Page  364 ï~~Connectionism and Musiconomy Selected for Extended Format by the Editors, ICMC-91 D. Gareth Loy 509 Barbara Ave., Solana Beach, CA 92075, dgl@frox.com Introduction When musicians speak of the form of a musical work, we traditionally mean its structural form--that is, its analytic deep structure and the syntactic rules relating to it; we tend not to mean the methods governing its actual fabrication and composition process--what we might call procedural form. The roots of the structural analysis of common-practice music come from such Renaissance theorists as Zarlino (1558), Baroque theorists Fux (1725), and the Baroque composers Marpurg, Rameau, and others. They developed compositional rule systems that captured the most regular aspects of the music of their times. However, even modern versions of these music theories are not complete: they are a necessary, but not a sufficient, guide for a composer who wants to learn to write music in a well-known style such as J.S. Bach's. In practice, besides performing structural analysis of the target style, a composer must "absorb" the target style by spending many hours with the corpus of works created in that style, and additional hours doing approximation trials with pencil and eraser. It is not well understood-except at a preconscious level in the mind of a suitably trained composer--how to absorb the essence of another composer's style. Some pioneering work has focused directly on the question of how humans perform this compositional learning task (such as Laske 1973) but progress has been slow, awaiting a suitably powerful understanding of learning, such as is emerging from connectionism now. So we are left with the questions: From where does a composer's procedural knowledge derive? How can we access it? How can we model it? Is it possible that a more rigorous structural analysis can succeed in becoming a complete formal theory incorporating procedural knowledge, or not? But before going on, the reader deserves an explanation for the invented term "musiconomy. In casting about for a term that successfully characterized the kind of music research discussed in this paper, and how it differs from, for instance, traditional musicology and such other topics as the psychology of music, I noted that Webster's Dictionary describes a nomy as a "...sum of knowledge regarding a (specified) field," and a logy as a "doctrine." For instance, compare astronomy and astrology. Then it occurred to me that there was a terminological hole that I could fill as follows: if astronomy followed from astrology, what follows from musicology? Using Automata to Create Original Works At least since the time of Guido of Arezzo (or d'Arezzo f. 1026) (Guido 1955, Loy 1989), there has been an attraction to the notion of automatic music generation and the use of structuring devices to constrain the space of possible choices. Guido's system, a simple table-driven scheme, is rare in that it actually stipulates a procedural method. All composers have used structural systems, such as isorhythm, rondo form, sonata ICMC 364

Page  365 ï~~form, etc., to frame compositions (Kirchmeyer 1963). But most are structural systems, not procedural ones like Guido's. Structural forms frame, but only incidentally do they facilitate the composer's task. Formal generative techniques, on the other hand, can actively facilitate the note-by-note production of compositions. In the 1930's, Joseph Schillinger proposed the then-radical notion that theories of art should not simply be observation-based tools of art historians and musicologists, but should be so rigorously analytical that they are also useful in the production of art (Schillinger 1948). He attempted to create such theories for the visual arts and music based on a quasi-mathematical foundation. Schillinger would undoubtedly have answered in the affirmative to our query whether a more rigorous structural analysis could succeed in becoming a complete formal theory incorporating procedural knowledge. Most if not all recent music theories, such as Lerdahl and Jackendoff (1983), have as a criterion of their success that they are generative as well as analytical. In the traditional realm of computer music, Laske (1973), for instance, has proposed a comprehensive system for analysis, composition and performance, called OBSERVER, using artificial intelligence techniques. The system, though never finished, was to have served as a platform for synthesis by analysis of compositions. Other such systems are described below. Composing with Computers The formalization of the musical process did not come to the forefront in music until computers became available to composers and music researchers in the mid-1950's. Computing not only facilitated the formalization of musical composition, it required it: only with