Music theorists have become familiar with David Lewin's elegant formulations, which use networks to model music analytically (Lewin 1987, 1993), and with various kinds of grammars (Chomsky 1971), which have generally been used to represent variants of Schenkerian tonal theory and to model music analytically in that way. Researchers in artificial intelligence, including some music theorists (Gjerdingen 1991), have used quite different kinds of networks in a quite different way, not to model music analytically so much as to process a representation of musical information to arrive at a result that represents other musical information, that is, to do something that analysts and other people do. These are ªparallel distributed processingº or ªneural netº models (Rumelhart and McClelland, 1988), and as with any artificial intelligence model, it is necessary to distinguish between the replication of a result or input-output function on the one hand, and on the other hand the mimicking of the process by which a human being might cognitively or otherwise achieve the same function. If a processing box cannot mimic the human function, it cannot be mimicking the human process. However, success at replicating the input-output function does not guarantee success at duplicating the process by which a human may transform that set of inputs to that set of outputs. What use, then, are artificial intelligence models to the human music theorist? A successful duplication of function may replace the analyst, without enlightening her. Even success at modelling the process may prove equally unenlightening, analytically. With so-called ªsymbolic processingº artificial intelligence models, conventionally thought to include grammars, the theorist may hope to capture formally some clarification of analytical musical reasoning; but not with the ªneural netº models, whose inner workings are often inscrutable, or if stricto sensu scrutable -- one can open the box and look -- a puzzling heap of jack- straws about as meaningful for music as a circuit diagram. Which is what they would be: they aim to trace the circuits of the cerebral machine. Suppose we had full information as to the flow of energies in someone's brain when listening to Mozart. Would this help us appreciate the particular Mozart piece in question, or to converse about it meaningfully? Would it help us to compose a similar piece, or to compose a very different piece of similar quality? Such questions are not necessarily merely rhetorical (Rahn 1993), but there is a tangled web of nets and grammars here that needs all sorts of clarification first. There seem to be great gulfs dividing the analytic models from the processing ones, and the symbolic from the sub- symbolic or non-symbolic. This paper will proceed to explicate nets through consideration of formal, semantic, and pragmatic questions: What are they, structurally? What do they mean? What are they good for? I hope to show that all sorts of nets and grammars can be described in the same formal framework both for their structure and for their formal semantics, facilitating their comparison pragmatically. I. The Formal: Nets are relations Let S be a set. Then S-squared is the Cartesian product of S with itself, which is the set of all ordered pairs whose elements are elements of S.S\2 = { | x,y ÎS } A binary relation R on S is any subset of S\2. This is a net, in all the senses mentioned above. We say that the elements of S in elements of R are the nodes and that an arc (or arrow) spans each pair of nodes in R. More generally, a n-dimensional net on S is any subset R of S\n. R = { < x/1, x/2, . . . x/n>, . . . } Each n-tuple or arc in R would be a point in n-dimensional space, except that in general we do not assume anything about S, or any further structure in R. For example, one kind of space would require that S be the field of reals -- the set of real numbers with addition and multiplication defined on them -- and that R have defined on it a metric function giving the distance or interval between each pair of points, and satisfying certain properties such as the triangle inequality. If R is a space, then we are talking geometry. Still, it is interesting to think of a space as a rigidly structured, non-deformable net with infinitely many nodes and infinite density (with respect to its metric), and to think of a net as a plastic, deformable space without a metric. In general, R is not a space. With no assumptions about S or R, we can ask R if it is 1. reflexive, 2. symmetric, 3. transitive, and 4. connected. For many of the nets considered here, R will be some sort of partial ordering. (A full and formal treatment of such matters may be found in any book on point-set topology, such as (Cech et al. 1966). However, in such texts, ªnetº may assume a somewhat more sophisticated though still conformable definition, e.g. (Cech et al. 1966, p. 257)). 