Page  00000325 Tonehole Radiation Directivity: A Comparison Of Theory To Measurements Gary P. Scavone1 Matti Karjalainen2 1Center for Computer Research in Music and Acoustics Department of Music, Stanford University Stanford, California 94305-8180 USA 2Laboratory of Acoustics and Audio Signal Processing Helsinki University of Technology FIN-02015 HUT, Espoo, Finland Abstract Measurements have been conducted in an anechoic chamber for comparison to current linear acoustic theory for radiation directivity from a cylindrical pipe with toneholes. Time-delay spectrometry using an exponentially swept sine signal was employed to determine impulse responses at points external to the experimental air column. This technique is effective in clearly isolating nonlinear artifacts from the desired linear system response along the time axis, allowing the use of a strong driving signal without fear of nonlinear distortion. The experimental air column was positioned through a wall conduit into the anechoic chamber such that the driver and pipe input were located outside the chamber while the open pipe end and toneholes were inside the chamber, effectively isolating the source from the pickup. Measured results are compared to both frequency-domain, transmissionnetwork simulations, as well as time-domain, digital waveguide calculations. 1 Introduction A variety of wind instrument tonehole models have been developed and implemented using both frequency- and time-domain methods (Keefe, 1990; Vilimiki et al., 1993; Scavone and Smith, 1997; van Walstijn and Scavone, 2000). These studies were primarily concerned with the internal scattering characteristics of toneholes in an air column model. Rousseau (1996) reported comparisons of a frequency-domain tonehole radiation model to a discrete set of two-dimensional polar frequency-domain measurements. A discrete-time implementation of this tonehole radiation model was discussed by Scavone (1999). The present paper compares both time- and frequency-domain tonehole radiation model results to a set of measurements conducted in an anechoic chamber using time-spectrometry techniques. 2 Tonehole Acoustic Theory Keefe (1982) presented the first detailed study of linear tonehole acoustic theory. Using frequency-domain, transmission-network methods, the tonehole junction is represented by a two-port element as shown in Fig. 1. The shunt impedance component, Zs, characterizes the properties of the tonehole branch, while the series impedances (Za) represent negative length corrections to the main air column due to the presence of the tonehole. Figure 1: T section transmission-line element representing the tonehole. Two-port digital waveguide implementations of Keefe's representation were discussed in Scavone and Smith (1997) and Smith and Scavone (1997). While the discrete-time implementation produced excellent agreement with Keefe's results, the model was limited in its ability to allow tonehole state changes. Subsequent models to implement a dynamic tonehole state were reported by Scavone and Cook (1998) and van Walstijn and Scavone 325

Page  00000326 (2000). In general, excellent agreement between the frequency- and time-domain results for internal wave propagation has been achieved, even when low-frequency approximations are used to simplify the implementations. Rousseau (1996) incorporated the results of Levine and Schwinger (1948) to model tonehole radiation and directivity from an unflanged cylindrical pipe. Figure 2 shows the radiation directivity calculated for a pipe of 2 centimeter radius. Radiation from an unflanged pipe (0.02 m radius) located inside the chamber, effectively isolating the driver and external microphones from one another. A Beyerdynamic DT880 headphone speaker was attached to the input of the pipe and driven directly from the output of a Power Macintosh computer soundcard. QuickSig DSP and measurement software was used for the signal generation and data acquisition, including source signal deconvolution (Karjalainen, 1990). The measurement setup presented significant problems because of low signal radiation from inside the cylindrical pipe to the external microphone. Measurement contamination due to nonlinear driver distortion was an additional concern. The use of an anechoic chamber minimized the ambient noise level, but a reasonable signal-to-noise ratio necessitated a powerful source signal. Despite having desirable source strength characteristics, maximum length sequence techniques were rejected because of weaknesses in the presence of system non-linearities. With these conditions in mind, a log-swept chirp signal of the form x(t) sin n(_) (e "() ) was chosen for this experiment (Farina, 2000). Chirp techniques are advantageous when nonlinear distortion may be present in a measurement system because linear and non-linear responses are clearly isolated in the time domain result via linear deconvolution. In addition, the