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Understanding Stefan Wolpe's musical forms.

IF NOTES ARE ASSEMBLED, a course of direction becomes tangible" ("Any Bunch of Notes: A Lecture," 296). (1) These words from Wolpe's lecture "Any Bunch of Notes" suggest two important directions in his thinking about musical form. The last word "tangible" reaches Out toward visual forms and the temporal aspect of painting, where each new brushstroke reconstructs and re-energizes a web of visual connections and the balance between form and empty space. The opening "If notes are assembled" (where the conditional includes a tantalizing if oblique reference to charged silence--what if notes are not assembled?) suggests the remarkable individuality of Wolpe's musical forms and his ideas about form, ideas as distinctive and charismatic as his ideas on pitch-class circulation, proportions, and the music itself.

This paper engages aspects of form in Wolpe's music, informed by his writings. It has three parts, devoted to conceptual issues, theory, and analysis. Part One explores Wolpe's ideas about form and some of their practical implications. Part Two develops a vocabulary and theoretic framework that support detailed analysis and critical study of Wolpe's musical forms. The reach of this theory is extremely broad, encompassing much contemporary music as well as common practice tonal music. Excerpts from Form for Piano (1959) illustrate these theoretic concepts, revealing subtleties of form that elude other approaches and providing a basis for stylistic comparison between compositions. Part Three presents a detailed analysis of several passages from Form IV (1969), the only other work Wolpe composed for piano solo after 1955. Closing remarks integrate Wolpe's ideas on form with their theoretic representations, and consider points of stylistic comparison and contrast between Form and Form IV based on aspects of asso ciative organization revealed by the analysis.


In music parlance, "form" has two common meanings. The first involves classification in terms of a repetition scheme, often articulated by tonal relationships as in ABA', sonata form, rondo, etc. Clearly, where Wolpe is concerned this will not get us very far. Large-scale returns are rare in his music, and on the few occasions they do occur it seems Wolpe did not think of them in the standard sense. (2) His writings avoid schema at all levels, even the terms "theme," "motive," and "gesture." In "Thinking Twice" (304), he instead refers to "organic modes," "genenc sets," "organic tasks" and "organic habits," (3) a phrase that connects nicely with "growth habit" in botany, a term that recognizes "the plant [as] a dynamic organism, forever augmenting and modifying its shape" (Bell 1991, 314).

If the musical work possesses a certain content, a significance, if it means something, its meaning is inherent in the work itself and is equally present in the whole. The content here cannot be external to what we call form; it is immanent in this form. ("To Understand Music," 9)

Wolpe's choice of words suggests that for him "form" is primarily a verb; it only becomes a noun when patterns of relation are abstracted from temporal process. This is closer to a second common meaning of "musical form": t he totality of events in a musical composition, or the "shape of a musical composition as defined by all of its pitches, rhythms, dynamics, and timbres" (New Harvard Dictionary of Music, 1986, 320). In his 1953 lecture "To Understand Music," Wolpe clearly identifies meaning in music with musical material, and form with content:

Significantly, though, Wolpe identifies musical form not with the events per se, but with their patterns of relation unfolding in time. His view of form--or, music formation--is eminently contextual and processive. Later in the same lecture he writes: understand a piece of music is to recreate its unique personality as it first emerged in the mind of the composer. This recreation does not require a memory capable of retaining the whole of the work from beginning to end. The synthesis proceeds progressively, moving with the flood of sound each moment of which thus bears in a sense the accumulated burden of the preceding moments, not because we remember them, but because we perceive each of them as direct functions of those which have preceded. ("To Understand Music," 17)

This idea that musical events are shaped and even constituted by their musical contexts is extremely important, and is a premise of my theoretic approach.

