Compositional Applications of Stochastic Processes in Computer Music

Compositional Applications of Stochastic Processes
Author(s): Kevin Jones
Source: Computer Music Journal , Summer, 1981, Vol. 5, No. 2 (Summer, 1981), pp. 4561
Published by: The MIT Press
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Kevin Jones
Compositional
Applications of
Stochastic Processes
The Music Department
The City University
London
England
Stochastic Processes
routines that permit the user to specify parametric
boundaries within a choice of differing degrees of
A stochastic process is a collection of random-varicontrol. Truax (1973) has incorporated structures
able quantities distributed in space or time. When
a to Koenig's tendency masks in his POD
similar
statistician makes use of a stochastic process, systems,
the
which offer possibilities for limited ranobject is to find basic patterns in a set of observed
dom variation. For digital microsound synthesis,
data that will provide more coherent information
Roads (1978) used a unit called an event, similar
about that data. In practice, a situation of complete
to a tendency mask, to encapsulate thousands of
randomness, where there is no order, is unlikely
to
short-duration
grains of sound. In spite of these acoccur. Indeed, the concept of absolute randomness
tivities, some skepticism has remained over the
turns out to be extremely difficult to define mathe"controllability" of stochastic techniques in com-
matically (Chaitin 1975). Mathematicians haveposition. This is a needless concern, since stochas-
classified various types of stochastic structures tic
as agenerative schemes may produce results that
basic framework for analysis of stochastic pro- sound far more ordered than what might be processes. When composers make use of stochastic
duced by a supposedly deterministic system.
structuring techniques in musical composition,
they are usually approaching the problem from the
other direction. The main interest is in a synthesis
Uses and Justifications for Stochastic Techniques
of a sequence of sound data within a structural
framework. A stochastic generative scheme isStochastic
a
techniques offer, first of all, a useful
means of setting up and manipulating stochastic
means of data reduction. Computer music systems
control structures. Although a composer has a difrequire accurate specification of all parameters conferent objective, mathematical techniques devel-cerned in defining a sound. This may be several
times
oped for analysis can be of great practical use in
the as much information as a common instruformal, structured environment of computer music
mental score would supply; a complex sound may
systems.
require vast amounts of input data and instructions.
By defining sets of parameter limits within which
actual values may be generated stochastically, one
Previous Work
can reduce significantly the amount of labor required. This is no abdication of composer responThe pioneering algorithmic composition experisibilities. In the past, composers have always relied
ments of Hiller (1958) and of Xenakis (1971) are
on performers' interpretations or on random enwell known. Their programs have made use of sim- vironmental influences for fine control of such paple stochastic constraints to generate output that
rameters as intonation, precise duration, timbre,
can be transcribed into traditional notation and
and intensity. As always, the choice of which paplayed by conventional instruments. At Utrecht, rameters will be specified manually and which will
Koenig (1971 a) has developed stochastic composing
be specified procedurally is up to the composer.
Stochastic techniques may also produce unantici-
Computer Music Journal, Vol. 5, No. 2, Summer 1981,
0148-9267/81/020045-17 $04.00/0
? 1981 Massachusetts Institute of Technology.
pated possibilities, where the bonds of a restrictive
and inaccurate acoustic theory and of a limited aural imagination may be broken. Such an approach
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only suits some composers, but the author has been
delighted by stochastic serendipity on many occasions. In addition, it is possible to use stochastic
techniques for overall structural control. For example, one can stochastically control the progressive
modulation from one structural type to another by
a gradual separation process plotted through a regulated sequence of steps. In this way, a composer is
following a precedent formed by centuries of tradition, since many composers in the past have made
how to program and test various probability distributions, so there is no need to enter into those
mechanics here. However, one definition should be
reviewed. In a program, an efficient way to use a
probability distribution is to stepwise-sum it, comparing a random number to the accumulated value
in an array. This array holds what Lorrain calls the
cumulative function (a stepwise-summed probability distribution).
use of formal calculations to structure their work.
