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A HYBRID COMPOSITIONAL SYSTEM: PITCH-CLASS COMPOSITION WITH TONAL SYNTAX.

I. THEORIES OF COMPOSITION AND THE HYBRID SYSTEM [1]

THEORIES OF COMPOSITION (i.e., theories for the generation of musical structures) form a spectrum whose range illustrates to what degree a theory determines [2] the organization of a composition's musical surface. At one end of the range, lie aleatoric theories, such as those developed by John Cage and employed in compositions such as Music for Piano (1952--56). [3] Since the materials and/or the procedures for organizing the materials are randomly produced with indeterminate composing theories, the specific nature of the musical surface cannot be predicted from the theory. [4] At the other end of the range, lie deterministic or algorithmic composing theories, such as those employed by, Lejaren Hiller, Larry Austin, and others. Since the materials and/or the procedures for organizing the materials are algorithmic, "once certain variables specified by the theory have been defined, a piece of music will emerge." [5] Tonal composing theories, such as the rules of counterpoint or the syntax associated with harmon ic progressions, lie somewhere in between the middle and the deterministic end of the range, because these theories determine many details of the compositional surface but they do not produce pieces. [6] Twelve-tone theories of composition are scattered throughout this spectrum of choice. A simple or minimal twelve-tone theory that only specifies an ordering for the twelve pitch classes along with a set of operations, such as transposition, inversion, and their retrogrades, does not specify how one row relates to any other. Since this theory determines very little about the organization of these musical structures (sequentially, polyphonically, or by any other musical parameter), very few details of a composition's musical surface (i.e., its score) can be predicted from the theory. Therefore, its position on the spectrum would be closer to aleatoric theories. A twelve-tone system incorporating a theory of hexachordal combinatoriality would move it closer to the middle of the spectrum, since the constraints im posed by hexachordal combinatoriality would limit the sequential and polyphonic organization of the twelve-tone rows on a composition's musical surface.

Theories of composition could also form a spectrum whose range illustrates to what degree a theory determines the organization of structural levels relating the musical surface to some background structure. At one end of the range lie completely bottom up approaches to composition, such as motivic composition. [7] Since motivic theories only generate surface structures, they say nothing about background structures or the relation of the surface to a background. At the other end of the range would be theories, such as a rational reconstruction of Schenker's analytical methodology, that would specify at every level the transformational relationships connecting the musical surface to a background structure. [8] Once again, a simple or minimal twelve-tone theory simply producing an ordering of the twelve pitch classes along with a set of basic operations determines very little about the organization of structural levels relating the musical surface to a background structure, presumably a single row form. Consequ ently, its position on the spectrum would be closer to the bottom up approaches to composition. A twelve-tone theory incorporating a theory of array structures, such as Morris, Composition with Pitch Classes, would move it closer to the middle of the spectrum, since it would specify more precisely the organization of structural levels relating the musical surface (the faster unfolding columnar aggregates of the array) to the background structure (the slower unfolding twelve-tone rows forming the linear aggregates of the array).

The hybrid compositional system, which will be outlined in this paper, lies somewhere in between the middle and the algorithmic end of the range of theories that specify the organization of a composition's musical surface, because the theory determines many details but it does not specify every detail of a composition's structure. The hybrid compositional system also lies somewhere in between the middle and the end of the range where theories specify at every level the transformational relationships connecting the musical surface to a background structure, since the system contains a set of transformational relationships connecting most of the musical surface to a background structure. The system is a composing language that combines structural elements from nontonal theories, such as twelve-tone arrays and self-deriving rows, and aspects of tonal theories, such as structural levels and prolongation. The system facilitates constructing complex and coherent musical structures, some of which function in a mann er similar or analogous to the musical structures in tonal systems.

Although the concept of incorporating structures usually associated with tonal music into nontonal systems appears contradictory, the contradiction is nominal, rather than structural. [9] If the diatonic set generates the tonal system's structural elements and relations, and if it is one member of a system of sets classified by their [T.sub.n]/[T.sub.n]I types, and if some sets in the system do not generate tonal structures, then in one sense, the labels tonal and nontonal simply denote sets possessing different structural properties. However, the labels do not imply or lead to the conclusion that differently labeled collections cannot generate similar or equivalent structures, and the labels do not imply or lead to the conclusion that every member of the system does not or cannot generate similar or equivalent structures. Discovering a syntactic connection between the tonal interpretation of the [T.sub.n]/[T.sub.n]I set [0,1,3,5,6,8,10] and other members of the [T.sub.n]/[T.sub.n]I system, and constructing a system based on those commonalties, would be a method of incorporating tonal structures into systems based on set classes other than the diatonic. David Lewin suggests this method of incorporating structures usually associated with tonal music into nontonal systems in his article "A Formal Theory of Generalized Tonal Functions." [10] Lewin transforms the arrangement of tonic, dominant, and subdominant triads linked by common tones (see Example 1) into an expression for constructing systems of tonal functions given a tonic pitch-class T, a dominant interval d, and a mediant interval m.

Each system consists of tonic, dominant, subdominant, mediant, and submediant triads. Lewin's generalization is not limited to any particular dominant or mediant interval, such as perfect fifth and major and minor thirds. Example 2 illustrates the triadic system determined by the ordered triple (C, 3,1) (d = 3 and m = 1). In the (C, 3,1) system, all the triads are [T.sub.n]/[T.sub.n]I [0,1,3] type trichords. Thus, Lewin shows a way to generalize tonal functions to sonorities not usually considered tonal.

Even if the tonal interpretation of the [T.sub.n]/[T.sub.n]I set [0,1,3,5,6,8,10] shared no syntactic features with sonorities not usually considered tonal, the diatonic set may not determine every syntactic property of the tonal system. [11] Since some structures, such as those associated with Schenker's conception of tonality, may be token representations of more general syntactic properties, another method of incorporating tonal structures into systems using [T.sub.n]/[T.sub.n]I sets might be creating a functionally analogous system based on generalizations of Schenkerian tonal syntactic features and interpreting those features within the context of a particular [T.sub.n]/[T.sub.n]I set type. For, example, concepts such as neighbor-note, passing note, prolongation, structural level, and Ursatz, may not be limited to the triads generated by the diatonic set. On the other hand, musical structures traditionally thought of as nontonal, such as tone rows and arrays, may be adapted to simulate tonal functions, or their structure could be the basis of determining tonally analogous functions.

The hybrid system integrates generalized tonal concepts with tone rows and arrays. The background structure of the system is a special type of array structure that contains a limited number of trichordal set classes (from 2 to 4 different classes) in the array columns. The array determines the background or long range voice-leading motions in a composition. The hexachords of the row underlying the array are independent "harmonic" areas, so during the course of a composition the large-scale "harmonic" motion moves from one area to its complement and then back to the original area. The transformations linking the background structure to the musical surface come from a modified and expanded version of John Rahn's definitional reconstruction of Schenker's analytical theory. The definitional portion of the theory generalizes tonal concepts, such as neighbor-note, passing note, prolongation, and structural level, so they can be incorporated into a system that has tone rows and arrays as its foundation.

Before proceeding from this brief overview of the hybrid system to a detailed investigation of its components, I would like to address several metatheoretical issues this theoretical approach to composition raises. Whether formalized or not, music theories are systems used to model (i.e., interpret the structure of) music. While formal logistic systems are concerned with preserving truth, music theories are mainly concerned with producing structural descriptions. Theories not only determine a structure's sense, but they also suggest how the elements of music, such as pitch classes, can be "sensibly" structured. To paraphrase Boretz, "[theories] constitute universes of "all the "things" that things can "be," and, in turn, they constrain (by implication) "all the things that can be a "thing." [12]

Theory's two different functions suggest that compositions and models of compositions are different sides of the same coin, since both view structure through a lens called theory. Either one starts with a theory and models a compositions's structure in the theory or one may create a compositional structure with an interpreted theory in mind. It is possible that a single theory could function as both a composing and parsing theory facilitating the creation or analysis of a composition. [13] The methodological distinction between composing and parsing functions is one of direction. In practice, the theory modeling a composition's structure is not always the same as the theory used to compose a work. Since structure can be viewed from different angles by a single theory, or viewed from each perspective by different theories, misconceptions can arise about the extent of a theory's parsing or composing functions, the connection or lack of connection between theory types, and theory's relationship to the composing process.

While some theorists may falsely conclude that parsing theories accomplish more than they do (leading to the belief that musical "laws" have been discovered), some composers falsely concluded that parsing theories have no connection with, and therefore nothing positive to offer, the composing process. [14] Since parsing and composing theories approach music from different directions (i.e., applied to structure as opposed to generating structure, or top-down versus bottom-up), the methodological distinction can be seen as an impenetrable barrier; like oil and water, the two types of theories do not mix.

A gap similar to that separating composing and parsing theories can also divide composing theories, which can be either one of two types. If the composing procedures are "intuitive," "bottom-up oriented," or partially specified (i.e., the procedures only reveal the most basic features of some undefined musical system), then the composing theory is "unexpressed." If the theory is explicit, "top-down oriented," or speculative (i.e., the systemic relations outlined in the theory apply to nonexistent compositions whose stylistic components are not specified by the theory, but, as Rahn states, the theory is presented "as an interesting possible way of organizing possible views of possible things"), [15] then the theory is "expressed." Some composers only favor the "bottom-up" or unexpressed approach, because composing with an expressed theory is seen as "theory" or "reason driven." [16] Sessions expressed this opinion in his discussion of Krenek's Uber neue Musik: [17]

creation--the end--is a subconscious process, while technique--the means--is the conscious or superconscious one; musical theory therefore that is before the fact can have no conceivable value to the musician, and can only be poisonous to him if he allows himself to be really exposed to it . . . musical theory is valid for the musician only insofar as it is practical and not speculative . . . it [Krenek's Ober nene Musik] stands fatally before the fact; its principles are, quite frankly and from the beginning, based on abstract reasoning rather than on concrete and demonstrable experience of effect.

The plausibility of concluding that expressed theories play a subordinate role in the creative process because compositional practice is antecedent to theory, or that music could be created without some theoretical notions, appears to be founded on three misconceptions. First, because unexpressed composing theories are not formalized, composing with an unexpressed theory fosters the illusion of "theory free" or intuitive composition. Second, the historical evidence supporting the popular notion that "theory always follows practice" is misinterpreted. Since parsing theories have a tendency to appear historically after the music that is intended to be modeled in the theory, the historical antecedence of compositional practice easily leads to the conclusion that theory does indeed follow practice; therefore, compositional practice does not employ theory. However, the popular aphorism could, perhaps, be more precisely stated as "parsing theories always follow unexpressed composing theories. "Finally, expressed c omposing and parsing theories might appear to be functionally identical, because they both appear to be "top-down" processes, which creates the illusion of equivalence between their very different functions.

