Some Remarks on Network Models of Music

John Rahn

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 ¬parallel distributed processing║ or ¬neural net║  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 ¬symbolic 
processing║ 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 ¬neural net║ 
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\2 = {  | x,y ╬S }
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, ¬net║ 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 
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 ¬real world║ relation being modelled by R.

Note that DEF 1 does not use F/sem to define the ¬real-world║ 
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 ¬things║ in the Sonata (S) and n 
¬dimensions║ 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 ¬things║ 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 ¬meaning║ 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 ¬real world║ 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 ╬ S as labels for the 
world-objects w/i ╬ 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 ╬ 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 ╬ S to map into an identical element 
of W, but not possible for distinct elements w/i ╬ 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 ¬meanings║ 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 
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 ¬real world║ 
relation being modelled by R; and 2. a semantic function such that for 
a╬R, 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 ¬relationally compatible║ 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 ╬S under F\nodes/sem is a 
4-tuple  (see eq 5), 

 F\nodes/sem(x/i) --> 

where F\squash is the ¬squashing function║ that combines the inputs 
from the other nodes, F\act is the ¬activation function║ 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 ¬weight║ 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 ¬relationally compatible║ 
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 ¬learning rule║ 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 ¬knowledge.║

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 ¬relationally 
compatible║ 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 ¬hidden nodes,║ which it must to escape 
certain severe formal limitations on its power, there is said to be an 
¬internal║  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 
¬transformations.║) 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 ¬I║ with 
the string ¬I-V-I.║ 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 
¬verb phrase.║ 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 ¬prolongs║ 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 
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 
¬terminal string║ 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 ¬low-
level║ 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 ¬middle C║). In some real sense, then, this net 
is doing symbolic processing, and no processing model could do 
To take another example: Robert Gjerdingen has used neural nets of 
the self-organizing, ¬adaptive resonance║ 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 
¬specific, low-level musical features║ 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, ¬memories of 
critical feature patterns that resemble not simple chords but the harmonic-
contrapuntal complexes referred to by music theorists as voice-leading 
combinations║ (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 ¬symbolic processing,║ 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, ¬knowledge║ 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 ¬hidden.║ 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 ¬processing║ 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 ¬analytical║ 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 ¬time series,║ 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 (¬points║ in n 
¬dimensions║) 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 ¬ordered in║ 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 ¬dimension║ 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 = { | si╬Si}

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




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. ¬Using Connectionist Models to Explore 
Complex Musical Patterns.║ In Music and Connectionism, eds. Peter Todd and 
Gareth Loy. Cambridge, MA: MIT Press.

Laden, Bernice. 1994. ¬A Parallel Learning Model of Musical Pitch 
Perception.║ Journal of New Music Research, vol. 23, forthcoming.

Lewin, David. 1982-3. ¬Twelve-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.  ¬Klumpenhouwer 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. ¬Logic, Set Theory, Music Theory."  College Music Symposium 19, 
no. 1 (Spring): 114-27.

___.  1993.  ¬Le 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 
#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 ¬less than or equal to║ sign
in   Pn/i is a series product sign, Greek capital pi with i directly 
below it and n directly above it