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Page 154 ï~~Implementation of a Variable Pick-Up Point on a Waveguide String Model with FM/AM Applications Scott A. Van Duyne and Julius 0. Smith, III Center for Computer Research in Music and Acoustics (CCRMA) Music Dept., Stanford University, Stanford, CA 94305 email: email@example.com Abstract The waveguide string model can be extended through the addition of a movable pick-up point along the string. As the pick-up slides along the string, a flanging effect results. Accelerating the pick-up produces glissandi effects. Modulating the placement of the pick-up sinusoidally at audio rates produces modified FM sidebands around each partial on the string, with control over the presence of even and odd sidebands. This non-physical effect can be used to enrich the timbre of physical models, taking advantage of the wellunderstood theory of FM synthesis. 1. Background and Description of the Model The bidirectional traveling displacement waves on an ideal string can be represented by a waveguide synthesis model. The actual displacement of the string at any point is determined by adding the displacements associated with the left- and right-going traveling waves in the bidirectional delay lines. The ideal reflection of the traveling waves at each end of the string is modeled by multiplying the incoming wave by -1 and sending it through in the other direction. Figure 1 illustrates the ideal string model. To listen to the vibrations on this string, a pick-up must be placed at some point on the string. For our purposes, we would like to place a pick-up at any continuous point along the string. This necessitates an interpolation of some kind. If the pick-up is to be placed exactly at a sample point, the values of the upper and lower delay lines at that point are simply added. If the pick-up is to be placed. at a fractional distance, a, between two sample points, then a weighted average of adjacent sample values must be performed. In this initial research, linear interpolation is used. The results are excellent, but there is room for a better interpolation method in extreme cases of pick-up point motion. Figure 2 shows a close-up of the implementation of an interpolated pick-up point. Sure I- cr 2 y(n) 154
Page 155 ï~~2. Moving the Pick-Up Point Acoustically, the position of a pick-up on a string determines the relative magnitudes of the partials heard. If it is placed at one of the end points, no sound will be present since the end points do not move. If the pick-up is placed in the exact center of the string, only odd harmonics will be present in the spectrum because all even harmonics will have a node at that point. In general, placing the pick-up at a position 1/k across the string will zero out partials with harmonic number a multiple of k, and reduce levels of near multiples of k. The placement of a pick-up is therefore essentially a comb filter. Sliding the pickup along the string slowly creates a flanging effect as the nodes of the various harmonics are crossed. Figure 3 shows a sonogram view of the spectrum of a sliding pick-up point. In this picture, the time axis is horizontal, and the frequency axis is vertical. The magnitude is indicated by relative lightness and darkness. Here, the pick-up slides across the length of the string at constant speed over the duration of the sound. The evolution of the first twelve harmonics on the string are shown in this sonogram view. Fig u. Figure 4. I.. 11. ___,__ __.. _ __ __ __ __.___,-_ __ __, When we slide the pick-up along the string, we are, in effect, catching up with one of the waves and losing ground with the other. This results in a separation of their frequencies due to Doppler shift in opposite directions. In an alternative view, since sliding the pick-up across the string at a constant rate modulates the amplitude of each harmonic on the string sinusoidally at a frequency of 0.5 k s, where 0.5 k is the number of periods of the kt harmonic standing on the string, and s is the number of string lengths per second that the pick-up is moving, one can view sliding the pick-up point as a ring modulation splitting each harmonic, fk, into two partials, fk Â~ 0.5 k s. Figure 4 shows a sonogram of a pick-up making discontinuous speed increases. Notice that the higher harmonics spread more than the lower in this linear frequency plot. This phenomenon is short-lived, however, if we are bounded by the string end points. It is convenient to contrive a theoretical placement of the pickup point beyond the ends of the string by extrapolation of the standing wave on the string. To generalize pick-up point motion we must define what it means to place the pick-up somewhere off the end of the actual string, that is, to extend the string in some way without discontinuity of the traveling waves in the delay lines. We do this by reading the separate upper and lower traveling waves back through the end point filters (here perfect reflections) and around backwards on the other delay line. To read at a pick-up located theoretically at some 1+8, where the string length is 1, and 5 is a positive number less than 1, we have the upper displacement, y+(1 + 5) = -y7(1- 5), and the lower displacement, y-(l +5b) = - y+( 1 - 5). Therefore the composite displacement of the string, y, at pick-up point, 1+5, is as follows: l+5) = y(l+(+)R+ y-(1 +S) = -y-(l -S)- y+(1-S) -- -(1 -5) In other words, to obtain a hypothetical displacement beyond the end of the string, just flip the sign and read backwards along the string. This is consistent with the "image method" for computing the displacement from traveling waves (Morse 1936). Now that we can maintain a continuous motion of the pick-up in one direction indefinitely, we may consider an arbitrary constant speed and acceleration of the pick-up point along the string. Figure 5 shows a string loaded with two harmonics at 155