Perhaps the equivocation of the processes results from the fact that since expressed composing theories move from theory (i.e. theory of structure) to the generation of a composition (i.e., the generation of structure), they are categorically more similar to parsing theories, which move from theory (i.e. theory of structure) to the analysis of a composition (i.e., the analysis of structure), than to "compositional practice" (i.e. unexpressed composing theories), which begin with data and no apparent theory of how that data is to be structured. In expressed composing theories, the move from theory to composition appears deterministic in the same way that a parsing theory determines what structures are available for analysis. On the other hand, unexpressed composing theories appear unrestricted by theoretical constraints, since the composing process only appears driven by data, with no a priori awareness of a relationship to a theory that structures the data. Although each type of composing theory appears to a pproach structure from opposite directions (theory versus data or top-down versus bottom-up), their apparent methodological differences should not be equated with the top-down/bottom-up methodology that identifies a theory as either parsing or composing. Parsing theories are top-down in that they are applied to an already existing structure. Consequently, the goal of parsing theories is to achieve as best they can a one-to-one relationship between theory and composition. For each structure posited by the theory a corresponding, structure will exist in the group of compositions under investigation. Ideally, the group of compositions should not contain structures unaccounted for by the theory. Furthermore, a by-product of parsing theories is they often specify the stylistic component of the music under investigation.

Unexpressed and expressed composing theories are both bottom-up, however, in the sense that they both generate a particular structure. Therefore, the goal of an expressed composing theory does not have to be achieving a one-to-one relationship between theory and composition. In fact, the relationship between expressed composing theories and composition is often a many-to-one mapping. The theory presents a range of possibilities from which the specific structures of a particular composition will emerge. Expressed composing theories simply either narrow the range more than unexpressed theories or they simply make the range of possibilities known. Unfortunately, an expressed composing theory is considered top-down, because the first step in the composing process is the theory's construction, while an unexpressed composing theory is considered bottom-up, because the theory of the data's structure is "constructed" in the process of composing. Unexpressed and expressed composing theories are actually variations on the same process. The former approach distinguishes itself from the latter by taking a nearly simultaneous process and turning it into a two-step process (Example 3). The movement in each case is from theory to the generation of structure, which is not the same as the movement from theory to the analysis of structure. [18] Based on these distinctions, it seems reasonable to conclude that expressed composing theories are not any more categorically similar than unexpressed composing and parsing theories, and expressed composing theories are no more "before the fact" then unexpressed theories. Furthermore, equating expressed composing and parsing theories not only misconstrues an expressed theory's relationship to structure, it misconstrues the nature of an unexpressed theory.

According to Benson Mates, "abstractly considered, any group of sentences concerning a given subject matter may be regarded as constituting the assertions or theses of a deductive theory, provided only that one very minimal condition is satisfied: all consequences of theses shall be theses, if they concern the relevant subject matter." [19] Any group of sentences constituting the assertions of any procedure for structuring music may be considered a music theory. Even the assertions that follow from a personal axiom, such as "I only compose what is in my heart," could be a theory, since those assertions could function as a plan or procedure for the organization and structuring of music. That is, the assertions following from "what is in the heart," like any theory, would limit the choice of musical structures that are consequences of those assertions. Since any conscious or unconscious procedure used to structure music can be considered a music theory, it seems reasonable to conclude that unexpressed and expr essed theories are similar conduits for the same process flowing in the same direction (but not originating from the same points, as Example 3 illustrates), and all composition is to some degree informed by theory. Therefore, if the label "theory-driven" can be applied to music generated by either unexpressed or expressed theories, it cannot be pejoratively applied to one theory type without reflecting pejoratively on the other type.[20]

In fact, there is much to be gained from working with expressed theories, because they allow composers to work directly with the boundary-setting elements of a composition, such as the rules governing voice leading, and they are useful tools for exploring the compositional possibilities within an expanded range of procedures.[21] A writer born, raised, and educated in Manhattan, for example, will have very little trouble writing about a writer born, raised, and educated in Manhattan. Our writer has a wealth of personal and intuitive knowledge to draw upon. Authors of fiction, on the other hand, might find it necessary to spend considerable time in speculative play creating a new or possible scenarios and narratives. As an author creates a theory of an alien world, he or she learns what creatures might inhabit it, or what effect this alien environment would have on the human species. Once the boundaries and limits of the world are set, characters and events can freely develop. In creating a possible world, on e is creating a context for imaginative play.

One might falsely infer, however, that "theory-driven" music is solely "about" structural relations. While I believe that structure is a necessary component of music, I do not want to imply that structure is both a necessary and sufficient determinant of a composition. [22] As well as being an end in itself structure can also be a means to an end. While the result of a cathedral's blueprint is the cathedral, the result of the cathedral is a religious/aesthetic experience. Since a cathedral's blueprint is about the cathedral's structure, it does not "say" [23] the same thing as the cathedral. However, one can better comprehend (but not necessarily reproduce) aspects of the cathedral experience from a sufficiently detailed model. For instance, one may wonder why the altar is in such a compelling location and then discover in the blueprint that it is the focal point of converging lines providing a context for comprehension. The concept of structure as a necessary but not sufficient component of music simply exp ands the domain of a composition's "richness" beyond considerations of structure alone. Therefore, the domain of analytical inquiry can include (but not necessarily be limited to) investigating elements that lie beyond considerations of structure alone, investigating the interaction between those elements that lie beyond considerations of structure and a composition 's structural foundation, and only investigating a composition's structural foundation. However, investigations focusing solely on a composition's structure or the structural foundations of a compositional system do not have to imply a disregard for those elements that may lie beyond the their reach. Investigations of structure may simply be limiting themselves to contemplating one aspect of a composition's or compositional system's "richness."

II. THE HYBRID SYSTEM

The hybrid system integrates generalized tonal concepts with tone rows and arrays. The foundation of the hybrid system is a class of rows I discovered that generate an array I call a uniform trichordal array. The 228 rows capable of generating these arrays are a subclass of the class of twelve-tone rows known as self-deriving. [24] (Example 4a.) A self-deriving row is capable of forming a [T.sub.n], [T.sub.n]I, [T.sub.n]M, or [T.sub.n]IM transformation of itself through a process of extraction. When a series of related rows are polyphonized [25] to form a two-lyne array, each extracted part as well as the columnar aggregates are transformations of each other. Each two-lyne array can be further polyphonized to form a four-lyne array. [26] (Example 4b) Self-deriving rows capable of generating uniform trichordal arrays can also generate uniform trichordal arrays containing from one to four discrete trichord types. In Example 4b for instance, only two [T.sub.n]/[T.sub.n]I trichord types, [0,3,7] and [0,1,4], occ upy the columns produced by partitioning the polyphonized rows ([T.sub.0] and [T.sub.0]R) by their discrete trichords. The pitch-class content of each trichord type in Example 4b is also fixed; pitch-class collections {0,4,7} and {1,6,t} produce the [T.sub.n]/[T.sub.n]I [0,3,7] trichord type, and pitch-class collections {2,3,e} and {5,8,9} produce the [T.sub.n]/[T.sub.n]I [0,1,4] trichord type.

Although tone-rows form the hybrid system's foundation, it is not a system of composition such as Babbitt's or Perle's. Rather than moving through aggregates, within the hybrid system, the hexachordal areas formed by pairs of trichord columns (the areas designated by the letters A and B in Example 4b) are independently composed out. [27] By fiat, a uniform trichordal array functions as a high-level schema for a composition, similar to Schenker's Ursatz. The array is a background schema that summarizes the numerous lower-level harmonic and voice-leading schemata that provide the temporal expansion or prolongation of the abstract background into a perceptible musical foreground that is the temporal elaboration of a high-level schema's harmonic and voice-leading features.

However, a uniform trichordal array must undergo a transformation to function as a high-level schema; its lynes and columns must be interpreted in pitch and time (Example 5). Each array lyne is assigned a register such that the pitch classes of a single lyne interpreted in pitch space do not exceed the range of an octave, and the four array lynes do not exceed the range of five octaves. [28] The registral lyne assignments are successive from bottom to top, with the bottom lyne assigned to the lowest and the top lyne assigned to highest registers roughly corresponding to bass, tenor, alto, and soprano voice ranges. The relationship "sooner than--later than" is sufficient to temporally interpret the array, that is, the trichords in each column temporally occur either sooner than or later than the trichords in the columns that precede or follow it. Enumerating the columns with the symbols [ST.sub.1] ... [ST.sub.n], (where ST stands for Schema Time) beginning with leftmost column is a convenient way to express this relation. Although the pitches in one column precede or follow the pitches in other columns, the pitches within each column are temporally undifferentiated. Consequently, the columnar row forms from a uniform trichordal array disappear in a high-level schema.

Every uniform trichordal array can produce eight permutations of its lyne ordering without destroying any of the row relationships shared by its columns and lynes. [29] Changing an array's lyne order can be a useful compositional property, because it highlights different aspects of an array's structure. The schema in Example 5, for example, reverses the top two lynes of the array. Permuting the lynes of a uniform trichordal array generates a high-level schema in which structural voice pairs, such as outer and inner voices, contain the same hexachords in one part of the schema and exchange hexachords with another pair of structural voices in another part of the schema. The first eight columns of Example 5, for instance, have the same linear hexachords in the outer and inner voices. The same is true of the second half of the schema, but now the inner voice hexachords have become the outer voice hexachords and vice versa. Overall, the schema only contains two pitch-class or pitch-distinct complementary hexachor ds.

The interpretation of a uniform trichordal array goes beyond simple pitch and time assignments, as is intimated by the slurs, beams, and note values in Example 5. These structural indicators reveal several very important schema features. A double bar bisects the schema, because each half contains the same hexachordal schema lines. These schema lines are indicated by the beamed groups of notes. The beams also outline and link together the pitches of the trichord in the first column embedded in each schema line. Since the schema-line hexachords are transformations of the columnar hexachords, they contain the same [T.sub.n]/[T.sub.n]I trichord types as the schema columns. The significance of the slurs like the significance of the beams also goes beyond their role of indicating the trichord structure of schema lines. A line's notes are the voice-leading goals of the harmonic structures determined by the schema columns; that is, they are the structural tones or goals around which melodic motion takes place. The a ggregate high-level schema line in Example 5, for example, indicates the voice-leading motions of structural tones over an entire composition, while the trichordal slurs and hexachordal beams indicate structural divisions within the aggregate line.

If schema lines determine the voice-leading adjacencies of structural tones, what are the connections in positions, such as [ST.sub.4] in the soprano line, where no structural tone is specified? In the alto voice of Example 5, a beam connects a solid notehead at [ST.sub.1] to an open notehead at [ST.sub.8]. The solid notehead is an octave doubling of a note from another schema line. Doubling a pitch from another schema line can fill any blank schema position. In Example 5, the structure of the inner-voice hexachordal schema lines from the schema's second eight columns determined the choice of [C.sub.4] and [C.sub.3] as the pitches to double. The hexachordal schema lines in the alto voice now both begin with the same pitch, while the tenor hexachordal schema lines both end with the same pitch. Since the outer voice hexachordal schema lines are identical to the inner voice lines, adding [C.sub.5] to the soprano and [C.sub.2] to the bass unifies the structure of all hexachordal lines. The compositional value of octave doublings may now be more evident; doublings can enhance the schema's structural bisection by further unifying the structure of hexachordal schema lines. Octave doublings can also be useful for weighting a pitch, thereby laying a foundation for the functional differentiation of doubled pitches. A schema with octave doublings already contains two levels, as is indicated by the solid noteheads filling blank schema positions distinguishing those pitches from pitches specified solely by the schema.