rigorously formal languages can one program computers. The early work in computer composition consisted of developing artificial procedural systems, mostly without attempting to relate them to any model of human processes. We may call this computer-mediated composition. The speed and power of a computer for quickly exploring the consequences of different compositional strategies was exhilarating to those with the inclination and the necessary access to such tools, such as Xenakis (1971). Gill (1963) wrote computer programs to generate music using the artificial intelligence technique of backtracking. Mathews and Rosier (1968) described a model of composing based on mathematical function composition. Many computer languages, programs, and systems have been written for representing the compositional process (Loy 1989). Gottfried M. Koenig (1970a, 1970b) combined computer-mediated and computer-generated composition with the development of his Project I and Project II programs. They are, in some respects, experiments in how to share the compositional process between man and machine. Another approach is to ask the question of whether (Kugel 1990, Myhill 1952) and how (Hiller and Isaacson 1959) the compositional process might be modeled as a computation. The former question is properly in the domain of philosophy. The latter is more a computer science question: the goal is not the generation of compositions so much as the design of machines to carry out this task. The question, "how does a human compose?" is properly a cognitive science question. Models using this approach would not only have to compose music, but also would have to account for the available knowledge about how composers actually carry out the task of composing. ICMC 365

Page  366 ï~~Influenced strongly by cybernetics and information theory, Lejaren Hiller and Leonard Isaacson conducted some experiments in the computability of the compositional process (Hiller and Isaacson1959). They thought of composition as the problem of using musical rules to select acceptable musical sequences out of the space of all possible combinations of musical sequences. They developed a computer model of the process of composing common-practice music using stochastic and Markov-chain techniques. Apart from the fact that it obeys certain compositional rules, Hiller and Isaacson's system does not describe the method of the typical human composer: composers usually do not employ a generate-and- test methodology. If this technique is extended to include a probabilistic analysis of a corpus of musical works, the result is a system that can mimic an arbitrary musical style. Probability distribution functions are used to specify, on a statistical rather than a prescriptive rule basis, the likelihood of a note being acceptable within a particular musical context, given a particular corpus of examples as guide. This method is a type of Markov-chain process. The technique has had widespread use and also widespread criticism (Lewis 1991). Prescriptive Rule-Based Composition Since the advent of computers, it has been a widespread fantasy of generations of music undergraduates studying species counterpoint to extend Fux's nearly formal rule set into a computable method for generating species counterpoint--a rule-based expert composing system. A similar fantasy concerns the rules of chorale harmonization used by Bach--another subject often inflicted on music undergraduates. Modern examples of emulating historical styles using prescriptive rules include VIVACE, a melodyharmonizing program by Thomas (1985). Ebcioglu (1984) describes a very elaborate system using traditional artificial intelligence techniques to solve chorale harmonization in the style of Bach. An attempt at solving the species counterpoint problem was done by Schottstaedt (1984), and his experience was typical. He set out to implement a full expert system for composing counterpoint, but what he found when he was done implementing the classical rules (Fux 1725) was that there were many remaining ambiguities. There were places where rules conflicted. Elsewhere the rules seemed incomplete. As a consequence, the results did not sound as well-formed as compositions created by trained humans. He struggled on until he had nailed down many more rules, supplied a means of reconciling conflicting constraints, and fine-tuned the weights of the various rules. The rules which he had to add, numbering over one hundred, rivaled the number of rules he got from Fux. Still the program could not be said to write wonderful species counterpoint. It seemed to perform about as well as a typical freshman composition student. So one can say Schottstaedt succeeded in creating an expert system for composing undergraduate counterpoint exercises. It is, of course, still a far cry from an undergraduate