log-swept chirp provides a better measure of a system's lowfrequency response, where measurement difficulties are typical, because of greater low-frequency signal energy. The microphone positions used during the measurements are illustrated in Fig. 3. All measureInternal Pickups External Pickup r_0 Angle from axis of pipe (degrees) Figure 2: Radiation directivity vs. angle from pipe end. Toneholes are typically partly flanged and partly unflanged, so that an intermediate model between these two cases would be most appropriate. However, an appropriate analytic solution for real tonehole geometries would be a non-trivial endeavor. This study makes exclusive use of the Levine and Schwinger model. Directivity filters are designed for specific tonehole-to-pickup angles, as reported in (Scavone, 1999). Wave propagation in free space is based on a lossless, near-field model, though the standard 1/r pressure dependence is incorporated in the models. 3 Data Acquisition The directivity measurements for this study were conducted in an anechoic chamber at the Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology using an experimental setup consisting of an aluminum cylindrical pipe of five meters length, 0.021 meter inner diameter, and 0.025 meter outer diameter. Two toneholes, each of 0.009 meter diameter, were drilled 0.1575 and 0.0945 meters from the output end of the pipe. The pipe was inserted through a duct in the chamber wall such that its input end was located outside the chamber and the output end was Figure 3: Internal and external pickup locations. ments were made using the same microphone (1/8 inch, Sennheiser KE4-211-2) and preamp (Unides Design). Two internal measurements were taken at the pipe input (0.0655 meters) and near the pipe center (2.0015 meters). External measurements were taken at a fixed radius (0.1285 meters) from the hole lattice center at 30 degree intervals within a plane bisecting the toneholes and pipe end, as illustrated in Fig. 4. Measurements were made for four different fingerings, varying both the tonehole 326

Page  00000327 90' / Hole 3 \ Hole 2 180' Hol 0*1 e 1 used in the results reported here because viscothermal losses were already incorporated in the models. 5 Measurements vs. Models The processed measurements represent the radiated impulse responses of the experimental system recorded externally at different pickup points. In the following plots, these responses are compared with the results obtained using a discretetime, digital waveguide model and a frequencydomain, transmission-network model of the same system. The measured data shows a consistent "notch" above 9 kHz, which is consistent with the first cross-mode for a pipe of this radius. Because the models are based on propagation in just onedimension, the frequency-domain plots are limited to a range of 10 kHz. 270' Figure 4: plane. Fixed radius external measurement and pipe end states. The computer sampling rate was 44100 Hz. 4 Data Processing Results were obtained by deconvolution (division in the frequency domain) of the recorded external response by the log-swept source signal (Karjalainen et al., 1995). An internal measurement, shown in Fig. 5, was made near the pipe input to capture the combined responses of the computer soundcard, headphone driver, and microphone system. This measurement system response was then deconvolved from the external results. While difficult to discern in Fig. 5, the measurement system produced reliable low-frequency results down to about 150 Hz. 0.06 0.04 0.02 0 -0.02 1 o -20 -30 -40 -50 2 -60 15 Time (ms) ~ ~ ~ 1 ~,- -----c' 70 1 0 1000 2000 3000 4000 5000 6000 7000 8000 Frequency (Hz) 9000 10000 Figure 6: Open holes: 1; Closed holes: 2 & 3. Measurement angle: 210'. 0.3 0.2 0.1 - 0 --0.1 0 0.5 1 1.5 2 2.5 3 Time (milliseconds) -20 -40 -60 -80.100 0 0.5 1 Frequency (Hz) In general, there is good "first-order" agreement between the measurements and models, both in terms of temporal features and frequency magnitude contours. For example, the response of the 3.5 4 4!5 5 system with all holes open is shown in Fig 8. In the top plot, the pickup is located directly above the toneholes. The pickup is located directly "behind" the toneholes in the bottom plot. In the time do- main response, the radiated impulse components from the toneholes are first recorded, followed by radiation from the open end of the pipe. A few consistent discrepancies can be observed. 1.5 2 X10 In particular, "rear angle" radiation appears to be overestimated using the Levine and Schwinger ponse near pipe model (compare the top and bottom plots of Fig. 7-9). This can be partly attributed to the use of the unflanged tonehole model. Relative sponse measure- phase differences between tonehole and open-end lation and possi- "events" can be partially explained by the fact that losses occurring the models implement a one-dimensional system. m the pipe input That is, the toneholes are represented by pointirement was not sources located at their midpoint along the prin Figure input. 5: Measured internal res The second internal pipe re4 ment was made to allow the calcu ble compensation of viscothermal for internal wave propagation froi to the first tonehole. This measi 327