The hexachordal structure of a high-level schema also determines the pitch-class structure that is the basis of compositional form in the hybrid system. The form generated by the schema in Example 5 is A-B-B-A-A-B-B-A, since it consists of repeating primary and secondary hexachordal areas. A double bar bisects the schema, because each section contains the same hexachordal schema lines, and pairs of schema lines contain the same hexachord. Because a schema's structure is retrograde symmetrical and redundant, pruning its columns to create shorter schema lines and shorter forms, such as A-B-B-A, results in no loss of essential information. The schema in Example 6 has half as many columns, but it contains the same twelve-tone and basic hexachordal structure as its counterpart in Example 5. Although the truncated array appears to destroy the linear twelve-tone structure of the schema, this is not the case. Technically, the nontruncated schema does not maintain strict twelve-tone structures in every line. Only the soprano and alto lines are complete twelve-tone rows related to each other by retrogression. The tenor and bass lines are hexachordal rotations of the soprano and alto. [30] Consequently, a nontruncated schema already contains autonomous hexachordal lines. Furthermore, in the uniform trichordal array that generates the nontruncated schema, the row forms of columns one through four and five through eight are [T.sub.0] and [T.sub.0]R, respectively, the same row forms underlying columns nine through twelve and thirteen through sixteen. Since either row form is sufficient for determining a schema's generating row, reducing a truncated array to its underlying twelve-tone row is still possible.

Producing an even simpler version of the schema is possible by pruning schema lines to trichord length, which eliminates the repeated B section resulting in the form A-B-A (see Example 7). The linear trichords are still transformations of the columnar trichords, while the final A section is simply a retrograde repetition of the first A section. Eliminating columns five through fourteen of the nontruncated schema, which would place columns fifteen and sixteen adjacent to column four, is an alternative method for generating the trichord-schema line. Of course, this method would result in slightly different schema voice leading. The retrograde repetition of the first A section, however, is a natural choice for completing the trichord line schema, since it preserves the linear trichords. Polyphonizing a single row into the beginning of a uniform trichordal array is the origin of the first four columns in a trichord line schema, so reducing a truncated array to its underlying twelve-tone row is still possible.

Because a uniform trichordal array determines many of the syntactic relations for trichords, trichords are a fundamental compositional structure in the hybrid system. Since a schema begins and ends with the same trichord, that trichord has an important function within the context of the schema and its own hexachordal area in establishing a foundation for tonal functions; it is the closure trichord. Since the relationship of other trichords in the array to the closure trichord determines the function of those trichords, the closure trichord is also the primary trichord. Trichord adjacencies are consistent throughout a schema, so a trichord's proximity to the primary trichord determines its status. Because the trichord {3e2} in Example 4b precedes or follows the primary trichord, it is the secondary trichord. The secondary trichord moves away from or leads back to the primary trichord. The primary and secondary trichords are analogous to I and V or the tonic and dominant triads in tonality. The hexachordal are as of a schema are in the same structural relationship as the primary and secondary trichords, so the criteria that determine the function of the primary and secondary trichords also determine the function of the primary and secondary hexachordal areas. The hexachord occupying schema columns one and two has the primary function of being the closure hexachord. Since hexachordal adjacencies are consistent throughout a uniform trichordal array, the complementary hexachord's proximity to the primary hexachord determines its function as the secondary hexachord that precedes the return of, or initiates movement away from the closure hexachord. The secondary hexachord also consists of two trichords complementary to the hexachord. The trichord in array column three is the primary or closure trichord of the secondary hexachordal area, while the trichord in array column four is the secondary trichord of the secondary hexachordal area. Since they do not share any common tones, the move from the primary to the secondary trichord within a hexachordal area produces the maximum pitch-class content variation. Hexachordal areas mirror the complementary relationship of trichords.

The central component of the hybrid system's engine for composing out the trichords within hexachordal areas is an expanded version of John Kahn's basic formalized tonal theory as presented in his article "Logic, Set Theory, Music Theory." [31] His theory is a rational reconstruction of many of the concepts from Schenker's analytical methodology. Rahn's theory, however, is useful to both analysts and composers, since it can be used to analyze or generate structure. Using the considerably powerful tools of finite set theory, Kahn constructs a tonal theory consisting of ten definitions or definitional schemata (see Appendix 1). [32] As well as clarifying any ambiguities in the means of expression, formal definitions also remove any ambiguities about what can satisfy their conditions. Furthermore, the theory is not limited to contexts usually associated with tonal music. Since the generality of a concept's definition expands a concept's range (i.e., the more specifically a concept is defined, the smaller is the number of objects fulfilling its definition, but the more generally a concept is defined, the greater is the number of objects fulfilling its definition, and, consequently, generality expands a concept's range), structures usually associated with tonal music can be incorporated into other contexts. Therefore, incorporating the definitions into the hybrid system was simply a matter of producing structures that fulfilled their conditions.

For example, although Definition IV in Rahn's theory defines the conditions for pitch adjacency in the tonal system, it also allows series other than major, minor, and chromatic scales to become the basis for the definition of neighbor note, neighbor-note prolongation, arpeggiation, and arp-prolongation. Definition IV actually contains eight definitions partitioned into three sets. IVC forms the first set, since it is a definitional schema that generates the second set, definitions IVD through IVH, by substituting values for the variable C. The third set consists of definitions IVA and B, which are special cases of definition IVC. For example, under the most general version of the definition (the definitional schema IVC), the series C, D, E, D#, G, B, C (formed by cyclically ordering the pitch classes of the primary hexachordal area of the high-level schema in Example 5), produces the pitch-class adjacencies (C, D), (D, E), (E, D[sharp]), (D[sharp],G),(G, B), and (B, C).

The group of definitions, VA, B, and C, define neighbor note based on the definition of pitch adjacency from clauses IVA, B, and C, respectively, and the time-adjacency definition from clause III. Each version of definition V defines neighbor note relative to the collection forming its adjacencies. The general version of definition V (the definitional schema VC, which in a slightly modified version outlined below defines the neighbor note function within the hybrid system) says x and y are neighbors with respect to C if and only if x and y are pitch-adjacent with respect to a cyclical ordering as defined in IVC and time adjacent. Therefore, the pitch-class adjacencies (C, D), (D, E), (E, D[sharp]), (D[sharp, G), (G, B), and (B, C) produced by the cyclical ordering C, D, E, D[sharp], G, B, C in Example 8 are all potentially neighbor notes.

Definitions I-V are the foundation for the definitions of neighbor-note prolongation, VIA, B, and C, arp-prolongation (definition VII), background (definition VIII), level (definition X), and level analysis (definition IX). The general version of definition VI (the definitional schema VIC, which in a slightly modified version outlined below defines neighbor-note prolongation within the hybrid system), says that x and y neighbor-note prolong z if and only if x and y are neighbors with respect to C (definition VC) and z is a note whose pitch is equal to either x or y and whose initiation (value of [T.sub.1]) is the earliest initiation of x or of y and whose release (value of [T.sub.2]) is the latest release of x or of y. Example 9 illustrates a foreground realization of the secondary trichord embedded within the cyclic ordering C, D, B, D[sharp], G, B, C. The [D[sharp].sub.4] fulfills definition I, x is a note, since it has pitch value, an initiating time point, and a terminating time point. The [D[sharp].sub.4] and [E.sub.4] f ulfill definition III, x and y are time adjacent, since x and y are both notes, and one note begins where the other leaves off. The [D[sharp].sub.4] and [E.sub.4] also fulfill definition IVC, since they are pitch-adjacent with respect to C, where C is defined as the series C, D, E, D[sharp], G, B, C. The [D[sharp].sub.4] and [E.sub.4] are also neighbors with respect to C (definition VC), since they are pitch adjacent and time adjacent. And finally, the [E.sub.4] neighbor note prolongs the [D[sharp].sub.4], because they are time adjacent in the realization and neighbors with respect to the series C.

The definition for arp-prolongation defines how A arp-prolongs B, but it does not specify the pitch-class content of A and B. One set of notes or rests A arp-prolongs another B if and only if A and B have the same pitches and all the initiation times in B are equal to each other and equal to the earliest initiation time in A and all the release times in B are equal to each other and equal to the latest release time in B. The generality of the definition allows, for example, set A to be the secondary trichord embedded in the cyclical ordering C, D, E, D#, G, B, C being arpeggiated in the foreground realization of Example 9. Under the definition, the circled foreground realizations in Example 9 are each arp-prolonging one of the chords in the measure following the foreground realization. For example, the second quarter of the foreground realization (set A) has the same pitches as the second chord in the measure following the realization (set B), and the earliest initiation time in A and all the release times in B are equal to each other and equal to the latest release time in B. Furthermore, the two chords in the measure following the foreground realization are themselves arp-prolonging the single chord in the final measure of Example 9. Definition VIII says that a set of notes A is a next-background level with respect to some other set B under the following conditions: there is a partition of A and a partition of B and there is a one-to-one correspondence between the elements of the partition of A and the elements of the partition B such that for each pair of sets of notes related by that correspondence, either the sets are equal or the set from B prolongs the set from A (by neighbor note or arp-prolongation). Under the definition, the measure following the foreground realization in Example 9 is its next-background level, and the final measure is a next-background level to the second measure. [33]

As the preceding discussion outlines, the portion of Rahn's theory incorporated into the hybrid system consists of definitions I through III, VII through X, and IVC, VC, and VIC. My constraints on the core theory reduce the generality of the definitions, so they only specify structures relevant to the hybrid system. They include adding new definitions, adding clauses to existing definitions, and modifying definitions (see Appendix II). For example, clauses 1, 2, and 3 incorporated into definition [IX.sub.h] (level analysis) abstracts from a composition's foreground one of the high-level schemata forms. The algorithm for the generation of a high-level schema run in reverse abstracts the schema from a uniform trichordal array, and collapsing the array's first aggregate column abstracts the twelve-tone row whose pitch classes, partitioning segmental trichords, and hexachordal collections are identical to the foreground hexachordal areas and the primary and secondary trichords from each area. [34] While definiti on IVC simply says C is a cyclic ordering of pitch classes and x and y are pitch-adjacent with respect to C if and only if x and y belong to pitch classes that are adjacent in C, the modified version, [IVC.sub.h], specifies a set of size n as C and x and y are n(set)-proximate with respect to C if and only if x and y belong to pitch classes that are proximate in C. [35] The modification to definitions VC and VIC constrains their collectional reference to the C formed with the n(set) in [IVC.sub.h]. As Example 8 illustrates, the n(set) in [IVC.sub.h] will be an ordering of the hexachord from the row generating the uniform trichordal array; and therefore, the constraints on definitions VC and VIC specify neighbors as pitch-class proximities within the series.

