Time-domain simulation of sound production of the sho - Acoustical ...

Time-domain simulation of sound production of the sho Takafumi Hikichi NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Naotoshi Osaka NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan

Fumitada Itakura Graduate School of Engineering, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya, Aichi 4688603, Japan

共Received 10 April 2002; revised 31 October 2002; accepted 11 November 2002兲 A physical model based on the sound production mechanism of the sho is proposed with intention of applying it to sound synthesis. Time-domain simulation was done using this model, and effects of the tube length and blowing pressure on the sounding frequency and sounds spectra were investigated. The reed vibration, pressure variation inside the tube, and threshold blowing pressure for oscillation were measured by artificially blowing air into the sho. The experimental results are in acceptable agreement with simulation results in terms of the relationships between tube length and threshold pressure and between tube length and the sounding frequency. In addition, recorded sound waveforms and simulated ones have a common feature in the sense that high-frequency components of their spectra increase with increasing blowing pressure. Further, it is concluded that a sho reed acts as an ‘‘outward-striking valve.’’ © 2003 Acoustical Society of America. 关DOI: 10.1121/1.1534605兴 PACS numbers: 43.75.Pq 关NHF兴 I. INTRODUCTION

Physical modeling is attracting much attention in attempts to synthesize the sounds of musical instruments. Physical models excel in naturally controlling musical instrumental sounds compared with other conventional methods, such as additive synthesis and sampling synthesis. Dynamic features, such as attack, transience, and frequency variation, can be realized because it has inherent mechanism that changes sound features like real instruments. In addition, artificial instruments can have the same parameters as real ones, which allows users to control the instruments intuitively. Many artificial instruments have recently been developed and offered to computer music composers.1–3 However, our understanding is not sufficient to make full use of physical models’ abilities. In particular, little attention has been paid to instruments other than Western orchestral instruments. Departing from this trend, this paper treats the sho, a free reed mouth organ used in traditional Japanese court music called ‘‘Gagaku.’’ The purposes of this paper are to model the sound production mechanism of the sho, and, through simulations based on that model, show the possibility of physical modeling that synthesizes realistic sounds. Figures 1共a兲 and 共b兲 show a sho, the sho disassembled. The sho is mainly composed of a mouthpiece, cavity, and seventeen bamboo pipes. Metal reeds are glued with resin to the lower side of the bamboo pipes. Figure 1共c兲 shows a close-up view of a reed. The sho is played by holding it in front of the face upright. When a player blows through the mouthpiece and closes the small finger holes on the tubes, oscillation commences and the reeds start sounding. The 1092

J. Acoust. Soc. Am. 113 (2), February 2003

tubes whose finger holes remain open do not sound even if air is supplied, so the player can control sounding by fingerings. Some tubes have one slot besides the finger hole, which determine the effective lengths. Each pipe has a different length and a different reed which gives a different pitch. The pitch range of a typical sho covers A4 共430 Hz兲 to F]6 共1451 Hz兲. Note that the standard pitch for playing gagaku music is A4⫽430 Hz, which is different from one normally used in Western music. Table I shows position, name and pitch of each pipe of a sho, and Fig. 2 shows top view of the cavity 共wind chest兲 part of the sho. One characteristic of the sho is that it can be played not only by blowing but also by drawing. In typical playing style, the fingers close five or six finger holes simultaneously, and multiple pipes sound in chords. The instrument is said to originate in the 3rd or 4th century in China. There are other musical instruments, mostly found in Eastern and Southern Asia, that work on a similar mechanism. These include Chinese sheng and Laotian khaen. It is said that the instrument was introduced to Japan around the 8th century, and that its structure has remained virtually unchanged to the present day. In spite of such a long history, its mechanism is not yet fully understood and has not been applied to synthesis. Acoustically, it can be described as free reeds coupled to pipe resonators. Let us call this structure as ‘‘sho-type.’’ The sho-type instrument is categorized as a free reed instrument like the harmonica, accordion, and reed-pipe organ. However, unlike these Western free reed instruments, the reeds of the sho-type instrument are approximately symmetrical, so that the same reed vibrates on both blowing and drawing. Some measurements of sho-type instrument have been re-

0001-4966/2003/113(2)/1092/10/$19.00

© 2003 Acoustical Society of America

FIG. 2. Top view of a wind chest of the sho. Small circles with numbers show holes where pipes are mounted.

