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Page 00000001 A Physical Model of the Sho and Its Application to Articulation Synthesis Takafumi Hikichi*, Naotoshi Osaka*, Fumitada Itakura*2 *1 NTT Communication Science Laboratories, NTT Corporation *2 School of Engineering, Nagoya University email@example.com Abstract This paper describes our recent development of a physical model of the sho, the Japanese free reed mouth organ. In ourprevious study, a basic model was proposed and its propriety was confirmed in terms of physical properties. In this paper, we present some improvement of the model considering the effects of reed thickness and radiation. This new model was used in the synthesis of sounds with various articulation, and it is confirmed that vibrato and other articulations, which can not be played on real instruments, can be synthesized. This shows the possibility of extending sho timbre beyond the traditional one. 1 Introduction We have been studying the sound production mechanism of the sho, the Japanese mouth organ, with the intention of applying it to a sound synthesis technique (Hikichi and Osaka 2001)(Hikichi and Osaka 2002). Shos, as well as shengs in China and khaens in Laos and Thailand, are categorized as free reed instruments with pipe resonators. They originated in South Asia, and have been played for more than a millenium. It is said that these instruments are the origins of Western free reed instruments, such as harmonicas, accordions, and reed-pipes. In spite of such a long history, not much attention has been paid to these instruments so far. Acoustical research on several free reed instruments has been reported in the literature (Cottingham 2001), but a synthesis model has not been proposed yet. The use of the sho in computer music has been reported in Nagashima and Ito (1999) as an interface device. We take a physical modeling approach to utilize these instruments in computer music. The sho is a mouth organ with 17 bamboo pipes fitted into a wind chamber; 15 of these have small metal reeds inside the chamber. When the player blows the instrument while closing a finger hole, thereby changing the acoustical length of the pipe, that pipe sounds. It should be noted that this instrument is played not only by blowing (exhaling) but also drawing (inhaling), because reeds are placed almost symmetrical to the slots. The sho is one of the instruments normally used to play the traditional court music of Japan, called Gagaku. In typical Gagaku music, the sho produces four- and five-note clusters. Generally, the sho does not produce sounds with pitch variation like vibrato or glissando, unlike other woodwind instruments, because a player can not control mechanical properties of the reed and pipe length while playing sounds. One of our motivations to build a physical model is to give players more control parameters than those they have now. This will open the possibility of extending sho timbre. We will begin by explaining the basic physical model of the sho briefly. Then, some modification of the model, which improve accuracy, is described. This new model was applied to synthesize various timbre with articulations such as vibrato and tonguing. Then, we shortly describe a prototype system implemented using the model. 2 Physical Model 2.1 Basic Model Our physical model of the sho is shown below (see Fig. 1). A more detailed derivation is described in Hikichi and Osaka (2001). The basic physical model is described by the following equations: d2x x r dx 2 1.5WL d + --+ - = -- (p(t) -p2(t)), (1) dt2 Q dt r M SU(t) 2 P ( (2) 2 CF(x) ] CF(x) (2) F(x) = W[x2 + b2]~ + 2L[0.16x2 + b2] p2(t) = ZoUi (t) + r(t) * (p2(t) + ZoUin(t)), (3) dr Uin(t) = U(t) + 0.4WLddt Equation (1) describes the motion of the reed when pressure inside the wind chest p(t) is applied, where
Page 00000002 x(t) 1 v-- 2.2 Consideration of the Reed Thickness In our former sho model, the reed model was based on the paper (Tarnopolsky et al. 2000). In their work, free reeds with initial displacement were investigated, which means reeds do not normally vibrate into the slots. As a result, effect of the reed thickness during vibration was not considered. However, for sho-type instruments including shengs and khaens, it is essential that reeds go into slots when they vibrate, because the reeds are chiseled from a single piece of metal plate. So, we modify the calculation of the area F(x) to mouth windchest bamboopipe Figure 1: Simplified model of the sho. x! h= b L Figure 2: Close view of the reed. p2 (t) is the pressure just under the reed, x the displacement at the tip of the reed, Q the resonance Q value, and,r the angular frequency. W, L and m are the width, length, and mass of the reed, respectively. Figure 2 shows configuration of the reed. Nonlinear coupling between the reed and the pipe is described by Bernoulli's equation, i.e. eq. (2), where U(t) is the volume velocity through the slit, F(x) the