Although [IVC.sub.h] defines n(set)-proximity with respect to a cyclic ordering C, it does not specify which of a hexachordal area's 6! orderings to choose. Example 10 illustrates 6 of the 6! possible orderings for the hexachord [0w4372]. [IVC.sub.h] also does not specify which of the six possible pitch interpretations of C becomes the area's canonical series. Clauses [IVC.sub.h]1 through 3 are criteria for selecting a single cyclic ordering. Clause 1, for example, says no members of the primary or secondary trichords can be adjacencies in C. [36] Clauses [IVC.sub.h]4 through 9 are criteria for selecting C's pitch interpretation by determining consecutive pitch interval size, assigning order numbers to C, assigning the order number 0 the meaning of "first and last pitch class of the scale," interpreting the order numbers as a strict simple ordering where the earliest note has the lowest pitch and the last note has the highest pitch, assigning a member of the primary trichord to order number 0, and restrictin g C to the interpretation that most resembles a set in normal form. [37]

Example 11 illustrates the pitch-space interpretation of the canonical series for the high-level schema in Example 6 generated by definition [IVC.sub.h]. Besides their quirky distribution of intervals, they contain one ordered interval whose direction is opposite to the other intervals in the series. Since it occurs where the larger intervals of the series begin, it structurally demarcates the intervallic shift and bisects the series, producing two tetrachords linked by a common pitch with a similar but not identical intervallic structure. The nonconformist interval also produces unusual neighbor note structures. Four pitches, 0,2, 7, and e, have upper and lower neighbors, pitch 4 has two lower neighbors (a lower and a next-lowest neighbor) and no upper neighbors, and pitch 3 only has two upper neighbors. Like the bisection of the series, double-lower or upper neighbor notes can be a compositional asset, generating new and interesting musical structures. In Example 6, the inner voice schema lines without oct ave duplications would begin with pitch 4 and end with pitch 2, the lowest-lower neighbor to pitch 4. The neighbor notes 2 and 4 might function exclusively as middleground structures referring to the schema line, while the closest neighbors in pitch, 3 and 4, could function exclusively in the foreground.

Although every canonical series for a high-level schema produces a unique pattern of ordered intervals, all canonical series are members of one of three classes determined by the number of out-of-order pitch-class pairs it contains. [38] All canonical series containing one out-of-order pitch-class pair form the first class. If a system's primary and secondary trichords were (0,1,2) and (3,4,5), [IVC.sub.h]1-9 would generate the canonical series which contains the out-of-numerical-order pitch-class pairs [less than]3,1[greater than] and [less than]4,2[greater than] (see Example 12). The series produces two pitches with upper and lower neighbor notes, two pitches with double lower neighbor notes, and two pitches with double upper neighbor notes. The class of canonical series with two out-of-direction intervals least resembles a "scale-like structure" therefore, a Type II canonical series functions in the foreground as a seven-pitch tone row. A Type 0 canonical series is equivalent to a hexachordal area's [T.su b.n]/[T.sub.n]I type in normal form, because it contains no out-of-numerical-order pitch-class pairs. In Example 13, [IVC.sub.h]1-9 generate a canonical series that is equivalent to the hexachord's [T.sub.n]/[T.sub.n]I type in normal form, [0,1,4,5,7,9]. Each pitch in a normal-form canonical series will have one upper and one lower neighbor note. [39]

The canonical array, which is derived from a canonical series (Example 14), reinforces the functions trichords derive from a high-level schema's structure, because it too is a series of alternating primary and secondary trichords. The canonical array generates additional functions, such as cadential formulas and local or hexachordal area voice leading. Every pitch in a canonical series is the base for either a primary or secondary trichord in the canonical array, and the lines formed by the middle and upper notes from each trichord in the array are rotations of the canonical series, so one voice-leading criterion could be move the pitches of the primary trichord to the pitches of the secondary trichord that are its neighbors in the canonical array. Furthermore, since each pitch in the canonical series is a neighbor to its proximities, each trichord in the canonical array built on a the base pitch of a canonical series functions as a neighbor chord in relation to its trichordal proximities.

Although the ordering that produces a uniform trichordal array's columnar row forms is not an overt structural feature of a high-level schema, the array's columnar trichords do maintain their spatial ordering. Since a canonical array, its corresponding hexachordal area in a high-level schema, and any repetitions of that hexachordal area all contain the same pair of trichords in different spatial orderings, spatial ordering becomes a structural marker of a trichord's position within a high-level schema and a canonical array. Because the canonical array in Example 14 begins and ends with the primary trichord in the spatial ordering [less than]0,4,7[greater than], it is the [less than]0,4,7[greater than] ordering that functions as the closure trichord for the canonical array. Furthermore, if the n! orderings of a set of pitch classes are no longer considered members of the same equivalence class, each spatial ordering could be the basis for establishing a specific cadential function for that ordering. A spatial ordering supraclass is an equivalence class that contains all spatial orderings of a trichord type that share the same base pitch. The spatial orderings [less than]2,3,e[greater than] and [less than]2,e,3[greater than], for example, are members of the same supraclass. The upper voices of supraclass members (e.g. the upper voices of the spatial orderings [less than]2,3,e[greater than] and [less than]2,e,3[greater than]) are rotations of each other, so we can label the equivalence-class tokens the open and close positions of the supraclass trichord type. [40]

Schemata also contain different spatial orderings of the primary and secondary trichords in the first occurrence of a hexachordal area and its repetition. While schema cadences (i. e., a cadence that ends on a primary trichord in the spatial ordering determined by a high-level schema) and the main canonical cadence trichord are identical in the primary hexachordal area of Example 14, they will be dissimilar in the second occurrence of the primary hexachordal area (columns 7 and 8), because the schema's spatial ordering of the primary trichord is different than the ordering in the main canonical cadence. When the primary ordering of the primary trichord from the canonical array and the spatial ordering of the primary trichord in the high-level schema are identical, a cadence on that ordering could signal the end of an important phrase, section, or the end of the composition. When the primary spatial ordering of the primary trichord from the canonical array and the spatial ordering of the primary trichord in t he high-level schema are not identical in the final hexachordal area of a high-level schema, the closure cadence for the work will be the schematic cadence. The main canonical cadence may end a main section, but the schema cadence brings the composition to its final close, in a coda for example.

An analysis of a foreground realization will give some of these relations a more concrete shape and sound. The trichord progression in Example 15a opens the first movement of my composition Tetralogy, a work for large chamber ensemble. The piano performs an obbligato function, so its part contains a realization of the entire progression (Example 15b). The primary trichord in its schematic registration from the primary hexachordal area and the canonical array begins the progression. Each line in the progression is a portion of the canonical series. Consequently, the first, third, and fifth trichords are primary types, while the second and fourth trichords are secondary types. Each of the secondary trichords is a neighbor chord prolongation of the primary trichords. Removing them in a reductive analysis would leave only three instances of the primary trichord. The predicate arp-prolongation would abstract from the passage the schema registration of the primary trichord.

The spatial ordering of the primary trichord ending the progression produces a localized cadential structure. The low degree of schematic voice-leading motions in the cadence (e.g., the only schematic reference is the (e-7) motion in the tenor voice) reinforces its status as a local cadence. The local nature of the cadence is also highlighted by the neighbor notes prolonging the [E.sub.4] and [G.sub.3] in measure two. The overall progression, however, does have important schematic implications. The beam connecting the half-note [G.sub.5] that begins the soprano line to the [C.sub.5] indicates a relationship or connection between the progression line and the soprano schema line. Since both lines begin and end with [G.sub.5] and [C.sub.5], the local line sets the stage for the global motion to be composed out over the course of the piece. [41] Some of the foreground figurations also reinforce schematic functions. The incomplete neighbor note, [B.sub.5], prolonging the opening [G.sub.5] on beat one of the sopra no, and the incomplete neighbor note, [G.sub.5], prolonging the [B.sub.5] at the end of the measure are foreground instantiations of the first schematic voice-leading motion in the soprano line of the high-level schema. The [B.sub.3]/[G.sub.3] neighbor note "trill" in the cadence is a literal reiteration of the neighbor note motion in the soprano voice of measure one.

Example 16b illustrates one of many possible progressions that could follow the initial progression. The second progression begins with the secondary trichord from the primary hexachordal area in its schematic registration, and only the soprano line completely duplicates a portion of the canonical series. The other lines consist of canonical series segments and neighbor note motions. The first, third, and fifth trichords are secondary trichords, while the second, fourth, and sixth trichords are primary trichords. The second and fourth primary trichords are passing chord prolongations of the secondary trichords leading to the fifth secondary trichord; the final pair of trichords in the progression form the cadence. Therefore, unlike the first progression, the second progression moves from the secondary to primary trichord. Furthermore, by running the definitions in reverse, the trichords that begin 16a and b could be abstracted from the overall progression to form the first progression of the high-level schem a. Several local details in both progressions support the schematic connections. The first soprano voice-leading motion in 16b is [B.sub.5] to [G.sub.5], which locally duplicates the schema voice-leading motion [G.sub.5] to [B.sub.5]. The [B.sub.5]/[G.sub.5], motion also connects with the very local neighbor prolongations of 16a. Therefore, the B-G schema voice leading occurs at three distinct levels within the progression. Overall, the progression's bass voice ascends from [C.sub.2] to [C.sub.3]. The ascent outlines the schema voice leading [C.sub.2] to [E[flat].sub.2], since 16a begins on [C.sub.2] and 16b begins on [E[flat].sub.2] The bass voice in 16 makes this connection more apparent, because it duplicates the connection between phrases within its phrase.

Although every trichord progression could consist of only primary and secondary trichords, varying the base progression by the inclusion of other trichords is also a possibility. Including a hexachordal area's total trichordal subset content as potential compositional resources creates a tertiary trichord level that generates a greater variety of trichord progressions. Example 17 lists the 20 possible trichords for the hexachordal set class generating the rows for Example 17. The structure of the particular subsystem, the [T.sub.n]/[T.sub.n]I types of the tertiary trichords, and the [T.sub.n]/[T.sub.n]I types of its primary and secondary trichords determine the value of a hexachord's trichords as a potential compositional resource. While the primary and secondary trichords do not share any common tones, they each share at least one and at most two common tones with the other eighteen trichords. Common tones provide one voice-leading criterion for connecting a primary or secondary trichord to a tertiary trich ord. For example, keeping the common tones shared by two trichords in the same voice. Moreover, the primary and secondary trichords do not contain pitch adjacencies from their canonical series as trichordal members, but all tertiary trichords will contain at least one pitch-class pair that is also an adjacency in the canonical series. Therefore, one way to view tertiary trichords is as trichords formed by suspended voice-leading motion between the primary and secondary trichords. [42]

Common tones or the lack of common tones also establish some functions for tertiary trichords. At the most general level, any of the eighteen trichords ultimately can function as a tertiary trichord, because any one of them could arise out of voice-leading motions between the primary and secondary trichords. The following criteria, therefore, are just some of the many possible rationales for introducing tertiary trichords into a hexachordal area. When the primary or secondary trichords share two common tones with a tertiary trichord, the tertiary trichord or trichords in the progression function essentially as melodic embellishments. When the primary trichord moves to the secondary trichord, the move from one trichord to another results in a complete change of pitch-class content. If the maximum pitch-class constancy between two trichords produces an essentially melodic progression, then the maximum pitch-class change between two trichords produces a "harmonic" progression. That is, the total lack of common tones disambiguates the identity of the two trichords, because one trichord does not overlap the other. Tertiary trichords sharing one common tone with the primary or secondary trichords produce progressions containing both harmonic and melodic elements.