That Wolpe identifies musical form with the temporal process of content is not in itself a particularly distinctive point of view. Heinrich Schenker, for example, also equated form with content. (4) But many of Wolpe's ideas about form expressed in his writings and music do challenge and even overturn aspects of form in tonal practice too often presumed to generalize beyond the repertoire in which they developed. Two common assumptions involve the relationship between units of form and time: that units of form must be temporally discrete, and that they must be temporally adjacent to combine into larger units. These conjoin as prerequisites to a third common assumption: that musical form denotes a well-formed hierarchy of inclusion relations among parts, as in 19th-century Formenlehre and Schenkerian theory alike. An additional, related and often hidden assumption about musical form inherited from tonal contexts is that form is necessarily teleological, purposefully directed toward a climax, return, or final resolution.

Wolpe overturns all of these common assumptions about musical form to predicate his form-sense on simultaneity rather than succession, associative heterarchies rather than hierarchy, and fortuitous coincidence or juxtaposition rather than teleology. In the "Lecture on Dada," he says:
The concept of simultaneities is one of the most truly fascinating
Today it is a workable element, a concrete element you work with,
  like a drill or a hose.
    It is a tool.
It means what repeatedly happens anyway,
 what happens in a street situation, what happens on a canvas,
 what happens to every six, seven, twenty people of different
  [who] find themselves at a moment of junction.
What I call the panorama of activities,
where, within certain confinements, spatial confinements
 certain things absolutely opposed, absolutely dissociated,
 that have nothing to do with each other,
  are brought together. ("Lecture on Dada," 210)

For aspects of a single idea, this second, more interesting, sense of simultaneity is closely related to multidimensionality: it amounts to recognition of the musical object as multidimensional, and on two levels. First, it is a property of musical space transferred to the objects that reside there: musical objects form not in one musical dimension at a time (pitch, time, loudness, etc.), but in many dimensions at once. So each "object" is actually a simultaneity of characteristics and potentials, described in many--often functionally independent--dimensions. Second, multidimensionality engages musical objects as holistic entities, exposing different aspects over time. At this level, the idea is no longer comprised of dimensions--it has "dimensions," aspects heard successively but recognized as parts of one thing. Remarkably, Wolpe locates the synthesis of aspects that constitutes the object not outside or among these aspects, but within how we perceive them, positing a dense perception or "experience of den sity involving multiple aspects of the same thing" ("Lecture on Dada," 212). (5) For Wolpe, "The object exists always in a state of simultaneity. Simultaneity means that it really comprises all levels at any given moment" (Wolpe 1999, 402).

When the object is a simultaneity, each time-slice becomes a coincidence of events, and each "event" a nexus among moments or musical segments. These need not be temporally adjacent or proximate, but may be dispersed throughout a composition, with profound implications for musical organization over longer temporal spans. In "Thinking Twice," Wolpe epigrammatically asks, "Who is going to rule over the sequences of a hundred released simultaneities?" ("Thinking Twice," 306). As object-recognition gathers these moments into significant formal structures, it implies a view of form that is non-sequential and non-hierarchic. Again, Wolpe:
. . . the same thing can exist under a variety of conditions,
as we can exist under a variety of conditions--
        only that we bring this into one focus.
. . . one revises the concept of sequences.
Sequence didn't exist, because each instance of the sequence had to
  with all other instances of other sequences at the same time.
So what you had virtually was nothing but multiple instances
  knotted together. ("Lecture on Dada," 210-1)

Multiple instances knotted together. That is the crux of the issue: it suggests that for Wolpe's music (and much other contemporary music as well), the concept of form requires a radical rethinking. Form is not schema, not one privileged, best-nested, view of the web of relationships. But nor does it evaporate into the totality of events in a composition. For Wolpe's music, form involves discernible musical objects, knotty webs of relation, and the possibility that listeners and analysts might navigate through these webs in many ways. (6) The challenge to music theorists is to find a way to understand and model form in these terms.

