Modal analysis of a complete clarinet M. Facchinettia,c, X. Boutillonb and A. Constantinescuc aLaboratoire d’Hydrodynamique, CNRS - Ecole Polytechnique, 91128 Palaiseau Cedex, France bLaboratoire d’Acoustique Musicale, CNRS - Université Paris 6 - Ministère de la Culture, 11 rue de Lourmel, 75015 Paris, France cLaboratoire 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 This is a continuation of an initial work by Pinard and Laine (unpublished reports of the Ecole Polytechnique) on isolated reeds.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 k (Fig. 1). 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. The reed geometry has been carefully measured. Reed cane is considered as an elastic transversely isotropic, ho- mogeneous material. Since we are concerned with in- dividual modes of the reed, losses are ignored. They INTRODUCTION would have to be taken into account in the actual dy- namics of the instrument. Five parameters are needed The classical view of a clarinet associates a linear res- to describe the material (in parenthesis, the values used 3 onator - the pipe - and a nonlinear excitor - the reed. The in the computation): density ρ (450 kg=m ), longitudinal purpose of our approach is to include the reed in the linear and transverse Young’s moduli EL (10 000 MPa) and ET part and to restrict the nonlinear aspects to boudary con- (400 MPa), transverse to longitudinal shear modulus G LT ditions: the air flow from the mouth to the pipe through (1 300 MPa), and longitudinal-transverse Poisson coeffi- the reed-slit and contact forces between the reed and the cient νLT (0.22). The reed is considered rigidly clamped curved lay of the mouthpiece. The present analysis deals on the section corresponding to the ligature and having a with the linear aspects of the ensemble of the reed cou- stress-free boundary elsewhere. This model has been im- pled to the pipe by means of a modal investigation of the plemented using linear Love-Kirchoff plate elements in instrument. the Castem finite-element code (www.castem.org:8001). The simplest reed model - a spring - has been ap- Three of the first modes of a reed are presented in Fig. 2. proached experimentally [1, 2, 3] and used in numeri- Acoustical studies of the clarinet have so far repre- cal simulations which were successful in describing basic sented the mouthpiece of a wind instrument by its equiv- features of the dynamics of a clarinet [4, 5, 6]. The fur- alent volume. To go beyond this approximation and to ther step in complexity is that of a single oscillator with compute the 3-D distribution of the pressure in the upper various possible sophistications [7, 8, 9, 10]. Modeling part of the instrument, a coupled fluid-solid model has the reed as a true continuous system is the current state been used. The air volume inside the mouthpiece and the of research. Several examples of modal analysis of iso- barrel is modeled with linear tetrahaedric and prismatic lated clarinet reeds have been presented over the recent finite elements of compressible elastic fluid. The acous- years using holographic interferometry [11, 12, 13, 14]. tic pressure at points of the open air surfaces is considered Two examples of finite-element modeling based on mea- to be zero as well as the normal derivative of the acoustic surements of the mechanical properties of cane have been pressure (corresponding to air flow) on the walls of the reported [15, 16]. The present model treats the reed as a mouthpiece and the barrel. The boundary condition cou- complex continuous system in association with the air- pling the reed and the mouthpiece involves the stress in column: fluid and solid finite element models (FEM) for the solid and the velocity of the fluid. The precise for- FIGURE 2. Modes at 2417, 4158, and 7020 Hz of an isolated reed.

mulation is given in [17]. Computed modes in the cou- ume. It appears that variations in eigenfrequencies due to pled situation match well the modal shapes on real reeds the model change are significant with regard to the align- as measured by holographic interferometry. The modal ment of resonances, even at low frequencies. In other acoustic pressure at an eigenfrequency of 4119 Hz is dis- words, the misalignment of peaks in either model is of played in Fig. 3. In this mode, the reed undergoes torsion the same order of magnitude as the frequency shifts due with a characteristic distance smaller than the half the to the presence of the reed and the prismatic shape of the

wavelength in air at that frequency (λ  10 cm); the re- mouthpiece. sulting acoustical short-circuit prevents any efficient cou- pling of the reed to the air in the mouthpiece. This ex- plains the fairly uniform acoustic pressure for this mode, REFERENCES except very near to the reed. 1. C. J. Nederveen, Acoustical aspects of woodwind instru- VAL − ISO >−2.51E−04 < 2.51E−04 ments 2nd ed., Illinois University Press, Dekalb, 1998. −2.47E−04

−2.24E−04 −2.00E−04 2. J. Gilbert, Ph.D. thesis, Université du Maine - Le Mans, −1.77E−04

−1.53E−04 −1.30E−04 1991. −1.06E−04

−8.24E−05

−5.89E−05 −3.53E−05 3. X. Boutillon and V. Gibiat, J.A.S.A. 100, 1178Ð89 (1996). −1.18E−05