Common tones, however, do not determine which of a hexachordal area's eighteen additional trichords can function as tertiary trichords. The introduction of tertiary trichords moves along two planes. A tertiary trichord moving along the first plane will be the same [T.sub.n]/[T.sub.n]I type as the primary or secondary trichords, but its pitch-dass content will be different. On the second plane, a tertiary trichord's pitch-class content and its [T.sub.n]/[T.sub.n]I type will both differ from the primary and secondary trichords. One tertiary trichord function is prolonging or delaying the move to the primary or secondary trichords in a progression. A tertiary trichord whose [T.sub.n]/[T.sub.n]I type is identical to the secondary trichord's type functions particularly well in this capacity, because it maintains the type of the secondary trichord while gradually introducing its pitch classes. Example 17 illustrates a detailed trichord vector for the primary hexachordal area of the high-level schema in Example 16. The hexachord contains three [0,1,4] type trichords besides the [0,1,4] trichord forming the subsystem's secondary trichord. The progression in Example 18 contains all the tertiary [0,1,4] [T.sub.n]/[T.sub.n]I-type trichords, gradually introducing the secondary trichord's pitch classes. The final [0,1,4] trichord in the progression is the secondary trichord, and it forms the main canonical cadence with the primary trichord. A second plane tertiary trichord whose [T.sub.n]/ [T.sub.n]/ type is not identical but closely related to the secondary trichord also functions well in this capacity, because, as well as gradually introducing the secondary trichord's pitch-class material, it also delays the move to the secondary trichord's [T.sub.n]/[T.sub.n]I type.

The hexachordal area also contains three [0,1,5] [T.sub.n]/[T.sub.n]I-type tinchords. Since the [0,1,5] trichord is very similar to the secondary trichord's [T.sub.n]/[T.sub.n]I type, it will function very nicely as a substitute for the secondary trichord. The progression in Example 19 contains three of the hexachord's [0,1,5] trichords functioning as tertiary trichords that gradually introduce the secondary trichord's pitch classes. The pitch class not introduced by the [0,1,5] type trichords appears when the cadential [0,1,4] trichord appears. The first two [0,1,5] trichords, (e,4,0) and (7,e,0), each share two common tones with their neighbors, so their function is more melodic. The third [0,1,5] trichord, (e,4,3), shares only one common tone with its neighbors. Its close proximity to the cadential [0,1,4] and its voice exchange relationship with the cadential [0,1,4] give the (e,4,3) tertiary trichord more of a harmonic function. That is, it will sound more like a change of harmony than a melodic embelli shment.

A graph of trichordal connections based on similarity criteria would be a helpful tool for determining a tertiary trichord's function. Morris's similarity index for pitch-class sets works very well. [43] However, tailoring the tool to the requirements of the hybrid system requires an additional constraint on its machinery, since the similarity index is a general measure of intervallic difference between two set classes. The index must correlate sets with specific pitch-class content, not just set classes. A graph generated by the tool should directly correlate tertiary trichords sharing one common tone and tertiary trichords sharing the smallest intervallic difference (i.e., a lower similarity index) with either the primary or secondary trichords. In general, every-other-node connections will indirectly correlate sets sharing two common tones and larger similarity indices (and therefore less intervallic similarity between the sets). Example 20 is a tertiary trichordal graph for the secondary trichord (2, 3, e) included in the primary hexachord (0, 2, 4, 3, 7, e). The lea side of the graph lists the [T.sub.n]/[T.sub.n]I types of the trichords included in the primary hexachord (0, 2, 4, 3, 7, e) along with the similarity indices (the numbers 0, 2, 4, and 6 in parentheses) relating the tertiary trichords to the secondary trichord. The top node in the graph is the secondary trichord, and the similarity indices and common-tone relations determine the connection between the tertiary and the secondary trichords. Example 21 illustrates a progression in which substituting an entire path from the graph (the path beginning with the (0,2,7) trichord moving to the (2,4,e) trichord) transforms a basic progression consisting of alternating primary and secondary trichords into a more variegated progression.

One of the most important structural joints within a high-level schema is the division between primary and secondary hexachordal areas. The structures linking hexachordal areas can serve a purely transitional function, they can function as tertiary areas for compositional development, or they can function as both transitions and tertiary areas for compositional development. Creating primary and secondary hexachordal areas also establishes a context for the introduction of chromatic pitch classes into either hexachordal area. [44] The introduction of chromatic pitch classes into the primary hexachordal area, for example, can prepare, facilitate, or foreshadow the move to the secondary hexachordal area. The schematic structure of Example 6 suggests one very important strategy for modulating the transition from the primary to the secondary hexachordal area. The union of the secondary trichord from the primary hexachordal area and the primary trichord from the secondary hexachordal area produces the [T.sub.e] tr ansposition of the primary hexachordal area ([T.sub.e] of (0,4,7,3,e,2) produces the set (e,3,6,2,t,1)). Since the [T.sub.e] transposition shares three pitch classes (e,3,2) with the primary and three pitch classes (6,t,1) with the secondary hexachordal areas, establishing it as a tertiary hexachordal area reduces the abruptness of the changes. [45] Moreover, the careful introduction of the three new pitch classes from the tertiary hexachordal area into the primary hexachordal area, or a carefully constructed transition from the primary to the tertiary hexachordal area, would smooth over half the rough transitional edges.

Although tertiary hexachordal areas do not have to be transpositionally or inversionally related to the primary or secondary hexachordal areas, one benefit of using transpositionally or inversionally related hexachords is that they maintain all the properties of the main hexachordal areas. They would, for example, maintain an intervallic consistency across the divide. They would also maintain the voice-leading structure of the schema. If, in Example 6 for instance, the [T.sub.e] transposition of the primary hexachordal area is unordered, it can produce twenty pairs of complementary trichords. Of course, one pair of complementary trichords will be the secondary trichord from the primary hexachordal area and the primary trichord from the secondary hexachordal area, since the union of those trichords produces the [T.sub.e] transposition of the primary hexachordal area. Consequently, the content of the tertiary area's complementary trichords can remain (e,3,2) and (6,t,1). The tertiary hexachordal area simply co mbines two trichords from different hexachordal areas (one from the primary and one from the secondary hexachordal areas) in the tertiary hexachordal area, the unordered [T.sub.e] transposition of the primary hexachordal area. [46]

As an ordered transposition of the primary hexachordal area, however, the content and the [T.sub.n]/[T.sub.n]I types of the secondary trichord from the primary hexachordal area and the primary trichord from the secondary hexachordal area will change ([T.sub.c] of the ordered set [less than]0,4,7,3,e,2[greater than] is [less than]e,3,6,2,t,1[greater than]). [47] The ordered transposition produces a new pair of complementary trichords for the tertiary area, (e,3,6) and (2,t,1), that the schema can easily incorporate. Since each of the new trichords shares two pitches with its neighbors in the other hexachordal areas, the structure of the schema lines with the common tones does not change (see Example 22). Although the new complementary trichords each share two pitches with the old ones, changing a single pitch in each trichord is sufficient to change the [T.sub.n]/[T.sub.n]I types of the original trichord pair. Nevertheless, even the [D.sub.4]/[F[sharp].sub.5] pitch exchange responsible for the content change does not alter the overall structure of the alto schema line (see Example 22). The [D.sub.4]/[F[sharp].sub.5] motion is simply expanded by repetition. The repetition maintains the schema voice-leading goal of [D.sub.4] moving to [F[sharp].sub.5]. [48] Inserting the new pair of complementary trichords into the schema in Example 22 leaves the schema lines virtually unchanged. [49]

With the new pair of complementary trichords, however, the structure of the tertiary area mirrors the structure of the primary and secondary areas, since it contains the same [T.sub.n]/[T.sub.n]I-type trichords in the same order as its neighbors. With a structure mirroring that of its neighbors, the tertiary hexachordal area becomes a transitional area that is more than a transition. It becomes an area suitable for compositional expansion. Moreover, the connection between the primary and tertiary area and the connection between the tertiary and secondary area remains the same. A [0,1,4] [T.sub.n]/[T.sub.n]I-type trichord moves to a [0,3,7] in the former connection, while a [0,1,4] [T.sub.n]/[T.sub.n]I-type trichord moves to a [0,3,7] in the latter connection. However, since the connection to the tertiary area in both cases involves common tones, it structurally distinguishes itself as an indirect connection. The ordered transposition of the primary hexachordal area produces a new pair of complementary tricho rds for the tertiary area that varies the content and [T.sub.n]/[T.sub.n]I types of the tertiary area's original trichords without destroying the underlying schema structure. Therefore, a tertiary hexachordal area related to the primary and secondary areas by transposition establishes a strong syntactic connection between a schema's primary and secondary hexachordal areas as well as creating a solid foundation for variation.

Even when the new complementary trichords elevate the tertiary area's primary function to compositional expansion, it will still function nicely as a transition. Only a single pitch-class change, for example (substituting pitch class 6 for pitch class 2), is required to transform the secondary trichord of the primary area into the primary trichord of the tertiary area. A carefully constructed string of trichords can very quickly execute the transition from the primary to the substitute tertiary hexachordal area. By carefully introducing one new pitch class at a time, the string in Example 23 requires only seven trichords to complete the transition. Furthermore, the transition reflects the trichord structure of the schema in Example 22, since it consists of alternating [0,3,7] and [0,1,4][T.sub.n]/[T.sub.n]I trichord types.

Between a high-level schema and the compositional surface are translevel structures. [50] Incorporating an entire schema line into lower structural levels produces the most overt and complete connection. Translevel structures incorporating an entire schema line also move along two planes, because they incorporate one of the high-level schema's lines or its transposition. A translevel structure incorporating a schema line will always be a chromatic event, since each line contains pitch classes from both hexachordal areas. The excerpt in Example 24 from the first movement of Textralogy incorporates the soprano and bass lines from the high-level schema. The progression temporally compresses and summarizes the global voice-leading motions of the schema's soprano and bass voices to produce a mid-level or translevel structure that outlines the structure of the composition's underlying high-level schema. The chromatic portion of the progression strongly suggests the secondary hexachordal area, and it suggests the o verall A-B-A structure of the high-level schema. The translevel progression exactly duplicates the [T.sub.n]/[T.sub.n]I-type structure and, for the most part, it duplicates the pitch content of the underlying high-level schema. Each of the translevel progression's chromatic pitches generates either a [T.sub.n]/[T.sub.n]I-type [0,3,7] or [0,1,4] trichord with two pitches from the primary hexachordal area.

