Patrizia

WIND INSTRUMENTS

Modal analysis of a complete clarinetM. Facchinettia,c, X. Boutillonb and A. Constantinescuc

aLaboratoire d’Hydrodynamique, CNRS - Ecole Polytechnique, 91128 Palaiseau Cedex, FrancebLaboratoire d’Acoustique Musicale, CNRS - Université Paris 6 - Ministère de la Culture, 11 rue de Lourmel, 75015

Paris, FrancecLaboratoire 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 thepipe, and a lumped-element model of the rest of the pipe. This is a continuation of an initial work by Pinard and Laine (unpublishedreports of the Ecole Polytechnique) on isolated reeds. The eigenmodes of the complete system are computed and the results lead to adiscussion 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.

m

r

k

FIGURE 1. Lumped elements model for the pipe, FEM for thebarrel, the mouthpiece, and the reed of the clarinet.

INTRODUCTION

The classical view of a clarinet associates a linear res-onator - the pipe - and a nonlinear excitor - the reed. Thepurpose of our approach is to include the reed in the linearpart and to restrict the nonlinear aspects to boudary con-ditions: the air flow from the mouth to the pipe throughthe reed-slit and contact forces between the reed and thecurved lay of the mouthpiece. The present analysis dealswith the linear aspects of the ensemble of the reed cou-pled to the pipe by means of a modal investigation of theinstrument.

The simplest reed model - a spring - has been ap-proached experimentally [1, 2, 3] and used in numeri-cal simulations which were successful in describing basicfeatures of the dynamics of a clarinet [4, 5, 6]. The fur-ther step in complexity is that of a single oscillator withvarious possible sophistications [7, 8, 9, 10]. Modelingthe reed as a true continuous system is the current stateof research. Several examples of modal analysis of iso-lated clarinet reeds have been presented over the recentyears using holographic interferometry [11, 12, 13, 14].Two examples of finite-element modeling based on mea-surements of the mechanical properties of cane have beenreported [15, 16]. The present model treats the reed as acomplex continuous system in association with the air-column: fluid and solid finite element models (FEM) for

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

MODELS AND RESULTS

The reed geometry has been carefully measured. Reedcane is considered as an elastic transversely isotropic, ho-mogeneous material. Since we are concerned with in-dividual modes of the reed, losses are ignored. Theywould have to be taken into account in the actual dy-namics of the instrument. Five parameters are neededto describe the material (in parenthesis, the values usedin the computation): density ρ (450 kg=m3), longitudinaland transverse Young’s moduli EL (10 000 MPa) and ET

(400 MPa), transverse to longitudinal shear modulus G LT

(1 300 MPa), and longitudinal-transverse Poisson coeffi-cient νLT (0.22). The reed is considered rigidly clampedon the section corresponding to the ligature and having astress-free boundary elsewhere. This model has been im-plemented using linear Love-Kirchoff plate elements inthe 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 repre-sented the mouthpiece of a wind instrument by its equiv-alent volume. To go beyond this approximation and tocompute the 3-D distribution of the pressure in the upperpart of the instrument, a coupled fluid-solid model hasbeen used. The air volume inside the mouthpiece and thebarrel is modeled with linear tetrahaedric and prismaticfinite elements of compressible elastic fluid. The acous-tic pressure at points of the open air surfaces is consideredto be zero as well as the normal derivative of the acousticpressure (corresponding to air flow) on the walls of themouthpiece and the barrel. The boundary condition cou-pling the reed and the mouthpiece involves the stress inthe 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-pled situation match well the modal shapes on real reedsas measured by holographic interferometry. The modalacoustic pressure at an eigenfrequency of 4119 Hz is dis-played in Fig. 3. In this mode, the reed undergoes torsionwith a characteristic distance smaller than the half thewavelength in air at that frequency (λ 10 cm); the re-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,except very near to the reed.

VAL − ISO

>−2.51E−04

< 2.51E−04

−2.47E−04

−2.24E−04

−2.00E−04

−1.77E−04

−1.53E−04

−1.30E−04

−1.06E−04

−8.24E−05

−5.89E−05

−3.53E−05

−1.18E−05

1.18E−05

3.53E−05

5.89E−05

8.24E−05

1.06E−04

1.30E−04

1.53E−04

1.77E−04

2.00E−04

2.24E−04

2.47E−04

FIGURE 3. Computed eigenmode at 4119 Hz in a mixed solid-air situation - Acoustic pressure inside the mouthpiece and bar-rel.