1.18E−05 3.53E−05 4. R. Schumacher, Acustica 48, 73Ð85 (1981). 5.89E−05

8.24E−05

1.06E−04 1.30E−04 5. M. Mcintyre, R. Schumacher, and J. Woodhouse, J.A.S.A. 1.53E−04

1.77E−04 2.00E−04 74, 1325Ð45 (1983). 2.24E−04

2.47E−04 6. C. Maganza, R. Causse, and F. Laloe, Europhysics Letters 1, 295Ð302 (1986). FIGURE 3. Computed eigenmode at 4119 Hz in a mixed solid- 7. B. Gazengel, J. Gilbert, and N. Amir, Acta Acustica 3, 445Ð air situation - Acoustic pressure inside the mouthpiece and bar- 72 (1995). rel. 8. E. Ducasse, Journal de Physique 51, 837Ð40 (1990). In order to simulate the modal behavior of the com- 9. S. Stewart and W. Strong, J.A.S.A. 68, 109Ð20 (1980). plete clarinet, the FEM of the top of the pipe is associated 10. S. Sommerfeldt and W. Strong, J.A.S.A. 83, 1908Ð18 with lumped elements representing the rest of the pipe. (1988). Simulations such as presented in Fig. 3 show that the 11. P. Hoekje and G. Roberts, J.A.S.A. 99, 2462 (A) (1996). acoustical field consists essentially of plane waves. The 12. I. Lindevald and J. Gower, J.A.S.A. 102, 3085 (A) (1997). rest of the pipe can therefore be adequately represented by its acoustic input impedance. The lumped-element os- 13. F. Pinard and B. Laine, “Etude préliminaire d’une anche de clarinette,” Ecole Polytechnique (1997). cillators are coupled to the finite elements by means of a rigid plate with negligible mass. We used measurements 14. B. Richardson, personal communication. (1999). provided by Vincent Gibiat. Each impedance peak is as- 15. D. Casadonte, J.A.S.A. 94, 1807 (A) (1993). sociated with an oscillator represented in its generic form 16. B. Laine and F. Pinard, “Etude numérique des modes in Fig. 1. The eigenmodes and eigenfrequencies have propres d’une anche de clarinette,” Ecole Polytechnique been computed for several fingerings of the instrument. (1998). We have then revisited the classical question of the har- 17. M. Facchinetti, X. Boutillon, and A. Constantinescu monicity of the eigenfrequencies. The traditional model (2001), submitted to the J.A.S.A. of the mouthpiece is that of a cylinder of equivalent vol- The Sounding Pitches of Brass Instruments D. M. Campbella,J.Gilbertb and A. Myersa aDepartment of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, U.K. Email: [email protected] bLaboratoire 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 to the lips is described in terms of the input impedance (in the frequency domain) or the input impulse response (in The long-term goal of current research in the acoustics the time domain). Experimental methods are available for of lip-reed excited instruments is to find a physical model measuring these properties of real instruments [5]. of the complete system including the player’s windway The assumption is often made that in a wind instru- and lips, the instrument resonator and the radiated sound ment only the lowest frequency plane wave mode can field. The generation of sound in a brass instrument oc- exist. This is not strictly true; in the vicinity of a bore curs through aeroelastic coupling between the mechan- discontinuity or a side hole evanescent higher modes are ical resonances of the lips and the acoustic resonances present, and in the flaring bell of a trumpet or trombone of the instrument’s air column [1]. The model should higher modes can propagate. These higher modes can af- correctly predict the threshold mouth pressure at which fect the reflection coefficient for the plane wave mode [6], destabilisation of the coupled system occurs and the play- and therefore influence the coupling with the lips. ing frequency just above threshold. It should also be able The characterisation of the air column in terms of its to reproduce the large amplitude behaviour characteristic input impedance is based on the assumption of linear of fortissimo playing in brass instruments, for which the sound wave propagation in the air column. It is now internal acoustic pressure can be greater than 180 dB. In well established that this linearity breaks down in very addition, features of brass playing technique such as the loud playing on the trombone, leading to the formation player’s ability to pull the frequency of the sounded note of shock waves [7]. The resultant transfer of energy from above or below the frequency of an acoustic resonance low to high frequency acoustic modes dramatically alters ("lipping") should be demonstrated by the model. the timbre of the sound, and by affecting the interaction One approach to this formidable task is to attempt to with the lips could in principle also modify the sounding develop separate models of the different subsystems, and pitch. 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 res- MODELLING THE LIPS onator, since the coupled system formed by these two ele- ments essentially determines the playing pitches of the in- In the nineteenth century Helmholtz modelled the strument [2, 3, 4]. The present paper describes some cur- brass player’s lip as a mechanical oscillator with a sin- rent work in this vigorous and expanding research field. gle 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 ade- AIR COLUMN RESONANCES quately represent its vibratory behaviour. Indeed, stro- boscopic observation has revealed that the motion of a Most models of the resonating air column of the in- brass-player’s lip is often far from one-dimensional [9, strument treat it as an acoustic waveguide, terminated at 10]. Nevertheless, the Helmholtz "swinging door" or the upstream end by the lips and at the downstream end "outward-striking" model describes many of the prop- by the radiation impedance of the open bell. The coupling erties of real lips, and physical models of trumpets and trombones using single degree of freedom lip models non-linear energy loss mechanisms at side holes and bore have generated remarkably realistic sounds [11, 12]. discontinuities if the model is to be capable of predicting It has also been shown that important aspects of the the sounding pitch and timbre at realistic playing levels. 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 ACKNOWLEDGMENTS from a trumpet or trombone which are difficult to distin- guish from notes played by a human performer [14]. The authors are grateful to many colleagues at Edin- A tentative conclusion which may be drawn from these burgh and Le Mans for helpful discussion and collabo- observations is that, while the complexities of the human ration. Financial support from EPSRC, CNRS and the lips must influence the more subtle aspects of brass in- Royal Society is acknowledged. strument 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 mechan- REFERENCES ical response can be readily measured. Such measure- ments on embouchures capable of sounding a trombone 1. S. J. Elliott and J. M. Bowsher, J. Sound Vib. 83, 181-217 have shown that the lips typically display several mechan- (1982). ical resonances in the frequency range of the sounding 2. S. Adachi and M. Sato, J. Acoust. Soc. Am. 99, 1200-1209 pitches [4]. Some of these show the classic Helmholtz (1996). "outward striking" character, while others in contrast 3. F. C. Chen and G. Weinreich, J. Acoust. Soc. Am. 99, 1227- have the "inward striking" behaviour usually associated 1233 (1996). with woodwind reeds. These experimental observations 4. J. S. Cullen, J. Gilbert and D. M. Campbell, Acustica 86, are useful in guiding the choice of lip model and in sup- 704-724 (2000). plying realistic values of model parameters. 5. D. B. Sharp, Acoustic Pulse Reflectometry for the Measurement of Musical Wind Instruments, PhD Thesis, University of Edinburgh (1996) LIP-RESONATOR COUPLING [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 Septem- The simplest model of the interaction between the lips ber 2000, 314-317. and the resonator is one in which a single mechanical lip mode is coupled to a single acoustic mode of the air col- 7. A. Hirschberg, J. Gilbert, R. Msallam and A. P. J. Wijnands, J. Acoust. Soc. Am. 99, 1754-1758 (1996). umn. It has been shown that this model is incapable of displaying the lipping technique [4]. On the other hand, 8. H. J. F. Helmholtz, On the Sensation of Tone 1877, trans. linear stability analysis of a model which simultaneously A. J. Ellis, repr. Dover 1954. couples two mechanical lip modes and one acoustic mode 9. D. C. Copley and W. J. Strong, J. Acoust. Soc. Am. 99, has shown that it can exhibit behaviour equivalent to lip- 1219-1226 (1996). ping. A model in which two lip modes with constant pa- 10. R. D. Ayers J. Acoust. Soc. Am. 108, 2567 (2000). rameters are coupled to a single mode of a cylindrical res- 11. R. Msallam, S. Dequidt, R. Caussé and S. Tassart, Acustica onator of variable length predicts a threshold playing fre- 86, 725-736 (2000). quency which changes continuously from below to above 12. C. Vergez and X. Rodet, Acustica 86, 147-162 (2000). the acoustic resonance frequency as the tube length is in- 13. J. Gilbert, S. Ponthus and J.-F. Petiot, J. Acoust. Soc. Am. creased, in qualitative agreement with experiments using 104, 1627-1632 (1998). the artificial lips [15]. 14. J. Gilbert and J.-F. Petiot, Proceedings of International A potential application of brass instrument modelling Symposium on Musical Acoustics, Edinburgh August is in the evaluation of the playing properties of historic 1997: Proc. Inst. Acoustics 19(5), 391-400 (1997). instruments [16]. Museum curators are increasingly un- 15. M. A. Neal, O. Richards, D. M. Campbell and J. Gilbert, willing to allow early instruments to be blown by human Proceedings of the International Symposium on Musical players because of the destructive effect of warm, moist Acoustics, Perugia, September 2001 (in press). breath. A measured input impulse response could be cou- 16. A. Myers, Characterization and Taxonomy of Historic pled to a realistic lip model to allow the instrument to be Brass Musical Instruments from an Acoustical Standpoint, virtually played, giving valuable information to makers PhD thesis, University of Edinburgh (1998). of reproductions. It will be necessary to take into account 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 two- dimensionalized 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 z- axes 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.