Transpositions of schema lines formed from the pitch classes of either the primary or secondary hexachordal area produce less direct translevel connections. They will not be pitch-class identical to the high-level schema lines, but they will have identical ordered interval content. The [T.sub.4] transposition of the soprano schema line in Example 25a produces a line whose pitch-class content is identical to the primary hexachordal area. The transposed schema line can begin anywhere pitch class e is present in the primary hexachordal area of the composition. As Example 25c illustrates, if the schema pitch [B.sub.5] is the goal of a progression, then the arrival at the goal pitch could simultaneously initiate a progression appended to the main progression containing the transposition of the soprano schema line. Since the pitch [B.sub.5] begins the transposed mid-level schema line, and since it is in the same register as and contains three pitches from the soprano schema line, it could underscore the structural significance of the pitch [B.sub.5] by creating a link between the [B.sub.5] as part of a mid-level schema line and the high-level schema. Example 25c illustrates a harmonization that underscores the mid-level schema line's connection to the underlying high-level schema. The secondary trichord harmonizes the pitches [B.sub.5] and [D.sub.6], while the primary trichord harmonizes the pitches [C.sub.6] and [E.sub.5]. Each chord is prolonged by a tertiary trichord whose [T.sub.n]/[T.sub.n]I type is identical to the primary or secondary trichords. A tertiary [T.sub.n]/[T.sub.n] type [0,3,7] trichord harmonizes the [E.sub.6], and a tertiary [T.sub.n]/[T.sub.n]I-type [0,1,4] trichord harmonizes the G5. Although the overall trichord progression does not exacdy duplicate the order of the trichord types in the high-level schema, it does consist of an alternating pattern of [T.sub.n]/[T.sub.n] type [0,1,4] and [0,3,7] trichords.

Although I have demonstrated one method of constructing a hybrid system, I have not addressed the question why do it? Pursuing a reactionary agenda does not have to be the motivation for or the subtext of an investigation that attempts to integrate tonal and nontonal concepts or structures into a single system. That is, the desire to integrate these domains is not an implicit admission of failure on the part of nontonal systems to create coherent and complex musical structures. The twentieth century is replete with examples of uncompromisingly atonal or serial compositions whose beauty demonstrates and whose beauty is, in part, the result of their systems' power. The desire to create a hybrid system stems more from the intuitively motivated and conscious realization that such a world is possible and the curiosity aroused by imagining what new creative possibilities and what new musical experiences would be the consequence of inhabiting a world that is at once familiar and alien.

[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]

CIRO SCOTTO is a composer, conductor, and theorist who is currently Assistant Professor of Music Theory at the Eastman School of Music in

Rochester, New York. He formerly taught at the University of Texas- Austin, and the University of California-Santa Barbara.

NOTES

(1.) Readers who want to examine the technical details of the hybrid system first may return to part I after reading part II.

(2.) The meaning of "determines" in this context is the degree to which choice is limited by the compositional system. This includes both the choice of musical structures allowed by the compositional syntax and the arrangement of those structures in a composition. For example, common practice tonality does not allow a [T.sub.n]/[T.sub.n]I[012] trichord to function as a tonic triad. Consequently, the choice of entities functioning as a tonic is limited by the system. If a composer decides to use a cadential 6/4 in a common practice composition, the system limits where that structure can occur within the sequence of events. Aspects of registration, rhythmic placement, doubling, and resolution are all determined by the system. If composers do not like the limitations, they are free to choose to use some other structure in its place, or they can choose to ignore the dictates of the system and "break a rule." The "musical surface" in this context is a traditional score for performance (i.e., a score that does not allow any additional choice with regard to the sequential arrangement of the musical structures it contains). A score represents the end of choice or the conclusion of sequentially arranging compositional musical structures.

(3.) In this composition, Cage allowed imperfections in the manuscript paper to determine the identity of the notes on the staff.

(4.) With aleatoric theories one cannot get an aural image, fuzzy or clear, of what the score might contain, since the musical structures and their sequential arrangement in a score cannot be anticipated from the theory. To make the point another way, the theories used in astrophysics allow one to anticipate quite precisely where and when a planet will appear in the night sky. Aleatoric theories essentially remove choice from the composing process or present unlimited choice, depending upon your perspective.

(5.) Robert D. Morris, Composition with Pitch-Classes (New Haven and London: Yale University Press, 1987), 3. That is, once the composer chooses the materials and the procedures, setting the algorithm in motion produces a score, since the algorithm chooses all aspects of the score. Consequently, one can get a clear image of what the score will contain from examining the algorithm. To make the point another way, if one was given a set of prefabricated parts and a set of instructions for assembling the parts, one could predict from the parts and the instructions that following the procedure would produce a desk, for example.

(6.) That is, transforming the rules in a harmony textbook, for example, into an algorithm would probably not generate scores that we would label "pieces of music." In this case, the algorithm would be under-specified, since many of the choices needed to produce a score from the rules are not specified by the theory.

(7.) Brian Ferneyhough's Lemma--Icon--Epigram for solo piano is one example of a motivically generated composition. For an analysis exploring the motivic generation of the work see Richard Toop, "Brian Ferneyhough's Lemma--Icon--Epigram," Perspectives of New Music 28, no. 2 (Summer 1990): 52-100.

(8.) The methodology presented in Fred Lerdahl's article "Atonal Prolongational Structure," Contemporary Music Review 4 (1989): 65-87, would be an example of this approach. With only slight modifications, a composer could adapt the concepts presented in the article for the generation of structure.

(9.) The distinction I am drawing is similar to one method of conceptualizing the mind-body distinction in philosophy. Dualists adhere to the notion of the mental and physical as different substances, so mentalistic and physicalistic expressions differ in meaning and reference. In a dualist theory, the mind and body are different structures, so explanations of mentalistic expression can't explain physicalistic expressions and vice versa. Materialists, however, adhere to the notion that the mental and physical are not two separate substances or structures. Although materialist offer many accounts of the relation between the mental and physical, the most pertinent to the present discussion is a materialist proposal called identity theory. Identify theorists use the distinction drawn by Frege between sense and reference. For example, the expression "the morning star" and "the evening star" have different senses (the star one sees in the morning as opposed to the star one sees in the evening), but they have the sam e denotation or reference, the planet Venus. For identity theorists, mental and physical expressions have different senses, but they have the same reference, physical phenomenon. The empirical facts to support an identity theory could come from studies of the brain, in which stimulating an area of the brain physically produces mental states in a subject. Therefore, for a materialist proposing an identity theory the distinction between the mind and body originates in language. That is, the structures underlying the distinction are not different, but linguistic usage creates the division. Similarly, linguistic usage (more than structural elements and relations) foster the apparent contradiction of incorporating structures usually associated with tonal music into nontonal systems.

(10.) David Lewin in "A Formal Theory of Generalized Tonal Functions," Journal of Music Theory 26, no. 1 (Spring 1982): 23-60.

(11.) Another example would come from scale theory. It demonstrates the unique properties possessed by the diatonic scale with regards to well-formedness that give rise to many tonal syntactic features (well-formedness is informally defined by Carey and Clampitt as "scales generated by consecutive fifths in which symmetry is preserved by scale ordering"). The majority of the 220 set classes in [T.sub.n]/[T.sub.n]I system as outlined by Forte, Kahn, and Morris do not possess the well-formedness property; however, even scale theory may not limit tonality to the diatonic set. Norman Carey and David Clampitt, "Aspects of Well-Formed Scales," Music Theory Spectrum 11, no. 2 (Fall 1989): 187-206.

(12.) Benjamin Boretz, "The Logic of What?" Journal of Music Theory 33, no. 1 (Spring 1989): 107-16.

(13.) Although the structural description underlying both a composing and parsing theory may be the same, the acts of analysis and composition are not the same. While analysis is deterministic, composition is indeterminate. The structural description in the context of each act, analysis or composition, becomes a different type of theory.

(14.) For example, in his book The Listening Composer (Berkeley and Los Angles: The University of California Press, 1990), 112, George Perle considers Forte's set tables "ludicrously useless."

(15.) John Kahn, "Lines (of and About Music)," (Ph.D. diss., Princeton University, 1974), 36.

(16.) Another way to view the problem is to say that "reason driven" theories are different from "intuitively driven" theories. Fred Lerdahl takes this view in his attack on serial composing theories in his article "Cognitive Constraints on Compositional Systems," Generative Processes in Music, edited by John A. Sloboda (Oxford: Clarendon Press, 1988), 231-259: "it becomes quite possible for the 'compositional grammar' to be unrelated to the other rules, the 'listening grammar' and 'intuitive constraints.' If this happens, the 'input organization' will bear no relation to the 'heard structure.' Here, then, lies the gap between compositional system and cognized result ... this gap is a fundamental problem of contemporary music. It divorces method from intuition...," 234-35.

(17.) Roger Sessions, "The Function of Theory," in Roger Sessions on Music, Collected Essays, ed. Edward T. Cone (Princeton: Princeton University Press, 1979), 264-266.

(18.) Robert Morris makes a similar point in Chapter One of his book: "In fact, such a theory emphasizes the mutuality between doing composition and thinking about it--each activity provides problems and solutions to the other." Robert D. Morris, Composition with Pitch-Classes, 3.

(19.) Technically, only one predicate is needed to produce a formal deductive theory: "T is a deductive theory formalized in the first order predicate calculus if and only if T is a pair of sets [less than][delta],[gamma][greater than] where [delta] is a set of non-logical constants of [pound] containing at least one predicate of degree [greater than or equal to] ...." Benson Mates, Elementary Logic (New York: Oxford University Press, 1972), 183.

(20.) My ultimate goal is not to demonstrate that the process of composition is completely theory-driven. I do agree with Sessions that "creation ... is a subconscious process." The point being made here is that expressed composing theories are no less connected to the subconscious process of creation than unexpressed theories, which appears to be an assumption underlying Sessions's criticism of Krenek's theory of composition. By demonstrating that unexpressed theories engage the same process, creation of a composition, from the same direction, it appears safe to conclude both theory types are informed by intuition (i.e., the subconscious). In spite of their similarities, one could still object to employing expressed composing theories as part of the compositional process, because 1) an expressed composing theory does not necessarily translate into what is heard, and 2) an expressed composing theory mechanizes the process removing choice. In many ways, one goal of this paper is to answer the first objection, an d readers will decide for themselves whether or not that particular objection has been successfully addressed. My answer to the second objection is that "precomposition" of the type employed in the hybrid system is composition. The system simply limits choice, it does not remove it.

(21.) Another benefit of speculative systems is the possibility of expanding experience beyond the "demonstrable experience of effect," as Sessions called it.

(22.) Although it could be argued that knowing a composition's structure is not a necessary component of appreciating a composition, I have found a high degree of correlation between the compositions I most appreciate and compositions with interesting structures. Consequently, studying a composition's structure for me is another way of appreciating or relating to a composition.

(23.) The distinction between what a cathedral's blueprint and a cathedral "say" is analogous to the distinction drawn by Scott Soames between what instances of schema K and a semantic theory that pairs object language sentences with propositions "say." (For a complete explication of Soames's argument Scott Soames, "Semantics and Semantic Competence," Philosophical Perspectives, 3; Philosophy of Mind and Action Theory (Atascadero, California: Ridgeview Publishing Company, 1989), 575-96.) According to Soames, semantic theories should reveal what information is encoded by sentences relative to contexts. Competent speakers correctly pair sentences with their contexts, and they grasp the information encoded in sentences. Therefore, it seems reasonable to assume that a semantic theory is equivalent to a theory of competence. Soames claims that knowledge of meaning is knowledge of that which is expressed by instances of schema K: 'S' expresses the proposition that P relative to the context C.