In order to simulate the modal behavior of the com-plete clarinet, the FEM of the top of the pipe is associatedwith lumped elements representing the rest of the pipe.Simulations such as presented in Fig. 3 show that theacoustical field consists essentially of plane waves. Therest of the pipe can therefore be adequately representedby its acoustic input impedance. The lumped-element os-cillators are coupled to the finite elements by means of arigid plate with negligible mass. We used measurementsprovided by Vincent Gibiat. Each impedance peak is as-sociated with an oscillator represented in its generic formin Fig. 1. The eigenmodes and eigenfrequencies havebeen computed for several fingerings of the instrument.We have then revisited the classical question of the har-monicity of the eigenfrequencies. The traditional modelof the mouthpiece is that of a cylinder of equivalent vol-

ume. It appears that variations in eigenfrequencies due tothe model change are significant with regard to the align-ment of resonances, even at low frequencies. In otherwords, the misalignment of peaks in either model is ofthe same order of magnitude as the frequency shifts dueto the presence of the reed and the prismatic shape of themouthpiece.

REFERENCES

1. C. J. Nederveen, Acoustical aspects of woodwind instru-ments 2nd ed., Illinois University Press, Dekalb, 1998.

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

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

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

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

6. C. Maganza, R. Causse, and F. Laloe, Europhysics Letters1, 295–302 (1986).

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

8. E. Ducasse, Journal de Physique 51, 837–40 (1990).

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

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 declarinette,” 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 modespropres d’une anche de clarinette,” Ecole Polytechnique(1998).

17. M. Facchinetti, X. Boutillon, and A. Constantinescu(2001), submitted to the J.A.S.A.

The Sounding Pitches of Brass InstrumentsD. M. Campbella, J. Gilbertb and A. Myersa

aDepartment of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, U.K.Email: D.M.Campbell@ed.ac.uk

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 aircolumn. The air column can be treated as an acoustic waveguide, but both non-planar modes and non-linear sound propagation maybe of significance. Some success has been achieved in modelling the lips as a mechanical oscillator with one degree of freedom, butstroboscopic observation has revealed that lip motion is relatively complex, and attention is now directed towards more sophisticatedmodels. Linear stability analysis has shown that a model coupling two mechanical lip modes and one acoustic mode can reproduce theessential features of the "lipping" technique. One application of brass instrument modelling is the prediction of the sounding pitchesof historic brass instruments based on non-invasive input impedance or input impulse response measurements.

INTRODUCTION

The long-term goal of current research in the acousticsof lip-reed excited instruments is to find a physical modelof the complete system including the player’s windwayand lips, the instrument resonator and the radiated soundfield. The generation of sound in a brass instrument oc-curs through aeroelastic coupling between the mechan-ical resonances of the lips and the acoustic resonancesof the instrument’s air column [1]. The model shouldcorrectly predict the threshold mouth pressure at whichdestabilisation of the coupled system occurs and the play-ing frequency just above threshold. It should also be ableto reproduce the large amplitude behaviour characteristicof fortissimo playing in brass instruments, for which theinternal acoustic pressure can be greater than 180 dB. Inaddition, features of brass playing technique such as theplayer’s ability to pull the frequency of the sounded noteabove or below the frequency of an acoustic resonance("lipping") should be demonstrated by the model.

One approach to this formidable task is to attempt todevelop separate models of the different subsystems, andthen to incorporate these sub-models in a global model ofthe coupled system. In recent years much of the effort hasconcentrated on modelling the lips and the acoustic res-onator, since the coupled system formed by these two ele-ments essentially determines the playing pitches of the in-strument [2, 3, 4]. The present paper describes some cur-rent work in this vigorous and expanding research field.

AIR COLUMN RESONANCES

Most models of the resonating air column of the in-strument treat it as an acoustic waveguide, terminated atthe upstream end by the lips and at the downstream endby the radiation impedance of the open bell. The coupling

to the lips is described in terms of the input impedance (inthe frequency domain) or the input impulse response (inthe time domain). Experimental methods are available formeasuring these properties of real instruments [5].

The assumption is often made that in a wind instru-ment only the lowest frequency plane wave mode canexist. This is not strictly true; in the vicinity of a borediscontinuity or a side hole evanescent higher modes arepresent, and in the flaring bell of a trumpet or trombonehigher modes can propagate. These higher modes can af-fect 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 itsinput impedance is based on the assumption of linearsound wave propagation in the air column. It is nowwell established that this linearity breaks down in veryloud playing on the trombone, leading to the formationof shock waves [7]. The resultant transfer of energy fromlow to high frequency acoustic modes dramatically altersthe timbre of the sound, and by affecting the interactionwith the lips could in principle also modify the soundingpitch.