FIG. 1. 共a兲 The sho. 共b兲 Disassembled. 共c兲 Close-up view of a reed.

ported by Cottingham.4 – 6 For khaen, the relationship between the pipe length and sounding frequency has been reported, and it was shown that the vibrating frequency of the reed could be pulled to match the pipe resonance. The input impedance of the khaen has also been measured, and it was concluded that the reed acted as ‘‘outward striking.’’ On the TABLE I. Position, name, and pitch of each pipe of a sho. Position is shown in Fig. 2. ‘‘Ya’’ and ‘‘mou’’ do not have the reeds so they do not contribute to sounding. Position

Name

Pitch

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

sen jyu ge otsu ku bi ichi hachi ya gon shichi gyo jyo bou kotsu mou hi

F]6 G5 F]5 E5 C]5 G]5 B4 E6

J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

C]6 B5 A5 D6 D5 A4 C6

relationship between blowing pressure versus sounding frequency, it was concluded that sounding frequency decreased with increasing blowing pressure. Although sound production mechanism of the sho-type instrument has not been proposed, it is expected that research for other wind instruments may give us hints to model this instrument. According to a standard theory for wind instruments, the sound production mechanism comprises a generator and a resonator. For instruments of this type, the generator corresponds to reeds and airflow, and the resonator to an air column. As for reeds, the behavior of a free reed has been investigated by Tarnopolsky et al. both experimentally and theoretically.7 A reed was treated as a pressure-controlled valve in a simplified manner, and good correspondence between theoretical and experimental results was found. So, we adopt their formulation to describe the motion of the sho reed. For modeling of the pipe resonators, we can utilize Schumacher’s well-known work on woodwind instruments to carry out time domain simulations.8 Section II describes the experiment carried out to ascertain the basic physical mechanism of the sho-type instrument. In Sec. III, our sho sound production system is formulated and the oscillation condition is analyzed based on linear theory. The simulation results obtained using our physical model are discussed and compared with experimental results in Sec. IV. Section V summarizes the results and concludes the paper.

II. EXPERIMENTS

Measurements were made on variations in threshold pressure and sounding frequency with tube length to get sufficient understanding on sound production mechanism of the sho-type instrument. Hikichi et al.: Sound production of the sho

1093

FIG. 3. The experimental setup.

A. Experimental condition

Figure 3 shows details of the experimental arrangement used. We focus on the ‘‘ichi’’ tube 共B4, 483.7 Hz兲 because it does not have any slot on the side wall and its configuration is the simplest among the tubes. The ‘‘ichi’’ tube was taken from a disassembled sho and mounted in an acrylic chamber made for this experiment. A pressure sensor was connected to the chamber to measure the difference between the pressure inside the chamber and the atmospheric pressure. We will refer to this as blowing pressure. A probe microphone was inserted into the tube via the finger hole, and the hole was closed with rubber tape. A condenser microphone was mounted 5 cm from the open end of the pipe. The instrument was artificially played by a compressor 共blow兲 or by a vacuum cleaner 共draw兲, and blowing pressure was adjusted manually using a regulator. Blowing pressure was in the range of 0 to 1 kPa, which was static with time. The reed vibration was measured by a laser displacement meter, and pressure variations were measured by microphones. All the recorded signals were transferred to computer via an A/D converter. The sampling frequency was set to 24 kHz. The cutoff frequency of the probe microphone was set to 3.6 kHz, and that of the displacement sensor and the condenser microphone were set to 10 kHz.

FIG. 4. Tube resonance frequency f p versus threshold pressure obtained by experiment. Blow means positive, and draw means negative pressure.

probably the small reed asymmetry. Another may be the difference in the configuration of upstream/downstream sides of the reed. In addition, hysteresis was found in the experiment, i.e., threshold pressures obtained by gradually increasing the blowing pressure were slightly different from those obtained by decreasing the pressure. C. Tube resonance frequency and sounding frequency

It was expected that sounding frequency would vary with tube length. To clarify this, the relationship between the tube resonance frequency and the sounding frequency under constant pressure 共positive: 0.8 kPa, negative: 0.5 kPa兲 was examined. Figure 5 shows experimental results for positive 共blow兲 and negative 共draw兲 pressures. Integer multiples of the pipe resonance frequency are shown by dashed–dotted lines. The dashed line shows the reed frequency and is in-

B. Tube resonance frequency and threshold behavior

First, the tube length dependence of threshold pressure was investigated. The length of the tube was varied and the threshold pressure was measured. The tube length was made longer by simply adding a piece of tube to one end, and blowing pressure was gradually increased until oscillation began or it reached 1 kPa. Figure 4 shows the relationship between tube resonance frequency and threshold pressure for positive 共blow兲 and negative 共draw兲 pressures. Here, the resonance frequency of the tube was calculated from its length under the assumption of one-fourth wavelength resonance, since the pipes can be regarded as having one end closed. There is a strong dependency on pipe length, i.e., there is a range in which the threshold is low, and also a range in which the reed cannot oscillate 共200–250 Hz兲. Further, the threshold pressure for negative pressure was lower than for positive pressure. One of the reasons for this is 1094


