a declarative model of atonal analysis · characteristics of atonal analysis in "atonal"...

Music Perception Fall 1988, Vol. 6, No. 1, 21-34

© 1988 BY THE REGENTS OF THE UNIVERSITY OF CALIFORNIA

A Declarative Model of Atonal Analysis

TOHN ROEDER University of British Columbia

Most computational models of musical understanding have focused on procedural aspects of analysis, suggesting techniques for parsing, com- paring, and transforming various representations of a piece, or adapting discovery procedures of artificially intelligent (AI) inference systems, which plan and follow agendas and goals. Much contemporary AI re- search, however, also focuses on declarative aspects of knowledge, at- tempting to define data representations and relations that are commen- surate with human cognition. Naturally, musical analysis has both procedural and declarative aspects: the declarative determines what the form of the analysis is, and the procedural determines how the analysis is obtained. However, a predominantly procedural analysis risks sacri- ficing the form of musical understanding to obtain efficiency or compat- ibility with a particular computer language. In this article I argue that, for a significant body of twentieth-century music, a declarative system models the structure of analytical understanding better than do existing procedural programs, and I present a functioning declarative system that infers complex musical structures from the elementary musical relations that it identifies.

Characteristics of Atonal Analysis

In "atonal" music, in the broadest sense of the term, pitch is structured without reference to a controlling key. The works to which the term is com- monly applied, composed by Schoenberg, Webern, and other composers in the first two decades of this century, are more specifically characterized by extreme and rapid contrasts of timbre, register, texture, pitch, pitch class, and durations, and negatively by the lack of sustained melody, regular pulse, and consonance. These features, and the lack of themes, keys, and the phrase structures and forms associated with more traditional means of pitch organization, lead analysts to hear the music in many different ways. However, most of them agree, expressly or tacitly, upon certain fundamental issues, including (1) the nature of musical analysis, (2) the nature of musical structures, (3) the nature of the events that make up those structures, and (4) the nature of musical meaning. For the present discussion these points of agreement warrant a brief summary.

Requests for reprints may be sent to John Roeder, School of Music, University of British Columbia, 6361 Memorial Road, Vancouver, British Columbia V6T 1W5, Canada.

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22 John Roeder

(1) Music is an object for contemplation, not simply immediate experi- ence. The analytical understanding that Milton Babbitt (1972) calls "re- flective, contemplated, and total" must be based on a detailed knowledge of every event in the piece, gained from "listening to the music very carefully and noting various structural perceptions" (Hasty, 1981). Such a synoptic understanding is manifestly not acquired in the course of a single listening, but rather by the coordination of multiple hearings and mental rehearsals, in which the events of the piece, even temporally distant ones, are learned in detail and associated in various ways.

(2) The structural components of a musical work are collections, or "segments" of associated events (Forte, 1973). An analysis expresses a "segmentation" which attributes to every musical event membership in at least one significant collection of events. These segments possess "a unitary value in some domain," (Hasty, 1981) that is, their identity and coherence arise from the perceived properties of the basic musical events of the piece. So a segment may be a collection of events whose temporal adjacency defines a melodic line, or whose simultaneity defines a chord; a segment may also be articulated by other musical properties, depending upon the style of the music under consideration.

(3) The only properties of musical events that are significant to musical structure are those that define segments; analysts do not ordinarily consider extramusical connotations of the music in determining its segmentai structure. The properties cited in most analyses are those that are perceived in the intensive listening described above: rhythm, essentially the attack time and duration of the events; timbre, such as the instrumental type and ar- ticulation; pitch; and loudness. Furthermore, the relations among segments derive from these same perceived properties of their events. For instance Babbitt (1972) claims that the operations of inversion, transposition, and retrogression are "familiar and rudimentary notions which depend upon only the most uncontrovertible [sic] and essential facts of musical perception: the capacity to recognize pitch identity and nonidentity, and interval- lie value under transposition in a semitonal system." Similarly, Hasty (1981) identifies some segments by the collective properties- such as pat- terns of intervals, attacks types, or contour- which they have in common with other segments. The structural importance of each musical domain varies from composer to composer and between or even within pieces, but their comparative brevity and untraditional construction compels the analyst to discover many relations among the events.