Page  00000328 - Measurement - Digital Waveguide - - Transmission-Network 0 C 14 14.5 15 Time (ms) 15.5 16 0. us 0.0 -0.0 rn8 -6 -7 -0 -S-20 -P-30-~ us-40' 500 ~ -60 --77 0.1 0.05 -us 0 --0.05 - 1000 2000 3000 4000 5000 6000 7000 8000 9000 1C Frequency (Hz) -Measurement jl- Digital Waneguide -- Tranemission-Network / 1 - Measurement - Digital Waveguide Ii i - - Transusissiun-Netwurk i'(/ 14 14.5 15 15.5 1 0, Time (ins) 00 1000 2000 3000 4000 5000 6000 7000 8000 9000 1001 Frequency (Hz) 8 -Measurement 6 - Digta Waeguieewr 4 2 6 1 14 14.5 15 15. 16 Time (ns) 0.0 0.0 - 0.0 S0.0 us -0.0: -0.0 -0.0' I 14.5 15 15.5 16 Time (ms) ta--is a -20 a -30 us 40,50 ~ -60 7 - - - (-I us -10 a -20 -30 s -40 ~ -60 K 3 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 7: Open holes: 1 & 2; Closed holes: 3. Measurement angles: 60' (top), and 300' (bottom). cipal axis of the pipe. Depending on the pickup location, the models will overestimate the propagation delay by an amount corresponding to half the hole diameter. In addition, when the pickup is located "below" a tonehole (angles between 210. and 330'), propagation delay across half the pipe diameter is neglected (as well as delay attributable to the pipe curvature). These affects are visible in Figs. 7 and 8. Closed-hole reflections are evident in the model results but more difficult to discern in the measurements (see Fig. 6). 6 Summary The measurement technique applied in this study appears to provide good results over a wide frequency range, with particular improvement at low frequencies. The measurements and model results show good "first-order" temporal and spectral agreement. Some discrepancies are apparent, most of which can be explained in terms of model simplifications. It appears that the unflanged open-hole model over-estimates rear-angle directivity. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 8: All holes open. Measurement angles: 90' (top) and 270' (bottom). References Farina, A. (2000). Simultaneous measurement of impulse response and distortion with a sweptsine technique. Audio Engineering Society. Presented at the 108th AES Convention, Paris, France. Karjalainen, M. (1990). DSP software integration by object-oriented programming: a case study of QuickSig. IEEE ASSP Magazine, pp. 21-31. Karjalainen, M., Huopaniemi, J., Viilimiiki, V., and Hernoux, B. (1995). Explorations of wind instruments using digital signal processing and physical modeling techniques. J. of New Music Research, 24(4):301-317. Keefe, D. H. (1982). Theory of the single woodwind tone hole. J. Acoust. Soc. Am., 72(3):676 -687. Keefe, D. H. (1990). Woodwind air column models. J. Acoust. Soc. Am., 88(1):35-51. Levine, H. and Schwinger, J. (1948). On the radiation of sound from an unflanged circular pipe. Phys. Rev., 73(4):383-406. 328

Page  00000329 0.05 Measurement uide Viilimiiki, V., Karjalainen, M., and Laakso, T. I. 0.04 Digital Waveguide 03- 0 Transmission-Network (1993). Modeling of woodwind bores with finger 0.02 - holes. In Proc. 1993 Int. Computer Music Conf., S.01 - - -pp. 32-39, Tokyo, Japan. Computer Music As_o. F- sociation. 14 14.5 15 15.5 16 Time (ms) van Walstijn, M. and Scavone, G. P. (2000). The wave digital tonehole model. In Proc. 2000 -o Int. Computer Music Conf., pp. 465-468, Berlin, - Germany. Computer Music Association. U -30 \.-40 B I1 2 -60 -70 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) 0.1- Measurement 0.08 'I - Digital Waveguide 006 I - Transmission-Network 0.04 - /.0 0.02 -0 Y -0.02 14 14.5 15 15.5 16 Time (ms) ~ -20- - N. -50 2 -60 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure 9: Open holes: 2 & 3; Closed holes: 1. Measurement angles: 0' (top) and 240' (bottom). Rousseau, A. (1996). Modelisation du rayonnement des instruments a vent a trous lateraux. Tech. rep., Institut de Recherche et Coordination Acoustique/Musique, Departement d'acoustique instrumentale (responsable: Rene Causse). Scavone, G. P. (1999). Modeling wind instrument sound radiation using digital waveguides. In Proc. 1999 Int. Computer Music Conf., pp. 355 -358, Beijing, China. Computer Music Association. Scavone, G. P. and Cook, P. R. (1998). Real-time computer modeling of woodwind instruments. In Proc. Int. Symp. on Musical Acoustics (ISMA98), Leavenworth, WA, pp. 197-202. Scavone, G. P. and Smith, J. O. (1997). Digital waveguide modeling of woodwind toneholes. In Proc. 1997 Int. Computer Music Conf., Thessaloniki, Greece. Computer Music Association. Smith, J. O. and Scavone, G. P. (1997). The one filter Keefe clarinet tonehole. In Proc. IEEE Workshop on Applied Signal Processing to Audio and Acoustics, New York. IEEE Press. 329