2. Semantics: Meaning To give an interpretation of a net R is a problem in labelling. Normally, it would suffice to involve the nodes in a labelling relation, that is, to define a semantic function F/sem from S to W where W is a set of objects in the world under consideration. DEF 1 An interpretation of a relation R on a set S is a semantic function F/sem(x/i) --> w/i assigning each x/i in S an object w/i in the world W in such a way that for every ordered n-tuple in R (for i ranging from 1 to n), the corresponding n-tuple of images is in relation RW where RW is the ªreal worldº relation being modelled by R. Note that DEF 1 does not use F/sem to define the ªreal-worldº relation RW, which may be a superset of the set of images of R under F/sem: the model may only partially mirror the reality. Moreover, DEF 1 implicitly constitutes a binary metarelation MR = {< , >, . . .} between as it were points in the formal domain of F/sem and points in the world. This metarelation may be treated as a simplified extensional specification of the interpretation. Since each semantic function F/sem from S to W is determinative for such a metarelation, and there are #S#W possible semantic functions, there are #S#W possible interpretations of a net R. Interestingly, but only if S=W, we can ask the same questions of MR that we asked of R: Is it reflexive, symmetric, transitive, connected, some kind of partial ordering? Is it a space, even? If S=W, the meaning of the net R is intra- rather than extra- . This might be called Liszt-processing: one legend has it that, when asked to explain his B-minor Sonata, Liszt played it again. We can imagine many paths from Sonata to Sonata, many ways of interpreting a net R by another net R-prime on the same underlying set S. In fact, if there are P ªthingsº in the Sonata (S) and n ªdimensionsº to the relation R (n things in each point), then there are P\n possible points in the set S\n so that #R <= P\n. Assume that the net R contains only half the possible points, so that #R is only one-half this number. Since the metarelation MR is a function on elements of R, every point in R has a meaning in RW=R\prime. The number of possible intra- Sonata interpretations under these conditions is #R\2, that is, about one- half P\n, squared, a formidable number in the case of the Liszt Sonata -- assuming that the decomposition of the Sonata into ªthingsº has anywhere near a decently fine granularity, and that there are more than a few musical things in each relational object. However, the normal definition of the interpretation of a relation, given in DEF 1, is not adequate for networks: the situation is more complicated. We have to consider not only the nodes, but also the arcs. The ªmeaningº of each arc from x/a to x/b (taking the two-dimensional case for simplicity) is in the first instance simply that x/aRx/b, so that w/aRWw/b under the semantic mapping F/sem. In this way of thinking, every arc expresses the same relation R, the relation which is the net as a whole. R gives the structure of the net, its topology of interconnection. The interpretation of the net translates R into the ªreal worldº relation RW, but under this interpretation again, every arc expresses the same holistic relation, RW. It makes sense to think of the various nodes x/i Î S as labels for the world-objects w/i Î W. Indeed, a label must be a sign, and the interpretation of S through F/sem makes its elements signs, but there is no guarantee that w/i Î W are also signs -- not every world is a world of signs. So it is not a matter of putting real-world labels on the nodes, but pasting nodal labels on real-world objects. Since F/sem is a function from S to W, it is possible for distinct x/i Î S to map into an identical element of W, but not possible for distinct elements w/i Î W to have an identical pre-image in S. We may paste more than one label on an object, but we can't paste the same label on several different objects. This is to say that one object may have several different structural ªmeaningsº in R, but that there must be an unambiguous real-world meaning (object) for each node, that is, each label. Consider now the arcs of a net. Each arc in R needs to have exactly one real-world meaning in addition to its expression of RW. Two different arcs may have the same meaning, but no arc may have more than one meaning. So in parallel with the semantic function on nodes, which we will now notate F\nodes/sem, there is a semantic function on arcs F\arcs/sem which maps each arc in R to some object wa/(i,j) in some world for arcs, WA. This world WA and function F\arcs/sem must be relationally compatible with F\nodes/sem, the world of nodes W, and the relation RW; that is, it must be a consistent extension of DEF 1, which is now amended to DEF 2: DEF 2 An interpretation of a net R on a set S is 1. a semantic function F\nodes/sem(x/i) --> w/i assigning each x/i in S an object w/i in the world W in such a way that for every ordered n-tuple in R (for i ranging from 1 to n), the corresponding n-tuple of images is in relation RW, where RW is the ªreal worldº relation being modelled by R; and 2. a semantic function such that for aÎR, F\arcs/sem(a) --> wa, where wa is some object in a set WA which is the world for arcs, and where F\arcs/sem is relationally compatible with R and F\nodes/sem. The sense of ªrelationally compatibleº is left unspecified in order to accommodate all the various cases, in which it will assume various precise meanings. Let us look at some of the cases. 