Without sequence, without hierarchy, what becomes of teleology? Not surprisingly, Wolpe abandons teleology as an aesthetic principle, and instead adopts a position that revels in the depth and breadth of the moment as simultaneity and the propulsive energy created by fortuitous coincidence or juxtaposition among moments. From the "Lecture on Dada":
. . . you have in this kind of music . . .
 the concept of unforseeability,
  you cannot explain.
  It means you cannot infer what is going to happen.
That means that every moment events are so freshly invented,
 so newly born,
 that it has almost no history in the piece itself
  but its own actual presence.
 It has its presence, its now situation,
  and then the now situation is joined with another,
   with the next now--
    an unfoldment of nows! ("Lecture on Dada," 213-4)

Temporal process is no longer a play of expectations formed by a directed sequence of events that resonate with theoretic implications outside the piece, but the locus of attention in the particularity of events and their associations or fortuitous relations with one another. Where parts of the web of association are stronger or weaker, or seem to gravitate toward one moment through changing strengths of association realized in sequential time, one might indeed construe or construct a kind of teleology. But it is important to recognize that this is not teleology in the standard sense: rather, it is specific to a set of associations, their temporal disposition, and a certain analytical perspective. In Wolpe's music, one can only identify the goal, or even the direction, of such teleologies through the music itself.

The five aspects of musical form we have considered so far--form as the process of content, temporal disjunction, temporal adjacency, their combination to produce hierarchy, and teleology--have something important in common: they all approach musical form explicitly as it unfolds in time. Where traditional views of musical form focus on the sequence of discrete events in a single, linear, time, Wolpe's writings suggest a spatial concept with extension in multiple dimensions, inspired by his contact with visual artists in the Bauhaus, Dada, and later the New York School. (7) Three basic and more overtly spatial principles of formal construction recur throughout his writings: symmetry, asymmetry, and opposition. (8)

Symmetry and asymmetry are complementary instances of Wolpe's "proportions," basic perceptual categories appropriate to both music and the visual arts. (9) Symmetry amounts to equivalence under transformation. In painting, the transformation is rotation either within a plane, or translation or inversion about an axis within the plane; in music, it appears as reflection around an axis in the linear dimensions of pitch and time. Symmetries by pitch inversion and temporal retrograde are extremely common in Wolpe's music as they are in Webern's, with whom Wolpe studied for three months in 1933 (Wolpe 1999, 380). Wolpe's interest in symmetries and asymmetries as musical proportions also intersects with concepts of visual balance and basic forms explored by Bauhaus artists. In Wolpe's late music, symmetries become a characteristic organic mode or habit and might well be seen as an anchor for his other ideas about form. The equivalence that associates parts of a symmetry defines a musical moment, a clear and cognit ively distinct "now" that may be one aspect of a simultaneity, or a premise for opposition in "an unfoldment of nows."

On asymmetry, Wolpe writes: "Asymmetry is not born in sleep....This means that symmetrical midpoint formations, too, are necessary means to stir up quantities" ("On Proportions," 170). Whether this is a statement about compositional practice or about perception, it suggests that the route to asymmetry is indirect; symmetry is prerequisite. Symmetry and asymmetry might be seen as figure and ground; the equivalence that associates parts of a symmetry can serve as the basis for modeling asymmetry, subject to inclusion relations that indicate added or missing elements. Where transformational equivalence between parts of a symmetry renders it closed and static, asymmetry opens; it is kinetically charged. Wolpe's phrase "the asymmetrical tug of groups" ("On Proportions," 148) suggests the force of cognitive dissonance between an inferred symmetry and the actual asymmetry presented. (20)

Depending on musical context, asymmetry may sound like a skewed symmetry or an incomplete one. Two common ways asymmetries arise in Wolpe's music are through shifting axes of symmetry and by superimposing two or more symmetries. Wolpe notes that superimposed symmetries have an interesting perceptual effect: "With the abundance of increasing symmetrical nestings, the clear perception of symmetrical structures in a particular region as symmetrical correspondences within the total region diminishes, in favor of a tightly condensed and completely centralized mass sound" ("On Proportions," 140). A third technique involves change in one dimension (e.g., note values) superimposed on strict symmetry in another (e.g., pitch retrograde). On the variety of means to asymmetry and their implications for larger-level organization, Wolpe muses: "Perhaps there is no hierarchy in the succession of asymmetrical occurrences. Who, rather than counting on variety, would catalogue cloud forms, or count stars, grass, raindrops?" ( "On Proportions," 151).