(4) The meaning of a musical event depends, according to Boretz (1970a, compare Roads, 1984), upon its "multiplicity, or multivalence, of reference. That is, a simultaneity of multiple implications of the same entity, each one of which is cognitive and specifiable, and no two of which are con-

A Declarative Model of Atonal Analysis 23

tradictory . . . [There are] several perfectly clear but distinct 'meanings' attachable to single events." The segments to which an event belongs thus provide a meaningful context for that event. Similarly, in Hasty's (1981) model of atonal musical processes, the meaning of each event is dynami- cally modified and augmented due to the continually changing relations that successive events manifest with it and its predecessors.

These four points of agreement about atonal analysis underlie the pitch- class (pc) set analyses of Allen Forte (Forte, 1972, 1973, 1974, 1985; see also Beach, 1979; Rahn, 1980). Their synoptic comprehensiveness neces- sitates their presentation on modified scores that represent temporal, pitch, and, to some extent, timbrai properties of every musical event. Circles and brackets on these scores indicate the structural components of the work, which belong to segments of two basic types. A primary segment is a "con- figuration that is isolated as a unit by conventional means, such as a rhyth- mically distinct melodic figure" (Forte, 1973); other examples of primary segments include a rest-delimited melodic fragment and a chord. A more complex kind of segment, which Forte calls "composite," is "a segment formed by segments or subsegments that are contiguous or that are other- wise linked in some way." Forte's analytical statements assert pitch-class and interval-class content relations among segments, demonstrating in ef- fect a network of abstract relations among pitch-class motives. For example, the same label given to two different segments indicates that their pc content is related by transposition or inversion, and that they have the same interval-class content. Thus the meaning of an individual event derives from its membership in various segments in a complex network of related segments.

Procedural Analyses of Atonal Music

Despite the inherently relational nature of pc-set analyses, Forte con- ceives of segmentation as a process. In fact he attempted to automate segmentation by means of a computer program that parses a score into segments and classifies them (Forte, 1966). As its representation of music, Forte's program uses DARMS, which encodes a score, part by part, as a continuous string of alphanumeric characters. The programming language, SNOBOL, in which the system is realized is especially well suited for such standard operations upon strings as concatenation, search, and compari- son. The program employs these operations to analyze the string representation of the music without regard to how they correspond to human cognitive processes. For instance, the parser identifies just one type of primary segment, consisting of rest-delimited instrumental parts. Secondary seg-

24 John Roeder

ments are produced by combining these primary segments into pairs according to the relative temporal positions of their first and last events; this arbitrarily binary combination procedure also seems motivated by the operations available in SNOBOL, rather than by the more subtle analytical procedures Forte describes in his other writings. The resulting segments are

similarly combined in pairs, and redundant results removed. Although this

procedure is well defined, the arbitrarily restrictive definition and representation of primary segments, and the arbitrarily binary combination procedure make it "overselective" (Alphonce, 1980).

Another formalized procedure more closely modeled upon human an-

alytical behavior was proposed by Laske (1984) to produce "a systematized set of examples for newly synthesized concepts."1 Like Lenat and Harris's (1978) scientific discovery system, Laske's system represents a small set of

given concepts as "frame" structures, then plans and executes an agenda to find significant relations among segments. As evidence for his procedural model, the author cites a transcription of a student's analysis of Debussy's Syrinx, which does seem to involve finding examples for musical concepts. Ironically, however, that analysis also points out a crucial omission from Laske's description: the specific and logical representation of musical relations (Smoliar, 1986). Without it the analytical system cannot tell which

concepts are new. For instance, the student analyst describes a "redun-

dancy" of motives in the piece. Laske claims that this concept is newly cre- ated; logically, however, it would seem to be prior to and implicit in the definition of one of the given concepts, the "basic cell." Whatever validity Laske's system may possess as a model of musical discovery, it, like Forte's

program, would benefit from a consistent and logical representation of musical relations obtained through the "painstaking exploration of the cognitive processes specific to the structuring of music" (Alphonce, 1980).

