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Page 00000001 A physically informed model of a musical toy: the singing tube Stefania Serafin, Juraj Kojs Medialogy Aalborg University Copenhagen sts @ aue.auc.dk Abstract The use of physically modeled musical toys for interactive performances in proposed and discussed. By modeling musical toys using physical principles, and using them in interactive computer music, it is possible for a larger audience to become interested in new music. 1 Introduction Physical models of musical instruments and sounding objects in general are interesting from different perspectives: from the point of view of an acustician, the use of physically based simulations allows to understand if a specific physical phenomenon has been well understood. From the point of view of a computer scientist, the development of efficient algorithms which allow to simulate in real-time physical phenomena is an interesting challenge. From the point of view of composers and performers, the appeal is to use such models in order to create sonorities which cannot be achieved with traditional instruments. Although physical models are appealing from the perspectives described above, there is still the problem of the acceptance of physical models from a general audience. In this paper we propose the use of corrugated tubes as musical instruments to be used in interactive performances. By creating a physical model of such instruments which can interact in real-time with its real counterpart, it is possible to allow different people to be active and participate in human-computer music performances. Physical models of sounding toys become therefore interesting for: 1. Pedagogical purpose: models can be used together with the real instrument to illustrate and explain to students physical and acoustical phenomena. 2. Performance purpose: models can be used together with the real instrument in interactive performances where the human player does not need to be a skilled performer, since such musical toys have a very steep learning curve. 3. Entartainment purpose: we observed that the use of such musical toys in interactive performances allows people to enjoy being involved in new music performances, which usually are not appealing to a large audience. Attractiveness of everyday objects such as plastic corrugated tubes lies in their potential for musical expressiveness. Rich timbral context can be achieved through a variety of performing modes. For example, the tube player can manipulate the instrument in the following ways: whirling in the air, blowing inside, tapping on one of the tube's ends, and scratching its surface. Motoric interaction of the player with the instruments is crucial in such a performance. The players enjoy the physical contact with the instrument and their immediate sonic response. In comparison with most of the traditional musical instruments, performing with the tubes is easy to learn. Moreover, these instruments are inexpensive and can be found in any toy store. The simplicity of the instruments limits their expressive possibilities. This may be overcome when the instruments are used in an interactive performance with the computer. In such case, they are combined with their physical models. The models extend the sonic properties of the real instruments and enhance their expressive potential. Effective communication between the real instrument and model result in creation of a new instrument with has a rich timbral pallet. 2 The singing tube The Voice of the dragon is the name given to a group of Japanese children twirling flexible plastic tubes above their heads. The burst of tones that emerged from each musical pipe soared and dropped with rotational velocity. A corrugated plastic tube, infact, produces pleasant sonorities while rotating in a circular motion. Singing corrugated tubes became popular in early 70s when a toy called hummer was introduced to the market. The hum Proceedings ICMC 2004
Page 00000002 mer is a corrugated plastic tube about one meter long, similar to the one shown in figure 1. When whirled in the air, the tube produces a series of pitches, starting from the first harmonic to its overtones. Mark Silvermann, after a visit to r d=000.6m = 0.019 m L = 1.05 m Figure 1: A singing plastic tube and its relevant physical dimensions. Japan where he heard the Japanese children's performance, studied the acoustics of such tubes (Silverman 1989). He mounted the corrugated tube to a thin slab attached to a wheel free to rotate in a vertical plane, with a counterweight fixed at the opposite end of the slab so that the center of mass of the system laid on the axis of the wheel. A motor whose velocity was varied using a rheostat was moving the wheel. Using a microphone, a stroboscope and a counter, the tones produced were recorded and the spin rates measured. The data were analyzed by Fourier analysis. Also Crawford studied the acoustics of corrugated tube using a different device (Crawford 1974). He attached the tube to his car and took it for a ride. Different tones were produced according to the car's velocity. Both researchers made the following observations: whirling the