canon etude to a major work of counterpoint. These systems represent some success in the formalization of the composition process, to be sure. However, it must be kept in mind that what they have succeeded with are the most elemental aspects of compositional structure: common-practice harmonization and melody formation, i.e., the most well-known and regular aspects of the most well-known and regular of Western musical styles. One likely difficulty is that prescriptive rule-based systems do not handle ambiguity gracefully, and are difficult to operate meaningfully in parallel. And yet, as discussed in the next section, structural ambiguity is very common in most forms of music, due in part to the inherent parallelism (as for instance, between the melodic, harmonic, rhythmic, and timbral dimensions) of music. ICMC 366

Page  367 ï~~The Problem of Formal Specification of Music The efforts described above to model music using rule-based techniques face the difficulty of satisfactorily handling the numerous structure by the means of expectancy violation, for at least two reasons. First, as I've mentioned before, expectancy violation (that is, violation of what Bharucha and Todd 1991 call "schematic expectancy") is necessary to keep the listener's attention engaged. Second, expectancy violation can, in the hands of a great artist, result in the communication of new ideas that reward our attention. It is generally the case that composers are more interested in establishing a formal device as a point of departure than as an end in itself. Taking an example from J.S. Bach, consider his rules of harmonic resolution and motivic sequencing. In the episode of a fugue, for example, he might sequence a fixed motive of four notes through a strictly determined set of transpositions based on the circle of fifths. At first glance it seems that this simple process could be formalized easily, and applied to automatically generate part of a fugal episode by rule. But simply putting note after note was not a main goal of the better Baroque composer; it was rather the attempt to keep the listener off balance, and hence mentally engaged in the unfolding composition. To this end, composers such as Bach might, after sequencing a motive through two and a half transpositions, prematurely break off the sequencing. This was accomplished, e.g., by shifting the compositional focus from sequencing (a horizontal musical dimension) to harmonic resolution (a vertical dimension). After two sequences of a motive the listener has a sense of expectation; after two and a half. a confirmation; then the listener is tempted to stop paying attention, thinking that the remainder of the phrase will probably just continue the pattern. To break off at precisely this point with a cadence is to deceive this expectation and reengage the listener's attention. This is in line with the theory that music is organized by the interweaving of expectation and surprise, developed by Meyer (1956); his ideas were strongly influenced by the work of Shannon and Weaver in information theory. Thus, the tension between one dimension of formal structure in a composition (e.g., a motivic plan) is played off against another (a harmonic plan) to achieve the higher aim of sustaining attention to the larger musical gesture. The position of the composer in this process is therefore really in the cracks between categories of formal description. Describing the compositional process within the framework of a strictly formal representation will necessarily miss this dimension of the composer's art. And yet, a computable approach is perforce necessarily strictly formal. This is the dilemma shared by all computer models of music. Why is this so hard? What strikes me most about the problem described so far is the disparity between the ease with which an average undergraduate can learn to analyze and compose in an evolved style (a process lasting an academic year, perhaps), and the degree of difficulty of finding suitably objective knowledge to represent composing, such that a computer can do as well (a process that has not yet been shown to halt). It seems that traditional techniques for modeling intelligence do not fare well when applied to deep humanistic problems, such as music. This concern parallels the one expressed generally in the field of connectionism (e.g., Rumelhart and McClelland 1986) which observes the failure of traditional artificial intelligence techniques to lead to satisfying solutions to deep problems in human cognition. ICMC 367