FIG. 5. Tube resonance frequency f p versus sounding frequency f s obtained by experiment. Blow means positive, and draw means negative pressure. Dashed–dotted lines show integer multiples of the pipe resonance frequency and dashed line shows the reed frequency. Hikichi et al.: Sound production of the sho

FIG. 7. A close-up schematic view of a reed.

p(t) as an input to our system, and neglect effects of the flow, cavity vibration, etc., on the pressure. As for reeds, the behavior of an outward-striking valve has been investigated by Tarnopolsky et al.7 and Fletcher.9 The formulation in Secs. III A and III B below are based on their work. A. Modeling the reed vibration

The vibration of the reed is that of a simple cantilever. Let pressure at the upstream of the reed 共i.e., pressure inside the cavity兲 at time t be p(t), and pressure at the downstream 共pressure inside the pipe兲 be p 2 (t). As shown in the Appendix, the equation of motion of the reed has the form d 2 x ␻ r dx 1.5WL ⫹ ␻ r2 共 x⫺x 0 兲 ⫽ ⫹ 共 p 共 t 兲 ⫺ p 2 共 t 兲兲 , dt 2 Q dt m

FIG. 6. 共a兲 Configuration of an actual sho. 共b兲 Simplified physical model.

cluded to make the data easy to understand. Here we will divide whole frequency range into three parts for explanation. First, when tube resonance frequency is low, the sounding frequency increases quickly with increasing pipe resonance until it reaches about 200 Hz. Around this frequency, the pitch reached 570 Hz. Then, there is a no-oscillation range of 200–250 Hz. And then again, there is a range where oscillation occurs around 480 Hz. From these results, we found that 共1兲 oscillation commenced above the reed frequency 共dashed line in the figure兲, and 共2兲 oscillation commenced above the first or third multiple of the pipe resonance frequency 共shown by the dashed– dotted lines兲. These frequency relationships suggest that the reeds in a sho operate in an ‘‘outward-striking’’ manner.

III. FORMULATION

From the experimental results mentioned above, it was shown that the reed behaves as an outward-striking valve. Based on this finding, sound production system of the sho is formulated. Figure 6 shows the actual configuration of the sho and its simplified model. A player blows the air into the wind chest as shown in Fig. 6共b兲, and as a result, the pressure inside the cavity p(t) increases. We consider the pressure J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

共1兲

where x is the displacement at the tip of the reed, Q the resonance Q value, ␻ r the angular resonance frequency, and x 0 the initial displacement. W, L, and m are the width, length, and mass of the reed, respectively. B. Airflow through the reed

From Bernoulli’s equation, the relationships among p(t), p 2 (t), and volume velocity through the slit U(t) are as follows: p 共 t 兲 ⫽ p 2共 t 兲 ⫹

冋

册冋

册

␳ U共 t 兲 2 ⳵ ␳U共 t 兲␦ ⫹ , 2 CF 共 x 兲 ⳵ t CF 共 x 兲

共2兲

where C is the flow contraction coefficient, ␳ the air density, ␦ the inertia parameter. The flow contraction coefficient represents the effect of the slit configuration. The area of the slit F(x) is described as F 共 x 兲 ⫽W 关 x 2 ⫹b 2 兴 1/2⫹2L 关 a 共 x 兲 2 ⫹b 2 兴 1/2, where a(x) is the average displacement of the sides of the reed, b the clearance gap around the reed. Figure 7 shows the reed configuration in detail. For a sho reed, we assume x 0 ⫽0. Considering it displaces both ways, and from the form of the mode function, it follows that 共see the Appendix兲, a 共 x 兲 ⬇ 兩 x 0 ⫹0.4共 x⫺x 0 兲兩 ⫽ 兩 0.6x 0 ⫹0.4x 兩 ⫽0.4兩 x 兩 . C. Reflection function of the tube

The acoustical characteristics of the tube are specified by the input impedance Z in( f ). Input impedance is defined by the ratio of sound pressure and volume velocity at the end Hikichi et al.: Sound production of the sho

