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Page 00000468 Modeling the Woodstock Gamelan for Synthesis Lydia Ayers and Andrew Homer Department of Computer Science, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong email: layers @cs.ust.hk and email@example.com ABSTRACT: This paper considers the spectral properties of the Woodstock Gamelan, a 3-octave set of tubular chimes in just intonation with a "tonic" (1/1) of G3. Throughout the instrument's pitch range, each chime has only three to five significant partials, and the frequency ratios of the partials are the same in the low and high registers. We have developed a Csound implementation for synthesizing Woodstock Gamelan tones. With simple changes to the model parameters, it also produces attractive related timbres. 1 Introduction The Woodstock Gamelan is a set of tubulong chimes inspired by the justly tuned instruments of Harry Partch [1-3], Erv Wilson, Dean Drummond [4-5] and Lou Harrison . Woodstock Percussion in upstate New York began making wind chimes in 1979. One of the authors heard these wind chimes at a crafts fair, where to her they sounded magical. She bought a set of "gamelan" chimes supported in a rack. The original number of tubes in the set was ten, forming the same three pentatonic scales as the wind chimes. But ten tubes was not enough! She ordered more and more custom-tuned tubes until the total instrument included more than 200 tubes (including 75 tubes in the middle octave). The Woodstock Gamelan has two types of aluminum tubes. Racks support the tubes down to about Eb4 (128/81, a frequency ratio relative to G3, the lowest tube on the gamelan). The length of the tube determines the pitch, and the diameter ranges from 25 to 30 mm. The highest tubes have a vibraphonelike sound. The lowest of the rack tubes sound more damped, because the partials above the fundamental stop vibrating after about.3 seconds. The hanging tubes do not have this damped quality. The hanging tubes have larger diameters than the rack tubes. The sound of the 26 hanging tubes is similar to that of orchestral chimes (tubular bells), the main differences being that they are made of aluminum instead of brass or steel and use softer mallets. Previous work on computer modeling pitched percussion instruments primarily focused on simulating large Western bells -, while instruments such as the vibraphone and orchestral chimes have received much less attention -. This paper investigates the spectral properties of the Woodstock Gamelan and develops a model for it. 2 Spectral Properties of the Woodstock Gamelan We performed a phase vocoder analysis on the Woodstock Gamelan to estimate the amplitudes and frequencies of the partials. The phase vocoder implements a bank of bandpass filters centered on the harmonics of a fundamental frequency . Beauchamp  gives more details on the application of this technique. We made some simple modifications to adapt Beauchamp's program to combine the frequency bins containing inharmonic partials . We analyzed eight tones over the pitch range of the instrument: G3-G6. We used all the G's and Db's of the gamelan's three-octave range, and since the quality of the low hanging tubes changes too much from G3 to Db4, we included Bb3 to fill the gap. The frequency ratios are not harmonic, but are very similar from one tone to another. The average frequency ratio of the second partial is 2.69, the average third partial is 5.15, the average fourth partial is 8.38, and the average fifth partial is 12.08. The fifth partial only appears in the low tones (G3-Db4) with any significant strength. The fourth partial is significant in the tones G3-G4, and the third in G3-G5. All the strong partials have a flat trajectory without any noticeable frequency modulation. -468 - ICMC Proceedings 1999