(a) 9 (b) 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 trajectory 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 vy- toward the negative y direction is estimated as 5.3/2.6 = 2.0 m/s. The propagation velocity vz+ 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 vz- 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 1/2 isotropic soft tissues can be approximated as cs = 0.955 (G ) , where G and are the shear modulus and the 3 3 3 2 density of the lip, respectively. Since is close to be 1.1 × 10 kg/m , G is evaluated as 1.21 × 10 cs . Next problem is to determine which velocity obtained in our measurement should be the most relevant to cs . The averaged value vz = (v z+ + v z- )/2 seems to be the best for cs since this vz 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 2 3.5 × 10 N/m for player B with respect to F2 tone. It was difficult to define an appropriate wave velocity for non- advanced 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. 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 reed- pipe 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 experimental work discussed in this paper involves the khaen and the sheng. Impedance curves The Asian free reed mouth organs employ metal free have been measured for several pipes from a sheng and reeds mounted in resonating pipes. Unlike the free for simulated khaen tubes constructed from PVC pipe. reeds found in Western instruments such as the reed organ, accordion, and harmonica, the reeds of these IMPEDANCE CURVES instruments are approximately symmetric, so that the same reed can operate on both vacuum and pressure For khaen pipe impedance measurements, the pipes (inhaling and exhaling). The best known of these are used were constructed from PVC tubing with perhaps the sheng of China and the sho of Japan. For a dimensions similar to actual khaen pipes. Each of the detailed survey of the Asian free reed instruments see PVC khaen pipe impedance curves was measured the article by Miller [1]. following the method of Benade and Ibisi [3]. A piezo Figure 1 shows typical reeds from a sheng and an buzzer was positioned in place of the reed. A small American reed organ. The khaen (also spelled kaen, hole was drilled, just large enough for a probe khene, caen) is a free reed mouth organ indigenous to microphone tip to be inserted so that the probe tip was northeastern Thailand and Laos and is perhaps the most inside the tube near the position of the reed. With the important musical instrument of the Lao people of this microphone connected to a spectrum analyzer, spectra region. The sheng, sho, and khaen all employ one reed were obtained using a swept sine wave. For the sheng, per pipe, thus requiring a separate pipe for each pitch. 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 These measurements appear to be the first input organ (right) from Gellerman [2]. impedance measurements on khaen pipes. Some typical results are shown in Figures 2 and 3. Previous The sheng employs a free reed at the end of a closed observations and measurements had concluded that pipe resonator of conical-cylindrical cross section, such a free reed pipe will sound at a frequency above while in the khaen the free reed is placed at the reed frequency and close to a resonant frequency of approximately one-fourth the length of an open the pipe [4,5]. This resonant frequency is normally the cylindrical pipe. In both cases the effective length of fundamental, but if the length of the tube places the the pipe is determined by one or two tuning slots cut in fundamental below the reed frequency, the reed-pipe the pipe. These slots not only shorten the effective can sound near the second harmonic of the pipe [4]. 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 . FIGURE 2. Impedance curve (dashed) and sound spectrum FIGURE 4. Impedance curve (dashed) and sound spectrum (solid) of a 59 cm PVC khaen tube with no end sections. (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