1095

of the tube with frequency f. Reflection coefficient R( f ) is defined by the input impedance Z in( f ) and characteristic impedance of the tube Z 0 as Z in共 f 兲 ⫺Z 0 R共 f 兲⫽ . Z in共 f 兲 ⫹Z 0

dx U in共 t 兲 ⫽U 共 t 兲 ⫹0.4WL , dt

共3兲

where the asterisk denotes convolution and U in(t) is the net volume velocity input to the tube. The quantity 0.4WL(dx/dt) is approximately the volume velocity displaced by a cantilever of width W and length L when its tip is moved by (dx/dt). The pipe shape is relatively simple. We have measured the resonance frequency of the ‘‘ichi’’ pipe, and have concluded that the pipe could be approximated as a cylinder.10 So, we use a simple reflection function of Gaussian type and adapt Schumacher’s method to calculate the pressure at the entrance of the pipe. D. Linear theory of oscillation

In the preceding sections, the sound production system was formulated with two linear elements and their nonlinear interaction. To gain insight into this system before simulation, this section investigates the condition under which selfoscillation continues within linear theory when the amplitude of oscillation is small. For brass instruments, self-oscillation condition based on linear theory has been discussed by Adachi and Sato.11 The derivation shown here is based on their work. First, let the dc component and ac component of the ¯ and U ˜ , ¯x and ˜x , etc. Assuming variables be described as U the input pressure p is static, the dc components satisfy the following stationary conditions:

␻ r2¯x ⫽ ␳

p⫽

1.5WLp , m

冋册 ¯ U

共4兲

2

,

2 CF ¯

共5兲

which are derived from Eqs. 共1兲 and 共2兲. ¯ , ¯F , ¯x are described as follows: The dc components U ¯x ⫽

1.5WLp m ␻ r2

,

¯F ⫽W 关¯x 2 ⫹b 2 兴 1/2⫹2L 关 0.16x ¯ 2 ⫹b 2 兴 1/2, ¯⫽ U 1096

冑

2p ¯. CF ␳


冉冑冊 2␳

2⫹

Reflection function r(t) is defined by the inverse Fourier transform of R( f ). It represents the pressure waveform, which is reflected back to the end of the tube when impulse pressure is injected at t⫽0. Using this reflection function, we calculate the pressure inside the tube p 2 (t) as p 2 共 t 兲 ⫽Z 0 U in共 t 兲 ⫹r 共 t 兲 * 共 p 2 共 t 兲 ⫹Z 0 U in共 t 兲兲 ,

Equation 共2兲, which governs the nonlinear airflow dynamics, is linearized as

共6兲共7兲共8兲

p

␦

d

˜U

dt U ¯

⫽⫺

p2 p

⫹2

˜F ¯F

共9兲

,

where airflow velocity is much greater than that of the reed. In the ordinary range of parameter values, the inertia of airflow can be neglected. Therefore, we omit the second term on the left-hand side of Eq. 共9兲. ¯ ⫹x ˜ into Eq. 共1兲, where the angular freSubstituting x⫽x quency of the reed ␻ (⫽2 ␲ f ), we have

冉冊

˜x ␻ 1.5WL ⫽⫺ ⌳ , p2 ␻r m ␻ r2

共10兲

where ⌳共⍀兲 is defined as ⌳共 ⍀ 兲⫽

1 . 1⫺⍀ ⫹ j 共 ⍀/Q 兲 2

˜ ( f )/p 2 ( f ), Reed compliance G is defined as G( f )⫽F similar to the previous work on brass instruments,11 where ˜F ( f ) and p 2 ( f ) denote Fourier components of ˜F and p 2 at frequency f, respectively. From Eq. 共10兲, G becomes G共 f 兲⫽

冉冊

˜F 共 f 兲 ␻ 1.5WL 共 W⫹0.8L 兲 ⫽⫺ ⌳ . 2 p 2共 f 兲 ␻r m␻r

共11兲

Equation 共11兲 shows that 兩 G( f ) 兩 takes its maximum near the reed resonance frequency f r , and that ⬔G( f ) shows ␲ decrease from the lower to the higher side of f r . Substituting the reed compliance G( f ) and input impedance Z in into the linearized equation 共9兲, we obtain the self-oscillation condition as follows: K共 f 兲⬅

冑

2p ¯ 共 f 兲 Z in共 f 兲 ⫽1, CG ␳

共12兲

¯ ( f ) is defined as where G ¯ 共 f 兲 ⫽G 共 f 兲 ⫺ G

¯F . 2p

¯ is also maximized near the Note that the magnitude of G ¯ has the reed resonance frequency and that the angle of G same sign as the angle of G. Oscillation happens under the condition that there exists a frequency f such that K( f ) is real and larger than 1. The magnitude condition K( f )⬎1 requires that the ¯ and Z in be large. This indicates that magnitudes of both G the oscillation has a frequency f that is near one of the resonance frequencies of the pipe as well as near the reed resonance frequency. ¯ ⫹⬔Z in The phase condition is written as ⬔K⫽⬔G ¯ ⫽0. Because ⬔G is positive, the phase condition is satisfied only if ⬔Z in is negative for oscillation. This indicates that the oscillation happens on the higher frequency side of the input impedance peak. According to Fletcher’s classification based on consideration of a simple pressure-controlled valve,12 the sho reed is classified as ‘‘outward-striking’’ valve. Hikichi et al.: Sound production of the sho

FIG. 8. The magnitude and phase of K( f ) defined by Eq. 共12兲 with blowing pressures of 50, 100, and 150 Pa.