3. Formal description and interpretation of different cases 3.1 Neural nets -- a sampler For the general case of some kinds of neural nets (Rumelhart and McClelland 1988), the value of each node x/i ÎS under F\nodes/sem is a 4-tuple (see eq 5), F\nodes/sem(x/i) --> where F\squash is the ªsquashing functionº that combines the inputs from the other nodes, F\act is the ªactivation functionº that computes the state of the node from its squashed inputs, Act/(i,t) is the value of the current activation state of that node at time t, and F\out is the output function. The activation state at time t is the value of the activation function at time (t-1) (see eq 6>): Act/(i,t) = F\act(F\squash(input/(i,t-1))). Time is measured in units equal to the tick time or cycle time of the net, which is necessarily discrete when running on a digital computer. The output of the node at time t is: Out/(i,t) = F\out(Act/(i,t)). The overall relation R among nodes gives the topology of connections RW among these 4-tuples (eq 8): x/iRx/j IFF w/iRWw/j (w/i sends its output directly to w/j). Each arc is mapped under the semantic function on arcs F\arcs/sem to a number wa/(i,j) which is the ªweightº of the connection between w/i and w/j. The nodal squashing function F\squash uses these weights to combine all the inputs received by the node. Therefore, the meaning of ªrelationally compatibleº in DEF 2 varies even within this general model of neural nets: the activation states and output values may have various ranges, for example, binary or real. The weights on the arcs will generally be real-number valued, since they generally are used to multiply the incoming signal along the connection, but the weights must coordinate with the rest of the net to produce the desired result. Indeed, the whole trick of using neural nets is to find a setup that can employ a good ªlearning ruleº that will modify these weights so as to come ever closer to a system which produces the desired transfer function from input environment to output. Once this has been achieved, it is in the set of weights that the system has stored its now re-usable ªknowledge.º 3.2 Lewin nets For the kind of networks described formally in other terms by David Lewin (Lewin 1987), F\nodes/sem maps nodes in S to musical objects in W, such as sets of pitch-classes or tonal chords. F\arcs/sem maps arcs to members of a mathematical semigroup of transformations or group of operations: this is then the world or arcs, WA. The overall relation R will set up an arc between two nodes x/i and x/j just in the case that the function on that arc maps w/i to w/j, that is (eq 9), wa/(i,j) (w/i) --> w/j. (I use the two-dimensional case for simplification.) The network of functions must be consistent: If there is a path from w/p to w/q then the function that maps w/p to w/q must be the functional composition of all intervening functions on the path. Here, the sense of ªrelationally compatibleº in DEF 2 comprises these group constraints. Note the differences between the treatment of musical representations by neural nets and by Lewin nets. A neural net doing music takes an input vector, processes it, and outputs another vector. Both input vector and output vector are some kind of representation of music. In addition, if the neural net has ªhidden nodes,º which it must to escape certain severe formal limitations on its power, there is said to be an ªinternalº representation of music that evolves among these hidden nodes while the net is learning its task. However, the internal representation evolves along lines that make it efficient but not meaningful in the sense that a human music theorist might look at it and be enlightened about the music. (The internal representation is the settled set of weights along the arcs among the hidden nodes.) The input vector also is, again for reasons of efficiency (getting the job done well) often a distributed representation, which may not make much intuitive sense to people, or be of much use to our musical thinking, and this may also be the case for the output representation. On the other hand, Lewin's nets evolved from the tradition of music theory and analysis. The musical objects that are the meaning of the nodes are carefully chosen by the analyst using a Lewin net for representation to embody in the way they combine together by the arc functions of the Lewin net a representation of the structure of the musical piece which gives musical enlightenment to another analyst. 