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 FIGURE 3. Impedance curve (dashed) and sound spectrum holes, the sounding frequency can be pulled by the pipe (solid) of a 59 cm PVC khaen tube with end sections. resonance nearly an octave above the reed frequency [5].) Previous measurements of reed motion using a Another earlier observation [4] confirmed and to laser vibrometer verify that, for these instruments, the some extent explained by these results is the difference free reed tongue makes only slight excursions in the in tone quality observed in comparing khaen pipes with upstream direction, thus approximating the outward and without the end sections. This difference in tone striking reed model [8]. quality seems closely related to the differences observed here in the input impedance curves for the REFERENCES two cases, illustrated by comparing Figures 2 and 3. 1. Miller, T.E., "Free-Reed Instruments in Asia: A Preliminary Classification," in Music East and West: Essays in Honor of Results for the sheng pipes Walter Kaufmann, Pendragon, New York, 1981, pp. 63-99. 2. Gellerman, R.F., The American Reed Organ and the The sheng pipes are cylindrical over most of their Harmonium, Vestal Press, 1996, pp. 3-4. length, but the lower portion in which the reed is 3. Benade, A.H. and Ibisi, M.I. , J. Acoust. Soc. Am. 81, 1152- 1167, (1987). mounted is conical. The frequencies of the resonances 4. Cottingham, J.P. and Fetzer, C.A., "Acoustics of the khaen," identified by the impedance peaks are not harmonic. Proceedings of the International Symposium on Musical Calculations have been made of the impedance maxima Acoustics, Leavenworth, Washington, 1998, pp. 261-266. modeling the sheng pipe as a compound conical- 5. Cottingham, J.P, "Acoustics of a symmetric free reed coupled to a pipe resonator," Proceedings of the Seventh International cylindrical horn following Fletcher and Rossing [6]. Congress on Sound and Vibration, Garmisch-Partenkirchen, These give qualitative results similar to the measured Germany, 4-7 July 2000, pp. 1825-1832. impedance curves, but do not accurately predict the 6. Fletcher, N.H. and Rossing, T.D., The Physics of Musical nd frequency ratios of the impedance maxima. The Instruments, 2 ed., Springer, New York, 1998, pp.. 216-217. pattern observed for the sheng is that the sounding 7. Fletcher, N.H., Acustica 43, 63-72 (1979). 8. Busha, M. and Cottingham, J.P., J. Acoust. Soc. Am. 106, frequency of the pipe is again above that of the first 2288, (1999). 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 Each frame in a moving picture of the total disturbance is a sinusoidal function of x. A few of