Page  368 ï~~Probabilistic systems only adapt distribution functions to a corpus of examples without discovering or acting upon any deep-level correlations in that corpus. Rule-based systems do not spontaneously generalize knowledge from their input, but only analyze according to the domain-specific rules supplied, which must still be developed heuristically. Learning with generalization is not addressed by either approach. It is possible to absorb unconsciously a particular style; but it is not clear that the rules of that style can then be made wholly conscious, and therefore formalizable, and therefore computable. Is Composing a Rule-Solving Problem? Let's conjecture for a moment that it is possible by some means to achieve a complete rule set for some artistic style; what then? Would it produce Art? I submit that simply satisfying the rules is only the beginning of art, not its goal. So the notion of an art form as a kind of rule-solving problem must be discarded. However, we must make sure we do not slip into the kind of thinking that Schillinger so correctly attacked: that the composer is Genius, and that Genius is above rules. What other approaches are there? Let us look again at the function of music in culture. It would seem that the problem of emulating existing styles grows even larger when we must consider that a musical style does not exist in a vacuum, but is based on the consensual experience of a whole culture. However, it is precisely this cultural context which provides the artist with the necessary hooks into the minds of his or her listeners. Expectation is the foundation of music; without it the listener hears only noise. But while awareness of the consensual rules is necessary, it is not sufficient. Were it sufficient, art would consist entirely of saying things people already understand. It is additionally necessary for the artist to play off different levels of structure by the means of expectancy violation, for at least two reasons. First, as I've mentioned before, expectancy violation (that is, violation of what Bharucha and Todd 1991 call "schematic expectancy') is necessary to keep the listener's attention engaged. Second, expectancy violation can, in the hands of a great artist, result in the communication of new ideas that reward our attention. Therefore, it is not likely that musical styles can be isolated successfully by simple heuristics and introspection, nor can they be readily modeled as a rule-solving problem. More powerful theories and techniques are necessary, ones which can model expectation and surprise, and which can generalize from their experiences to novel arrangements. Connectionism and Music As I have said, the challenge to rule-based models of music is to allow the rules to interact with each other in novel situations. But Bharucha and Todd show (1991) that it is possible for an artificial neural network to be trained to understand and anticipate common musical practices, and yet to learn to expect the exceptions to the norm that are embodied in actual compositions. That is, a connectionist network model can simultaneously memorize and generalize, these being two key elements in learning. Perhaps connectionism can show the way to techniques that do not have the liabilities of strictly formal systems. What becomes of a student's understanding of music theory after music theory classes are over? One would think that here, if anywhere, is the preeminent occasion for taking precognitive musical knowledge and raising it to consciousness. But the common experience is that the knowledge acquired originally in childhood by memorization and its attendant intuitive abstraction may be said to be sharpened and validated ICMC 368

Page  369 ï~~by formal training, though it is certainly not replaced by it. The fact that even mature theories of music are informal is strong evidence that the performer, the listener, and composer do not operate principally as rulebased problem-solvers. Instead, imagine that strong association networks are formed in the musical mind such that, given a partial input (as during a performance) the pattern-completion mechanisms that are a natural consequence of the neuronal structure of the listener's mind consider the possible completions of that input against the expectancies of the current style. Through pattern-completion structures inherent in the brain, the listener anticipates the possible outcomes. The composer, understanding this predictive mechanism, manipulates these expectancies to his or her artistic aim. The neuronal nature of the brain also helps explain how we manage to keep up with the number of simultaneous dimensions necessary to appreciate music. The brain is a massively parallel processing system. The conventional wisdom is that developing an expertise in music consists of acquiring a rule set which then receives progressively more rules with experience. If the brain followed music by evaluating rule systems (which must to a certain extent be a sequential process), then the more experienced a musician became, the more constraints he or she would be required to resolve for