3.3 Grammars There are of course various kinds of grammars, each with its own formal properties, capabilities, and limitations. Chomsky's classic Syntactic Structures (1971) divides them into three classes of increasing power: finite-state machines, phrase-structure grammars, and transformational grammars. (These last have no relation to Lewin's ªtransformations.º) Traditionally, grammars are not thought of as nets, but they do fall into the scheme formulated here. Consider phrase-structure grammars, which are often used to model music. They consist of a vocabulary V and a set P of Post productions (named after the mathematician Emil Post). Each Post production replaces a string or substring of vocabulary elements with another, usually longer such string. For example, the production (eq 10) I --> I-V-I would allow replacing any instance of the singleton string ªIº with the string ªI-V-I.º The overall result of such productions is a tree structure. In natural language grammars, there is a distinction between terminal nodes, which are elements of the target language, and non- terminal nodes, which are theoretical constructs in the grammar such as ªverb phrase.º No such distinction is forced upon phrase-structure grammars that model music, though it may be retained if useful. We discard it here to simplify. Now obviously a tree is a net, in particular a kind of partial order. Let S=V and let R/i be a tree producible by P. There are infinitely many such trees in the grammar but if, as commonly used, each tree represents an analysis of a piece of music, then there are only finitely many trees resulting in that piece. (Some would have it that there is uniquely one such tree; this would then be a feature of the particular grammar they might construct.) More precisely, W is a world of musical objects and F\nodes/sem maps S into W. An arc is in R just in case x/j is an immediate subnode of x/i in the tree, so that x/j is an element of a string produced from x/i by a production in P. The interpretation of the arcs is the semantic function F\arcs/sem which maps each arc into the Post production rule that produced x/j from x/i in that tree. The common musical interpretation of R dates back to Michael Kassler: if w/iRWw/j then w/j ªprolongsº w/i. The tree moves from background to foreground in the sense borrowed from Schenker, or in the direction of successively greater musical elaboration. The Post productions are then musical transformations from background to foreground. However, it is also possible to use a grammar purely as a processing device, for its results rather than for the analysis its derivations embody. In such a case, the meanings of the arcs are Post productions which may not carry much semantic freight other than their formal function. There are many possible ways to qualify or inflect a phrase-structure grammar. There is a hierarchy of types of these grammars from least to most powerful: less powerful is the context-free grammar, in which every production rule has a singleton string (a single sign) as its left-hand side. More powerful is the length-increasing context-sensitive grammar, in which the left side of each production must be a shorter string than its right side. The most powerful phrase-structure grammars have no restrictions on the relative string lengths of their productions. In addition to this hierarchy, observe that a derivation in a phrase- structure grammar applies one production rule at each step. Thus the derivation generates a string of rule-applications (see example 1). For systems in which there are many possible derivations for each final result -- for example, many different analyses of the given musical piece under the theory, in the musical interpretation -- there will be at each stage of the derivation a number of possible choices of production rule to get to the next stage. It is possible to define a metagrammar to constrain these choices of production rule in the object grammar. In the simplest case, define an nth-order finite-state machine to produce strings of production rules (see example 1). A more complex approach would generate a ªterminal stringº of production rules in the object grammar using a metagrammar which is itself a phrase-structure grammar. One can keep on recursing (a meta-metagrammar and so on) if there is any use for it. 4. Pragmatical matters 4.1 Symbolic vs non-symbolic processing At first glance the distinction between symbolic processing and non- symbolic processing seems opaque. An example of non-symbolic processing might be a neural net that models the perception of pitch from acoustic data. When fed a representation of a set of sinusoidal frequencies, the net has been