Two undamped sinusoidal plane waves of the same these are shown at an interval of Dt = T / 12 from -T / 2 wavelength l propagate in opposite directions in a to 0, with T = the period. The height of a crest in the uniform pipe. All acoustic pressures are normalized to total wave ( V) must vary as it shifts in order to squeeze the amplitude of the stronger wave, which travels to the through each node. The curve traced by a crest is right (+ x direction). The amplitude of the weaker wave 2 2 1/2 is R, the pressure reflection coefficient at the origin, pcrest(x) = (1 - R ) / [1 + R - 2Rcos(2kx)] . (2) and here R is limited to positive values. Thus at time t = 0 both pressure waves have crests at x = 0. At each The velocity of the total wave varies, whether one point in the region x < 0, where the waves overlap, the focuses on a crest or on a point of contact with the total acoustic pressure varies sinusoidally in time. The upper envelope curve (temporal maximum, O). For amplitude of that temporal variation is either type of feature the average velocity over a half period is v = l / T to the right. Within that interval the 2 1/2 Ap(x) = [1 + R + 2Rcos(2kx)] , (1) variation can be considerable, as shown in Fig. 2. Crests travel rapidly through nodal regions and slowly where the propagation number k = 2p / l. Imperfect near antinodes, while temporal maxima show the nodes and antinodes are visible in the amplitude opposite behavior. The two normalized velocities have envelope, which is the pair of dotted curves ± A p (x) in the same time dependence except for a shift by T/4: Fig. 1. (The value R = 0.5 results from dividing the 2 2 area of the pipe’s cross section by three for x > 0.) vx, feature / v = (1 - R ) / [1 + R ± 2Rcos(2w t)], (3)

with + for a crest, - for a contact point, and w = 2p / T. 1 3

0

1.5

-1

Relative Velocity Relative Acoustic Pressure -0.5 -0.25 0 0 -0.5 -0.25 0 x / l t / T

FIGURE 1. Amplitude envelope (dotted) and lurching wave FIGURE 2. Normalized feature velocities: solid = crest; plots (solid) for R = 0.5: V = crest; O = contact point. dotted = contact (temporal maximum). The behavior of the normalized particle velocity are just the squares of the normalized acoustic is similar to that of the acoustic pressure. In this pressure and particle velocity, respectively. The example its reflection coefficient at x = 0 is just potential energy density and its envelope are -R, so we can adapt the results obtained for shown in Fig. 3(a). The kinetic energy density has pressure. The amplitude envelope for x < 0 gets the same envelope shape, but shifted by l / 4. The shifted by l / 4, placing a node at x = 0, and the signs total energy density and its envelope are shown in for the two velocities in Eq. (3) are exchanged. Fig. 3(b). Finally, Fig. 4 shows the envelope and The antinodes and nodes of the two amplitude instantaneous plots for the acoustic intensity, envelopes are the only locations where acoustic pvx,particle, the power flux per unit area. The pressure and particle velocity are in phase with horizontal lines at 0 and 1 are the upper limits for each other, as in a single traveling wave. At those the two envelope curves, independent of the value points the lurching wave can be coupled without of R. reflection to an incident wave in an upstream pipe of appropriate diameter. For the odd integer multiples of l / 4 between a node and an antinode, IMPLEMENTATION AND USES the diameter of the lurching wave segment must be the geometric mean of the diameters on either side The software was developed in MATLAB,â of it to avoid reflection. For the integer multiples version 6. Scrollbar controls on the screen for R of l / 2 between two antinodes or two nodes, the and other parameters make this a hands-on tool. diameters upstream and downstream from the Students can check their solutions to analytic lurching wave segment must be equal. problems by writing program lines for calculating the dependent variables. Those who are just learning to use complex exponentials can verify ENERGY AND INTENSITY that they give the same results as real sinusoids. This software is also useful for studying waves in The potential and kinetic energy densities also higher dimensions, or those with damping in show lurching behavior. Their normalized values propagation. Input impedance, pressure reflection coefficient, and effective length, all treated as (a) functions of axial position as well as frequency, 2 can also be examined in non-uniform air columns.

1 ACKNOWLEDGMENTS

This work has been supported by the Paul S.