Consequently, it is reasonable to assume that "a correct semantic theory will entail instances of schema K, knowledge of which explains semantic competence" (580). Soames constructs a semantic theory that correctly pairs object language sentences with propositions. When his semantic theory is supplemented with a theory of object language syntax and an interpretation of its vocabulary, the theory produces theorems of the following sort:

(11) '[backwards E x Lx,n' expresses (with respect to every context) the proposition which is the ordered pair whose first coordinate is the property SOME, of being a non-empty set, and whose second coordinate is the function g which assigns to any object o the proposition which is the ordered pair whose second coordinate is the relation of loving and whose first coordinate is the ordered pair the first coordinate of which is o and the second coordinate of which is Nixon.

Although Soames has developed a semantic theory that correctly pairs object language sentences with propositions of the right sort, he claims that theorems, such as (11) are not instances of schema K, such as (12):

(12) '[backwards E] x Lx,n' expresses the proposition that something loves Nixon (with respect to every context C).

He claims that in instances of schema K, such as (12), contain a singular term t which denotes the proposition expressed by '[backwards E] x Lx,n'. Although theorem (11) of the semantic theory contains a singular t' which refers to the same proposition as t, "t and t' are distinct, non-synonymous expressions." (585). (11) and (12) "say" different things, because t is 'the proposition that something loves Nixon', and t' is the complicated definite description given in (11) (585). Although it seemed reasonable to assume that semantic theories are equivalent to semantic competence, Soames's argument demonstrates that semantic competence is not the result of knowing semantic theorems, such as (11).

The reasons (11) and (12) "say" different things are analogous to the reasons why a cathedral's blueprint and a cathedral "say" different things. The blueprint is a complicated definite description of the "proposition" a cathedral expresses, while the cathedral expresses the "proposition." However, if the end of a blueprint was a self-referential building, and the end of a self-referential building was to be a reflection of its own structure, then the blueprint and the building would both express "propositions" that said the same thing.

(24.) A sampling of articles exploring multiple order functions and self-deriving arrays includes David Lewin, "On Certain Techniques of Re-ordering in Twelve-Tone Music," Journal of Music Theory 10 (1966): 276-87; Donald Martino, "The Source Set and its Aggregate Formations," Journal of Music Theory 5, no. 2 (1961): 224-73; Robert Morris, "On the Generation of Multiple Order Function Rows," Journal of Music Theory 21, no. 2 (Fall 1977): 238-63; Philip Norman Batstone, "Multiple Order Functions in Twelve-Tone Music," Perspectives of New Music 10, no. 2 (1961): 60-72 and 11, no. 1 (1972): 92-111; and Peter Westergaard, "Toward a Twelve-Tone Polyphony," Perspectives on Contemporary Music Theory, ed. Benjamin Boretz and Edward T. Cone (New York: Norton and Co., 1972), 90-112; Daniel Starr, "Derivation and Polyphony in Twelve-Tone Music," Perspectives of New Music 23, no. 1 (Fall-Winter 1984): 180-257; David Kowalski, "The Construction and Use of Self-Deriving Arrays," Perspectives of New Music 25, nos. 1 & 2 (Wint er-Summer 1987): 286-361; and Ciro G. Scotto, "Can Non-Tonal Systems Support Music as Richly as the Tonal System?" (D.M.A. diss., University of Washington, 1995), Chapters 2 and 3.

(25.) The term polyphonization as it pertains to twelve-tone rows was coined by Daniel Starr, "Derivation and Polyphony," Perspectives of New Music 23, no. 1 (Fall-Winter 1984): 180-257.

(26.) The combinatorial relationships generating a simple two-lyne or four-lyne array are extensible to partitions other then [6.sup.2] and [3.sup.4]. See Scotto, "Can Non-Tonal Systems Support Music ... Chapter 2.

(27.) The rows in a row table or matrix can be grouped according to the pitch-class content of their hexachords. When the discrete hexachords of a row and its transformation are identical with regard to their unordered pitch-class content, the rows form a harmonic area. Therefore, harmonic areas are distinguished by the pitch-class content of the hexachords of the rows in the area. The first eight measures of Milton Babbitt's Three Compositions for Piano are an excellent example of the use of harmonic areas in a composition. Each measure contains a pair of hexachords divided between the right and left hands. The pitch-class content of the hexachords is identical but the ordering of the pitch classes is different, because the hexachords are all members of different rows from the same harmonic area. David Lewin coined the term "harmonic area."

(28.) As will be shown later in the paper, the "soprano" schema line is analogous in function to an Urlinie. The assignment of a schema line to a single octave sets up the condition that schema lines will be completed in the octave in which they begin.

(29.) Ciro G. Scotto, "Can Non-Tonal Systems Support Music as Richly as the Tonal System?" Chapter 3.

(30.) Treating the hexachords of a twelve-tone row independently has often been a compositional strategy employed by composers, such as Schoenberg (Third String Quartet) and Babbitt (Sextets).

(31.) College Music Symposium 19, no. 1 (1979): 114-127.

(32.) Of course, Rahn's definitional system itself contains terms, such as pitch and note, that at a higher level need to be defined. One source for the higher level definitions would be Benjamin Boretz, "Metavariations: Studies in the Foundations of Musical Thought," Perspectives of New Music 8, no. 1 (Fall-Winter 1969): 1--74; 8, no. 2 (Spring-Summer 1970): 49--111; 9, no. 1 (Fall-Winter 1970): 23-- 42; 9, no. 2 and 10, no. 1 (1971): 232--70; 11, no. 1 (Fall-Winter 1972): 146--223; 11, no. 2 (Spring-Summer 1973): 156--203.

(33.) John Roeder tested the validity of Rahn's definitions by translating them into the syntax of the Prolog programming language for his logic-programming model of music, in "Logic-Programming Models of Music: A Semiotic Evaluation," Music and Science, A Symposium at the University of Washington School of Music, 19--23 February 1991 16-36 (Seattle: University of Washington, 1991). Although Roeder's evaluation of Rahn's definitions justifies their function on a formal basis, one may still question the validity of the approach according to the conditions outlined by Joseph N. Straus in "The Problem of Prolongation in Post-Tonal Music," Journal of Music Theory 31, no. 1 (Spring 1987): 1--21. In Example 9, the neighbor note to D #, E, forms an interval class 1, which is also an interval formed by two chord members, E[flat], and D. Specifically, Example 9 appears to violate Straus's "condition #4," the harmony/ voice leading condition: A clear distinction between the vertical and horizontal dimensions. However, one sh ould not draw a more general conclusion from Straus's condition that no context could be established that would disambiguate whether a specific interval class 1 was harmonic or melodic in nature. In fact, Example 9 demonstrates just such a context. Of course, the context established in Example 9 is more entity based than the tonal system (demonstrating one way in which the tonal and hybrid systems differ), since it is the interval class 1 between pitch classes D# and E that is a neighbor function and the interval class 1 between pitch classes D (E[flat]) and D that is harmonic.

(34.) Using the hybrid systems definition for analytical purposes would demonstrate, as the discussion intimates, the coherent relationships connecting a compositional surface to the array functioning as the compositions high-level schema and to the row generating the array. In other words, using the hybrid systems definitions in an analytic capacity would demonstrate the connection between a composition's underlying twelve-tone array and the compositional surface. The analysis of Example 15 will demonstrate this idea.

(35.) The change from adjacency to proximity is intended to avoid any confusion between the use of the word in musical acoustics to refer to scale step, which in the literature refers to major and minor seconds.

(36.) This criterion eliminates orderings a, b, and f from Example 10.

(37.) These criteria eliminate orderings d and e from Example 10 leaving only ordering c, which is shown in its pitch interpretation in Example 11.

(38.) Scotto, "Can Non-Tonal Systems Support Music..., Chapter 5.

(39.) More specifically, all canonical series function as seven-note rows. The designation "scale-like" simply indicates that the numerical ordering of the pitch-class names is isomorphic to the ordering of the pitch classes in the canonical series. When the orderings are isomorphic, the canonical series is "scale-like." As the two orderings start to diverge, the canonical series become less "scale-like." Furthermore, although the canonical series is essentially a seven-note row, it does not function as traditional rows function on the surface of a composition. To be sure, both a traditional twelve-tone row and a canonical series function in the background as referential orderings for partial orderings on the surface of a composition, and to be sure, a twelve-tone row can function exclusively in this role (Babbitt's Second String Quartet, for example), but a canonical series is always exclusively a referential ordering for partial orderings, it establishes functions for partial orderings, and it determines pitc h relations.

(40.) Its base pitch determines the order position of each spatial ordering within the canonical array. The labels [less than]0 ... [less than]6 indicate the order position of a canonical series pitch upon which a spatial ordering is built. Since the labels do not have suffixes, they also indicate that the spatial orderings are all in close position, while the labels [less than]0+ ... [less than]6+ indicate the open position members of the supraclass.

(41.) Over the course of the entire composition, the soprano schema line will progress from [G.sub.5] to its end goal of [C.sub.5]. The local soprano line in Example 15a progresses from the same [G.sub.5] and leads to the same goal, [C.sub.5]. Since the overall motion in each line has the same origin ([G.sub.5]) and goal ([C.sub.5]), the local line, which unfolds over a shorter time span than the schema line, foreshadows a motion that will also take place over the course of the entire composition. In the present case, the internal motions connecting [G.sub.5] and [C.sub.5] in the schema line and the local line are different. However, as Example 25 will demonstrate, the internal motions connecting the first and last pitches of a schema and local line can also be identical, thereby strengthening the local and global connections.

(42.) Since tertiary trichords could be a product of voice-leading motions, they will not have the same structural status as the primary and secondary trichords within the hybrid system.

(43.) Robert Morris, "A Similarity Index for Pitch-Glass Sets," Perspectives of New Music 18, nos. 1 & 2 (1979-80), 445-60.

(44.) Chromatic pitch classes in the context of the hybrid system are defined as pitch classes from the complementary hexachordal area.

(45.) Robert Morris has discussed the compositional implications of set classes that share subsets in several of his works, such as Composition with Pitch Classes cited earlier. Exploring these works might be useful to readers who wish to explore alternative methods of relating hexachordal areas. Robert Morris, "Combinatoriality without the Aggregate," Perspectives of New Music 21, nos. 1 & 2 (1982-1983), 478 and Robert Morris, "Compositional Spaces and other Territories," Perspectives of New Music 33 (Winter and Summer 1995): 328-358.

(46.) By keeping the pitch-class content (as well as the set class) of the primary and secondary trichords intact in a Tertiary hexachordal area, the transition between the primary and secondary hexachordal areas will maintain the structural trichords of the high-level schema rather than introducing new trichords. However, the function of these trichords will change. The secondary trichord from the primary hexachordal area becomes the primary trichord and the primary trichord from the secondary hexachordal area becomes the secondary trichord in the tertiary hexachordal area. I view this as an analogue to the changing function of the dominant of the tonic key area after a modulation to the dominant key area.