Such representations have been proposed by Boretz (1969, 1970b), who constructed an analytical language from formal definitions of perceivable event relations, and by Rahn (1979), who proposed a collection of definitions to describe a hierarchical analysis of tonal music. Both of these formal

systems are declarative rather than procedural, concerned with the logical definition of musical events and relations rather than the process by which

they are perceived. However, recently developed programming languages have made it possible to construct a declarative system which not only represents those musical properties and relations specific to atonal analysis, in

logical predicates analogous to the declarative statements of Boretz and Rahn, but which can also function to produce a pc-set segmentation.

1. Similarly, Hasty (1981) states that in the second step of analysis "rules are devised to form a theory which might account for these [structural] perceptions."


The Structure of a Declarative Analytical System

Following the form of atonal analysis outlined above, the system consists of a collection of predicates that describe the formal structure of a segmentation. The most primitive statements are the "facts" of a piece, which describe the properties of every event in sufficient detail to support analytical statements. Representing a more complex level of musical understanding are predicates that express how events may associate in various kinds of segments. Still higher-level predicates specify how segments are related in a segmentation. The system attributes meaning to events by identifying their membership in segments that have significant set-theoretical relations. Thus the analytical understanding that could be represented procedurally by the results of a segmentation process is represented instead declaratively by the instantiation of musical relations among events and segments.2

To attribute declarative meaning to a musical event in a simple pc-set analysis, the system needs information about just four properties of the event: its pitch, the instrument that plays it, its attack time, and its duration. Each event is a set of specific values for each of those four attributes.

event(Pitch, Instrument, Attack, Duration).

Any collection of such events forms a context in which the events may have meaning, and all event relations and structures are associated with musical contexts. The system only recognizes structures and relations of events if the events belong to the musical context under consideration. A context may be a segment, a section, an entire piece, or even a collection of pieces, and each segment possesses its own local structures and relations.

Since an event only has meaning in a context of which it is part, the system must recognize the membership of an event in a context. This is accom-

plished by the following declarations:

element(E,[E|T],T). An event E is an element of a context that begins with E and ends with the context T.

element(E,[Y|T],[Y|Tl]):- An event E is an element of a context that begins element(E,T,Tl). with an event Y and ends with the context T if E

is an element of the context T.

These and subsequent relational declarations are expressed in the Edin-

burgh syntax of the programming language Prolog (Clocksin & Mellish, 1984), and correspond to Horn clauses in first-order predicate logic. The

2. Along with these formal specifications, the design of the system was also constrained to avoid metalogical constructs (such as cuts and asserts in Prolog), in order to be as declarative as possible within the limitations of a procedural machine architecture.

26 John Roeder

Fig. 1. Segmentation of Webern's Op. 11, No. 3 (transcribed from program output). Copy- right 1924 by Universal Edition. Copyright renewed 1952 by Anton Webern's Erben. Re- printed by permission of European-American Music Distributors Corporation, sole U.S. agent for Universal Edition.

symbol:- stands for "if," the comma stands for "and," and the semicolon stands for "or." The system similarly represents the subset relation of a collection of events to a context that contains them, along with the remainder of the events in the context. (The definitions of subset and other clauses omitted in the main text can be found in the Appendix.)

A piece is the relation of a composer, a title, and a collection of events. Figure 1 shows the score of Webern's piece for cello and piano, Opus 11, No. 3. It can be represented declaratively as follows:


piece (webern,op 1 I_no3, Webern's Opus 11, No. 3 is a collection of events [event(27,cello,0,12), including an Et 3 and a Ft 3 played by the cello event(28,cello,0,12), at the start of the piece for the duration of 12 event(48,cello,12,8), triplet sixteenths, a cello C4 12 triplet sixteenths event(38,piano,15,15), ...]). after the first event of the piece, a D3 played by

the piano 15 triplet sixteenths after the first event of the piece for the duration of 15 triplet sixteenths, etc.

This representation will allow the system to use information about one work to direct its analysis of another.