Rel. P.E. Density 0 Veneklasen Research Foundation and the CSULB -0.5 -0.25 0 Scholarly and Creative Activities Committee. We are grateful to Nader Inan for his assistance. (b) 1

2

1 stic Intensity

0 0 -0.5 -0.25 0 Rel. Tot. Energy Density Relative Acou x / l -0.5 -0.25 0

FIGURE 3. Lurching behaviors of energy densities and x / l their envelopes: (a) potential energy density, (b) total energy density; R = 0.5 and Dt = T/12 for -T/2 < t < 0. FIGURE 4. Acoustic intensity and envelope. 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 the width H of the two channels are kept constant: h=1mm and H=20mm. The foot is supplied with a Edgetone oscillations have been experimentally mixture of N2O2 and CO2 from a 50Bar bottle of and theoretically investigated by several authors, see compressed gas via a regulator. The mean foot for instance Holger and Crighton [1,2]. All studies pressure is measured using a manometer DIGITRON have been carried out on a geometry with a long 2020P. channel insuring a fully developed flow before the The labium consists in a metal plate on which flue exit. Powell [3], studied aero-acoustical sources strain gauges are glued, allowing force measurements at the labium of the edgetone in terms of Lighthill’s [6]. The metal plate is clamped on one end in a heavy analogy [4]. Using ad hoc assumptions, he shows steel jaw. The edge of the plate has 90¡ cut (±45¡ that the force term is dominant and he relates from the channel axis) and is placed at a distance radiated sound to force induced at the labium by the W=9mm from the flue exit. In the case of the channel jet movement. He confirmed it experimentally using with chamfers at the end, the distance between jet a setup where the jet is issuing from a long flue separation points and the labium is increased of channel. 0.7mm. Previous experimental observations carried on a The radiated sound is measured using a B&K 1/4” complete recorder flute configuration [5] have shown microphone connected to an FFT analyzer, the that small geometrical modifications of the flue microphone is placed at a distance x=23cm above the channel geometry could have great effects on the jet labium. st nd rd stability. Therefore these parameters, known as Measurements are carried on the 1 2 and 3 crucial by recorder makers, are suspected to oscillating modes of the edgetone. Force influence sound production. In this paper we discuss measurements presented in this paper are limited to st st in an edgetone configuration the effect of the channel the 1 mode. The frequency of the 1 mode ranges geometry on the sound source, through from 400Hz to almost 2kHz. The Reynolds number measurements of the jet force on the labium as well varies within a range 300-2500. Undesirable as measurements of the radiated sound. acoustical reflections are limited as much as possible by placing glass wool blocks in the directions of EXPERIMENTAL SETUP maximal radiation of the edgetone.