MODELLING THE LIPS

In the nineteenth century Helmholtz modelled thebrass player’s lip as a mechanical oscillator with a sin-gle degree of freedom [8]. The human lip is a complexmass of tissue and muscle, and it seems unlikely thatsuch a highly simplified mechanical model could ade-quately represent its vibratory behaviour. Indeed, stro-boscopic observation has revealed that the motion of abrass-player’s lip is often far from one-dimensional [9,10]. Nevertheless, the Helmholtz "swinging door" or"outward-striking" model describes many of the prop-erties of real lips, and physical models of trumpets and

trombones using single degree of freedom lip modelshave generated remarkably realistic sounds [11, 12].

It has also been shown that important aspects of thevibratory behaviour of human lips can be experimentallyreproduced using artificial lips made from water-filledrubber tubes [13]. The artificial mouth can obtain notesfrom a trumpet or trombone which are difficult to distin-guish from notes played by a human performer [14].

A tentative conclusion which may be drawn from theseobservations is that, while the complexities of the humanlips must influence the more subtle aspects of brass in-strument playing, the essential features of the non-linearinteraction with the air column can be modelled using amuch simplified lip model.

An advantage of the artificial lips is that their mechan-ical response can be readily measured. Such measure-ments on embouchures capable of sounding a trombonehave shown that the lips typically display several mechan-ical resonances in the frequency range of the soundingpitches [4]. Some of these show the classic Helmholtz"outward striking" character, while others in contrasthave the "inward striking" behaviour usually associatedwith woodwind reeds. These experimental observationsare useful in guiding the choice of lip model and in sup-plying realistic values of model parameters.

LIP-RESONATOR COUPLING

The simplest model of the interaction between the lipsand the resonator is one in which a single mechanical lipmode is coupled to a single acoustic mode of the air col-umn. It has been shown that this model is incapable ofdisplaying the lipping technique [4]. On the other hand,linear stability analysis of a model which simultaneouslycouples two mechanical lip modes and one acoustic modehas shown that it can exhibit behaviour equivalent to lip-ping. A model in which two lip modes with constant pa-rameters are coupled to a single mode of a cylindrical res-onator of variable length predicts a threshold playing fre-quency which changes continuously from below to abovethe acoustic resonance frequency as the tube length is in-creased, in qualitative agreement with experiments usingthe artificial lips [15].

A potential application of brass instrument modellingis in the evaluation of the playing properties of historicinstruments [16]. Museum curators are increasingly un-willing to allow early instruments to be blown by humanplayers because of the destructive effect of warm, moistbreath. A measured input impulse response could be cou-pled to a realistic lip model to allow the instrument to bevirtually played, giving valuable information to makersof reproductions. It will be necessary to take into account

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

ACKNOWLEDGMENTS

The authors are grateful to many colleagues at Edin-burgh and Le Mans for helpful discussion and collabo-ration. Financial support from EPSRC, CNRS and theRoyal Society is acknowledged.

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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, 1227-1233 (1996).

4. J. S. Cullen, J. Gilbert and D. M. Campbell, Acustica86,704-724 (2000).

5. D. B. Sharp, Acoustic Pulse Reflectometry forthe 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 of5ième Congrès Francais d’Acoustique, Lausanne Septem-ber 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 Tone1877, trans.A. J. Ellis, repr. Dover1954.

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, Acustica86, 725-736 (2000).

12. C. Vergez and X. Rodet, Acustica86, 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 InternationalSymposium on Musical Acoustics, Edinburgh August1997: Proc. Inst. Acoustics19(5), 391-400 (1997).

15. M. A. Neal, O. Richards, D. M. Campbell and J. Gilbert,Proceedings of the International Symposium on MusicalAcoustics, Perugia, September 2001 (in press).

16. A. Myers, Characterization and Taxonomy of HistoricBrass Musical Instruments from an Acoustical Standpoint,PhD thesis, University of Edinburgh (1998).

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

The experimental data were obtained when five (two advanced, two medium, and one beginning) players blew a commonnatural horn equipped with a transparent mouthpiece for the French horn. Three-dimensional understanding of the upper lipmotion 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-Dtrajectory of the wave crest on the upper lip surface illustrates the lip-wave propagation. These 3-D lip motion and 2-D lip-wavepropagation are superposed each other. The lateral wave propagation (with the speed of about 2 to 4 m/s) creates the outward lipopening in the lowest F2 (78 Hz) tone. Non-advanced players cannot generate a distinct lip wave. The vertical wave propagationgoverns the upward lip opening in higher-mode tones. The application of the Rayleigh-type surface-wave assumption and theaveraged vertical wave velocity (about 1.8 m/s) to the estimation of lip tissue elasticity approximates the shear modulus asabout 4000 N/m2. The generation and propagation of the lip wave was first defined quantitatively by the present stroboscopicvisualization.