each note heard, a process which would eventually exceed the capacity of the brain to follow a musical thought in real time. And since the constraints that we do know about in music mix levels in imprecise ways, there would necessarily be an explosion of rules to cover all realistic cases. But it is a common experience that musical judgement for many tasks, such'as improvisatory harmonization, actually becomes quicker, even as the musician gains experience, as more and more rules are learned. This could not be the case if human musical cognition required any substantial sequential arbitration among competing rule systems. Musical Knowledge through Self-Organization Besides providing an interesting model of human musical cognition, network techniques can also help us further the work in the formalization of music by analysis of examples. Network techniques may be able to help discover musical rules in a way that will make them available to a formalization process. In this approach, connectionism would be seen as continuing to derive procedural knowledge about composition from automatic analysis of existing works, like some of the traditional techniques already discussed. But it would be doing so with a minimum of pre-specified knowledge and bias, and with tools that can resolve and generalize about multidimensional correlations. As Gjerdingen shows (1991), neural networks have a demonstrated ability to passively acquire musical knowledge simply from the principles of self-organizing systems, such as those developed by Stephen Grossberg (1982). Bharucha and Todd (1991) have furthermore demonstrated that we need only postulate general principles of self-organization in order to model the otherwise seemingly contradictory practices of memorization and abstraction. Conclusions Theories can fail for a variety of reasons. A good sign that a theory is in danger is when one observes an explosion of rules when trying to explain a problem within the bounds of the theory. Copernicus observed ICMC 369

Page  370 ï~~the weakness of the theory which attempted to describe the orbits of planets by an infinite summation of epicycles. Newton formulated a mathematics of elliptical orbits based on gravitational attraction, and the epicycles, and the theories that created them, vanished. Connectionist techniques are able to avoid the combinatoric explosion problem associated with rule-based techniques of musical knowledge. They can also provide many other useful features as side effects, such as spontaneous generalization, pattern completion, and graceful degradation. Thus, the theories of connectionism may be able to eliminate what we might call musical epicycles introduced by conventional models of musical knowledge. Connectionism is a very pragmatic theory, in that it takes as its model something basic and incontrovertable, namely the neuronal basis of the brain. Clearly it is still a research domain in its infancy. A criticism has been laid to connectionism that it may not scale any better than traditional artificial intelligence techniques as the problems to which it is applied grow more complex (Papert 1988). And for problems of any size, as Scarborough, Miller, and Jones (1991) describe, it is often difficult to investigate and interpret the results of a network, either to improve its performance or even to understand it at all. Still, the inherent advantages of the connectionist approach are compelling, because one need only postulate general principles of selforganization with little or no domain-specific knowledge in order to begin modeling arbitrary musical processes. Thus we can proceed with an objectivity heretofore difficult to obtain. A Further Artistic Question Up to now, I have described automata as unreflective tools of composers or researchers. I have used such verbs as "produce" and "generate" to describe compositions made by automata, and the verb "create" for human compositions. But is there really a categorical difference between compositions created by humans and those produced by automata? Could there be an automaton that could pass a Turing test for music? Alan Turing proposed a test for determining whether a system had artificial intelligence: a subject sitting at a terminal is asked to determine whether he or she is communicating with another human in the next room, or with a computer. The subject can ask any questions. If subjects consistently guess that the responding entity is a human, but it is in fact a computer, a reasonable person would be forced to consider that the machine had intelligence. A musical Turing test might be easier for a computer to pass someday, since music is a very abstract artistic medium. But even if a system passed a musical Turing test, that would prove no more than that a reasonable facsimile of human musical functioning could