The jet is issuing from a straight flue channel made RESULTS of a removable block embedded in a fix volume, the foot. Three blocks are used providing three different Force measurements are presented Figure 1 as a channelgeometries: a long straight channel of length function of the mean jet speed Uj. The force is made L=18mm, for which a Poiseuille flow is expected in dimensionless using the total jet force: the flow at the flue exit, a long channel with 1 h /2 chamfers at its exit (where edges are cut at ~45¡ on F = (ρ H ∫ U (y) 2 dy) (1) a 2 0 −h / 2 s 1mm length) and a short channel for which a nearly square velocity profile is expected. The height h and ρ where 0 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 10 20 30 0.1 Uj (m/s) Figure 2: Comparison of the force measured at the 5 10 15 20 30 labium ( ) and the force deducted from radiated sound measurements (--) as a function of the mean Uj (m/s) jet speed Uj. Long channel, no chamfers. Figure 1: Force measured at the labium as a function profile change at the flue exit, the two collapse. of the mean jet speed Uj for a flue exit/labium distance of 9mm. (-) short channel, (--) long channel, When using a short channel, the edgetone oscillation no chamfers, (-.) threshold is lower. This behavior is also present on oflong the channelchannel length [5]. An withexample of comparisonchamfers. frequency measurements and is the consequence of a between the measured force and the force deduced higher propagation velocity on the jet when the from radiated sound measurement using Powell’s velocity profile is sharp [5]. The transition velocity is approximation is shown Figure 2, in the case of the also lowered. Ségoufin&al. [5] have shown that long channel without chamfers. The two curves fit when the channel is shortened, the jet flow is more very well with an evolution pattern that is common to complex and exhibits numerous developed vortical the three configurations. There is however a slight structures. Maybe this results in turbulence appearing oscillation of the force deduced from radiation that at a lower jet velocity. It has to be noted that despite could be related room resonances, although we tried the presence of turbulence before the labium, the to limit it to the maximum. force term induced by the jet movement is still dominant. DISCUSSION Adding chamfers to the flue exit of the long channel reduces dramatically the sound production. This effect cannot be explained in terms of velocity Results obtained for our long channel confirm Powell's data and validates our experimental setup. profile modification as it is assumed that chamfers don’t modify it. We don't have any explanation for The driving pressure range has been extended and sound production exhibits a sharp decline. We this effect at the time being, and further investigations would involve numerical simulation. suspect this is due to turbulence appearing before the jet reaches the labium. This can be checked by flow REFERENCES visualization. The same good fit between measured and deduced force was observed in the three configurations, which means that the dipolar source 1. A.W. Nolle, J. Acoust. Soc. Am., 103, 3690-3705, 1998 2. D. Crighton, J. Fluid. Mech., 234, 361-391, 1992 induced by the jet movement on the labium is always 3. A. Powell, J. Acoust. Soc. Am., 33, 395-409, 1961 dominant. 4. M. J. Lighthill, Proc. Roy. Soy. Lond., A 211, 564-587, The configurations can be more carefully compared 1952 & A 222, 1-32, 1954. using Figure 1. The dimensionless force exhibits a 5. C. Ségoufin, B. Fabre, M.P. Verge, A. Hirschberg and similar aspect in all the cases, being approximately A. P. J. Wijnands, Acta Acustica, 86, 649-661, 2000 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 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, The instruments studied are similar but not identical oboe, clarinet and saxophone, a chromatic scale may to those studied previously [6]. An acoustic current be played over at least one octave simply by opening was synthesized having frequency components from tone holes, one at a time, starting at the downstream 0.2 to 3 kHz with 2 Hz increments, and input to the end. Mechanisms that couple the keys allow the embouchure hole via a short pipe whose impedance player to cover more tone holes than s/he has fingers. approximates the radiation impedance that normally On the recorder, and on the ancestors of the modern loads the instrument at this point when it is played. flute, oboe and clarinet, opening successive finger To measure the frequency dependence of propagation holes produces a diatonic scale. The missing notes in of the standing wave, a probe microphone was the chromatic scale are achieved by cross fingering. In inserted at the embouchure, and also at the open or both old and new instruments, a second register is closed tone holes The technique for the measurement obtained by overblowing, using nearly the same of transfer functions is described elsewhere [6], except fingerings. Third and higher registers involve cross that a probe microphone (external diameter 1.0 mm) fingering. was used for all measurements. 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 RESULTS AND DISCUSSION sufficiently low, the shunt approximates an acoustic short circuit. So, for instruments with large holes The first plot shows the input impedance spectrum of (flute and saxophone) cross fingering has relatively a reproduction of a 19th century flute (chosen because little effect in the lowest register. Various authors of its relatively small tone holes) for the fingering have modelled tone holes with a range of used to play E4 or E5. At low frequencies, the complications[1-5]. spectrum is not very different from that of a shorter Although the standing wave is attenuated by the version of the bore, as if it were terminated a little tone hole, it still extends past it along the bore, beyond the first open hole. The flute, being played giving an extra length or end effect that varies with with the embouchure open to the radiation field, frequency. The end effect decreases as the size of the operates at minima in impedance. For most tone hole increases, and depends (to a lesser degree) fingerings, the first few impedance minima are on the height of its chimney, the extent of covering harmonically spaced. (See [6-9] for detailed by keys, and the degree of undercutting (i.e. discussions.) At higher frequencies, the minima are chamfering of the junction between the tone hole and weaker (due to wall losses) and, in the example the bore). shown here, they also occur at frequencies lower than In this study we measure the frequency dependence harmonic multiples of the fundamental, due to the of the propagation of the standing wave beyond the greater propagation of the wave beyond the first open first open tone hole for simple and cross fingerings. tone hole. __|Z| acoustic pressure increases with length past the first MΩ closed hole. This increases the effective length of the 10 bore with frequency, and so the first minimum is 5 flattened, and the second a little more, etc. This allows the production of flattened notes, but it also 2 produces inharmonic minima, which do not support 1 the high harmonics in a strongly non-linear vibration rŽgime, and hence contribute to the darker timbre of .5 cross fingered notes. .2 Musically useful discussion requires much greater detail. Discussions of fingering effects and a much .1 larger set of results are given on our web site [9]. .05 f/kHz __|Z| MΩ FIGURE 1. The acoustic impedance spectrum, in 10 MPa.s.m−3, for a simple fingering that plays E4/5. 5 E 678 2 1 .5 6 .2 .1 7 .05 f/kHz 8 FIGURE 3. The impedance for a cross fingering.