Making perceptual and analytical sense of Wolpe's emphasis on opposition as a premise for musical form poses more of a challenge. In the 1953 lecture "To Understand Music," Wolpe identifies musical understanding with the perception of unity: "To understand a melody, a phrase, a musical work is to perceive its unity" ("To Understand Music," 16). If this is so, what musical understanding can come from opposition? What cognitive or analytical activity concerned with relationships can grasp opposition on its own terms? The very idea of opposition as a means for musical form seems paradoxical: how can opposition construct musical form, be form-making rather than form-breaking?

Once again I think it is important to remember Wolpe's connection with the visual arts. Opposition was the crux of Joharnnes Itten's pedagogical approach in the Bauhaus basic course Wolpe took in the early 1920s (11) and an aesthetic premise of Dada. Based on a survey of Wolpe's writings, and particularly his 1962 poetic soliloquy "Lecture on Dada," I posit three ways of understanding opposition as a premise for musical form.

The first correlates with multidimensionality An opposition implies a point of contact, a basis for opposition between two things: high and low in pitch, or soft and loud in volume. The opposition extends this point of contact into a line, which then defines a polarity and a dimension within the totality of action within multi-dimensional musical space. Reading "the double content of a single phase of action" as "opposition," this is essentially what I hear in Wolpe's words:

the double content of a single phase of action points to the multidimensional space. Its multiple dimensions are simultaneously in motion, meeting and intersecting at points of highly diverse and diverging directions. The total spatial traffic is engaged. The zigzags and curves and multispatial moves, the manifold opposites, with all their multiple modifications--isn't this the oneness of space in action? ("Thinking Twice," 299)

A second interpretation allies opposition with complementation, and therein to harmony and unity. This idea is intrinsic to Itten's color theory, where harmonious dyads, triads, and tetrads are sets of pigments that when mixed together yield a neutral gray. Itten's statement "Harmony implies balance, symmetry of forces" (Itten 1970, 19) recommends opposition itself as a basis for harmony, and means to unity. This idea is also evident in Schoenberg's practice of hexachordal combinatoriality, where complementation effects aggregate completion. As Edward Levy (1971), Martin Brody (1977), Christopher Hasty (1978), and Austin Clarkson (1993) have explained, the pitch-class structure of Form for Piano (1959) is based on complementation between members of the Z-related set classes 6-3[012356] and 6-36[012347]. But in general Wolpe's own ideas about chromatic completion and focusing or refocusing individual pitch-class cells in a circuitous trip through the aggregate tend to limit clear examples of pitch-class comple mentation in his music.

The third interpretation addresses the perceptual question: what does the mind, the ear, the eye, do with opposition? In the "Lecture on Dada," Wolpe offers this answer--surprising in that it so deftly resolves the paradox of opposition and musical form. He writes:
I remember the experience of the adjacencies of opposites.
    That was a new concept--
      a new experience which then became a concept...
           that all things are in the reach of the human mind,
              and that to connect is a mental act,
                 depending on the will to connect. ("Lecture on
Dada," 205)

And, later:

[Dada] has become the formal element which allows the two opposites,
  two completely disparate things to be brought together...
Now that leads to great consequences, namely,
  that the opposites as such disappear. ("Lecture on Dada," 209)

In other words, given opposition, the mind seeks connection--the mind will make a connection. Opposites attract, rather than repel; they form rather than divide. In Wolpe's words:
But that two things coexist in a totally isolated, dissociated form,
  and you still receive a form,
  the sensations of a strange relationship of estrangements,
    that was fantastically new to me. ("Lecture on Dada," 209)

This interpretation resonates nicely with the idea of opposition as a means to unity.


