics

SESSIONS

Tones and Wave Forms of the Shakuhachi S. Takahashi and T. Matsui a

Department of Engineering, Kogakuin University, 163-8677 Tokyo, Japan

The sound of the shakuhachi makes one wonder about the relationship between the physical characteristics of sounds and the mental feelings they arouse. In this paper, we outline a preliminary study of the characteristics of the sound of the shakuhachi in terms of the relationship between sounds and mental feelings. This general area of study could become important for the manmachine interfaces of the future. We recorded the scale of a shakuhachi and the various timbres of its sounds on a digital audiotape (DAT). We were able to determine several characteristics of the shakuhachi’s sound analyzing these recorded sounds.

INTRODUCTION When one of the authors was beginning to learn to play the shakuhachi, he listened to the long tones of the shakuhachi, with their changing timbres, at a master player’s recital. This was about twenty years ago, and the player is no longer with us. The cassette tape the author recorded at the time doesn’t accurately reproduce the variation timbres because of the bad recording conditions. In playing the shakuhachi, a common technique is to play with a constant pitch and varing timbre. The timbre makes the listener experience bright and then dark feelings, as the tone with its initially bright timbre fades out. The method of playing is truly fascinating. The variation in timbre gives the listener the feeling that the bright and dark timbres are as authentic the perception red and blue as warm and cold colors. The sound of the shakuhachi brings the listener to wonder about the relationship between the physical characteristics of sounds and mental feelings. The physical characteristics of shakuhachi sound have been studied [1]. In this paper, we outline a preliminary study of the characteristics of the sound of the shakuhachi in terms of the relationship between sounds and mental feelings. This general area of study could become important for the manmachine interfaces of the future.

THE PROCESS OF THIS RESEARCH The scale of a shakuhachi and the various timbres of its sounds were recorded to digital audiotape (DAT) in an anechoic room. The several pitches are produced by two fingerings, and each pair of them are the same pitch and different timbres. We recorded changes in timbre on long tones. Studying

the relationships between waveforms and sounds is part of an attempt to make artificial sounds that have many analogies with the sounds produced by real shakuhachis. If the artificial sounds seem to be real, we can consider that we have determined the characteristics of shakuhachi sounds.

SOME RESULTS By investigating recorded sounds of a shakuhachi, we were able to determine several characteristics of the shakuhachi’s sound. The shakuhachi is characterized by the beginnings and endings of the sounds it is made to make. At high pitches, the shakuhachi is not very different from other wind instruments. The sounds of the shakuhachi look like sine waves. At low pitches, however, a loud sound often alternates between two wave shapes that are different from each other, as shown in Figure 2(1). This means that listener hears loud sounds as low pitches due to the large amplitudes of the harmonics despite the small low-frequency fundamental component. The alternation between two wave shapes seems to reduce the maximum amplitude. The timbre usually changes with the pitch or loudness of the sound produced by the shakuhachi. When playing the shakuhachi, it is very difficult to obtain varied timbers with the same pitch. Sounds in and around the middle of the shakuhachi’s pitch range are easier to play with more stability, so it is a little easier to vary the timber. Figure 1 shows a long tone produced by a player endeavoring to change the tone’s timbre while maintaining the same pitch of “g”, which is notated as by shakuhachi players.We happened to find two waveforms with similar spectra but with the harmonic components in different phases as shown in Figure 2.

SESSIONS

Amplitude (dB)

Time (sec)

Frequency (Hz)

FIGURE 1. The Spectra of a Shakubhachi Sound With the Same Pitch and Varied Timbre

(1)

(2)

FIGURE 2. Two Waveforms with the Same Spectra and Different Phases.

These waveforms are described by the following functions. sin wt + sin( 2w t + 2p / 3) + 0.6 sin 3wt (1)

sin wt + sin 2wt + sin 3wt

(2)

phase which will be perceived as the same sound. Figure 3 shows two typical waveforms from the long tone used to produce Figure 1. We hear the typical loud shakuhachi sound of low pitch as dark, and the sine-wave-sound like as bright. We tried to make an artificial sound that was like this long tone. The tone was, however, satisfactory.

CONCLUSION Some characteristics of shakuhachi tones were obtained. Our next work will be an investigation of the relationship between spectra and feelings in more detail.

ACKNOWLEDGMENTS 1. 068

1. 07

1. 072

1. 074

1. 076

1. 078

1. 08

1. 082

We are very much obliged for the help given to us by Prof. Toyama and to the members of his laboratory at Kogakuin University. 3. 13

3. 132

3. 134

3. 136

3. 138

3. 14

3. 142

3. 144

FIGURE 3. Typical Waveforms of the Sound Used to Produce Fig.1.

The well-known unawareness of the phase of sound can be confirmed by listening to two with different

REFERENCES 1. Y. Yasuda, New Edition, The Acoustics of Instruments, Tokyo: Ongakunotomosha, 1996, pp. 79-85(in Japanese).

SESSIONS


 
 
 
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