The visualization such as stroboscopic viewing is essential to know real motion of a vibrating body such as thelips of brass players. Particularly it is very useful to observe real motion from various directions. Such observationmakes 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 naturalhorn (made by the Lawson Brass Instruments Inc.). An acrylic transparent mouthpiece was used for observation.

The x-, y-, and z-axes respectively tend to the outward direction almost parallel to the mouthpiece axis, the lateral direction alongthe lower lip surface, and the vertical direction. Note that a x-z plane for side-view analysis vertically cuts theupper lip at the center of the lip.

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.

3-D IMAGE OF THE UPPER LIP MOTION

(a)

(b)

Abstract: A slow-motion picture of the upper lip motion was taken by the stroboscope from the frontal and side directions.

The coordinate system for motion analysis of the lip opening is also indicated in left frames.

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.

the upper-lip tip [ in Fig. 1(b)] and the wave crest [ in Fig. 1(b) and cf. Fig 2(b)]. Particularly, when the lipmotion for higher modes is examined, we may recognize the x-z traj ectory of wave crest as an effective measure ofplayer'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-zplane 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 inx-z plane with lip-opening shape in y-z plane (see Fig. 2). The wave crest propagates almost along the y-axis fromframe #1 to #11 in Fig. 2(b) because the x-axis overlaps on the y-axis in our (slightly slant) side viewing. The crestappears near the lip edge in frame #1 and propagates toward the lip center in frame #11 [cf. Fig. 1(b)]. Thispropagation distance from the lip edge to the lip center may be estimated as about 5.3 mm from Fig. 2(a). Also, thetime 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- towardthe negative y direction is estimated as 5.3/2.6 = 2.0 m/s. The propagation velocity vz+ in the positive z direction isdirectly given as 2.7 m/s from the distance 4.1 mm and the interval 1.5 ms between frame #11 and #17. Similarlyvz- 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 tosuppose the Rayleigh-type surface wave by omitting any discussion on this validity, the surface-wave velocity ofisotropic soft tissues can be approximated as cs = 0.955 (G ) 1/2 , where G and are the shear modulus and thedensity of the lip, respectively. Since is close to be 1.1 × 103 kg/m3, G is evaluated as 1.21 × 103 cs

2 . Nextproblem is to determine which velocity obtained in our measurement should be the most relevant to cs . Theaveraged value vz = (v z+ + v z- )/2 seems to be the best for cs since this vz can minimize the influences of the blowingpressure and surface tension due to the embouchure. As a result, G is estimated as 4.1 × 103 N/m2 for player A and3.5 × 103 N/m2 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 waspostponed, our new information mentioned above will serve to establish proper lip modelling and simulations

(

In our experiment five French horn players, who consist of two advanced(A and B), two medium(C and D), and

observed. Although the motion of the upper lip in Fig. 1(a) looks 2-D, that of a beginning player E looks 3-D becauseof 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

in more realistic situations.

one beginning (E), played notes F2 to C5 with a common natural horn. The individuality of players is well

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Pipe Impedance and Reed Motion for a free ReedCoupled to a Pipe Resonator

James P. Cottingham

Physics Department, Coe College, Cedar Rapids, IA 52402, USA, jcotting@coe.edu

The Asian free reed mouth organs employ metal free reeds mounted in resonating pipes. Measurements of the input impedancecurves have been made for some representative pipes from these instruments. These complement earlier measurements of reedposition and phase using a laser vibrometer system, as well as measurements of sound spectra produced when the pipes areplayed. 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 firstimpedance peak of the pipe.

INTRODUCTION

The Asian free reed mouth organs employ metal freereeds mounted in resonating pipes. Unlike the freereeds found in Western instruments such as the reedorgan, accordion, and harmonica, the reeds of theseinstruments are approximately symmetric, so that thesame reed can operate on both vacuum and pressure(inhaling and exhaling). The best known of these areperhaps the sheng of China and the sho of Japan. For adetailed survey of the Asian free reed instruments seethe article by Miller [1]. Figure 1 shows typical reeds from a sheng and anAmerican reed organ. The khaen (also spelled kaen,khene, caen) is a free reed mouth organ indigenous tonortheastern Thailand and Laos and is perhaps the mostimportant musical instrument of the Lao people of thisregion. The sheng, sho, and khaen all employ one reedper pipe, thus requiring a separate pipe for each pitch.

FIGURE 1. Reeds from a sheng (left) and an American reedorgan (right) from Gellerman [2].

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

The experimental work discussed in this paperinvolves the khaen and the sheng. Impedance curveshave been measured for several pipes from a sheng andfor simulated khaen tubes constructed from PVC pipe.

