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Page 194 ï~~Composing the means with which I compose: Waveforms with multiple paths and cycle-lengths Arun Chandra Computer Music Studio School of Music University of Illinois ABSTRACT To describe a sound in terms of the structure of its waveform, and not in terms of its acoustic appearance, is the compositional premise of these programs. As a music composer, I am interested in describing precisely the path of transformation I desire, then discovering what its acoustic consequences are, instead of describing precisely the acoustic consequences I desire, then doing what is needed to achieve them. To this extent, these projects remain experimental ones. TrikTraks is a time-based synthesis programs that allows for the structural specification of waveform states and how they change. All aspects of the resulting sound (pitch, volume, color, and gestural behavior) are interdependent, continuously changing, and controlled by relatively few variables. This report describes some results of constructing states consisting of multiple cycle lengths, and some possibilities for using multiple paths. 1. Overview The grammar for waveform specification and transformation is conceptually simple. The state of a waveform is defined as a sequence of segments. Each segment has at least two variables: number of samples (length) and amplitude (height). (Other segment types, having more variables, can also be defined, see [4, 5].) The state is iterated until the desired duration is reached. Upon each iteration, each variable changes by its specified amount. TrikTraks allows the above variables to be given a unique path that determines their changing magnitude over time. Each path can be specified as: 1) a sine wave; 2) a triangle wave; 3) a sawtooth wave; 4) a square wave; 5) a polynomial between degrees 3 and 10; 6) or as an amplitude modulated FM wave. Every variable of every segment can be given a path that is independent of all other paths. Upon each iteration of the sequence of segments, each variable of every element changes its magnitude as determined by its path of transformation. 2. Predecessors The significant predecessors of TrikTraks are wigout (designed and written by the author), SAWDUST, designed by Herbert Brin and written by Gary Grossman, Jody Kravitz, and Keith Johnson at the University of Illinois, and SSP (Sound Synthesis Program), designed by Gottfried Michael Koenig, and realized by Paul Berg, Robert Rowe, J.D. Banks, and David Theriault at the Institute for Sonology, Utrecht. These programs can be classified as "non-standard sound synthesis programs," following S. R. Holtzman's definition. Paul Berg's description of SSP is appropriate for TrikTraks: "... [it is] best suited for a user who wants to define structures and listen to the results rather than a user who at all costs must have a certain sound." 3. Postulate By definition, a "tone" requires a repeating period. The "period" can be the consequence of a mathematical function (such as a sine function), or it can be constructed, sample by sample, and then repeated. The repetition of a sequence of samples, regardless of their amplitudes, gives that sequence the function "period," and will create a sound. The length of that sequence (the number of samples in it) determines the sound's base frequency, and the amplitudes determine its timbre. Changing the number of samples in the sequence changes the base frequency of the sound; changing the amplitude content of the sequence changes the timbre. If it is the case that "repetition" is what generates both frequency and timbre, then constructing a waveform with multiple repetitions should result in one waveform with multiple sounds. TrikTraks allows the exploration of waveforms with multiple transformation paths. 4. TrikTraks 4.1. Standard Paths "Standard path" refers to using a standard waveform to control the path of the variables amplitude and number of samples. Input to TrikTraks is from a datafile, where is specified 1) the path range (minimum and maximum values); 2) the control waveform type (sine, square, triangle, or sawtooth); 3) control waveform frequency; and Chandra 194 ICMC Proceedings 1996
Page 195 ï~~4) the control waveform phase. Note: "control waveform frequency" refers to the number of complete periods of the control waveform that will occur over the total time requested. So, if the duration = 10, frequency = 3.5, and type = 3, the path will be three and 1/2 periods of a sine wave over 10 seconds. As an example, here is an input datafile, specifying the variables for a sequence of 3 segments. The first row of each pair refers to the number of samples and the second to the amplitude. The first column sets the variable's maximum value, the second its minimum, the third the type of controlling waveform, the fourth the controlling waveform's frequency, and the last is the controlling waveform's phase. # TrikTraks: input file for standard waveforms duration: 10 # duration of sound in seconds 100 10 