We now establish a vocabulary and conceptual framework to support detailed analysis of form in Wolpe's music sensitive to the ideas outlined above. This work draws from a general theory for musical segmentation and music analysis I have developed more fully elsewhere (Hanninen 1996 and 2001). For the reader's convenience, a glossary appears at the end of the paper. My ideas on segmentation draw inspiration from the work of James Tenney (1986), Tenney and Larry Polansky (1980), and David Lewin (1986), as well as from Christopher Hasty (1978 and 1981b), who developed his theory of musical segmentation in the context of analytical work on Wolpe's music. But there are important differences, particularly with respect to musical form.

In the course of this theoretic exposition, we will apply individual concepts to the analysis of passages from Wolpe's Form for Piano. Form is an ideal subject for a preliminary study of aspects of form in Wolpe's music, with its diverse musical formations, modest means and scope, and relative familiarity due to the analytic attention of Levy (1971), Brody (1977), Hasty (1978), and Clarkson (1993). Some of my examples recall points made by one or more of these writers, but place them specifically in the context of form and a new approach to the study of form in Wolpe's music. I have selected individual passages partly for purposes of illustration, but primarily so that cumulatively, as a set, they suggest aspects of form in the composition as a whole.

We start with a definition: a musical segment is a grouping of tones an analyst recognizes as a readily audible unit. A segment is an element of form. Segments are supported by one or more criteria. A criterion is a rationale for cognitive grouping of musical events, a reason to hear one grouping of tones among or instead of others. I draw a fundamental distinction between two domains of music discourse, the sonic and the contextual, and between two basic types of segmentation criteria associated with these domains, sonic criteria and contextual criteria. (12)

Activity in the sonic domain involves psychoacoustic attributes of individual notes such as their pitch, duration, loudness, or articulation. The entire range of possible values an attribute might assume constitutes a sonic dimension. Pitch, duration, and loudness, for example, are three different sonic dimensions. A sonic criterion responds to changing attribute values within a sonic dimension. When the change in values--i.e., the interval--between two adjacent notes exceeds both that between the first note and its predecessor, and the second and its successor, that sonic criterion defines a boundary that implies a grouping. (13)

Example 1a shows the action of three sonic criteria in the opening measures of Form, translating observations made by Levy, Brody, Hasty, and Clarkson into my terminology. In measure 1, uniform durations and dynamics denies these dimensions any segmenting power, while the largest leap between F4 and Bb4 places a sonic boundary in pitch between the opening dyad and the tetrachord that follows. The "extra" quarter-note duration that separates the E4 in measure 1 from the Bb5 in measure 2 creates a sonic boundary in duration; this grouping is reinforced by coincident sonic boundaries in pitch created by the large interval from E4 to Bb5, and dynamics, as mezzo forte overtakes the opening pianissimo. (14)

In a remarkably detailed and perceptive analysis of these two measures, Hasty proposes a number of other groupings based on associations either evident within these bars, or between a grouping here and one later in the composition. (15) Such associations involve not attribute values of individual notes, but associations between groups of notes in their entirety. They lie in a different domain--what I call the contextual domain. Discourse in the contextual domain attends to equivalence or similarity between groups of sound-events, relations that arise only at the group level within a musical context containing both groups. Observations such as "same pc set," "same rhythm," "same set class," etc., all lie in the contextual domain. Individual contextual criteria identify the means for association, as in Example 1b, where both belong to set class 3-7[025] and pitch-spacing <23> is repeated by inversion. (16) I name contextual criteria with a shorthand notation: C stands for contextual criterion; the subscript giv es first the contextual subtype (e.g., set class), then the element that defines the association (e.g., set class 3-7[025]). While there are some guidelines for assessing criterion strength within contextual subtypes such as pc set or set class, and between nested dimensions (e.g., pitch and pitch-class), it is primarily the potential to associate two or more groups of tones within a particular musical context that determines the functioning strength of contextual criteria.

Individual criteria come to support segments through a complex web of coincidences and conflicts with the action of other criteria. This view roughly accords with those of Hasty (1978, 1981b) and Tenney and Polansky (1980). (17) A detailed account of points of intersection and difference would take us too far afield, but suffice it to say that in my view both segment recognition and segments' perceived relative strengths resist quantification and instead involve and convey analytical interpretation. As interpretations, they are intersubjective--that is, mutually intelligible--for reasons we may locate in musical particulars.