tube slowly initiates the first overtone; with increased velocity, the higher partials resonate. Obviously, the length of the tube determines the pitches that will sing. Blowing into a smooth tube or whirl it in the air, no sound is produced. However, whirling a corrugated tube results in a noticeable tone. In a corrugated tube open at both ends, a tone is produced when the "bump" frequency of the air flowing through the tube equals one of the resonant frequencies of the tube. Air velocity, tube length, corrugation and diameter sizes therefore influence the pitch and volume of the sound produced. The interest of toy makers as well as composers on singing tubes is due to the fact that the tubes' pure tones create an unusual ear-pleasing sonic zone easily discernible from the sound of the other bore-based instrument family members. In the following section the sound production mechanism of corrugated tubes is described in details. 2.1 Acoustics of corrugated tubes In order to understand how the corrugated tube produces sound, different components need to be taken into account. As suggested in (Silverman 1989) and (Crawford 1974), with both ends open the tube resembles a centrifugal pump. When whirling, the air is sucked in through the end closer to the hand and pushed out through the outer end. In order to make the vibrational modes resonate and thus produce pitch, some of the airflow energy is converted to excitation energy. It is interesting to investigate the dynamics of the tube first at a large scale and then at a microscale, which implies examining the role of the corrugations. At a large scale there exists a vortical flow centered in the stationary end of the tube and normal to the axis of the tube with tangential air velocity given by: V(s) = WUnS where Wmu is the angular velocity and s is the position along the tube (from 0 to L). Along the tube, the rotationally induced pressure difference between the two ends produces an axial flow with velocity v. In order to relate the angular velocity wm to the pressure difference p, assuming the flow to be incompressible and smooth, it is possible to use Bernoulli's principle, and consider that moving past the corrugations the air is perturbed at a frequency proportional to the axial air velocity and inversely proportional to the corrugation spacing. When the frequency of perturbation of the air coincides with a resonant frequency of the tube, the sound is amplified. The finally component of the acoustics of the tube that influences its sonorities is the role of the rotation. When the tube rotates a Doppler shift is perceived, i.e. an apparent change in frequency content which is due to the motion of the tube relative to the listener. As derived in most elementary physics texts, the Doppler shift is given by 1 + V1s l- s= 1 - L (1) where wýJ is the radian frequency emitted by the source at rest, wl is the frequency received by the listener, vis denotes the speed of the listener in the direction of the source, vsl denotes the speed of the source in the direction of the listener, and c denotes sound speed. 2.2 Analysis of a corrugated tube We analyzed the tube shown in figure 1, in order to validate the theory presented in the previous section. The corrugated plastic tube was 1.08 m long, and with a radius r 0.019 m. The corrugations were 6 mm long. Proceedings ICMC 2004
Page 00000003 Mode Theoretical Experimental Rotational number n Frequency Frequency Velocity 1 156 Hz 2 312 Hz 310 Hz 0.5 Hz 3 468 Hz 464 Hz 0.9 Hz 4 624 Hz 625 Hz 1.7 Hz 5 781 Hz 769 Hz 2.5 Hz 6 937 Hz 925 Hz 3.0 Hz 7 1093 Hz 1081 Hz 3.3 Hz 8 1249 Hz 1250Hz 4.2 Hz Table 1: Mode number, theoretical and experimental frequency and rotational velocity of the tube used for the simulations. The second column of table 1 shows the theoretical modes of our corrugated tube. The third column shows the experimental frequencies obtained by recording the tube while rotating at different velocities and analyzing the amplitude of its frequency response. The peak of the spectrum were easily detected, since one strong mode is always perceivable. Note the good agreement between the experimental and theoretical frequency. The fourth column shows the rotational velocities associated with each frequencies. Those velocities were calculated looking at the time domain waveform, and counting the number of rotations per second which are clearly visible as, for example, figure 2 shows. Each rotation, infact, corresponds to a modulation of the time domain waveform. ~: k i ii 0.08 0.06 - 2500 - iiii iiiiiiiii iiiiii iiii,, 1ooo !!!!!!!!iJi~i.......::ii~i~...................... iiiiiiiiiiiii.....iiiiiiiiiiii o 5 0 0..........................ii..... o 0.5 1 1.5 2 2.5 Time (s) Figure 3: Sonogram of the rotating tube while varying the rotational speed. Two arpeggios are obtained by varying the rotational speed. 