(47.) The ordering of a schema line can always activate the latent ordering of the schema columns, because the ordering of the rows in the columns is discernible from the ordering of the schema lines. Another way of varying the content of the tertiary area's hexachords while maintaining their [T.sub.n]/[T.sub.n]I types is to produce all the pairs of complementary trichords for the hexachord and see how many pairs vary the pitch-class content while maintaining the [T.sub.n]/[T.sub.n],I types of the original trichords.

(48.) Even in the context of purely "classical" twelve-tone composition, the repetition of a pair of pitch classes was never seen as destroying the structural integrity of the row.

(49.) Inserting a tertiary hexachordal area that leaves the schema lines virtually unchanged means that local motions within the tertiary hexachordal area can still highlight the global motions unfolding more slowly in the background.

(50.) The structures discussed below offer only a glimpse of the many possibilities in which a compositional surface can be connected to or be an expression of its underlying schema.
EXAMPLE 4A
SOURCE ARRAY FOR THE UNIFORM TRICHORDAL ARRAY
T0 [less than]O-4-7-3-E-2-T-1-6-8-9-5[greater than]
Source Row for Self-Deriving Row
[T.sub.7]   7          e2  t 6  9    5 8         1    3   40
[T.sub.7]R  04         3   1    8 5  9           6 t  2e  7
            [T.sub.0]                [T.sub.0]R
EXAMPLE 4B
A UNIFORM TRICHORDAL SELF-DERIVING ARRAY
[037]      [014]  [037]  [014]  [014]       [037]  [014]  [037]
A                 B
           2      6      9      5           1             4
7          c      T             8                  3      0
4                 1      5      9           6      2
0          3             8                  t      e      7
[T.sub.0]                       [T.sub.0]R
[037]      [037]      [014]  [037]  [014]  [014]       [037]  [014]
A
           0          3             8                  t      c
7          4          5      1      5      9           6      2
4          7          c      t             8                  3
0                     2      6      9      5           1
[T.sub.0]  [T.sub.0]                       [T.sub.0]R
00
[037]      [037]
A
           7
7
4          0
0          4
[T.sub.0]
EXAMPLE 13
UNIFORM TRICHORDAL ARRAY CONTAINING ONE NORMAL-FORM (TYPE 0) CANONICAL
SERIES
(a)
                         [0,3,7]  [0,4,8]  [0,3,7]  [0,3,6]
[T.sub.0]   [T.sub.5]bR  0        9        6
            [T.sub.5]a            5        t        2
[T.sub.0]R  [T.sub.5]aR  7                 3        e
            [T.sub.5]b   4        1                 8
            [T.sub.7]
(a)
            [0,3,6]     [0,3,7]  [0,4,8]  [0,3,7]
[T.sub.0]   8                    1        4
            e           3                 7
[T.sub.0]R  2           t        5
                        6        9        0
            [T.sub.7]R
EXAMPLE 17
Trichord Vector for the [T.sub.n]/
[T.sub.n]I-Type Hexachord [0,1,3,4,5,8]
[0,1,2]  [0,1,3]  [0,1,4]  [0,1,5]  [0,1,6]  [0,2,4]  [0,2,5]  [0,2,6]
(3,4,5)  (0,1,3)  (0,1,4)  (0,1,5)           (1,3,5)  (3,5,8)
         (1,3,4)  (4,5,8)  (0,1,8)                    (0,3,5)
                  (0,3,4)  (3,4,8)
                  (1,4,5)  (0,4,5)
[0,1,2]  [0,2,7]  [0,3,6]  [0,3,7]  [0,4,8]
(3,4,5)  (1,3,8)           (0,5,8)  (0,4,8)
                           (1,5,8)
                           (1,4,8)
                           (0,3,8)


A B[flat] C D[flat] E[flat]c F[sharp]

(C, D[flat], E[flat]) = [0,1,3] = tonic

(E[flat], E, F[sharp]) = [0,1,3] = dominant

(A, B[flat], C) = [0,1,3] = subdominant

(D[flat], E[flat], E) = [0,1,3] = mediant

(B[flat], C, D[flat]) = [0,1,3] = submediant

EXAMPLE 2: RIEMANN SYSTEM DETERMINED BY THE ORDERED TRIPLE (c, 3, 1) [composing-theory/composing][right arrow] composition

N.B. Although unexpressed and expresses composing and theories are being represented as either a one or two step linear process, in reality, the distinctions "top-down" and "bottom-up" are, perhaps, best represented as poles at either end of a continuum. The movement along the line is not unidirectional as much as it is an oscillation between poles. Consequently, neither approach is a completely topdown or bottom-up process, and neither approach is entirely before or after the other.

composing theoryu [right arrow] composingu [right arrow] composition

APPENDIX I: RAHN'S DEFINITIONAL TONAL SYSTEM

I x is a note

IFF x = [less than]z, [less than][T.sub.1], [T.sub.2][greater than][greater than] for some value of z, [T.sub.1], [T.sub.2].

II x is a rest

IFF x = [less than]s, [less than][T.sub.1], [t.sub.2][greater than][greater than] for some value of [T.sub.1] and [T.sub.2]. (s is a constant.)

III x and y are time-adjacent

IFF x and y are notes or rests and [T.sub.2] of x equals [T.sub.1] of y or [T.sub.2] of y equals [T.sub.1] of x. (One note begins where the other leaves off.)

IVA x and y are pitch-adjacent

IFF x and y are notes whose pitches are a minor, major, or augmented second apart.

IVB x and y are circle of fifths pitch-adjacent

IFF x and y are notes whose pitches are a perfect fourth or fifth apart.

IVC x and y are pitch-adjacent with respect to C

IFF C is a cyclic ordering of pitch classes and x and y are notes whose pitches are less than an octave apart and belong to pitch classes that are adjacent in C.

IVD x and y are chromatically adjacent

IFF x and y are pitch-adjacent with respect to the chromatic scale.

IVE x and y are diatonically adjacent

IFF x and y are pitch-adjacent with respect to a major scale.

IVF x and y are extended diatonically adjacent

IFF x and y are pitch-adjacent with respect to a major or harmonic minor or melodic minor scale.

IVG x and y are circle of fifths adjacent

IFF x and y are pitch-adjacent with respect to the circle of fifths.

IVH x and y are triad-adjacent

IFF x and y are pitch-adjacent with respect to any (cyclic ordering of a major or minor pitch-class triad.

VA x and y are neighbors

IFF x and y are time-adjacent and pitch-adjacent (IVA).

VB x and y are N* neighbors

IFF x and y are time-adjacent and circle of fifths pitch-adjacent (IVB) or circle of fifths adjacent (IVG).

VC x and y are neighbors with respect to C IFF x and y are time-adjacent and pitch-adjacent with respect to C (IVC).

VIA x and y N-prolong z

IFF x and y are neighbors and z is a note whose pitch equals the pitch of x or of y and whose initiation (value of [T.sub.1]) is the earliest initiation of x or of y and whose release (value of [T.sub.2] is the latest release of x or of y.

VIB x and y [N.sup.*]-prolong z

IFF x and y are [N.sup.*] neighbors and z is a note whose pitch equals the pitch of x or of y and whose initiation release (value of [T.sub.2] is the latest release of x or of y.

VIC x and y NC-prolong z

IFF x and y are neighbors with respect to C and z is a note whose pitch equals the pitch of x or of y and whose initiation (value of [T.sub.1] is the earliest initiation of x or of y and whose release (value of [T.sub.2] is the latest release of x or of y.

VII A arp-prolongs B IFF

A is a set of notes or rests and B is a set of notes and a pitch is an A IFF it is in B, and all initiations ([T.sub.1]) in B are equal to each other and equal to the earliest initiation in A, and all releases ([T.sub.2]) in B are equal to each other and equal to the latest release in A.

VIII A is a next-background to B IFF

A and B are distinct sets and for at least one set [A.sub.1] and at least one set [B.sub.1], [A.sub.1] partitions A and [B.sub.1] partitions B and there is at least one one-to-one correspondence, X, from [A.sub.1] to [B.sub.1], such that for every member of X, [less than]a, b[greater than], b=a or b NC-prolongs a or b arp-prolongs a.

IX A is a level-analysis of B

IFF B is a set of notes or rests and A is a set of sets of notes or rests and B is an element of A and every member of A except B is a next background to exactly one member of A.

X A is a level

IFF for some value of X and Y, A is an element of X and X is a level-analysis of Y.

APPENDIX II: CLAUSES FOR DEFINITION [IX.sub.h]

(1)...and the most background level contains either ST1 through ST4, ST1 through ST6, ST1 through ST8, or ST1 through ST16.

(2)...and for any timespan within some level that contains only pitches that belong to a certain hexachordal collection, some more background level must contain only pitches of the primary or secondary trichord of the hexachordal collection within that timespan.

(3)...and some level of A contains only the primary and secondary trichord of the primary hexachordal area and the primary and secondary trichords of the secondary hexachordal area in that order embedding a high-level schema.

[IVC.sub.h] x and y are n(set)-proximate with respect to C IFF C is a cyclic ordering of n different pitch classes and x and y are notes whose pitches are less than an octave apart and belong to proximate pitch classes in C.

Additional Clauses for [IVC.sub.h]

[IVC.sub.h]1 No two members of the primary trichord can be proximate in C, and no two members of the secondary trichord can be proximate in C.

[IVC.sub.h]2 In C, each pair of pitch classes forms an ordered pitch-class interval that is equal to or less than 6.

[IVC.sub.h]3 Exceptions: If none of the cyclic orderings meet both criteria [IVC.sub.h]1 and [IVC.sub.h]2, then if two members of the primary or secondary trichord are proximate in the cyclic ordering C that meets criterion [IVC.sub.h]2, reorder by generating the minimal number of order reversals to fulfill criterion [IVC.sub.h]1, even though the reordering breaks criterion [IVC.sub.h]2.

[IVC.sub.h]4 Interpreting a cyclic ordering C in pitch: each consecutive pitch distance is less than or equal to 6.

[IVC.sub.h]5 Number a cyclic ordering C of pitch classes.

[IVC.sub.h]6 The pitch class numbered "0" will carry the music-theoretical meaning of "first and last pitch class of the scale" which is the cyclic ordering.

[IVC.sub.h]7 Any interpretation in pitch and time of a cyclic ordering C with order numbers will be a strict simple ordering derived from the cyclic ordering C in which the earliest note has the lowest pitch and the last note has the highest pitch and order number 0 of the cyclic ordering C is the pitch class of the first and last notes.

[IVC.sub.h]8 The pitch class assigned order number 0 in the cyclic ordering C should be one of the pitch classes from the primary trichord.

[IVC.sub.h]9 Of the pitch interpretations of a cyclic ordering C that begin with pitch classes of the primary trichord, choose the ordering whose pitch intervals are most packed to the left.

[VC.sub.h] x and y are neighbors with respect to C

IFF x and y are time-adjacent and n(set)-proximate with respect to C ([IVC.sub.h]).

[VIC.sub.h] x and y NC-prolong z

IFF x and y are neighbors with respect to C (TVCh) and z is a note whose pitch equals the pitch of x or of y and whose initiation (value of [T.sub.2]) is the latest release of x or of y.
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Title Annotation:music
Author:SCOTTO, CIRO
Publication:Perspectives of New Music
Date:Jan 1, 2000
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