Page 156 ï~~I kHz and 5 kHz with an accelerating pick-up point to produce expanding glissandi. In this four-second example, the pickup point speed accelerates from 0 to 1000 string lengths per second. Since a string length is half the wavelength of the string fundamental, the downward gliding frequencies make it half way to DC. (There are also some aliasing sidebands in the picture relating to the error of linear interpolation.) In the four seconds of sound depicted in Figure 6, the pick-up point is being modulated sinusoidally in placement at a rate of 5 Hz, but its deviation from center is 10 string lengths. Where the double vibratos are at their furthest separation is when the pick-up point is flying past its center point of modulation at its greatest speed. Notice that the double vibratos in the upper harmonics cross over each other. iu 5FiguL 3. Modulating the Pick-Up Position at Audio Rates The next logical step is to modulate the pick-up point at audio rates. We define the pick-up point modulation as follows: xp(t) = Xpo+ d sin ox-,t where xp(t) is the resultant pick-up point placement measured in string lengths; xp0 is the center point of modulation; d is tge deviation measured in string lengths; andto is the frequency of the pick-point modulation in radians. A closed form expression for the resultant spectrum heard from this pick-up can be found as follows: t) Yky(t), where keH yk(t) = AkJO(Ik)sin4kcostokt + _AkJi(k)cos 4k[sin(ok +itoxp)t+sin(cok -ioxp )t] i=l i odd + AkJi(Ik)sin4k[cos(ok+iwoxp )t+ cos(cok-ioxp )t] i=l i even where k is the harmonic number; H is the set of harmonics on the string; Yk is the contribution to the total spectrum resulting from the ktd harmonic; A k is the amplitude of the kth harmonic; 1k = rkd is the modulation index; 4k = irkx0 is the phase offset determined by the position of the modulation center point; w0k is the frequency of the kth harmonic in radians; and txp is the radian frequency of the pick-up point placement modulation. This expression says that around each of the harmonics on the string, (Lok, sidebands are produced at plus and minus multiples of the pick-up modulation frequency, oXp, and that their magnitudes are dependent on Bessel functions of the first kind operating on an index based on d, the deviation of pick-up point modulation. These equations differ from ordinary FM, however, in that the odd sidebands are scaled.by cos 4k and the even sidebands are scaled by sin k where k = itkrp0 is, in effect, the relative position of the pick-up modulation center point measured along the standing wave components of' each of the harmonics on the string. When xpis at a harmonic node, that is at a position n/k on the string, where n is an integer and k is the harmonic number, then sin d k evaluates to zero and the even harmonics are zeroed out. On the other hand, whenXp is at an anti-node of a harmonic, that is, at a position (n +.5).k along the string, cos Ck evaluates to zero and the odd 156
Page 157 ï~~harmonics are zeroed out. Sliding the center point between a node and anti-node produces a gradual exchange of energy from the odd sidebands to the even sidebands. Figure 7 illustrates a four second sound created by sliding the center point of pick-up modulation from one end of the string to the other over the duration of the tone. The fundamental frequency of the string is 1000 Hz. Four harmonics were loaded onto the string at start-up. The modulation frequency was 100 Hz and the deviation was 0.1 string lengths. There are more side bands around the upper harmonics due to the greater effect of the deviation distance on the shorter wavelengths. Observe how the odd and even sidebands fade in and out as the center point of pick-up modulation glides over the nodes and anti-nodes of the various harmonics. Figure 8 shows how speed and modulation can be combined. Here the first and fifth harmonics were loaded onto a 1000 Hz string at start-up time. The center point of modulation accelerated across the string from a speed of 200 string lengths per second up to 500 string lengths per second. Meanwhile, with a modulation frequency of 200 Hz, the deviation ramped from 0 to.1 string lengths back down to 0 over the course of the sound. Note that we lose the nice feature of individual control of odd and even sidebands when using speed or acceleration in combination with modulation. Figure 7igure8....................,,....... '..............f.. y v.4 ~ re t,... arw Conclusion While it is true that the results in the simple case of the ideal string with perfect reflections at the end points could be duplicated with traditional FM means, this research brings up new possibilities of combining FM and physical modeling techniques. There are many possibilities when a low pass or other system filter is placed at the end of the string instead of the perfect reflection. Many other model parameters can be similarly modulated, unlike the corresponding parameters of natural instruments, such as string length, bridge coupling, body resonances, and so on. Waveguide models can be enriched by a modulating pick-up point, while on the other hand, traditional FM instruments can be rejuvenated by the loop filtering methods of physical modeling and waveguide synthesis. References Momose, H. Sonogram: An Acoustic Signal Analyzer/Editor. Software for NEXT, version 0.90(Beta), 1991. Morse, P. M. Vibration and Sound. Published by the American Institute of Physics for the Acoustical Society of America, 1976. (First edition 1936, second edition 1948). Smith, J. 0. and P. R. Cook. "The Second-Order Digital Waveguide Oscillator," elsewhere in this proceedings. See in particular the references at the end of that article regarding the digital waveguide approach.,. Smith, J. 0. "Waveguide Synthesis Tutorial," to appear in the Computer Music Journal, MIT Press, Spring 1993 (est.). 157