Every event relation has the same form, partitioning the context into related events and unrelated events.

sameJnstrument(event(Pl,I,Al,Dl), Two events are related if the same instru- event(P2,I,A2,D2),C, R) :- ment plays them both in the context C. The

subset([event(Pl,I,Al,Dl), remainder of the events in the context form event(P2,I,A2,D2)],C,R). R.

same_attack (event(Pl,Il,A,Dl), Two events are related if they have the event (P2,I2,A,D2), C, R):- same attack time in the context C. The re-

subset([event(Pl,Il,A,Dl), mainder of the events in the context form event(P2,I2,A,D2)], C, R). R.

Similarly, events may be temporally_adjacent, or sound_together, if they belong to the same context and have the appropriate temporal relations. These low-level predicates express the basic relations a listener may per- ceive among musical events. Accordingly, the most fundamental analytical statement the system can make about a context is the association of all pairs of events in all possible relations, such that several different relations obtain for every event.3

A primary segment is defined as a collection of events that are related in the same way:

primary([H,X|T], Context,Rem,Relation):- A collection of events containing Goal = ..[Relation,H,X,Context,R], events H and X is a primary segment call(Goal), under the stipulated Relation in a primary([X|T],[X|R],Rem,Relation). given Context if H is so related to X

in that Context, and if all the events in the collection except H are a primary segment under the same Rela- tion in the same Context.

primary ([H],Context,Remainder,_):- A single event H belonging to a con- element(H,Context,Remainder). text is a primary segment.

3. Competing interpretations of unorganized data also characterize the local organizing processes in Arbib's (1979) model of visual cognition.

28 John Roeder

That is, the cognitively based relations are the basis for the cognitively most important types of segments.

Most of Forte's "conventional" primary segments are covered by this definition. An "instrumental part," for example, is a collection of events associated by the relation sameJnstrument. A "chord" is a collection of events with the same attack. In a "melodic line" the events are temporally adjacent, that is, every event in the line is immediately preceded or followed

by another event in the line. Thus the formal structures of seemingly different types of segments are in fact identical: a segment of every type is a collection of events associated by one of the basic, and formally identical, event relations.4

Some of Forte's more complex segments can be expressed as primary segments of one type contained within the context of primary segments of another type. Consider, for example, a declarative definition of the rest- delimited melodic lines in the instrumental parts of a piece:

primary(IP,Context, _,same_instrument), A rest-delimited instrumental part is primary (RDIP,IP, _,temporally_adjacent). a primary segment RDIP of tempo-

rally adjacent events in the context of a primary segment, IP, of events played by the same instrument.

Although a comprehensive set of definitions of all types of primary segments is beyond the scope of this brief description of the system, the system similarly represents them all as collections of cognitively associated events in various contexts. The segmentation of contexts according to various defined musical relations constitutes the basic analytical capability of the system.

A somewhat more sophisticated analysis exposing the multiple function- alities of events can also be achieved using only the declarations cited above. Events belonging to more than one primary segment in the same context are describing by the conjunction of clauses, for example:

primary([E _],Context,same_attack), An event E is part of a chordal pri- primary([E _],Context,temporally_adjacent). mary segment and also part of a

melodic primary segment.

This collection of Prolog clauses thus constitutes a functional segmenter that can identify and relate many sorts of primary segments. A query by the user is expressed in the form of a goal, which the system satisfies by ap- plying the cognitively based relations it knows to the facts of the piece. Con-

4. Other types of primary segments recognized in atonal analysis may also be expressed declaratively . For instance, in one kind of primary segment all events are related in the same way to one event in the segment, but not necessarily to each other.


sider this declaration of an exhaustive partition of a context into primary segments of a single type:

primary _segmentation([S|R],Context,Relation):- A list of event collections is a pri- primary(S,Context,Rem,Relation), mary segmentation of a Context primary_segmentation(R,Rem,Relation) . for a specified Relation if the first

collection S is a primary segment for the specified Context and Re- lation, and if the rest of the list R is a primary segmentation of the rest of the Context under the same Relation.

primary_segmentation ([],[],_). An empty context has an empty primary segmentation.