sin 3.8 0.75 # segment 1: samples 10000 -10000 tri 1.5 0.75 # amplitude 0 0.1 0.2 0.3 0.4 0.0 0.6 0.7 1 0.6 0.0 50 20 squ 5 0.5 10000 -10000 tri 3 0.25 # segment 2: samples # amplitude 200 100 saw -3.5 0.75 # segment 3: samples 30000 -30000 sin 11 0.83 # amplitude The above is one sequence of segments that are iterated and written to disk until the total duration is reached. As in wigout, a sequence can have up to 64 segments. The first segment (in the above datafile) has samples that will vary in number between 100 and 10 (their "range"), the path will be that of a sine wave, 3.8 periods of the sine wave will be generated over the total duration (10 seconds), and the starting phase of the sine will be 0.75. The first segment's amplitudes, however, will range in value between Â~10000, the path will be 1.5 periods of a triangle wave over 10 seconds, whose phase is 0.75. Here is a normalized plot of both the sample and the amplitude paths for the first segment: The distortions in the plot are due to the procedure with which the lookup algorithm (used to generate the triangle and other waves) was called. They were allowed to exist, since they were the result of the proper operation of the lookup algorithm, and generated a few more resistances to periodicity and symmetry. By choosing distinct values for the type of path, its frequency and phase, each variable of each segment can have a path independent from all other segments' paths. The sounding frequency of this waveform is the sum of the segments' samples at every iteration. Since this number is constantly changing, the frequency is constantly changing. The square and sawtooth paths have a periodic jump from the minimum to the maximum value (or vice versa). This results in an immediate and drastic frequency change, when applied to the variable number of samples. The amount of the change depends on the user specified maximum and minimum values for that variable, and the content and behavior of the neighboring segments. The sine and triangle path, in obvious contrast, have smooth rises and falls. The compositions smear pulse no sneer and The thin red line of subject matter were composed using only the above waveform paths. 4.2. Polynomial Paths A consequence of using the standard paths was periodic oscillations of the variables between their maximum and minimum values. (The overall periodicity of the resulting sound was far more complex than the periodicity of any one path, but, nonetheless, the resulting regularity was noticeable.) Since periodicity is (in information theoretical terms) a greater degree of redundancy than needed to convey a message, my interests as a composer led me to look for ways in which it was minimized. Polynomials are used in Herbert Brin's SAWDUST system in the implementation of the VARY algorithm. I ICMC Proceedings 1996 195 Chandra
Page 196 ï~~decided to try to use them in TrikTraks by specifying equally spaced zero-crossings, then scaling them to their specified limits. The values are loaded into a table, and interpolating wave table lookup algorithms are used to extract values with the desired frequency and phase. This procedure allows for a polynomial to occur with a variable number of periods over the requested duration. From TrikTraks, a polynomial path is specified in the following way, here given for only one segment: # TrikTraks: input file for polynomials 200 100 poly 5 3 0 # samples 30000 -30000 poly 7 2.5 0 * amplitude The range is given first (maximum and minimum values), then the type of path ("poly"), followed by the degree of the polynomial to be generated, the number of periods over the sound's duration, and the initial phase of the polynomial. Polynomials can be requested of degree 3 to degree 10. Here is a plot of the paths for both the samples and the amplitudes, for the data given above: The postulate was that the change of rate of peaks and troughs would be determined by the FM function, and the change of magnitude of peaks and troughs would be determined by the AM function. Please note that the FM and AM functions are not being used to directly generate sound samples. Rather, they are being used to generate the path that will be followed by the variables in every segment. Specification of the amfm function is done in the following way: * TrikTraks: input file for amfm (1) 200 100 amfm 0.75 1 1.25 1.5 3 # samples 30000 -30000 amfm 0.5 0.4 0.6 0.8 # amplitude The first two numbers specify, again, the range of the path (maximum and minimum), followed by the type of path chosen ("amfm"). The following numbers represent: 1) fl, the frequency of the amplitude modulation; 2) 12, carrier frequency for FM; 3) f3 modulation frequency for FM; and 4) mi, modulation index for FM. This is a normalized plot of both the sample and amplitude paths for the above data: 0.90. \" 0.7 0.6 1 1 O 0.64 0.3 0 0.2 0.4 0.6 0.8 As you can see, the resulting complexity of the path is greater than that of either the polynomial paths or the standard waveform paths. The amfm path can be used in conjunction with the polynomial paths or the standard paths described above. Thus, an waveform can be constructed using standard waveforms, and polynomials, and the amfm function. The implementation of the amfm paths is similar to that described for the polynomial paths above: the function is evaluated, the output is scaled and loaded into a table, and output values are interpolated from the table. As an example of the variety of path combinations available with the amfm function, the input files and normal 0 0.2 0.4 0.6 0.8 Please note that the above are the paths for one segment of a waveform, and that a waveform can have up to 64 segments, and thus there can be up to 128 distinct paths. 