The Random Decaying Function
Finally, electronic and computer music is frequently criticized for sounding too fastidiously sterile, too regular, and hence too artificial. Stochastic The author has made use of a probability distribution defined by a random decaying function in
techniques may be used to produce fuzzy edges and
composing the 1979 orchestral movement Firelake,
to "humanize" computer-generated sounds.
where a random-integer generator was made to call
itself recursively. The function RND(N) will produce an integer between 1 and N. The function
Probability Assignment over an Event Space
RND(RND(N)) will therefore produce integers
When a stochastic structure is applied in musical weighted toward 1. So when used over an event
space of order n, event eI will occur most of all,
composition, it is necessary to define an event
space over which it operates. An event space is anevent e2 not quite so often, and so on, with event e,
ordered set of events that may consist of a natural hardly occurring at all. The assignment was applied
sequence of common musical material such as theover a number of different musical event spaces.
notes in the major scale of C, individually defined For example, pitches were derived from an event
space defined as the chromatic scale built upon the
sound complexes, selected natural sounds, or whatnote G, which served as a quasi-tonal center. The
ever basic compositional elements a composer may
require. The order of an event space is the number lengths of multiple events, the durations of pauses,
of events it contains.
and certain rhythmic patterns were all derived from
The most basic type of stochastic structure is a the same function.
simple probability distribution over an event space.
This is represented in software as an array that
specifies a set of probabilities corresponding to the Markov Chains
elements of the event space. The size of the array
In the systems so far described, the probabilities remust equal the order of the event space, and the
sum of the probabilities should be 1. If the proba- main constant over a period of time. More sophistibilities in the assignment are equal, then what can cated control mechanisms may be constructed by
be called an aleatoric process will ensue, with all using a Markov chain that takes into account the
events equally likely to occur. Over time, there will context of an event in a sequence making the probbe no evident pattern in any resulting generated se-ability of its occurrence depend on the event that
quence. Short sequences containing a very small preceded it. In this way a matrix of probabilities
number of events might, however, appear to have over an event space may be built up. The row of
some sort of predetermined quality, as is apparent probabilities corresponding to the first event is used
at the beginning of the author's 1977 tape composi- to derive the next event (Lorrain 1980). The row
tion Macricisum. Lorrain (1980) describes in detailcorresponding to the new event is used to derive
46
Computer
Music
Journal
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the next event, and so on for the required sequence
length; hence the chain aspect of this kind of stochastic algorithm.
When one defines a Markov chain for a specific
musical task, a number of properties of events
can be very useful for describing relationships and
classifications. These properties are examined in
greater detail elsewhere by the author, and brief intuitive proofs of some of the results are given (Jones
1980b). A more detailed mathematical treatment
which it communicates will also be transient. Thus
recurrence and transience are both class properties,
and any Markov chain can be divided into groups of
recurrent and transient classes of events.
Some important results follow from this classification that are of great value in analyzing and constructing Markov chains for musical composition.
Once a process has left a transient class of events it
cannot return to it. On the other hand, once a process has entered a recurrent class, it cannot move
may be found in the literature (Arthurs 1965;
out of it. It forms a closed set. Thus if a recurrent
summary of some of these properties follows.
class consists of just one event, it is called an absorbing event. A Markov chain must contain at
Lakacs 1966; Romanovsky 1970; Bhat 1972). A
Properties of Markov Chain Events
If an event e, can be followed by event e,, then e, is
said to be accessible from e,. If events e, and e, are
both accessible from each other, then they are said
to communicate. The communication relation has
least one recurrent class. It cannot contain tran-
sient classes only. A Markov chain consisting only
of one recurrent class (and no transient classes) is
irreducible. A Markov chain containing exactly one
recurrent class (and possibly some transient classes)
is known as an ergodic Markov chain. It is possible
to make useful predictions about its behavior and
eventual outcome. (It is also necessary that the
the three properties associated with an equivalence
Markov chain should not contain periodic classes.
relation. It is reflexive: event e, communicates with
For a further discussion and development of periitself. It is symmetric: if event e, communicates
odicity in Markov chains see writings by Jones
with event e,, then event e, communicates with[1980b] and Romanovsky [1970].) Whether a chain
is ergodic or nonergodic makes a significant differevent
ei.
It
is
transitive:
if
event
e,
communicates
with event e, and event e, communicates with ence for the type of musical task to which it may
be applied.
event ek, then event e, communicates with event
ek. Thus groups of communicating events can be
split into equivalence classes of events. The events
in one equivalence class will not communicate
A Musical
Example of Markov Chains
with any event in another equivalence class, but
an
event in one equivalence class may be accessible
A Markov chain of order eight is defined by the
from an event in a different equivalence class. In
stochastic matrix in Fig. 1. Next to the matrix is a
musical composition, separation into equivalence
corresponding event-relation diagram that helps
classes of communicating events provides a convenient way of grouping events into a temporalone
or to visualize the structure. (This Markov chain
is derived from an example in Kaufman's book
sequential hierarchy. This is clarified in the exam[1968].) The events are grouped into equivalence
ple explained later.