Next, a quantitative analysis is done for threshold pressure and sounding frequency using Eq. 共12兲. Threshold pressure is given as a pressure when 兩 K( f ) 兩 ⫽1 holds at a frequency f that satisfies ⬔K( f )⫽0. The magnitude and phase of K( f ) defined by Eq. 共12兲 are plotted in Fig. 8 with blowing pressures of 50, 100, and 150 Pa (Q⫽10). From Fig. 8, it is clear that, as the pressure increased, 共1兲 the phase of K( f ) changed so as to have a crossing point with zero axis, 共2兲 the crossing frequency shifted to the lower side, and 共3兲 the magnitude of K( f ) became larger near the reed frequency. So, threshold pressure was determined by gradually increasing blowing pressure until the magnitude of K( f ) exceeded 1 at a frequency f that satisfies ⬔K( f )⫽0, which was determined to be the sound frequency. Figure 9 shows 共a兲 threshold pressure and 共b兲 sound frequency with some values of Q and additional mass rate ␣ m determined by the above procedure. Here, the additional mass corresponds to a piece of lead attached to the tip of the reed. The amount of this additional mass was specified by a percentage of the original reed mass. In experiment, threshold pressures were 370 Pa for positive pressure and 90 Pa for negative pressure for normal pipe length (L p ⫽24.1 cm), as shown in Fig. 4. In Fig. 9共a兲, threshold pressure increases with increasing additional mass rate, and increases with decreasing Q value. Considering the experimental results, the ranges of Q⫽8 – 10 and ␣ m ⫽0 – 30% seem appropriate. By using these values, threshold pressure in the range of 140 to 350 Pa was obtained, which was considered to be reasonable. In Fig. 9共b兲, when Q⫽8 or 10 was used, the sounding frequency obtained by linear analysis was smaller than the 483 Hz obtained from experiment. It is expected that nonlinearity may play a role in shifting the sounding frequency to the higher value. Further, the effect of additional mass on the sounding frequency was found to be small when the natural frequency of the reed was kept constant: increasing the added mass slightly lowers sounding frequency. IV. SIMULATION

Equations 共1兲–共3兲 were discretized, and a simulation was done in 48 kHz sampling. Given p(t), the following J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

FIG. 9. 共a兲 Threshold pressure versus additional mass rate ␣ m obtained based on the linear theory analysis (Q⫽20,10,8,6). 共b兲 Sound frequency versus additional mass rate ␣ m obtained based on the linear theory analysis (Q⫽20,10,8,6).

three variables were calculated: displacement of the reed x(t), pressure inside the tube p 2 (t), and volume velocity U(t). Parameter values used in the simulation are shown in Table II. Simulation results will be compared with both experimental and theoretical results. A. Threshold behavior

The length of the tube was varied, and the tube length dependence of threshold pressure was investigated. Delay time in a reflection function was adjusted in inverse proportion to the tube length, and the blowing pressure was gradually increased until oscillation began or it reached 1 kPa. Figure 10共a兲 shows simulation results for some Q and additional mass values that were suggested by linear theory analysis. In the simulation, the same results were obtained for positive and negative pressure cases. A strong dependency on pipe length was observed, which is consistent with experimental results. There is a range in which the threshold is low, and also a range in which the reed cannot oscillate 共210–350 Hz兲. These characteristics can be interpreted from Hikichi et al.: Sound production of the sho

1097

TABLE II. Parameters used in the time-domain simulation. Variable

Symbol

Reed Length Width Thickness Displacement at the tip Initial displacement Gap between the reed and plate Resonance Q value Natural angular frequency Material density Additional mass rate Mass

L W h x x0 b Q ␻r ␳r ␣m m

Tube Pressure inside the tube First resonance frequency Radius Characteristic impedance

p2 fp rp Z0

Others Blowing pressure Volume velocity through the slit Channel length Flow contraction coefficient Air density Sound velocity

Value 10 mm 2 mm 0.3 mm 0 0.01 mm 8 –35 2 ␲ ⫻470 Hz 8⫻103 kg/m3 0–0.3 ␳ r WLh(1⫹ ␣ m )