4.3. Amplitude modulated FM functions Although the polynomial paths do generate a variety of peaks and troughs (as compared to the standard paths described above), because of the equidistant zerocrossings used to generate the polynomials, the rate of the peaks and troughs remains constant. One can have multiple periodicities, but they remain periodicities. As a result, I wondered what would happen if one used an amplitude modulated FM function for the generation of the paths: v = sin(2Pi f, t) * sin(2Pi f2 t + mi * sin( 2Pi f3 t) ) Chandra 196 ICMC Proceedings 1996
Page 197 ï~~ized plots for two segments are given below. # TrikTraks: input file for amfm (2) 50 10 amfm 1 1.25 1.5 1.75 10000 -10000 amfm 0.15 0.5 0.9 4 04 0 V o0.2 0.4 0.6 # TrikTraks" input file for amfm (3) 100 80 amfm 0.075 0.5 1 2 20000 -20000 amfm 0.1 0.02 0.03 20 # samples # amplitude samples # amplitude 00 0.0 0.7 0.6 0.s 0.4 0.3 0.2 0.1 regard to the acoustic consequences. As a result, a composer of music has to develop new ways of specifying what she wants with regard to the output. Relatively simple input can generate wildly complex output behavior. This potential richness is, more than anything else, the most seductive aspect of this project for this composer. 6. Acknowledgments I'd like to thank Jim Beauchamp and Sever Tipei, the co-directors of Computer Music Studios in the School of Music at the University of Illinois, for allowing me access to its facilities. Herbert Brin has been a constant and invaluable conversation partner in this, and many other, investigations. Conversations with Michael Brin helped develop the the AMFM function in TrikTraks. The ideas that generated wigout could not have been had without the work of Heinz von Foerster and W. Ross Ashby. I wrote TrikTraks in C, and it runs under NeXTStep 3.x, IRIX 5.3, and DOS 6.x. Up till now, six compositions have been written with them: An untitled poem in 16 stanzas by Keith Moore for trombone and tape (13 minutes, 1992), A Bit, A Curve, Alas, A Wave! for tape (8 minutes, 1993), smear pulse no sneer for tape (10 minutes, 1994), 700,000 Dead for voice and tape (7 minutes, 1995), The Last Statement for voice and tape (6 minutes, 1995), and the thin, red line of subject matter for tape (6 minutes, 1995). References 1. Ashby, W. Ross. An Introduction to Cybernetics. Methuen and Company, Ltd. London: 1956. 2. J.D. Banks, P. Berg, R. Rowe, D. Theriault. SSP: A Bi-Parametric approach to Sound Synthesis. Institute of Sonology, Utrecht: 1979. 3. Brin, Herbert. My Words and Where I Want Them. Princelet Editions. Urbana, Illinois: 1991. 4. Chandra, Arun. "The linear change of waveform segments causing non-linear changes in timbral presence" In Contemporary Music Review, vol. 10, part 2. Harwood Academic Publishers: Basel, Switzerland. 5. Chandra, Arun. "CounterWave: a program for controlling degrees of independence between simultaneously transforming waveforms" in Proceedings of the International Association of Knowledge Technology and the Arts, Osaka, Japan: September, 1993. 6. von Foerster, Heinz, editor. Cybernetics of Cybernetics. Biological Computer Laboratory, Report Number 73.38, University of Illinois. Urbana, Illinois: 1974. 7. Holtzman, S.R. "A Description of an Automatic Digital Sound Synthesis Instrument," in D.A.I. Research Report No. 59. Department of Artificial Intelligence, Edinburgh: 1979. 8. Roads, Curtis, editor. Composers and the Computer. William Kaufmann, Inc. Los Altos, California: 1985. 9. Shannon, G.E., and Weaver, W. The Mathematical Theory of Communication. University of Illinois Press. Urbana, Illinois: 1949. 5. Conclusions Since this has been an exploratory project, there are no conclusions per se, but rather indications for further work. Some possibilities are: 1. Allow for a variable in a waveform to have zero change over the course of a sound. 2. Allow for the amplitude changes to happen logarithmically rather than linearly. 3. Allow for the shifting from one waveform type to another in medias res. 4. Develop secondary control waveforms for the variables. This would begin to approach an elementary calculus of waveforms, in which the rates of change are themselves changing. This approach of structural modification of waveforms in time generates a high degree of unpredictability with ICMC Proceedings 1996 197 Chandra