f/kHz ACKNOWLEDGMENTS

FIGURE 2. The ratio of the pressure at the open tone We thank John Tann and Terry McGee. This work holes to the pressure at the embouchure. was supported by the Australian Research Council. 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 REFERENCES pressure at the 6th, 7th and 8th tone holes to that at the embouchure for the same acoustic current at the 1. J.W. Coltmann, J.Acoust. Soc. Amer., 65, pp. 499-506 (1979). embouchure. The sharp peaks coincide with the 2. W.J. Strong, N.H. Fletcher and R.K. Silk, J. Acoust. Soc. minima in Z: other frequencies do not generate a Amer., 77, pp. 2166-2172, (1985) strong standing wave for this fingering. At the first open hole (hole 6), the first several 3. D.H. Keefe, J. Acoust. Soc. Amer., 72, pp. 676-687, 1982. peaks (those corresponding to the harmonic minima in Z) have similar magnitudes, but they become 4. C J. Nederveen, J.K.M. Jansen and R.R. van Hassel, Acustica weaker at higher frequencies. Further down the bore 84, pp. 957-966, (1998). (holes 7 & 8), the penetration increases with 5. V. Dubos, J. Kergomard, A. Khettabi, J-P. Dalmont, D.H. frequency, up to the cut off frequency for the array of Keefe, C.J. Nederveen, Acustica 85, pp. 153-169 (1999). open tone holes (for this instrument around 2 kHz), above which there is little difference with position, 6. J. Wolfe, J. Smith, J. Tann, and N.H. Fletcher, J. Sound & and the standing wave propagates freely [8]. In Vibration, 243, 127-144 (2001). modern instruments with larger tone holes, (data not 7. N.H. Fletcher and T.D. Rossing, The Physics of Musical shown), the penetration increases less rapidly with Instruments. New York, Springer-Verlag, 1998 frequency and the cut off frequency is higher. When cross fingerings are used, the analogous 8. A.H. Benade J. Acoust. Soc. Amer. 32, 1591-1608 (1960). curves are more complicated. The wave propagates strongly down the bore, and for most frequencies, the 9. J. Wolfe, J. Smith and J. Tann. Flute acoustics www.phys.unsw.edu.au/music/flute Tones and Wave Forms of the Shakuhachi

S. Takahashi and T. Matsui

aDepartment 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 man- machine 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 the relationships between waveforms and sounds is part of an attempt to make artificial sounds that have many When one of the authors was beginning to learn to analogies with the sounds produced by real shakuhachis. play the shakuhachi, he listened to the long tones of If the artificial sounds seem to be real, we can consider the shakuhachi, with their changing timbres, at a that we have determined the characteristics of master player’s recital. This was about twenty years shakuhachi sounds. ago, and the player is no longer with us. The cassette tape the author recorded at the time doesn’t SOME RESULTS accurately reproduce the variation timbres because of the bad recording conditions. In playing the By investigating recorded sounds of a shakuhachi, we shakuhachi, a common technique is to play with a were able to determine several characteristics of the constant pitch and varing timbre. The timbre makes shakuhachi’s sound. The shakuhachi is characterized by the listener experience bright and then dark feelings, the beginnings and endings of the sounds it is made to as the tone with its initially bright timbre fades out. make. The method of playing is truly fascinating. The At high pitches, the shakuhachi is not very different from variation in timbre gives the listener the feeling that other wind instruments. The sounds of the shakuhachi the bright and dark timbres are as authentic the look like sine waves. At low pitches, however, a loud perception red and blue as warm and cold colors. sound often alternates between two wave shapes that are The sound of the shakuhachi brings the listener to different from each other, as shown in Figure 2(1). This wonder about the relationship between the physical means that listener hears loud sounds as low pitches due characteristics of sounds and mental feelings. The to the large amplitudes of the harmonics despite the physical characteristics of shakuhachi sound have small low-frequency fundamental component. The been studied [1]. In this paper, we outline a alternation between two wave shapes seems to reduce preliminary study of the characteristics of the sound the maximum amplitude. The timbre usually changes of the shakuhachi in terms of the relationship with the pitch or loudness of the sound produced by the between sounds and mental feelings. This general shakuhachi. When playing the shakuhachi, it is very area of study could become important for the man- difficult to obtain varied timbers with the same pitch. machine interfaces of the future. Sounds in and around the middle of the shakuhachi’s pitch range are easier to play with more stability, so it is THE PROCESS OF THIS RESEARCH a little easier to vary the timber. Figure 1 shows a long tone produced by a player endeavoring to change the The scale of a shakuhachi and the various timbres of tone’s timbre while maintaining the same pitch of “g”, its sounds were recorded to digital audiotape (DAT) which is notated as by shakuhachi players.We in an anechoic room. The several pitches are happened to find two waveforms with similar spectra but produced by two fingerings, and each pair of them with the harmonic components in different phases as are the same pitch and different timbres. We shown in Figure 2. recorded changes in timbre on long tones. Studying

Amplitude (dB)

Time (sec)

Frequency (Hz)

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

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 (1) (2) sine-wave-sound like as bright. We tried to make an artificial sound that was like this long tone. The tone FIGURE 2. Two Waveforms with the Same Spectra and was, however, satisfactory. Different Phases.

These waveforms are described by the following functions. CONCLUSION sint
sin(2t
2 / 3)
0.6sin 3t (1) Some characteristics of shakuhachi tones were 

 (2) sin t sin 2 t sin 3 t 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 REFERENCES Produce Fig.1. 1. Y. Yasuda, New Edition, The Acoustics of Instruments, Tokyo: The well-known unawareness of the phase of sound Ongakunotomosha, 1996, pp. 79-85(in Japanese). can be confirmed by listening to two with different

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