trained to output a representation of one of the pitches of the piano keyboard (Laden 1994). This is a typical ªlow- levelº perceptual task, to which neural nets are particularly well suited. Unless a person writes computer-synthesized music, she is unlikely to hear sounds explicitly in terms of their separate Fourier components, as well as by the usual method of hearing single sensations of pitch, each corresponding to a fused stack of harmonically related partials. This synthesis into pitch is not something most listeners spend much thought on. They do not pay much attention to the individual spectral components. Yet this neural net does nothing but process representations. It is fed representations of acoustic data and outputs representations of phenomenal data (such as ªmiddle Cº). In some real sense, then, this net is doing symbolic processing, and no processing model could do otherwise. To take another example: Robert Gjerdingen has used neural nets of the self-organizing, ªadaptive resonanceº kind to model musical cognition at a higher level (Gjerdingen 1991). The binary input vectors represent the presence or absence at a particular musical moment of each of 34 ªspecific, low-level musical featuresº such as scale degree (separately for melody, bass, and inner voices), melodic contour and inflection, and others (Gjerdingen 1991, p. 141). The net learns to organize these inputs into representations of the music at yet higher levels, ªmemories of critical feature patterns that resemble not simple chords but the harmonic- contrapuntal complexes referred to by music theorists as voice-leading combinationsº (Gjerdingen 1991, p. 146). My point is that the output is certainly a representation of material that is conceptual or symbolic in nature, and the input is not so very low-level either, feeding in musical concepts preformed that take the average Freshman ear-training class a while to learn. This net is doing symbolic processing. The difference between the situations of the first net for pitch perception and this one for musical analysis is that in the case of pitch perception, the human process being modelled may usually not involve much human representation or conscious thought. There is plenty of representation involved in both nets as they process, and plenty of higher-level representation involved in the input and output for the second net. Any music-analytical model, such as Lewin's analytical nets or Gjerdingen's processing-analytical nets, will involve higher level representations. 4.2 Analysis vs processing So what is the difference between an analysis such as a Schenker analysis or a Lewin-net analysis, and on the other hand a processing model such as Gjerdingen's ART networks or even Laden's pitch- perceptual network? Both do ªsymbolic processing,º since (as we have seen) the distinction between symbolic and nonsymbolic processing has to do with the domain being modelled rather than the modelling itself. But do both processing models and analytic models process? Let us return to Lewin nets as the very model of an analytic model (Lewin 1993). We see before us a net whose nodes are musically meaningful objects, a net throbbing with static dynamism as each musical object constantly and happily transforms itself along the arcs connecting and relating it to every other musical object in the net. The sense of energy bound in a knot in this net whose nodes are constantly going everywhere else in the net, a net which as a whole remains fixed, fully occupying the place of the musical piece it represents by analysis -- it is this sense of almost particular energy that makes us happy, as we recognize its resemblance to our experience of the music itself. Suppose someone came along who did not share our joy in this analytical experience, who was interested only in processing and its result, not in contemplating the sensual richness of the dyad consisting of the analytical object and the musical piece. Perhaps this person is a film composer or Musak manufacturer who just wants fast, automatic ways to make a musical product; or perhaps this Lewin net is going to model the automation of factories. Now imagine a Lewin net closely modelling a desirable kind of piece or system. Remove all the nodal content. The result is a processing machine: its connectivity is unchanged, and the meaning of its arcs is unchanged (they each mean a certain function in a group). The empty nodes are waiting, as it were, like nestlings with beaks gaping wide. The minute Mama drops one worm (any musical object) into one beak (a node) -- abandoning the avian simile now -- the entire net crystallizes into a new content computed by the functions on the arcs. The Lewin net has computed a new whole which is isographic to the first and to any other such crystallization. Just as in the neural net machine, ªknowledgeº is embodied in the Lewin net's connectivity (relation R) and in the meanings of the arcs. For neural nets, the semantic function F\arcs/sem maps arcs to numbers that serve as weights; for the Lewin net, F\arcs/sem maps arcs to members of a group of functions. To compare the two kinds of nets further as processing machines: in neural nets, the functions that compute reside in the nodes as nodal content, while data fed to these functions initiates in the input vector fed to the input nodes and filters through the weights that reside on the arcs, which are (I would argue) part of the data for that functioning net. (I do not count the activation states of the nodes as data, since each state is merely a stage in the composite filter-function that makes up the node.) For Lewin nets, the data (musical objects) reside in the nodes as nodal content and are fed to and result from the functions that reside on the arcs. A neural net takes as input a vector of values, one for each input node; this vector is a representation. The net processes this input representation and outputs a vector of values, one for each output node; this vector is also a representation. Only input nodes receive input, only output nodes put out, and any nodes that do neither are called ªhidden.º A Lewin net takes as input one (perhaps musical) object, which may be put in any of its nodes. The output of a Lewin net is the entire ensuing functionally related content of all the nodes in the net. Any Lewin-net node may be an input node, and all Lewin-net nodes together are the output. In the Lewin net, the content of each node is a representation, and the net as a whole is a representation. In the end, it seems that what distinguishes a processing model from an analytical model is not so much the model itself, but where the interest of the user lies. Models tend to be characterized as ªprocessingº ones when the user is interested in getting a result. The focus is on the transfer function from the set of inputs to the set of outputs. On the other hand, users of ªanalyticalº models are interested in what's going on there. They use their sense of what's going on in the model as one face of a coin whose obverse is what's going on in the musical piece. The same distinction can be made for research in AI: some researchers just want the result, others are interested in the process itself as a model for some other process, such as neural activity in the human brain. 5. Networks reconsidered: multidimensionality Two aspects of music make it a particularly difficult analytical study from a formal point of view. One is its intimately temporal nature; the other is its inherent multidimensionality. An analyst of stock-market fluctuations is interested in ªtime series,º harmonic analysis, and so on -- many of the tools are not inapt for musical analysis, too -- but in the end the market analyst is only interested in the behavior of one variable as a function of time. Musicians have to pay attention to many variables. Moreover, it would oversimplify, and falsify, to describe music as a function of time. Rather, time is one of a number of qualities which vary in codependence with one another. One of the problems with some models of music, such as grammars (as usually employed), is that they are concerned only with strings, which are one-dimensional in appearance. Now, if each element of a string is an n-tuple, and each place in the string is a value (if only an ordinal value) in some dimension such as the temporal, the the string of n-tuples translates into a relation in n+1 dimensions. We can look at any string of n-tuples (ªpointsº in n ªdimensionsº) as the projection of an (n+1)-ary relation onto one of its dimensions -- any one of its dimensions, so long as there is a strict simple ordering defined for that dimension. Suppose that a musical dimension does not have a strict simple ordering defined on it, but does have a partial ordering, perhaps a tree: then the (n+1)-ary musical relation can be displayed by a quasi-projection so that each node of the tree labels the n- tuple of the remaining values in each relational point. When we project musical objects onto one of their dimensions, and that one exhibits some order, we say that the objects are ªordered inº that dimension (Rahn 1975). Indeed, if a musical relation is in n dimensions each of which exhibits some order, the points in the relation are separately and simultaneously ordered in each of the n dimensions. This is part of the wonder of music. Implicit in all of this is a need to amend our original definition of relation in eq. 1 and 2. We can no longer assume that each ªdimensionº of R has the same underlying set S. Instead, we have the set product of n distinct sets Si, i=1,n (equation 11) Pn/i = { | siÎSi} so that the i-th element of each n-tuple is a member of set Si. Then the relation R is defined as before (mutatis mutandis) (equation 12). R = { , . . . } In the musical