area of the slit, C the flow contraction coefficient, p the air density, 6 the inertia parameter, and b the clearance gap around the reed. Equation (3) is applied to calculate pressure at the entrance of the tube P2(t), where Zo is the characteristic impedance of the tube, Uin (t) the net volume velocity input to the tube, r(t) the reflection function, and the asterisk denotes convolution. The quantity 0.4WL(dx/ is approximately the volume velocity displaced by the reed motion. We have reported the basic characteristics of this model, such as the relationship between pipe length and threshold pressure and between pipe length and vibration frequency in Hikichi and Osaka (2001), and acceptable agreement was obtained. In terms of sound quality, however, there was some discrepancy in the spectra, although the synthesized sounds were perceptually acceptable. An analysis of recorded sounds revealed that, in most cases, the fundamental frequency did not have the maximum energy and 2nd or 4th had high energies. There also seemed to have broad formant peaks around 2-3 kHz and 10 -12 kHz. However, our former model could not produce sounds with such spectra. The following subsections describe some modifications made to the model to get better results. F(x - h), F'(x) F(x + h), F(0), x > h, x < -h, Ix < h. (4) This equation takes the reed thickness into account. Figure 3 shows results obtained by the previous model (top) and by the modified one (bottom). The spectrum for the modified model shows formant peaks around 2 kHz and 7 kHz, and a dip around 4 kHz. Further, these formant peaks were found shift to high frequency when reed thickness was decreased. Moreover, although the sho reed is considered to be symmetrical, there is a small asymmetry of the reed inside the slot because back side of the reed is curved slightly (Masumoto 1968). This asymmetry can be represented by eq. (4) by simply setting a different h value for positive and negative sides. 90 80 70 -60 2o 30 20 I kl%,L IU Fq ny [kHz] Frequency [kHz] 2U Figure 3: FFT spectra of sounds obtained by the conventional model (top), and the new model considering the thickness (bottom). 2.3 Sound Radiation Let us consider sound radiation from an open end of a pipe. From theoretical consideration, the transformation function, defined by the ratio between the sound energy within the pipe and the total radiated energy, is known to rise 6 dB per octave. This function rises below the reflection cutoff frequency defined by ka = 2, where k is the wave number and a the radius of the
Page 00000003 pipe (Fletcher and Rossing 1998). In the case of the sho, cutoff frequency is about 22 kHz. So, we calculated the radiated sound by simply differenciating the volume velocity Uin (t) as a first approximation. As a result, although the radiated sound obtained by differenciating Uin(t) showed acceptable agreement with the recorded tone in the low-frequency range, unnatural high frequency components were found. Then, in order to get a better aural result, an extra lowpass filter was applied to the output signal. This signal is shown in Fig. 4 (top). It was found that the similarity with the recorded tone improved, as shown in Fig. 4 (bottom). This extra low-pass filter may be considered to represent the directional effect, which becomes predominant in the high-frequency range. Frequency [kHz] 3.2 Articulation Synthesis One of the merits of physical modeling is that we can chage model parameters, such as pipe length and the natural frequency of the reed, without physical limitations. Utilizing this merit, we tried to synthesize some typical ornaments, such as vibrato, portamento, tonguing, and tremolo effects. We used three parameters as control parameters: blowing pressure, pipe length, and the natural frequency of the reed (hereafter referred to as reed frequency). The blowing pressure is mainly used to control the amplitude envelop of sounds. It should be noted that there is some indirectness of amplitude control, which means the shape of control parameter tends to be somewhat different from the amplitude envelop of the sound itself (Jaffe and Smith 1995). Nevertheless, when oscillation happens once, amplitude varies almost precisely according to the blowing pressure. Then, both the pipe length and reed frequency affect pitch and timbre. It was found that different pairs of these two parameters produce different timbre at the same pitch. So, these parameters can be used to manipulate timbre. Some of our synthesis results are as follows. Vibrato Figure 5 shows a sound waveform with vibrato and corresponding parameters. Pipe length was fixed to an initial value, and pressure and reed frequency were varied with time. Blowing pressure was inferred from recorded sound samples of normal playing. Reed frequency was varied by sinusoidal function with increasing amplitude. The result is acceptable. Tonguing Figure 6 shows a sound waveform with the tonguing