IMPEDANCE CURVES

For khaen pipe impedance measurements, the pipesused were constructed from PVC tubing withdimensions similar to actual khaen pipes. Each of thePVC khaen pipe impedance curves was measuredfollowing the method of Benade and Ibisi [3]. A piezobuzzer was positioned in place of the reed. A smallhole was drilled, just large enough for a probemicrophone tip to be inserted so that the probe tip wasinside the tube near the position of the reed. With themicrophone connected to a spectrum analyzer, spectrawere obtained using a swept sine wave. For the sheng,bamboo pipes from a real instrument were used. In thiscase no hole was drilled, but the flexible probemicrophone tube was fed through the existing fingerhole, which was otherwise sealed with putty.

Results for the PVC khaen tubes

These measurements appear to be the first inputimpedance measurements on khaen pipes. Some typicalresults are shown in Figures 2 and 3. Previousobservations and measurements had concluded thatsuch a free reed pipe will sound at a frequency abovethe reed frequency and close to a resonant frequency ofthe pipe [4,5]. This resonant frequency is normally thefundamental, but if the length of the tube places thefundamental below the reed frequency, the reed-pipecan sound near the second harmonic of the pipe [4].

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

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 tosome extent explained by these results is the differencein tone quality observed in comparing khaen pipes withand without the end sections. This difference in tonequality seems closely related to the differencesobserved here in the input impedance curves for thetwo cases, illustrated by comparing Figures 2 and 3.

Results for the sheng pipes

The sheng pipes are cylindrical over most of theirlength, but the lower portion in which the reed ismounted is conical. The frequencies of the resonancesidentified by the impedance peaks are not harmonic.Calculations have been made of the impedance maximamodeling the sheng pipe as a compound conical-cylindrical horn following Fletcher and Rossing [6].These give qualitative results similar to the measuredimpedance curves, but do not accurately predict thefrequency ratios of the impedance maxima. Thepattern observed for the sheng is that the soundingfrequency of the pipe is again above that of the first

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

impedance peak, with the second harmonic of thesound spectrum apparently reinforced by the strongnearby peak in the impedance curve.

SUMMARY

The free reeds coupled to pipe resonators in theinstruments under consideration seem to behave as"opening" or "outward striking" reeds as discussed byFletcher [7]. The sounding frequency of the reed-pipecombination is above the natural frequencies of boththe reed and the pipe, regardless of the direction ofairflow. (For the bawu, a free reed pipe with fingerholes, the sounding frequency can be pulled by the piperesonance nearly an octave above the reed frequency[5].) Previous measurements of reed motion using alaser vibrometer verify that, for these instruments, thefree reed tongue makes only slight excursions in theupstream direction, thus approximating the outwardstriking reed model [8].

REFERENCES1. Miller, T.E., "Free-Reed Instruments in Asia: A Preliminary

Classification," in Music East and West: Essays in Honor ofWalter Kaufmann, Pendragon, New York, 1981, pp. 63-99.

2. Gellerman, R.F., The American Reed Organ and theHarmonium, Vestal Press, 1996, pp. 3-4.

3. Benade, A.H. and Ibisi, M.I. , J. Acoust. Soc. Am. 81, 1152-1167, (1987).

4. Cottingham, J.P. and Fetzer, C.A., "Acoustics of the khaen,"Proceedings of the International Symposium on MusicalAcoustics, Leavenworth, Washington, 1998, pp. 261-266.

5. Cottingham, J.P, "Acoustics of a symmetric free reed coupledto a pipe resonator," Proceedings of the Seventh InternationalCongress on Sound and Vibration, Garmisch-Partenkirchen,Germany, 4-7 July 2000, pp. 1825-1832.

6. Fletcher, N.H. and Rossing, T.D., The Physics of MusicalInstruments, 2nd ed., Springer, New York, 1998, pp.. 216-217.

7. Fletcher, N.H., Acustica 43, 63-72 (1979).8. Busha, M. and Cottingham, J.P., J. Acoust. Soc. Am. 106,

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 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) 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.)

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

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 pcrest(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.

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

-0.5 -0.25 0

-1

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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 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.

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.

IMPLEMENTATION AND USES 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.

ACKNOWLEDGMENTS 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.

FIGURE 4. Acoustic intensity and envelope.

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Experimental Results on the Influence of ChannelGeometry 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 fullydeveloped flow at its exit. In this paper, the influence of channel length and chamfers on sound production is investigatedexperimentally by studying aero-acoustical sources at the labium. The dipolar source induced by the jet movement on thelabium 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 experimentallyand theoretically investigated by several authors, seefor instance Holger and Crighton [1,2]. All studieshave been carried out on a geometry with a longchannel insuring a fully developed flow before theflue exit. Powell [3], studied aero-acoustical sourcesat the labium of the edgetone in terms of Lighthill’sanalogy [4]. Using ad hoc assumptions, he showsthat the force term is dominant and he relatesradiated sound to force induced at the labium by thejet movement. He confirmed it experimentally usinga setup where the jet is issuing from a long fluechannel.