be constructed using computational means. It would say nothing about several other questions, taken up next. There are really at least three questions here: first, can musical automata be constructed which exhibit human behavior to a sufficient degree as to exhibit intelligence? This would be the computer scientist's test. Second, can musical automata be constructed which successfully model our understanding of human musical cognition? This would be the cognitive psychologist's test. Third, can human musical cognition be represented computationally at all? This would be the philosopher's question. Whether computers can successfully approximate human musical cognition says next to nothing about the theoretical prospects of characterizing musical cognition via computations (Kugel 1990). Consider the difference between compositional automata and human composers in the light of what is communicated through music. The generic meaning of composition is the combining of elements in ICMC 370

Page  371 ï~~patterns, similar to the composing of mathematical functions. Clearly, by this definition, automata can compose. But where those patterns are the result of some stochastic or deterministic process, we would tend to say that the patterns were generated, or produced, because the meaning of the pattern is a function of the process used to produce it: analyze the music it produces and what you get is the reverse-engineered automaton. On the other hand, when those patterns are the result of the considered activity of a human mind we tend to say that the patterns were created because the meaning of the patterns is related to the considerations of their creator. Observing the patterns, we recover those considerations, and derive satisfaction from the insights thus shared. The considerations need not be intellectual, but can touch on any aspect of experience. The above points suggest that it is a reflexive quality based on meaning that distinguishes a creative act from a generative one. The meaning communicated by a formal system is, essentially, itself. The meaning communicated in a creative act lies beyond the created product itself. Its meaning is the tying together of previously unrelated ideas and experiences in the mind of the listener, and connecting them to ideas and experiences in the mind of the composer that led to the creation. So can automata--in particular, artificial neural networks--create in the sense developed above? The simplest answer is: Yes, if they have something to say beyond simply expressing their own formal structure; otherwise no. Unfortunately, this simple statement doesn't go very far, especially in the case of artificial neural networks, which can be modeled as having "experiences" and "ideas'. However, the reflexive aspects of current artificial neural systems are still dominant: networks "create" because they are designed to do so. To model creativity itself, creativity would have to be an emergent property. Otherwise, it is simply the vicarious expression of some other's creativity designed into the system. (See also Lewis 1991, for another perspective on this question.) These comments only focus on the philosophical level of machine models of human artistic expression. As I mentioned before, there are numerous other perspectives from which to view music and composition, e.g., from the point of view of computer science, psychology, musicology, etc., each of which comes with its own set of questions. From these varied viewpoints, computer-based -- and especially connectionist -- models of musical behavior may be very successful indeed. Therefore, it is not likely that musical styles can be isolated successfully by simple heuristics and introspection, nor can they be readily, modeled as a rule-solving problem. More powerful theories and techniques are necessary, ones which can model expectation and surprise, and which can generalize from their experiences to novel very abstract artistic medium. But even if a system passed a musical Turing test, that would prove no more than that a reasonable facsimile of human musical functioning could be constructed using computational means. It would say nothing about several other questions, taken up next. There are really at least three questions here: first, can musical automata be constructed which exhibit human behavior to a sufficient degree as to exhibit intelligence? This would be the computer scientist's test. Second, can musical automata be constructed which successfully model our understanding of human musical cognition? This would be the cognitive psychologist's test. Third, can human musical cognition be represented computationally at all? This would be the philosophers question. Whether computers can successfully approximate human musical cognition says next to nothing about the theoretical prospects of characterizing musical cognition via computations (Kugel 1990). ICMC 371