150– 600 Hz 3.5 mm ␳ c/ ␲ r 2p

p U

0–1.0 kPa

␦

1 mm 0.61 1.2 kg/m3 340 m/s

C ␳ c

the frequency relation between the pipe resonance frequency and natural frequency of the reed. It was also found that threshold pressure increased with Q and with additional mass. However, there is a considerable discrepancy between Fig. 10共a兲 and 4: there is a wider nonoscillation range, or simulation results are shifted to the higher frequency. It is suspected that this is partially due to incorrectness of parameter values, and partially due to the nonlinearity of the model. To get more reasonable results, the simulation was redone using different Q and reed frequency f r values, as shown in Fig. 10共b兲. By using more larger Q value, the agreement between experiment and simulation results was improved. Using the reed frequency f r ⫽400 also produced an acceptable result. Next, threshold pressure was compared in the original pipe length L p ⫽24.1 cm ( f p ⫽353 Hz) between simulation and theory. The threshold pressure by theory did not exactly coincide with the simulation. A Q value of 10 was suggested from linear theory analysis, but a larger value was shown to be appropriate in the simulation. It is inferred that this discrepancy is due to the nonlinear terms, which were neglected in the linear theory analysis. When Q⫽35, simulation and experiment showed a better correspondence, and the oscillation condition was also satisfied, i.e., 兩 K( f ) 兩 ⬎1 held. B. Sounding frequency

Then, the relationship between the tube resonance frequency and the sounding frequency under constant pressure was examined. Figure 11 shows simulation results using the same parameters used in the preceding section. In contrast to Fig. 10, the Q value and mass affect sounding frequency only 1098


FIG. 10. 共a兲 Tube resonance frequency f p versus threshold pressure obtained by simulation with two Q values and additional mass rate ␣ m used as parameters. 共b兲 Q values and reed frequency f r are changed to fit to the experimental result.

slightly. When f r ⫽400, the result was different from the others and from the experimental result, and was found to be inappropriate. The data in Fig. 11 are in fair agreement with those in Fig. 5. In the simulation, oscillation commenced above the reed frequency 共dashed line in the figure兲, and also above the first or third multiple of the pipe resonance frequency 共dashed–dotted lines兲, the same as for the experimental results. Pitch reached 570 Hz in the experiment, and 630 Hz in the simulation. Although the experimental result was limited to below 360 Hz, the sounding frequency rises with increasing tube resonance frequency in the simulation. In terms of the range of the pipe resonance frequency that can oscillate, Q⫽35 seems appropriate. In the original pipe length, L p ⫽24.1 cm, the sound frequency obtained by simulation was 477 Hz, and the difference between the experiment and simulation was about 1.2%. The sound frequency obtained from linear theory was Hikichi et al.: Sound production of the sho

FIG. 11. 共a兲 Tube resonance frequency f p versus sounding frequency f s obtained by simulation with two Q values and additional mass rate ␣ m used as parameters. Dashed–dotted lines show integer multiples of the pipe resonance frequency and dashed line shows the reed frequency. 共b兲 Q values and reed frequency f r are changed to fit to the experimental result.

417 Hz, so it shifted about 1.3% higher in the simulation based on the model that includes nonlinear terms. C. Sound spectrum

Next, simulated sound spectra were compared with those obtained in the experiment. Figure 12共a兲 shows spectra of recorded sounds with different blowing pressures, i.e., 0.4, 0.6, 0.8 kPa from the top. High-frequency components in the spectra increased with increasing blowing pressure. The same tendency was found in the negative case. Radiated sound pressure cannot be simulated using our model, but volume velocity signal can. According to Causse´’s method,13 given the pipe shape, the transfer function between the volume velocity at the entrance of a pipe and sound pressure at the exit of a pipe can be calculated. In the calculation of the transfer function, as a termination conJ. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

FIG. 12. Variation in spectra with different blowing pressure. 共a兲 Spectra of recorded sound 共positive, 0.4, 0.6, 0.8 kPa from the top兲. 共b兲 Spectra of simulated sound 共positive, 0.4, 0.6, 0.8 kPa from the top兲. Simulated sound was calculated by multiplying the simulated volume velocity by the transfer function of the pipe. The high frequency components increased with increasing blowing pressure, the same as for the recorded sounds.

dition, spherical radiation is assumed. Sound pressure at the exit of the pipe was then calculated by multiplying the simulated volume velocity with this transfer function. Figure 12共b兲 shows spectra of simulated sounds with blowing pressures of 0.4, 0.6, 0.8 kPa from the top. The high frequency components increased with increasing blowing pressure, the same as for the recorded sounds. The second and fourth harmonics of the spectra of the recorded sounds are enhanced, as shown in Fig. 12共a兲. It Hikichi et al.: Sound production of the sho