Previous experimental observations carried on acomplete recorder flute configuration [5] have shownthat small geometrical modifications of the fluechannel geometry could have great effects on the jetstability. Therefore these parameters, known ascrucial by recorder makers, are suspected toinfluence sound production. In this paper we discussin an edgetone configuration the effect of the channelgeometry on the sound source, throughmeasurements of the jet force on the labium as wellas measurements of the radiated sound.

EXPERIMENTAL SETUP

The jet is issuing from a straight flue channel madeof a removable block embedded in a fix volume, thefoot. Three blocks are used providing three differentchannelgeometries: a long straight channel of lengthL=18mm, for which a Poiseuille flow is expected inthe flow at the flue exit, a long channel withchamfers at its exit (where edges are cut at ~45° on1mm length) and a short channel for which a nearlysquare 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 amixture of N2O2 and CO2 from a 50Bar bottle ofcompressed gas via a regulator. The mean footpressure is measured using a manometer DIGITRON2020P.

The labium consists in a metal plate on whichstrain gauges are glued, allowing force measurements[6]. The metal plate is clamped on one end in a heavysteel jaw. The edge of the plate has 90° cut (±45°from the channel axis) and is placed at a distanceW=9mm from the flue exit. In the case of the channelwith chamfers at the end, the distance between jetseparation points and the labium is increased of0.7mm.

The radiated sound is measured using a B&K 1/4”microphone connected to an FFT analyzer, themicrophone is placed at a distance x=23cm above thelabium.

Measurements are carried on the 1 st 2nd and 3rd

oscillating modes of the edgetone. Forcemeasurements presented in this paper are limited tothe 1 st mode. The frequency of the 1st mode rangesfrom 400Hz to almost 2kHz. The Reynolds numbervaries within a range 300-2500. Undesirableacoustical reflections are limited as much as possibleby placing glass wool blocks in the directions ofmaximal radiation of the edgetone.

RESULTS

Force measurements are presented Figure 1 as afunction of the mean jet speed Uj. The force is madedimensionless using the total jet force:

Fa =12

ρ0H Us(y) 2 dy−h / 2

h /2∫( ) (1)

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

of the channel length [5]. An example of comparisonbetween the measured force and the force deducedfrom radiated sound measurement using Powell’sapproximation is shown Figure 2, in the case of thelong channel without chamfers. The two curves fitvery well with an evolution pattern that is common tothe three configurations. There is however a slightoscillation of the force deduced from radiation thatcould be related room resonances, although we triedto limit it to the maximum.

DISCUSSION

Results obtained for our long channel confirmPowell's data and validates our experimental setup.The driving pressure range has been extended andsound production exhibits a sharp decline. Wesuspect this is due to turbulence appearing before thejet reaches the labium. This can be checked by flowvisualization. The same good fit between measuredand deduced force was observed in the threeconfigurations, which means that the dipolar sourceinduced by the jet movement on the labium is alwaysdominant.

The configurations can be more carefully comparedusing Figure 1. The dimensionless force exhibits asimilar aspect in all the cases, being approximatelyconstant at the beginning and declining quickly at agiven transition velocity. The force exerted by the jetissuing from the short channel is higher compared tothe long channel, but when made dimensionless witha reference force taking into account the velocity

profile change at the flue exit, the two collapse.When using a short channel, the edgetone oscillationthreshold is lower. This behavior is also present onfrequency measurements and is the consequence of ahigher propagation velocity on the jet when thevelocity profile is sharp [5]. The transition velocity isalso lowered. Ségoufin&al. [5] have shown thatwhen the channel is shortened, the jet flow is morecomplex and exhibits numerous developed vorticalstructures. Maybe this results in turbulence appearingat a lower jet velocity. It has to be noted that despitethe presence of turbulence before the labium, theforce term induced by the jet movement is stilldominant.

Adding chamfers to the flue exit of the longchannel reduces dramatically the sound production.This effect cannot be explained in terms of velocityprofile modification as it is assumed that chamfersdon’t modify it. We don't have any explanation forthis effect at the time being, and furtherinvestigations would involve numerical simulation.

REFERENCES

1. A.W. Nolle, J. Acoust. Soc. Am., 103, 3690-3705, 19982. D. Crighton, J. Fluid. Mech. , 234, 361-391, 19923. A. Powell, J. Acoust. Soc. Am., 33, 395-409, 19614. 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

Figure 2: Comparison of the force measured at thelabium ( ) and the force deducted from radiatedsound measurements (--) as a function of the meanjet speed Uj. Long channel, no chamfers.