Page  372 ï~~Consider the difference between compositional automata and human composers in the light of what is communicated through music. The generic meaning of composition is the combining of elements in patterns, similar to the composing of mathematical functions. Clearly, by this definition, automata can compose. But where those patterns are the result of some stochastic or deterministic process, we would tend to say that the patterns were generated, or produced, because the meaning of the pattern is a function of the process used to produce it: analyze the music it produces and what you get is the reverse-engineered automaton. On the other hand, when those patterns are the result of the considered activity of a human mind we tend to say that the patterns were created because the meaning of the patterns is related to the considerations of their creator. Observing the patterns, we recover those considerations, and derive satisfaction from the insights thus shared. The considerations need not be intellectual, but can touch on any aspect of experience. The above points suggest that it is a reflexive quality based on meaning that distinguishes a creative act from a generative one. The meaning communicated by a formal system is, essentially, itself. The meaning communicated in a creative act lies beyond the created product itself. Its meaning is the tying together of previously unrelated ideas and experiences in the mind of the listener, and connecting them to ideas and experiences in the mind of the composer that led to the creation. So can automata--in particular, artificial neural networks--create in the sense developed above? The simplest answer is: Yes, if they have something to say beyond simply expressing their own formal structure; otherwise no. Unfortunately, this simple statement doesn't go very far, especially in the case of artificial neural networks, which can be modeled as having "experiences" and "ideas". However, the reflexive aspects of current artificial neural systems are still dominant: networks "create" because they are designed to do so. To model creativity itself, creativity would have to be an emergent property. Otherwise, it is simply the vicarious expression of some other's creativity designed into the system. (See also Lewis 1991, for another perspective on this question.) These comments only focus on the philosophical level of machine models of human artistic expression. As I mentioned before, there are numerous other perspectives from which to view music and composition, e.g., from the point of view of computer science, psychology, musicology, etc., each of which comes with its own set of questions. From these varied viewpoints, computer-based -- and especially connectionist -- models of musical behavior may be very successful indeed. References Bharucha, J.J., and P.M. Todd. Music and Connectionism Ed. Peter Todd and Gareth Loy. "Modeling the Perception of Tonal Structure with Neural Nets." Buxton, B., W. Reeves, R. Baecker, and L. Mezei. 1978. "The Use of Hierarchy and Instance in a Data Structure for Computer Music." Computer Music Journal 2(4): 10-20. Buxton, B., R. Sniderman, W. Reeves, S. Patel, and R. Baecker. 1979. "The Evolution of the SSSP Score-editing Tools." Computer Music Journal 3(4): 14-25. Byrd, D. 1984. "Music Notation by Computer." Ph.D. dissertation. Bloomington: Indiana University Department of Computer Science. Cage, J. 1961. Silence.\fP Cambridge, Mass.: MIT Press. ICMC 372

Page  373 ï~~Chafe, C., B. Mont-Reynaud, and L. Rush. 1982. "Toward an Intelligent Editor of Digital Audio: Recognition of Musical Constructs." Computer Music Journal6(1): 30-41. Clynes, M. 1984. "Secrets of Life in Music." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, pp. 225-232. Cope, D. 1990. "Pattern Matching as an Engine for the Computer Simulation of Musical Style." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, pp. 288-291. Dannenberg, R., and G. Bloch. 1985. "Realtime Computer Accompaniment of Keyboard Performances." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, pp. 279-290. Ebcioglu, K. 1984. "An Expert System for Schenkerian Synthesis of Chorales in the Style of J. S. Bach." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, pp. 135-142. Fux, J.J. 1725. Gradus ad Parnassum. Reprinted in 1943 as Steps to Parnassus. New York: W. W. Norton. Gabor, D. 1947. "Acoustical Quanta and the Theory of Hearing." Nature 159: 591-594. Gill, S. 1963. "A Technique for the Composition of Music in a Computer." The Computer Journal6: 129-133. Gjerdingen, R.O. Music and Connectionism. Ed. Peter Todd and Gareth Loy. "Using Connectionist Models to Explore Complex Musical Patterns." Grossberg, S. 1982. Studies of Mind and Brain: Neural Principles of Learning, Perception, Development, Cognition, and Motor Control. Boston: Reidel/Kluwer. Grout, D.J. 1980. A History of Western Music. New York: W.W.Norton. Guido of Arezzo. 