1099

FIG. 13. Spectral centroid versus blowing pressure. The plots 䊊 and ⫻ show experimental and simulation results, respectively.

seems that this is due to the 3/4 lambda and 5/4 lambda resonances of the pipe. In order to find out whether this inference is correct or not, the transfer function of the pipe was calculated, and the frequencies of the 3/4 lambda and 5/4 lambda resonances were 1054 Hz and 1764 Hz, respectively. On the other hand, the frequencies of the second and fourth harmonics are about 966 Hz and 1932 Hz. So, it is concluded that predominance of the second and fourth harmonics is not due to the peaks of the transfer function of the pipe. To evaluate changes in spectral components with blowing pressure quantitatively, spectral centroid defined by the following equation was calculated and compared: C⫽

N/2⫺1 兩 Y 共 k 兲兩 •k F s 兺 k⫽0 , N/2⫺1 N 兺 k⫽0 兩 Y 共 k 兲兩

where Y (k) is the discrete FFT spectra, F s the sampling frequency and N the FFT length. Figure 13 shows the spectral centroids of the recorded and simulated sounds. Although there is some discrepancy, both plots show similar trend: the spectral centroid gradually increases as blowing pressure increases. V. CONCLUSIONS

We investigated the effects of tube length on the threshold pressure and on sounding frequency by simulation, and the result was proved to coincide with that obtained by experiment. Recorded and simulated sounds have a common feature in the sense that high-frequency components of their spectra increase with increasing blowing pressure. Further, from the frequency relationships, it was concluded that the reeds of the sho act as ‘‘outward-striking valves.’’ Especially, noteworthy is that the frequency relationship also holds even for negative pressure. In terms of the relationship between the reed resonance frequency and one of the pipe resonance frequencies, the sound production mechanism can be explained as follows. First, the oscillation commences with a frequency f, which is near the reed frequency f r and above one of the pipe reso1100


nance frequencies. When the finger hole is opened, the effective pipe length is shortened and the first pipe resonance frequency f p becomes so high that the condition for oscillation is not satisfied. When the finger hole is closed, f p becomes lower than f r and the oscillation condition is satisfied. Our model does not yet predict the finer details of the experimental results. Here, the difference in threshold pressure is considered. In the experiment, the threshold pressure was different for positive and negative pressures, whereas in simulation it perfectly coincided. In our model, the reed is assumed to be symmetrical and its thickness is neglected. But, the back of the reed is actually slightly chiseled. As a result, the reed goes out of the slot more easily when drawing than when blowing. By incorporating this asymmetry, it is expected that the difference between positive and negative pressures can be represented. Measurements of the blowing pressure dependency of sounding frequency were also done, although we have not shown here. As stated in the introduction, previous studies on khaen reeds have shown a linear decrease of playing frequency with increasing blowing pressure. Cottingham et al. also reported same trends on harmonium-type reeds without a pipe resonator, and they showed theoretical calculations which agree well with the experimental data.14 However, our measurement results obtained from all tubes showed variations among tubes. Pipe configuration and particles that were spread on the reeds might affect reed oscillation, so more work should be done before drawing conclusions. We believe this dependency of sounding frequency on blowing pressure may play an important role in producing characteristic timbre of the sho. The sho is usually played in chords with gradual crescendo and decrescendo, which causes slight pitch variations on each note. In addition, traditional chords have many dissonant tones, from the Western tonal musical theory point of view, so sounds have inherent complex beats and keep changing with time. From the sound-quality point of view, sounds produced by our model have basic characteristics of real tones, although they are somewhat monotonous. The reason is due to the fact that we used only simple control parameters with time. It is expected that use of realistic control parameters may make the sounds more realistic. Other future work which was not mentioned above is to improve the simulation by including the effects of radiation, vortices, and wall vibration of the instrument in the model.

ACKNOWLEDGMENTS

The authors are grateful to Dr. Ken’ichiro Ishii, Director of the NTT Communication Science Laboratories, and Dr. Hiroshi Murase, Manager of the Media Information Laboratory for their support. They wish to thank reviewers for their helpful suggestions. They also thank CIAIR research members at Nagoya University for fruitful discussions. Part of this work is supported by the Center of Excellence 共COE兲 formation program of the Ministry of Education, Culture, Sports, Science and Technology of Japan 共No. 11CE2005兲. Hikichi et al.: Sound production of the sho

APPENDIX

The complete derivation of Eq. 共1兲 can be found in the Appendix in Ref. 7. Here, it is briefly summarized. First, the equation of motion of the cantilever is described as

evaluate the vertical component of the side opening a(x) when its tip opening is x. a 共 x 兲 ⫽x 0 ⫹ 共 x⫺x 0 兲

冕

L

0

⌿ 共 s 兲 ds⬇0.6x 0 ⫹0.4x.