If after an event e, has occurred it is at some classes
stage C,- C4. Classes C,, C3, and C4 consist of
transient events. C2 is the only recurrent class and
certain to occur again, then it is said to be recurrent. If there is a possibility that an event will is,
not
therefore, an ergodic Markov chain. It can be
that the process will always settle to the
recur, it is said to be transient. It can be shownseen
that
if an event e, is recurrent, then all events withevents in the equivalence class C2, when event e2
which it communicates are also recurrent. Simwill tend to occur twice as often as event e,. Five
ilarly if an event e, is transient, all events with possible event sequences that might be generated
Jones
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47
Fig. 1. A Markov chain of
order eight (a) and its corresponding event-relation
diagram (b).
(b) -
(a) Next Events
1
2
3
4
5
6
7
8
, .3
Current 1 0 1 0 0 0 0 0 0
Ce
2
.5
.5
0
0
0
0
0
0
3
.1
.1
.4
.4
0
0
0
0
4
0
.2
0
.8
0
0
0
0
5
0
0
.5
.5
0
0
0
0
6
.1
0
.1
0
.1
.7
0
0
7
0
0
0
8
0
.2
0
0
.4
.2
0
0
.3
0
.6
?
ii
4I
.6 4
/
C6
Events
. .3 2
C3,
e8
e.
\
.2
.
S1
=
this
e6
rC
e
e3
,
3
1
e2
el
1
matrix
e6
.
.5
2
.3
C
from
e
es
e4
0
\
Y-
5
.8
.7
e3
are
e4
a
e5
e2
S2 = e6 e6 e6 e6 e1 e2 e1 e2 e1 e2 e2 e, e1 e2 e1 e2 el
(Jones 1980a), in the second study, "Laitrapartial,"
e2 e2
S3 = e8 e4 e2 e2 el e2 e1 e2 e2 e2 e2 e1 e2 e2 e, e2 e,
of the author's composition Macricisum. The event
space consisted of the harmonics on a given fundamental. The process changed in steps of 12o sec.
S4 = e7 e8 e7 e8 e4 e5 e4 e2 e2 e2 e2 e1 e2 e2 e2 e2 e,
Twenty such processes were made to operate sie2 el e2 el e2
multaneously. For some generations a bias was inS5 = e8 e7 e8 e4 e5 e4 e5 e4 e5 e3 e2 e2 e1 e2 e e2 e2
troduced into the process, encouraging upward
e, e2 e2 e2
movement through the higher harmonics or, alternatively, downward movement toward the fundaWith the event space defined in Fig. 2, these five
mental. The aural result was to produce crystalline
sequences will transcribe as shown in Fig. 3. A
shimmering sheets of sound full of complex intercomparison of these sequences with the event-relamingling harmonics.
tion diagram of the original Markov chain (Fig. 1)
The simple probability assignments considered
helps to show the musical significance of the difinitially are also a special case of a Markov chain in
ferent equivalence classes of events.
which all the rows are equal; the probability of
e2
each event's occurrence will remain the same no
A Special Case
matter which event has preceded it.
A special type of Markov chain is the random-walk
Extensions of Markov Chains
process. Such a process can only move from an
event e, to an adjacent event e,, or e,_,. The matrix
The Markov chain structure itself may be extended
representation of this process has nonzero entries
immediately on either side of the main diagonal
in a number of useful ways. These include increasand zeros elsewhere. The random-walk process
ingwas
the order of the Markov system, turning events
scalar values to vectors, and introducing the
used to control very small incremental changesfrom
in
sonic structure, a form of adjunctive synthesisnotion of a finite-state grammar.
48
Computer
Music
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Fig. 2. An event space for a
simple clarinet piece. (Music printing by Leland
Smith.)
C,: es
mp
C,: e7 r ip e8
C,:
e,
e2
/f
f
Nth-Order Markov Chains
structures can become unwieldy and almost impos-
sible to use.