Page 00000469 Figure 1 shows the amplitude envelopes of the partials for the same eight tones. The third partial dominates the other partials in the low hanging tubes (G3, Bb3, Db4), while the fundamental is strongest in the higher rack-supported tubes. The fundamental increases in dominance and the higher partials weaken as the fundamental frequency increases. The fundamental of the rack tubes has a slower decay over the first second of the tone, though the overall decay time shortens with high fundamental frequencies, ranging from 21 seconds for the low hanging tubes, down to 4 seconds for the highest tube. Ideally, the precision-tuned tubes produce tones with minimal amplitude modulation, but scratching and denting them can cause them to beat slightly. Two of the tones (Bb3's third partial and G3's fundamental) have some amplitude modulation. R: G6 (1568 Hz) Db5 (550 Hz) G5 (784 Hz) Db6 (1100 Hz) G3 (196 Hz) Bb3 (235 Hz) Db4 (277 Hz) G4 (393 Hz) Figure 1. Amplitudes plotted against time for the first second of several Woodstock Gamelan tones. 3 A Synthesis Model for the Woodstock Gamelan We can use the parameters from the analysis to build a synthesis model for the Woodstock Gamelan. We use additive synthesis with three to five partials. Random amplitude and frequency deviations show up in the analysis of the original tones, and we include them in the model. Without the random deviations, the synthesized tones are noticeably less "life-like," so we regard them as necessary features. The random amplitude modulation uses interpolated random values between +5% of the overall amplitude. The interpolated values change 5-10 times every second. The random frequency modulation uses interpolated random values between ~.2% of the fundamental frequency. The interpolated values change 50-80 times every second. The amount and frequency of the random deviations approximate that of the original tones. We used the models to construct a Csound instrument design. The code includes a few extra lines to simulate damping on early releases. We used an automated breakpoint-picking program  to datareduce the amplitude and frequency envelopes for the Csound implementation. The program picked breakpoints at times.003,.006,.01,.03,.06,.1,.3,.6, 1, 3, 6, 9, 12, 15, 18 and 21 seconds (if the tone is shorter than any breakpoint time, a zero-value is used for that breakpoint). To avoid selecting outlying breakpoint values, the program averaged the amplitudes of points with time values within +20% of each breakpoint to determine the breakpoint value. Averaging also reduces undesired amplitude modulation. The Csound instrument contains the parameters from the eight original analyzed tones. Each tone has its own amplitude envelopes and scaling factors (and an amplitude normalization factor). The Csound instrument automatically selects the parameters of the original tone nearest in frequency to the fundamental frequency. The frequency ratios of the partials are set to 1, 2.667, 5.333, 8.533, and 12, which are the nearest just frequency ratios to those found in the previous section. These ratios can be changed to the average ratios (1, 2.69, 5.15, 8.38, and 12.08). ICMC Proceedings 1999 -469 -
Page 00000470 4 Results and Extensions The harmonic spectrum amplitude envelopes of the synthesized Woodstock Gamelan tones are good matches to those shown in Figure 1. The main difference is that our breakpoint picking algorithm has smoothed out the amplitude modulation of Bb3 and G5. In addition to simulating the Woodstock Gamelan, our design can also produce related sounds by changing the value of a brightness factor. The Csound instrument automatically selects the parameters of the original tone nearest in frequency to the fundamental frequency, or the user can force the instrument to use a particular parameter set. This allows a sort of "brightness" control, since the parameters of the lower tones include more partials with significant amplitudes. The Csound score controls brightness: a zero value indicates to use "nearest neighbor" parameter selection, while a value of 1 selects parameters of the highest tone (the least bright), and a value of 8 selects parameters of the lowest tone (the brightest). Figure 2 gives some suggested timbral semantic labels for different pitch ranges and brightness factors. The dots indicate the trajectory of the Woodstock Gamelan in this space, with an orchestral chime-like sound in the lower register gradually changing to one of the most attractive sounds, a "silky" vibraphonelike timbre in the high register for brightness (BR) 1 between G5-G7. Interesting surprises were a Javanese Gamelan sound, a Balinese trompong, and a damped steel drum. The strident orchestral chime sounds more "electric" above about G4 for brightness 6-8. Between G3-G4, there is also an African thumb piano timbre for brightness 3 that sounds like a low marimba for brightness 4 and 5. We have completed several original compositions using the colorful timbres of the Woodstock Gamelan Csound design, including Ontres de la Leune, Theme and Variations for