effect. Pipe length and reed frequency were fixed, and blowing pressure was modulated sinusoidally. By the changing modulation rate, we got various sounds from tongued tones to slower amplitudemodulated ones. Timbral Tremolo One of the interesting uses of these control parameters is "timbral tremolo", which means alternation and repetition between two short sounds with different harmonic structures. This is inspired by the "altenate fingering" technique in woodwind instruments, which produces tones of the same pitch using different fingerings. To obtain this effect, we started by finding two pipe length and reed frequency pairs, i.e., (leni, freqi) and (len2, freq2), that would produce the same pitch. Then, we controlled them simultaneously with repetition. Figure 7 shows FFT spectra of the resultant signal taken from different parts (top and middle), and the waveform of the transition (bottom). Timbral transition was obtained, but some distortion was heard. The distortion seems to come from discontinuity when chaging pipe length, i.e., changing the delay of the reflection function. To avoid this discontinuity, we need to include Figure 4: FFT spectrum of a simulated sound obtained by a high-pass and another low-pass filtering of the flow Uin(t), considering radiation effect (top), and that of a recorded sound (bottom). 3 Toward Expressive Synthesis Based on the model above, we will discuss how to control parameters to get expressive sounds. 3.1 How Does Human Play the Sho? A player controls a real instrument using some amount of blowing/drawing air and fingers to open and close finger holes. In the traditional playing style, it is said that the way of changing fingerings while blowing or drawing is highly sophisticated and stylized. However, the player can not change pipe length intentionally during a note by fingering. The natural frequencies of the reeds also can not be controlled by the player. That is, one could say that the pitch can not be changed intentionally, which is in contrast with other woodwind and brass instruments, where the pitch can be changed by the lips and fingers.
Page 00000004 some kind of interpolation operation. 1 S0.5 EQO 0a -0.5 PRESSURE REED FRED. 0.5 1 1. 2 S04 x1o ~0.5 Time [ci 1.5 2 Frequency [kHr] 100 90 80 70 so B so 40 30 20 o s io Is Frequency [kHr] 4UOUI I0 I, i i i I 1 _IL I i 1 I S0.5 1 Time [ci 1.5 2 Figure 5: Control parameters and resulting sounds with vibrato. 2000 0 TI V I 0.99 1 1.01 1.02 1.03 1.04 Time [s] Figure 7: FFT spectra of the signal calculated from different parts (top and middle). Waveform of the transition (bottom). Acknowledgements A part of this work is supported by the Center of Excellence (COB) formation program of the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 1 1CE2005). SC EC -0.5 KO Time [s] II I 1 I 0 - 0.5 1 Time Is] 1.5 2 References Figure 6: Control parameters and resulting sounds with tonguing. 3.3 Implementation A prototype synthesis system was implemented in C, with a simple graphical user interface implemented in Tcl/Tk, and running on Linux PC. The GUI panel offers pipe selection, control parameter modification, and synthesis of sounds with current parameter settings. Users can also see fingerings for traditional sho tone clusters. 4 Conclusions We discussed some modifications of our physical model of the sho in terms of reed thickness and radiation. Then, the synthesis of sounds with various articulations was examined. Based on the synthesis model, vibrato, tonguing, and timbral tremolo effects were obtained, and the possibility of a synthesis model for exploring timbre was shown. Cottingham, J. P. (2001). The Asian free reed mouth organs. In Proceedings of the International Symposium on Musical Acoustics, pp. 61-64. Fletcher, N. H. and T. D. Rossing (1998). The physics of musical instruments. New York: Springer-Verlag. Hikichi, T. and N. Osaka (2001). Time-domain simulation of sound production of the sho, the Japanese freereed mouth organ. In Proceedings of the International Symposium on Musical Acoustics, pp. 71-76. Hikichi, T. and N. Osaka (2002). Measurements of the resonance frequencies and the reed vibration of the sho. Acoustical Science and Technology 23(1), 25-27. Jaffe, D. A. and J. 0. Smith (1995). Performance expression in commuted waveguide synthesis of bowed strings. In Proceedings of the International Computer Music Conference, pp. 343-346. International Computer Music Association. Masumoto, K. (1968). Gagaku -A new approach to traditional music. Tokyo; Ongaku-no-tomo-sha (in Japanese). Nagashima, Y. and T. ]Ito (1999). It's SHO time -An interactive environment for SHO(Sheng) performance. In Proceedirngs of the International Computer Music Conference, pp. 32-35. International Computer Music Association. Tamnopoisky, A. Z., N. H. Fletcher, and J. C. S. Lai (2000). Oscillating reed valves -An experimental study. Journal of the Acoustical Society of America 1 08(1i), 400 -406.