case, some but not all of the underlying sets Si may be spatial as described earlier. Suppose four of the Si are spatial in the same way, that is, with the same metric function and so on. The these four make up four-dimensional space as part but not all of R. Suppose three other Si make up a different space. Then we see a four-dimensional space growing out of itself at three different right angles a qualitatively different three-dimensional space. Suppose further that there are five more Si that are only partially ordered and have no metric, and two more Si that are not even partially ordered. Then that heterogeneous seven-dimensional space (4 + 3) has growing at right angles five more heterogeneous, soft, wobbly, partially ordered dimensions, and two more amorphous blobs of nonspace. Feel familiar? Each of these partly-points, these heterogeneous n-tuples, is an arc as constituted by the relation R. Have we sufficiently appreciated that when we talk of a space of points of R, and the distance between two points and so on, we are talking about a space of arcs, the distance between two n- dimensional arcs in R? If we were to draw a figure in this n-space, we would be connecting a number of n-dimensional points that are themselves arcs into one n-dimensional figure. Since each relation R is a set of such n-dimensional points, each relation R is such a figure. The geometry and topology of the combination of such figures, their projections on various combinations of their dimensions, their interpenetrance and their shadows on each other, their equivalence under such mundane spatial manipulations as affine transformations, all that would constitute the technology of a music theory which was not content to model each piece of music as one relation, one network, but which employed in the service of music's magnificent multiplicity a science of relations of relations, a topology of swarms of nets, and the geometry of composite figures each betokening the whole, but all together figuring the diversity of coexistent wholes we know as music. -------------------------------------------- Example 1 S=all subsets of the set of integers mod 12 P={P1, P2, P3, P4} where P1: empty set --> {0} P2: X --> T1(X) where T1 is transposition by 1 P3: X --> X-{0, 2, 5} P4: X-{0, 2, 5} --> {0, 3, 7}-X-{0, 2, 5} A derivation: rule string axiom empty set P1 {0} P3 {0}-{0, 2, 5} P2 {0}-{1, 3, 6} P3 {0}-{1, 3, 6}-{0, 2, 5} P4 {0, 3, 7}-{0}-{1, 3, 6}-{0, 2, 5} P2 {1, 4, 8}-{0}-{1, 3, 6}-{0, 2, 5} string of productions: P1-P3-P2-P3-P4-P2 metagrammar (finite state): P2 is followed by P3 P3 is followed by P4 or P2 P4 is followed by P2 diagram: P3 P2 P1 P4 ---------------------------------------------------- References Cech, Eduard, Frolik Zdenek, and Miroslav Katetov. 1966. Topological Spaces (revised edition). New York: Wiley&Sons (Interscience Publishers). Chomsky, Noam. 1971. Syntactic Structures. The Hague: Mouton. Gjerdingen, Robert. 1991. ªUsing Connectionist Models to Explore Complex Musical Patterns.º In Music and Connectionism, eds. Peter Todd and Gareth Loy. Cambridge, MA: MIT Press. Laden, Bernice. 1994. ªA Parallel Learning Model of Musical Pitch Perception.º Journal of New Music Research, vol. 23, forthcoming. Lewin, David. 1982-3. ªTwelve-Tone Techniques in Atonal and Other Music Theories." Perspectives of New Music 21: 312-71. ___. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. ___. 1990. ªKlumpenhouwer Networks and Some Isographies that Involve Them." Spectrum 12/1: 83-120. ___. 1993. Musical Form and Transformation: 4 Analytic Essays. New Haven: Yale University Press. Rahn, John. 1975. "On Pitch or Rhythm: Interpretations of Orderings Of and In Pitch and Time." Perspectives of New Music 13, no. 2: 182-204. ___. 1979. ªLogic, Set Theory, Music Theory." College Music Symposium 19, no. 1 (Spring): 114-27. ___. 1993. ªLe Compositeur et ses AMIs: Remarques sur la CAO." Cahiers de l'Ircam / Recherche et Musique Bilan 1992, second trimester no. 3: 119-32. Rumelhart, David, and James McClelland. 1988. Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations. Cambridge, MA: MIT Press. --------------------------------------------------------- Notes for typesetting According to normal mathematical usage, all variables should be in italic (as specified in the Chicago Manual of Style or other style book). exx: R, S, W, F/sem, x/i, wa, and so on x/i and similar constructions mean x subscript i S\2 means S superscript 2, etc. F\arcs/sem and similar constructions mean F superscript arcs subscript sem #S means the number of elements in S, that is, the cardinality of S; could be typeset as |S| R\prime means R superscript prime sign --> means the mapping arrow sign <= means the ªless than or equal toº sign in Pn/i is a series product sign, Greek capital pi with i directly below it and n directly above it -------------------------------------------------------