This higher level predicate can be used to express analytical goals that may be satisfied in a variety of ways consistent with the cognitively based segmentation rules. For example, any chord in Webern's Opus 11, No. 3 is expressed declaratively as a primary segment by the conjunction of two clauses:

piece(webern,opll_no3,Context), SJist is a list of event collec- primary_segmentation(S_list,Context,same_attack). tions such that the events in

each collection belong to the context of Webern's Op. 11, No. 3 and are attacked at the same time.

The event collections satisfying this relation are listed above Figure 1. True to the declarative representation, no procedure forms or compares structures, the system simply recognizes the presence of primary segments based of the network of cognitively based relations in the data, and it will do so identically for all known relations. In fact, by rewriting this conjunction using other declared relations, such as sameJnstrument, temporally_adjacent, same-duration, and same_pc, we can represent interesting aspects of the segmentai structure of this piece, as shown below Figure 1. The first two lines below the score show the segments based upon the relations sameJnstrument and temporally_adjacent; these correspond to what we conceive to be the individual instrumental parts and the rest-delimited melodic lines, respectively. The last two lines show that interesting segments can also consist of nonadjacent events. For instance, nearly every event has the same duration as another event in the piece (Berry, 1976, pp. 397-400); the relation partitions the piece into several segments containing one, two, or three events. Also, nearly every event has the same pitch class as another event: the bottom line under the score, which indicates the segments

30 John Roeder

containing events with the same pitch class, reveals that the second half of the piece recapitulates the pcs of the first half (Wintle, 1975).

The general definition of a segmentation for an arbitrary list of relations is:

segmentation([X|Y],Context,[H|T]):- A list of primary segmentations is a primary _segmentation(X,Context,H), segmentation of a Context for a spec- segmentation(Y,Context,T). ified list of relations if the first pri-

segmentation([], _,[]). mary segmentation on the list, X, is a primary segmentation of the specified Context under the relation H, and if the rest of the list Y is a segmentation of the Context under the other relations.

The crucial analytical statements in a pc-set analysis assert that the pitch- class contents of two or more segments are identical under transposition or inversion, so that the segments belong to the same Tn/TnI-equivalence class (Rahn, 1980; Forte, 1973). The analyst normally determines the class, or type, of a pc collection by using a procedure to reduce the collection to a standard form that can be found in a table of set types. However, this set- classification procedure can be very simply declared as the relation of the collection to the standard form of the abstract set-type in a particular context:

set_type(Set,Type,Context):- A Set belongs to a certain Type in a intervaLnormal_form(Type,Int_Series), Context, if an Interval-Series associ- subset(Ordering,Set,[]), ated with that Type spans some Or- pcJntervaLseries(Ordering, dering of the Set (Regener, 1974).

Int_Series,Context) .

The clause pc JntervaLseries expresses the relation of an ordered collection of pitched events to the ordered series of pitch-class intervals that span them in a particular context. Consistent with this relation, the standard table of set classes is declared as a collection of relations among interval series and set-class labels:

interval_normal_form('3-l',[l,l]). The interval normal form of a set be- intervaLnormal Jorm('3-2',[1,2])... longing to class 3-1 is the series of pc in-

tervals [1,1], etc.

With these added relations the system can express the relation of segments to set-class labels, so that the following conjunction of clauses:

A Declarative Model of Atonal Analysis 3 1

piece(webern,opll_no3,Context), P is a primary segment belonging to primary(P,Context, _,temporally .adjacent), set type Type, and consisting of tem- set_type(P,Type,Context). porally adjacent events that belong

to the context of Webern's Op. 11, No. 3 and are attacked at the same time.

is satisfied by every rest-delimited melodic line P that belongs to the set-class Type in the Context of the Webern piece.

Note that the declarative definition of set-type is not restricted to primary segments; so the declaration

piece(webern,opl l_no3,Context), set_type(P, Type, Context).

is satisfied by any collection P of events having an interval normal form recognized by the system, whether or not P is a primary segment. Analysts of- ten consider such a complex segment as significant if it belongs to the same type as a primary segment. Forte's composite segments are a case in point. The collection of events in a composite segment are not uniformly related as they would be in a primary segment, but Forte stipulates that each event is "contiguous" with another event in the collection. Although Forte does not formally define continguity, his analyses suggest the following rule:

contiguous(X,Y,Context,Remainder):- Events X and Y are contiguous in sound_together(X,Y,Context,Rem); a Context if they sound together, temporally _adjacent(X, Y,Context,Rem) ; or if they are temporally adjacent, not between(X, _,Y,Context,Rem). or if there is no event between

them temporally in the context.