2.3 Modeling corrugated tubes We consider the tube as an organ pipe open at both ends. A one dimensional digital waveguide can therefore be used to model the tube resonator (Smith 1996). We model the corrugations by reproducing their effect rather than building a precise physical model, since the turbulence inside the rotating tube are complex and still not well understood. In our algorithm the distance between corrugations is known, and the axial air velocity v, of mode n is calculated. At each sample i, we calculate the frequency fen for which fen v,/d. Then we calculate the closest mode fa of the tube that corresponds to fen. This is the center frequency of a bandpass resonant filter whose role is to amplify fa. The last step of the tube simulation consists of modeling the rotation. The motion of the tube relative to the listener produces the well-known Doppler effect, i.e. an apparent change in frequency content of an acoustic signal. A simulation of the Doppler shift was proposed in (Takala and Hahn 1992) and (Savioja, Huopaniemi, Lokki, and Viininen 1999). Recently (Smith, Serafin, Abel, and Berners 2002) a detailed simulation of the Doppler shift was proposed, which uses time varying delay lines (Laakso, Vilimiki, Karjalainen, and Laine 1996). The Doppler shift was applied to the simulation of the circular rotation of a Leslie horn. Since there is a strong similarity between the rotation of the Leslie horn and the rotation of the corrugated tube, we use the same algorithm here. The model is driven by two kinds of parameters: 1. Physical parameters: Length of the tube L and distance between corrugations d. Figure 2: Time domain waveforms for the corrugated tube when rotating at 1.7 rot/sec. Notice the modulations given by the Doppler effect. Figure 3 shows the sonogram of the rotating tube while performing two arpeggios varying the rotational speed twice over time. High partials are amplified when the rotational speed increases, according to Table 1. Proceedings ICMC 2004
Page 00000004 2. Control parameters: Angular velocity Wjm and rotational radius rs. The angular velocity Wm and the rotational radius rs are used as a parameter for the Doppler effect simulation. A fractional delay line (Laakso, Vilimiki, Karjalainen, and Laine 1996) allows to continuously vary the discrete length N of the tube, where N = fs2L/c, where fs is the sampling rate and c is, as before, the speed of sound in air. A sharp second order resonant filter, tuned to the mode of the tube closest to wL/d, where d is the distance between corrugations, models the effect of corrugations as well as propagation losses. We noticed, infact, that adding an additional lowpass filter to account for propagation losses along the tube does not improve the quality of the synthesis. We therefore lump all frequency dependent losses into the bandpass filter that accounts for corrugations. 2.4 Simulation results Figure 4 shows the sonogram of the synthetic corrugated tube while increasing the rotational speed. Notice how, as in the case of the real tube, one strong mode appears in the spectrum when the rotational speed increases. piece Garden of Dragon by Juraj Koys for cellophane, corrugated tubes and computer. This piece presents a communication between the virtual and real singing pipes using Max/MSP. The Max/MSP physical model extends the possibilities of the real tubes in several areas: vibrato, pitch and amplitude control, and noise to pitch ratio. The acoustic tubes control the virtual choir of tubes using the fiddle~ pitch tracker (Puckette and Apel 1998) of Max/MSP. 4 Conclusions The use of physically modeled musical toys in interactive performances is proposed. Musical toys allow a wider audience to become interested in interactive computer music, giv the fact that they are easy and fun to play and at the same time can produce interesting sonorities while connected to a computer model. References Crawford, F. (1974, april). Singing Corrugated Pipes. American Journal of Physics 42, 278-288. Laakso, T. I., V. Vilimiki, M. Karjalainen, and U. K. Laine (1996, January). Splitting the Unit Delay-Tools for Fractional Delay Filter Design. IEEE Signal Processing Magazine 13(1), 30-60. Puckette, M. and T. Apel (1998). Real time audio analysis tools for Pd and Max/MSP. In Proc. International Computer Music Conference, Michigan, Ann Arbour, pp. 109-112. Savioja, L., J. Huopaniemi, T. Lokki, and R. Vaininen (1999, Sept.). Creating interactive virtual acoustic environments. Journal of the Audio Engineering Society 47(9), 675-705. Silverman, M. P. (1989). Voice of the Dragon: the rotating corrugated resonator. Eur. J. Physics 10, 298-304. Smith, J. (1996). Physical Modeling using digital waveguides. Computer Music Journal 2, 44-56. Smith, J., S. Serafin, J. Abel, and D. Berners (2002). Doppler Simulation and the Leslie. In Proc. Workshop on Digital Audio Effects (DAFx-02), Hamburg, Germany, pp. 188-191. Takala, T. and J. Hahn (1992). Sound rendering. Computer Graphics SIGGRAPH'92(26), 211-220. Zicarelli, D. (1998). An extensible real-time signal processing environment for MAX. In Proc. International Computer Music Conference, Michigan, Ann Arbour, pp. 463-466. 2500 1500 Time Figure 4: Sonogram of the synthetic increasing the rotational speed. corrugated tube while 3 Musical applications The corrugated tube physically informed model has been implemented as an extension to the real-time environment Max/MSP (Zicarelli 1998). The model has been used in the Proceedings ICMC 2004