A three-dimensional Markov chain may be used
where the occurrence probability of an event deMarkov Chains with Event Vectors
pends on the two preceding events. The probabilities are represented by a three-dimensional
A different extension is to use Markov chains
stochastic matrix. With greater dependence between events a tighter, more regular pattern strucwith event vectors. Each event, instead of being
ture will be defined. Three-dimensional Markov
considered as a single entity, is described by an
event vector that contains the values of a number
chains were used to generate rhythms in the third
study, "Skirtriks," of Macricisum. A two-event of controlling parameters, themselves defined by a
space consisting simply of "long" and "short" notes
set of event-parameter spaces. A different Markov
was used. Regular rhythmic patterns were produced
chain may be used to control each parameter, or a
in this way that occasionally shift or move out ofMarkov chain may control more than one paramstep, creating a fallible, spontaneous-sounding re- eter. Markov chains with event vectors were used
sult. Three-dimensional Markov chains may be throughout the composition Macricisum, but their
further generalized to n-dimensions, also called application is clearest in the first study of that
Markov chains of the nth order, where the occur-piece, "Sonatanos." Here, each event vector defined
rence probability of an event depends on the pre- a block of clustered points of sound in terms of the
ceding n-1 events. Taken too far, however, suchfollowing parameters: pitch range, overall pitch,
Jones
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49
Fig. 3. Five possible event
sequences formed by the
application of the Markov
chain of Fig. 1 over the
event space of Fig. 2. (Music printing by Leland
Smith.)
AFI3is3-r-
I
3I
313
1.1
I f A I A -J" f " I
f -P P--~- arff
Arw In L -- II-
2.
m.
.
mf
fI
mf
f
mf
f
Mf
ff
f I>f
3.
Ifmk
50
Computer
my
Music
f
a
Journal
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Fig. 3 (Cont'd)
r4. _
fM
-w-Z
5.
f 1- p m
V
I
I
T
1?
f mf i > mf mf
Fin
block
le
envelop
ments,
By
extension of Markov chains comes about when use
eters
c
specifie
is made of structures from formal linguistics. Fortween
t
mal grammars have been suggested as a powerful
ified,
means of specifying data in computer music sys-t
making
tems (Buxton 1978; Roads 1979; Holtzman 1979). A
cally
formal grammar consists of a set of w
symbols; a set
Markov
of terminals or events that correspond to the event
control
space in the schemes described previously; a set
Jones
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51
of production rules, which specify ways in which
symbols may be rewritten by combinations of symbols and terminals; and a starting symbol, which is
used to begin the generative process. The basic relationships in a Markov chain may be represented by
a type of grammar known as a finite-state grammar
or Chomsky Type-3 grammar (Chomsky 1963). In
such grammars, each production rule specifies that
a symbol may be replaced only by an event followed by one other symbol or by a single event that
terminates the process. In this way a linear structure is built up.
Stochastic Grammars
Thomason 1978, p. 189). If the grammar is inconsistent, it is likely to generate ad infinitum, and is
therefore of no practical use.
Space Grammars
What is called here a space grammar can operate
across many dimensions. Thus when such a grammar is applied in a musical context, the parameters
specifying simultaneously occurring events are
intrinsically related to one another as well as to
their temporal neighbors. All musical grammatical
events are computed in a logical rather than timesequential order.
More general context-free grammars, without the
linear, one-token-at-a-time restriction on produc- One-Dimensional Space Grammars
tion rules, can produce whole strings of symbols in
a single operation. For the types of musical applicaA simple stochastic grammar is defined as consisttion with which this article is concerned, a stochasing only of a single variable A and a single terminal
tic grammar may be used. A stochastic grammar a. A set R consisting of the following two producincludes a probability assignment over the orderedtion rules is defined.
set of production rules. It is thereby possible to set
A -- AA (1)
up a generative structure associating probabilities
A --+ a (2)
with each choice of generation possibilities. The
definition of a working stochastic grammar is rather
A probability array P = (p
difficult. Owing to the complex embedded struc- set R to determine which
tures that result, it is possible that a set of produc-be applied. The sum of p,
tions may never terminate unless provisions for
must be greater than p, t
termination are provided. As fast as one branch
a sequence of generations
terminates, another may split into a further set ofper se makes little struct
generations. It is necessary to apply a test for con- syntactic structure respo
sistency. This is done by setting up a first-momentstring may be preserved
matrix M similar to a stochastic matrix, with rows to divide up the space, in
and columns corresponding to the ordered set of sional straight line, such
symbols. Each entry mi, in the matrix is calculatedapplied, the space is split
by adding the probabilities associated with each halves to be subdivided by
production rule in which symbol V, is replaced by athe grammar, and whene
sequence including V,. The stochastic grammar issplitting process will cea
consistent if the modulus of the largest eigenvalue'tree of a sequence genera
of M is less than 1, in which case any set of generamar is shown in Fig. 4. T
tions can be guaranteed to terminate (Gonzales and
sically as a means of div
that will generate the rhy
scribed at the bottom of t
1. The eigenvalues of a matrix M are the set of solutions X
An increase
in variable
the value
the equation IM - XII = 0, where
X is a scalar
and
the identity matrix.
tions to split to a greater
52 Computer Music Journal
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Fig. 4. A spatial and rhythmic interpretation of the
derivation tree.
a
aaaa
a
aa
a
a
a
a
a
aaa
a
a
?I I W I I I JI I I III I I 1
t gJ In i iil I' J
rhythm with many notes of shorttion
duration.
