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 Stable URL: https://www.jstor.org/stable/3679879 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at https://about.jstor.org/terms The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to Computer Music Journal This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 Journal This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 Computer Music Journal This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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) 58 Computer Music Journal This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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 Computer Music Journal This content downloaded from 79.117.197.149 on Sun, 18 Jan 2026 14:40:29 UTC All use subject to https://about.jstor.org/terms 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. 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