1026. "Guidonis Aretini Micrologus." In J. Smits van Waesberghe, ed. Corpus Scriptorum de Musica Rome: American Institute of Musicology. Printed in 1955. Kirchmeyer, H. 1963. "On the Historical Constitution of a Rationalistic Music." Die Reihe 8:11-24. Koenig, G. M. 1970a. "Project One." Electronic Music Reports 2. Utrecht: Institute of Sonology. Koenig, G. M. 1970b. "Project Two." Electronic Music Reports 3. Utrecht: Institute of Sonology. Kugel, P. 1990. "Myhill's Thesis: There's More Than Computing in Musical Thinking." Computer Music Journal 14(3): 13-25. Hiller, L., and L. Isaacson. 1959. Experimental Music. New York: McGraw-Hill. Kohonen, T., P. Laine, K. Tiits, and K. Torkkola. Music and Connectionism Ed. Peter Todd and Gareth Loy. "A Nonheuristic Automatic Composing Method." Laske, O. 1973. "Towards a Musical Intelligence System: OBSERVER." Numus-West 1(4):11ff. Lerdahl, F., and R. Jackendoff. 1983. A Generative Theory of Tonal Music. Cambridge, Mass.: MIT Press. Lewis, J.P. Music and Connectionism. Ed. Peter Todd and Gareth Loy. "Creation By Refinement and the Problem of Algorithmic Music Composition." Loy, G. 1989. "Composing With Computers--A Survey of Some Compositional Formalisms and Music Programming Languages." In M.V. Mathews and J.R. Pierce, eds. Current Directions in Computer Music Research. Cambridge, Mass.: MIT Press, pp. 291-396. Mathews, M.V., and L. Rosier. 1968. "Graphical Language for the Scores of Computer-generated Sounds." Perspectives of New Music 6:92-118. Meyer, L. 1956. Emotion and Meaning in Music. Chicago: Chicago University Press. Moore, F.R. 1989. Private communication. Moore, F. R. 1990. Elements of Computer Music. Englewood Cliffs, NJ: Prentice-Hall. Moorer, J.A. 1972. "Music and Computer Composition." Communications of the ACM 15. Moorer, J.A. 1975. "On the Segmentation and Analysis of Continuous Musical Sound by Digital Computer." Ph.D. dissertation. Stanford University: Department of Computer Science. Myhill, J. 1952. "Some Philosophical Implications of Mathematical Logic: Three Classes of Ideas." Review of Metaphysics 6(2): 165-198. Papert, S. 1988. "One Al or Many?" Daedalus 118(1): 1-14. Perle, G., and P. Lansky. 1981. Serial Composition and Atonality. Los Angeles: University of California Press. Potter, G.M. 1971. "The Role of Chance in Contemporary Music." Ph.D. dissertation. Bloomington, IN: Indiana University Department of Music. Available through University Microfilms. Roads, C. 1979. "Grammars as Representations of Music." Computer Music Journal 3(1 ): 48-55. Rodet, X., Y. Potard, and J.-B. Barriere. 1984. "The CHANT Project: >From the Synthesis of the Singing Voice to Synthesis in General." Computer Music Journal 8(3): 15-31. Rumelhart, D.E., and J.L. McClelland, eds. 1986. Parallel Distributed Processing: Explorations in the Microstructure of ICMC 373

Page  374 ï~~Cognition.\fP Cambridge, Mass.: MIT Press/Bradford Books. Scarborough, D.L, B.O. Miller, and J.A. Jones. Music and Connectionism Ed. Peter Todd and Gareth Loy. "Connectionist Models for Tonal Analysis." Scheidt, D.J. 1985. "A Prototype Implementation of a Generative Mechanism for Music Composition." M.S. Thesis. Kingston, Ontario, Canada: Queen's University Department of Computer and Information Science. Schenker, H. 1906. Newe Musikalische Theorien undPhantasien. Universal Editions. (Published between 1906 and1935 in two volumes.) Schottstaedt, B. 1984. "Automatic Species Counterpoint." Technical Report STAN-M-19. Stanford, CA: Stanford University, Center for Computer Research in Music and Acoustics. Schillinger, J. 1948. The Mathematical Basis of the Arts. New York: The Philosophical Library. Schillinger, J. 1978. The Schillinger System of Musical Composition New York: Da Capo Press. Sundberg, J., and A. Friberg. 1986. "A Usp Environment for Creating and Applying Rules for Musical Performance." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, pp. 1-4 Tenney, J., and L. Polansky. 1980. "Temporal Gestalt Perception in Music." Journal of Music Theory 24(2): 205-241. Thomas, M.T. 1985. "VIVACE: A Rule-based Al System for Composition." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association: 267-274. Vercoe, B., and M. Puckette. 1985. "Synthetic Rehearsal: Training the Synthetic Performer." In Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, pp. 275-278. Winograd, T. 1968. "Linguistics and the Computer Analysis of Tonal Harmony." Journal of Music Theory 12:2-49. Xenakis, I. 1971. Formalized Music. Bloomington, IN: Indiana University Press. Zarlino, G. 1558. Instiutioni Harmoniche. Republished in 1968 as The Art of Counterpoint. New York: W. W. Norton. ICMC 374