Woodstock Gamelan, Merapi, and Sultan Agung's Music Box. P I T C H G7 G6 G5 G4 G3 G2 G1 GO Xylophone Crotales Glockenspiel with sustain Chinese Qing Chinese Qing Chinese Vibraphone * Xylophone struck stone) (struck stone) Triangle Xylophone Pengling Xylophone with sustain with sustain (small bell) Small M l Woodstock Woodstock Javanese Woodstock Woodstock W ock with sustainrs Gamelan Gamelan Homemade Bonang Gamelan Gamelan Gamelan it ssta with rubber with rubber metal bars Pots rng mallets mallets -Steel Large African Sustained Sustained ock Wood k Drums Javanese umb Low Low Wootock Woodstock Orchestral Drums Javanese Thumb Low Low Gamlan Gamelan Chimes with Soft Bonang Piano Marimba Marimbaan Ga lan mes Mallets Pots A Low Sustained Small Marimba Balinese Electric Javanese Low Javanese and Trompong Bass Gong Marimba Bonang Vibraphone (Pot Drums) Pots Cross -Plucked -Plucked -Plucked Large Plucked A Low -Plucked -Plucked -Plucked javanese String String String Bonang Sheet Javanese Bass Bass Bass Pots Metal Gong -Plucked String Bass 1 2 3 4 5 6 7 8 BRIGHTNESS FACTOR Figure 2. Woodstock Gamelan interpreted timbre labels for different pitch ranges and brightness factors. The dots indicate the trajectory of the Woodstock Gamelan in this space. - 470 - ICMC Proceedings 1999
Page 00000471 5 Conclusions The Woodstock Gamelan has only five exponentially decaying partials, with frequency ratios of about 1, 2.69, 5.15,8.38 and 12.08. The ratios are close to the just ratios of 1, 2.667, 5.333, 8.533 and 12. With appropriate amplitude envelopes, we built a model for the Woodstock Gamelan that produces tones that our listening tests show are nearly the same as the originals. Our model allows composers to conveniently resynthesize and extend the beautiful tuning and timbre of the Woodstock Gamelan without devoting a huge room to the instrument. Such modifications include changing the brightness factor and envelopes to manipulate the timbre, and changing the frequency ratios to control the inharmonicity of the sound. Acknowledgements Thanks to Stacey Bowers at Woodstock Percussion for a stimulating discussion and for answering all our questions. This work was supported in part by the Hong Kong Research Grant Council's Projects HKUST6073/97E, HKUST6136/98E, and HKUST6087/99E. We used James Beauchamp's excellent spectral analysis and display package Sndan for the spectral plots in this paper. References  H. Partch, Genesis of a Music. (Da Capo, New York, 1974).  H. Partch, Enclosure i: Four Historic Art Films by Madeline Tourtelot with Music by Harry Partch. [video recording] (Innova 400 Recordings, St. Paul MN, 1995).  H. Partch, The Bewitched. [CD] (Composers Recordings, New York, 1990).  D. D'ummond, Newband Play Microtonal Works by Partch, Cage, La Barbara and Drummond. [CD] (Mode, New York, 1990).  D. Drummond, Newband Play Microtonal Works by Harry Partch, Thelonius Monk, Mathew Rosenblum, Dean Drummond & James Pugliese. [CD] (Mode, New York, 1994).  L. Harrison, Lou Harrison: Gamelan Music. [CD] (Musical Heritage Society, Ocean, NJ, 1993).  J.-C. Risset, Introductory Catalogue of Computer-Synthesized Sounds. (Bell Telephone Laboratories, Murray Hill, NJ, 1969).  J. Chowning, "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation," Journal of the Audio Engineering Society, vol. 21(7), pp. 526-534 (1973).  J. Harvey, "Mortuos Plango, Vivos Voco: A Realization at IRCAM," Computer Music Journal, vol. 5(4), pp. 22-24 (1981).  T. Rossing, The Science of Sound, (Addison-Wesley, Reading, MA, 1990), pp. 264-266.  G.W. Baxter and K. Hagenbuch, "A Student Project on Wind Chimes -- Tuning in to Standing Waves," Physics Teacher, vol. 36, pp. 204-208 (1998).  B. Vercoe, Csound: A Manual for the Audio Processing System and Supporting Programs with Tutorials (MIT Media Lab, Cambridge, MA, 1992).  M. Dolson, "The Phase Vocoder: A Tutorial," Computer Music Journal, vol. 10(4), pp. 14-27 (1986).  J. Beauchamp, "Unix Workstation Software for Analysis, Graphics, Modification, and Synthesis of Musical Sounds," Audio Engineering Society Preprint No. 3479 (1993).  A. Homer, L. Ayers, and D. Law, "Modeling Small Chinese and Tibetan Bells," Journal of the Audio Engineering Society, vol. 45(3), pp. 148-159 (1997).  A. Homer and J. Beauchamp, "Piecewise Linear Approximation of Additive Synthesis Envelopes: A Comparison of Various Methods," Computer Music Journal, vol. 20(2), pp. 72-95 (1996). ICMC Proceedings 1999 -471 -