1e-4

5e-4

10e-4

Uj (m/s)7 10 20 30

50e-4F (N)

1

0.5

0.1

F/Fa

5 10 15 20 30Uj (m/s)

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

long channel with chamfers.

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, jrs@newt.phys.unsw.edu.au

Opening successive tone holes in woodwind instruments shortens the standing wave in the bore. However, the standingwave 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 theirnatural scales. The extent of propagation of the standing wave depends on frequency, so that different modes of the boreare affected to different degrees, giving different timbres to cross fingered and simply fingered notes. We measure thefrequency dependence of the transmission of waves in the bores of flutes in both the regions where tone holes are closedand open, and of the radiation from open holes of different sizes. We use these results to explain the different effects ofcross fingering in modern and traditional instruments, and the differences in timbre between cross fingered and normallyfingered notes.

INTRODUCTION

On modern woodwind instruments such as the flute,oboe, clarinet and saxophone, a chromatic scale maybe played over at least one octave simply by openingtone holes, one at a time, starting at the downstreamend. Mechanisms that couple the keys allow theplayer to cover more tone holes than s/he has fingers.On the recorder, and on the ancestors of the modernflute, oboe and clarinet, opening successive fingerholes produces a diatonic scale. The missing notes inthe chromatic scale are achieved by cross fingering. Inboth old and new instruments, a second register isobtained by overblowing, using nearly the samefingerings. Third and higher registers involve crossfingering. Opening a tone hole provides a low inertance shuntfrom the bore to the external radiation field. If thehole has a large diameter and if the frequency issufficiently low, the shunt approximates an acousticshort circuit. So, for instruments with large holes(flute and saxophone) cross fingering has relativelylittle effect in the lowest register. Various authorshave modelled tone holes with a range ofcomplications[1-5]. Although the standing wave is attenuated by thetone hole, it still extends past it along the bore,giving an extra length or end effect that varies withfrequency. The end effect decreases as the size of thetone hole increases, and depends (to a lesser degree)on the height of its chimney, the extent of coveringby keys, and the degree of undercutting (i.e.chamfering of the junction between the tone hole andthe bore). In this study we measure the frequency dependenceof the propagation of the standing wave beyond thefirst open tone hole for simple and cross fingerings.

MATERIALS AND METHODS

The instruments studied are similar but not identicalto those studied previously [6]. An acoustic currentwas synthesized having frequency components from0.2 to 3 kHz with 2 Hz increments, and input to theembouchure hole via a short pipe whose impedanceapproximates the radiation impedance that normallyloads the instrument at this point when it is played.To measure the frequency dependence of propagationof the standing wave, a probe microphone wasinserted at the embouchure, and also at the open orclosed tone holes The technique for the measurementof transfer functions is described elsewhere [6], exceptthat a probe microphone (external diameter 1.0 mm)was used for all measurements.

RESULTS AND DISCUSSION

The first plot shows the input impedance spectrum ofa reproduction of a 19th century flute (chosen becauseof its relatively small tone holes) for the fingeringused to play E4 or E5. At low frequencies, thespectrum is not very different from that of a shorterversion of the bore, as if it were terminated a littlebeyond the first open hole. The flute, being playedwith the embouchure open to the radiation field,operates at minima in impedance. For mostfingerings, the first few impedance minima areharmonically spaced. (See [6-9] for detaileddiscussions.) At higher frequencies, the minima areweaker (due to wall losses) and, in the exampleshown here, they also occur at frequencies lower thanharmonic multiples of the fundamental, due to thegreater propagation of the wave beyond the first opentone hole.

1

2

5

10

.5

.2

.1

.05f/kHz

|Z|__MΩ

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

f/kHz

6

7

8

6 7 8E

FIGURE 2. The ratio of the pressure at the open toneholes to the pressure at the embouchure.

Figure 2 shows the frequency dependence ofpropagation of the standing wave into the region ofopen holes, expressed as the ratio of the acousticpressure at the 6th, 7th and 8th tone holes to that atthe embouchure for the same acoustic current at theembouchure. The sharp peaks coincide with theminima in Z: other frequencies do not generate astrong standing wave for this fingering. At the first open hole (hole 6), the first severalpeaks (those corresponding to the harmonic minimain Z) have similar magnitudes, but they becomeweaker at higher frequencies. Further down the bore(holes 7 & 8), the penetration increases withfrequency, up to the cut off frequency for the array ofopen tone holes (for this instrument around 2 kHz),above which there is little difference with position,and the standing wave propagates freely [8]. Inmodern instruments with larger tone holes, (data notshown), the penetration increases less rapidly withfrequency and the cut off frequency is higher. When cross fingerings are used, the analogouscurves are more complicated. The wave propagatesstrongly down the bore, and for most frequencies, the

acoustic pressure increases with length past the firstclosed hole. This increases the effective length of thebore with frequency, and so the first minimum isflattened, and the second a little more, etc. Thisallows the production of flattened notes, but it alsoproduces inharmonic minima, which do not supportthe high harmonics in a strongly non-linear vibrationrgime, and hence contribute to the darker timbre ofcross fingered notes. Musically useful discussion requires much greaterdetail. Discussions of fingering effects and a muchlarger set of results are given on our web site [9].