1

⳵ 2␰ ⳵␰ ⳵ 4␰ ␳ r Wh 2 ⫹R ⫹K 4 ⫽W 共 p 共 t 兲 ⫺p 2 共 t 兲兲 , dt ⳵t ⳵s

共A1兲

where ␰ (s,t) is the displacement of the reed, s the distance from its clamped root, ␳ r the material density, W the width, h the thickness, K the bending stiffness of the reed, and R its damping coefficient. When the reed is assumed to vibrate in the normal mode, we can write ␰ (s,t)⫽ 关 x(t)⫺x 0 兴 ⌿(s), where ⌿(s) is the form of the mode function, normalized so that ⌿(L)⫽1, where L is the reed length. Multiplying both sides of 共A1兲 by ⌿(s) and integrating over the reed length L then gives

␥ WL d 2 x ␻ r dx ⫹ ␻ r2 共 x⫺x 0 兲 ⫽ 共 p 共 t 兲 ⫺p 2 共 t 兲兲 , 共A2兲 2 ⫹ dt Q dt m where Q is the resonance Q value and m is the effective mass of the reed as given by m⫽ ␳ r WLh

共A3兲

and

␥⫽

兰 L0 ⌿ 共 s 兲 ds 兰 L0 ⌿ 共 s 兲 2 ds

⬇1.5.

共A4兲

From a knowledge of the form of ⌿(s), we can also


C. Roads, ‘‘Physical modeling and formant synthesis,’’ The Computer Music Tutorial 共MIT Press, London, 1996兲, Chap. 7, pp. 263–315. 2 J. O. Smith, ‘‘Physical modeling synthesis update,’’ Comput. Music J. 20, 44 –56 共1996兲. 3 G. P. Scavone and P. R. Cook, ‘‘Real-time computer modeling of woodwind instruments,’’ Proceedings of the International Symposium on Musical Acoustics 共Acoustical Society of America, New York, 1998兲, pp. 197–202. 4 J. P. Cottingham and C. A. Fetzer, ‘‘Acoustics of the khaen,’’ Proceedings of the International Symposium on Musical Acoustics 共Acoustical Society of America, New York, 1998兲, pp. 261–266. 5 J. P. Cottingham, ‘‘The acoustics of a symmetrical free reed coupled to a pipe resonator,’’ Proceedings of the 7th International Congress on Sound and Vibration, 2000, pp. 1825–1832. 6 J. P. Cottingham, ‘‘The Asian free reed mouth organs,’’ in Proceedings of the International Symposium on Musical Acoustics 共Fondazione Scuola di San Giorgio, Venezia, 2001兲, pp. 61– 64. 7 A. Z. Tarnopolsky, N. H. Fletcher, and J. C. S. Lai, ‘‘Oscillating reed valves-An experimental study,’’ J. Acoust. Soc. Am. 108, 400– 406 共2000兲. 8 R. T. Schumacher, ‘‘Ab initio calculations of the oscillations of a clarinet,’’ Acustica 48, 71– 85 共1981兲. 9 N. H. Fletcher, ‘‘Autonomous vibration of simple pressure-controlled valves in gas flows,’’ J. Acoust. Soc. Am. 93, 2172–2180 共1993兲. 10 T. Hikichi and N. Osaka, ‘‘Measurements of the resonance frequencies and the reed vibration of the sho,’’ Acoust. Sci. Technol. 23, 25–27 共2002兲. 11 S. Adachi and M. Sato, ‘‘Time-domain simulation of sound production in the brass instrument,’’ J. Acoust. Soc. Am. 97, 3850–3861 共1995兲. 12 N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd ed. 共Springer-Verlag, New York, 1998兲, Chap. 13, pp. 401– 428. 13 R. Causse´, J. Kergomard, and X. Lurton, ‘‘Input impedance of brass musical instruments—Comparison between experiment and numerical models,’’ J. Acoust. Soc. Am. 75, 241–254 共1984兲. 14 J. P. Cottingham, C. H. Reed, and M. Busha, ‘‘Variation of frequency with blowing pressure for an air-driven free reed,’’ J. Acoust. Soc. Am. 105, Pt. 2, 1001 共1999兲.

Hikichi et al.: Sound production of the sho

1101