An in- (1) will di
rule
crease in the value of p, will produce
the reverse rule (2) wil
duction
effect, resulting in more long notes.
It is necessary also
P
=
(
p,
,
Two-Dimensional Space Grammars
It
P2
is not convenient
when operating suc
The grammar can be extended into
two Fig.
dimensions
but
5 demonstr
if one introduces an additional production
rule. The
possible
application
symbol /, will be introduced, indicating
a split in
When
used to div
the nth dimension. Thus the three
production
rules
of
productions
will
are now:
Fig. 6. If the proba
crease p, and favor
A - A/ A (1)
such as those in Fig
musically with tim
A -a (3)
pitches on the vert
If applied over the
two-dimensional
favor
sequential act
A --- A / A (2)
Jones
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53
Fig. 5. Application of a
two-dimensional space
grammar.
Stage 1 A - A A Rule (1)
StageA A Rules (2) and (1)
Stage
3
A
Stagea
A
A
Rules
(2),
(3),
(3),
Stage 4 AA A
aaA
A
(etc.)
Stage 5a [
-a
a
aa
a
a
a
a
a
short
Suc
p,
wil
use
notes
gra
group
rec
the
v
ing 1.
zonta
The procedure "note" merely writes the approprithe
ate Music V data statement to "play" a notesh
with
values of start, finish, and intensity applying when
it is called. By repeatedly calling itself, this short
"compose" procedure will generate an entire comAddin
positional structure. A considerable variety of output can be achieved by changing the probabilities to
vary the horizontal/vertical ratio, the overall soundg
The
addin
density, and the depth of detail. In the version of
serve
the procedure in Code Listing 1, all the probabilially
ties are equal. Figure 9 is a graphic score represent- f
with
ing a sound structure generated by this procedure
with the initial call
bility
previo
compose (1,10,1500);
the
re
54
Computer
Music
Journal
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Fig. 6. Stages in a sequence
of derivations in applica-
tion of a two-dimensional
space grammar.
Stage 1
Stage 2
Stage 3
Stage 4
Stage 5
Jones
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55
Fig. 8. A transcription of
Fig. 7(b) into common mu-
Fig. 7. The plane divided
by a two-dimensional
space grammar with a
sical notation. (Music
printing by Leland Smith.)
slight bias toward vertical
divisions.
(a)
(b)
Fig. 8
AtIL
9*cz
56
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Code Listing 1. Procedure
Fig. 9. A pitch/time score
compose.
of sound data generated
by a two-dimensional
space grammar.
procedure compose (var start, finish, intensity: real);
type
branch = (vertical, horizontal, sound, silence);
var
production: branch;
begin {work}
{ choose_branch is a procedure, which selects one element of the branch type each time it is called}
production: = choose_branch;
case production of
vertical: begin {v}
mid: = start + (finish - start)/2;
compose ( start, mid, intensity);
compose (mid, finish, intensity)
end {v}
horizontal: begin {h}
compose (start, finish, intensity/2);
compose (start, finish, intensity/2)
end {h}
sound: note (start, finish, intensity); {writes a note}
silence:
end {case}
end {work}
Fig. 9
Hz
8000
4000f___ f7-
f f fff-
2000
if
1000
iff
500
fif
150
125
15
0
1
2
3
4
5
6
7
8
9
10
Sec
Jones
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57
Fig. 10. Dividing a block
using a three-dimensional
space grammar.
(a)
(b)
Three-Dime
duced varies. If ps is very small, a swiss-cheese-like
structure that contains a few holes will result. If p4
addition
is roughly equal to ps, a structure with a spongelike
total of five alternatives:
quality will result. If p, is large, then a few blocks
will be left suspended in a largely empty space.
A - A/A (1)
The structures may be mapped into a musical
form in the same way as was done for a two-dimenA -- A/3A (3)
sional grammar. The block in Fig. 11(a), for example, may be separated into a series of slices across
A -- (5)
its width along an arbitrary "timbral plane" to prothe proto-score of Fig. 12.