1

2

5

10

.5

.2

.1

.05

|Z|__MΩ

f/kHz

FIGURE 3. The impedance for a cross fingering.

ACKNOWLEDGMENTS

We thank John Tann and Terry McGee. This workwas 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, Acustica84, 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 MusicalInstruments. 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 acousticswww.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 themental feelings they arouse. In this paper, we outline a preliminary study of the characteristics of the sound of the shakuhachi interms 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 digitalaudiotape (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 toplay the shakuhachi, he listened to the long tones ofthe shakuhachi, with their changing timbres, at amaster player’s recital. This was about twenty yearsago, and the player is no longer with us. The cassettetape the author recorded at the time doesn’taccurately reproduce the variation timbres because ofthe bad recording conditions. In playing theshakuhachi, a common technique is to play with aconstant pitch and varing timbre. The timbre makesthe listener experience bright and then dark feelings,as the tone with its initially bright timbre fades out.The method of playing is truly fascinating. Thevariation in timbre gives the listener the feeling thatthe bright and dark timbres are as authentic theperception red and blue as warm and cold colors.The sound of the shakuhachi brings the listener towonder about the relationship between the physicalcharacteristics of sounds and mental feelings. Thephysical characteristics of shakuhachi sound havebeen studied [1]. In this paper, we outline apreliminary study of the characteristics of the soundof the shakuhachi in terms of the relationshipbetween sounds and mental feelings. This generalarea of study could become important for the man-machine interfaces of the future.

THE PROCESS OF THIS RESEARCH

The scale of a shakuhachi and the various timbres ofits sounds were recorded to digital audiotape (DAT)in an anechoic room. The several pitches areproduced by two fingerings, and each pair of themare the same pitch and different timbres. Werecorded changes in timbre on long tones. Studying

the relationships between waveforms and sounds is partof an attempt to make artificial sounds that have manyanalogies with the sounds produced by real shakuhachis.If the artificial sounds seem to be real, we can considerthat we have determined the characteristics ofshakuhachi sounds.

SOME RESULTS

By investigating recorded sounds of a shakuhachi, wewere able to determine several characteristics of theshakuhachi’s sound. The shakuhachi is characterized bythe beginnings and endings of the sounds it is made tomake.At high pitches, the shakuhachi is not very different fromother wind instruments. The sounds of the shakuhachilook like sine waves. At low pitches, however, a loudsound often alternates between two wave shapes that aredifferent from each other, as shown in Figure 2(1). Thismeans that listener hears loud sounds as low pitches dueto the large amplitudes of the harmonics despite thesmall low-frequency fundamental component. Thealternation between two wave shapes seems to reducethe maximum amplitude. The timbre usually changeswith the pitch or loudness of the sound produced by theshakuhachi. When playing the shakuhachi, it is verydifficult to obtain varied timbers with the same pitch.Sounds in and around the middle of the shakuhachi’spitch range are easier to play with more stability, so it isa little easier to vary the timber. Figure 1 shows a longtone produced by a player endeavoring to change thetone’s timbre while maintaining the same pitch of “g”,which is notated as by shakuhachi players.Wehappened to find two waveforms with similar spectra butwith the harmonic components in different phases asshown in Figure 2.

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

FIGURE 2. Two Waveforms with the Same Spectra andDifferent Phases.

These waveforms are described by the followingfunctions.

ttt 3sin6.0)3/22sin(sin (1)ttt 3sin2sinsin (2)

1.068 1.07 1.072 1.074 1.076 1.078 1.08 1.082

3.13 3.132 3.134 3.136 3.138 3.14 3.142 3.144

FIGURE 3. Typical Waveforms of the Sound Used toProduce Fig.1.

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

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

CONCLUSION

Some characteristics of shakuhachi tones wereobtained. Our next work will be an investigation of therelationship between spectra and feelings in moredetail.

ACKNOWLEDGMENTS

We are very much obliged for the help given to us byProf. Toyama and to the members of his laboratory atKogakuin University.

REFERENCES

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

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