The relation of two contiguous primary segments in a composite segment can then be declared as:

composite(C,Context,Relations):- A composite segment C in a Con- segmentation(S,Context,Relations), text is the union of two primary element(Ll,S,_), element(L2,5,_), segments in the context such that element(Pl,Ll,_),element(P2,L2,_), an element of one primary seg- notPl=P2, ment is contiguous with an ele- element(El,Pl,_),element(E2,P2,_), ment of the other primary seg- contiguous(El,E2,Context,_), ment. union(Pl,P2,C).

But a more general type of contiguous segment can also be declared:

contiguous_segment(Seg,Context) : - A collection of events Seg is a con- all_contiguous(Seg,Seg,Context). tiguous segment in a Contextif ev-

all_contiguous([H|T],S,Context):- ery event in it is contiguous to an- element(X,S,_),not X = H, other event in it. contiguous(X,H,Context,_), all_contiguous(T,S,Context) .

alLcon tiguous ( [] ,_,_) .

32 John Roeder

Accordingly, the system can express the relation of any primary segment to all more complexly contiguous segments of the same type:

primary _segmentation(Segmentation,Context,Relation), element(Primary_Segment,Segmentation,_), In a given Context, Segment is any set_type(Primary_Segment,Type,Context), contiguous segment that belongs set_type(Segment,Type,Context), to the same set Type as a contiguous_segment(Segment,Context). Primary .Segment.

This conjunction of relations can be used in the Prolog system to list all composite segments that belong to the same set type as the primary segments. As an illustration, Figure 2 shows the composite segments the program finds that belong to the same set type (Forte number 3-3) as the first piano verticality, in which the events are related by same_attack (Williams, 1983). Each of these segments is perceptually coherent, in the sense that ev-

ery event in the segment is contiguous, in one of the three ways we have defined, to another event in the same segment. Interestingly, this composite segmentation provides a meaningful context for every event of the piece.

In all, this system exhibits some basic structural characteristics of atonal pc-set analysis. It represents abstract pc-set-analytical understanding in stages: more complex structures and relations are logical conjunctions of simpler ones, and an entire network of segmentai relations are demonstra- bly founded upon a few cognitively based event relations. Analytical knowledge is distributed throughout the system in the form of clauses that

Fig. 2. Segments consisting of contiguous events and belonging to the same set type (transcribed from program output). {x,y,z} identifies a pitch-class collection of type 3-3 (intervalnormal form <l,3>or<3,l>). [c] indicates which type of contiguity obtains between the corresponding events (see the Appendix: t means the events are temporally adjacent, s means the events sound together, and n means that there is no intervening event between the two events.


specify the structure of each musical event as a collection of audible properties. The clauses attribute the musical meaning of an event to the multiplicity of relations it bears to other events and to the multiplicity of structures to which it belongs. They relate musical events according to their audible properties, define the form of segments according to those musical relations, classify the segments by type, and relate different segments belonging to the same type. Since the system structure corresponds to theo- rists' conceptions of the structure of atonal analytical knowledge, and since the system supports the same kind of pc-set analytical statements that hu- mans make, it would appear to be a good model of human analytical understanding of atonal music, and it suggests that a declarative system might be a useful model of more general music-analytical knowledge as well.