A typical string duce
generated
with thes
examples here have been kept simple for
rules could be theThe following:
expository reasons and for clarity, but there is no
reason
why generations should not be continued
(4/3 ((6/12 a))/3 ((al/6 )/1 (a12l ))
to
a
very
detailed level and the parameter scales
By extending the previous operations, one may
expanded to produce very large and complex
use this to divide up a three-dimensional block by
structures.
slicing it in half across the width when rule (1) is
applied, vertically when rule (2) is applied, and
lengthwise when rule (3) is applied. Thus the preMulti-Dimensional Space Grammars
ceding string is equivalent to the block in Fig. 10(a).
Chunks terminated by rule (4) are shaded; chunks
The basic concept of a space grammar can be e
generated by rule (5), the null production, are not.
tended to an arbitrary number of dimensions
These empty chunks have been removed in Fig.
10(b), leaving only the block that has been generproductions of the form A --* A /, A. This is
and even necessary for sophisticated compo
ated by the grammar. Two additional blocks generapplications. Many dimensions can be usefu
ated by this same grammar are given in Fig. 11.
dimensions to specify spatial location; dime
Each block in this figure is the inverse of the other;
for pitch, intensity, duration; and more dime
a's have been replaced by i4's, and vice versa.
for representing various timbral indices. Code
If one varies the relation between the p4 and p,
ing 2 gives a general form for a multidimens
probabilities, the solid density of the structure proThe
A -- A /A (2)
A --- a (4)
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Fig. 12. A partial musical
Fig. 11. A block (a) and its
transcription of the block
of Fig. 11(a).
inverse (b) both generated
by a three-dimensional
space grammar.
(a)
(b)
Fig. 12
Pitch
Instrument 1
Pitch
Instrument 4
Time
Pitch
Time
Pitch
Instrument 2
Instrument 5
Time
Pitch
Instrument 3
Time
Time
Pitch
Instrument 6
Time
Time
Jones
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59
Code Listing 2. Procedure
composen
procedure compose_n (var L_1, U_1, L_2, U_2, ... L_n, U_n: real);
type branch = (dl, d2, ... dn, sound, silence);
var production: branch;
begin {compose_n}
{choose_branch a dimension (d variables) to split or a note or silent event}
production: = choose_branch;
case production of
dl: begin { 1 }
mid: = (L_1 + (L_1 + U_1)/2;
compose (L_1, mid, L_2, U_2, ... L_n, U_n);
compose (mid, U_1, L_2, U_2, ... L_n, U_n);
end { 1 }
d2: begin { 2 }
mid: = L_2 + (L_n + U_n)/2;
compose (L_1, U_1, L_2, mid, ... L_n, U_n);
compose (L_1, U_1, mid, U_2, ... L_n, U_n);
end
dn: begin { n }
mid: = L_n + (L_n + U_n)/2;
composer (L_1, U_1, ... L_n, mid);
compose (L_1, U_1, ... mid, U_n);
end
sound: note (L_1, U_1, L_2, U_2, ... L_n, U_n);
silence:
end {case}
end { compose_n}
space-grammar procedure. In this algorithm, the
variable L-i is the lower value limit for the dimen-
merely scratched the surface of what is an extensive and largely unmapped area of compositional
exploration.
sion i, while Ui is the upper limit for i.
An additional procedural parameter may be in"So vast is the scope that lies open to things far
corporated to monitor the depth of generations.
The "choose_branch" procedure selects the appropri- and wide without limit in any dimension."
ate production, making use of a probability array.
Lucretius, Book One
By changing simple control probabilities, large
amounts of material with a great variety of character may be generated. Macro and micro structures Conclusion
may be generated by one procedure that can operate
to as great a depth of detail as is desirable or practi-This article has described a number of different
cal. Further extensions of this skeleton space gram- types and applications of stochastic processes.
mar would be possible if one defined a larger set of Many of these have shown themselves to have pracvariables and alternative terminal functions, and
tical potential, and a great deal of research remains
made structural divisions at other than equal ratios, to be done in their exploration and development.
Stochastic approaches can be applied in music analfor example, at the Golden Section. This article has
60
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ysis, sound synthesis, and composition. There is
also much useful research to be done in the psychoacoustic perception of stochastic music structures.
the Conference on Computer Music in Britain.
London: Electro-acoustic Music Association of Great
Britain.
Jones, K. J. 1980b. "Computer Assisted Application of
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61