## 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 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
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.

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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

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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.

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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

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