Appendix

subset([],X,X). subset([H|T],X,R):-

element(H,X,D), subset(T,D,R).

sound_together(event(Pl, II, Al, Dl), event(P2, 12, A2, D2), Context, Remainder) :-

subset( [event(Pl,Il,Al,Dl), event(P2, 12,A2,D2)], Context, Remainder),

AKA2 + D2,A2<A1+D1. successive(event(Pl,Il,Al,Dl),event(P2,I2,A2,D2),Context, Remainder):-

subset([event(Pl,Il,Al,Dl),event(P2,I2,A2,D2)], Context, Remainder),

A2is Al + Dl. temporally_adjacent (X,Y,Context,Remainder) :-

successive (X, Y,Context,Remainder) ; successive ( Y,X,Context,Remainder) .

order(event(Pl,Il,Al,Dl),event(P2,I2,A2,D2),Context,Remainder):- subset([event(Pl,Il,Al,Dl),event(P2,I2,A2,D2)],Context,Remainder), AKA2.

order (First,Middle,Last,Context,Remainder) : -

order(First,Last,Context,Remainder), order (First,Middle,Context,_), order (Middle,Last,Context,_) .

between(First,Middle,Last,Context,Remainder):- order(First,Middle,Last,Context,Remainder); order(Last,Middle,First,Context,Remainder).

34 John Roeder

References

Alphonce, B.H. Music analysis by computer- A field for theory formation. Computer Music Journal, 1980, 4(2), 26-35.

Arbib, M. A. Local organizing processes and motion schémas in visual perception. Machine Intelligence, 1979, 9, 287-298.

Babbitt, M. Remarks on the recent Stravinsky. In Boretz & E. T. Cone (Eds.), Perspectives on Schoenberg and Stravinsky. New York: Norton, 1972, pp. 165-185.

Beach, D. Pitch structure and the analytic process in atonal music: An interpretation of the theory of sets. Music Theory Spectrum, 1979, 1, 7-22.

Berry, W. Structural functions in music. Englewood Cliffs, NT: Prentice-Hall, 1976. Boretz, B. Meta- Variations (I). Perspectives of New Music, 1969, 8(8), 1-74. Boretz, B. Conversation with Elliott Carter. Perspectives of New Music, 1970a, 8(2), 1-22. Boretz, B. Sketch of a musical system (Meta- Variations, Part II). Perspectives of New Music,

1970b, 8(2), 49-111. Clocksin, W. F., & Mellish, C. S. Programming in Prolog (2d éd.). New York:Springer-

Verlag, 1984. Forte, A. A program for the analytic reading of scores. Journal of Music Theory, 1966,

10(2), 330-364. Forte, A. Set and nonsets in Schoenberg's atonal music. Perspectives of New Music, 1972,

11(1), 43-64. Forte, A. The structure of atonal music. New Haven and London: Yale University Press,

1973. Forte, A. Analysis Symposium :Webern, Orchestral Pieces (1913), Movement I (Bewegt).

Journal of Music Theory, 197 r4, 18, 13-43. Forte, A. Pitch-class set analysis today. Music Analysis, 1985, 4(1 & 2), 29-58. Hasty, C. Segmentation and process in post-tonal music. Music Theory Spectrum, 1981, 3,

54-73. Laske, O. Keith: A rule-system for making music-analytical discoveries. In M. Baroni & L.

Callegari (Eds.), Musical grammars and computer analysis, Florence: Leo S. Olschki, 1984, 165-199.

Lenat, D. B., & Harris, G. Designing a rule system that searches for scientific discoveries. In D. A. Waterman & F. Hayes-Roth, (Eds.), Pattern-Directed Inference Systems. New York: Academic Press, 1978, 25-52.

Rahn, J. Logic, set theory, music theory. College Music Symposium, 1979, 19(1), 114-127. Rahn, J. Basic atonal theory. New York: Longman, 1980. Regener, E. On Allen Forte's theory of chords. Perspectives of New Music, 197 r4, 13(1),

191-212. Roads, C. An overview of music representations. In M. Baroni & L. Callegar (Eds.), Musical

grammars and computer analysis. Florence: Leo S. Olschki, 1984, 7-37. Smoliar, S. W. Review of Musical grammars and computer analysis. Journal of Music The-

ory, 1986, 30(1), 130-141. Williams, E. W. Jr. On Mod 12 complementary interval sets. In Theory Only, 1983, 7(2),

34-43. Wintle, C. An early version of derivation: Webern's Op. ll/III. Perspectives of New Music,

1975, 13(12), 165-177.

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