From individual segments as elements of form, we proceed to sets of segments as agents of musical form. Here we do not presume temporal adjacency, but instead ground our approach in contextual criteria and their power to associate segments over various temporal distances. This approach reconceives the study of musical form in terms of particular webs of association among segments, and how these associations are actually disposed in time in particular pieces.

Once again we start with a definition. An associative set is an unordered collection of two or more segments interrelated by contextual criteria. Each segment in an associative set relates to at least one other by one or more contextual criteria; conversely, every contextual criterion shown to contribute to the set must support at least two of its segments. We name associative sets with letters, e.g., associative set A; and individual segments with the set name followed by a number that reflects chronology in the score, e.g., Al, A2. (18) An associative set may have any number of segments; the segments one includes are a matter of analytic interpretation.

Segments within an associative set may be interrelated myriad ways. In some sets, contextual criteria create such a complex web of similarity that no further organization emerges. In others, they suggest subordinate, internal groupings of segments called associative subsets. An associative subset is a set of two or more segments embedded in a larger set and distinguished from its total membership by the action of additional, often stronger, contextual criteria. (19) Whether one models a certain group of segments as an associative set or an associative subset is a question of analytic interpretation.

Example 2a shows five segments from the first two measures of Form, named Al through A5. Example 2b organizes segments A2 thu A5 into a contextual association graph, a system of named nodes representing segments in an associative set, connected by arrows or boxes that indicate contextual associations among segments. Different weights of line and dotted lines indicate stronger and weaker segments according to the key at the upper right. Each arrow is labeled with the strongest supporting contextual criterion and a description of the transformation involved. The graph is basically atemporal, although segment names suggest chronology and arrows suggest temporal precedence within node pairs. The graph shows four associative subsets, partitioned into two pairs of subsets, [A2-A3] + [A4-A5]; and [A2-A4] + [A3-A5]. Each pair accounts for every note in the score. The A2-A3 and [A4--A5, pairing is reinforced by the strong sonic disjunctions in pitch, duration, and dynamics at the end of measure 1 noted earlier. The s econd pairing, A2-A4, A3-A5, relies on contextual criteria--specifically, repetition of the pc sets {F, A[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII], B[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]}, and {E, G, A}--subject to inclusion relations. This pairing is less audible for several reasons, most notably the lack of clear sonic support in measure 1. (20) Yet it illustrates an important technique in Wolpe's music: what is only tenuously presented at the outset comes into focus over time as subsequent, sonically stronger, segments articulate and activate grounds for association only latent in earlier, sonically weaker ones.

Temporal disposition of associative sets or subsets is another area of interest. We define two basic temporal dispositions: condensed and dispersed. In a condensed disposition, all or most segments of an associative set are temporally adjacent (perhaps overlapping) or proximate. In a dispersed disposition, all or most of the segments are temporally nonadjacent or relatively distant, such that the span of time over which the set is active well exceeds the sum of durations of its member segments.

Example 3a shows measures 27-9 of Form, the transition from the original pair of 6-3[012356]/6-36[1012347] hexachords to their transpositions by [T.sub.5]. Example 3b organizes the seven chords into three associative sets, each in condensed disposition. Sonic disjunctions in duration--both the distance between attack points and the changes in note values from one set to the next-and changes in articulation clarify the associative set structure in this passage. (21)

Example 4a shows four segments of an associative set E in dispersed disposition. E1 is the opening hexachord from measure 1 and corresponds to Al above. (22) E2, from measure 3, is a reordering of the opening hexachord that retains all pitches but E-natural in register. After a clear pause at the end of measure 10, E3 begins a new section in measure 11 with a recollection of El, again in even quarter-notes and at pitch level but with the first two notes reversed. E4 is from measure 59, where the opening tune returns an octave lower than before, with repeated notes added. Example 4b shows the contextual association graph: E1 associates closely with E3 and to P4, in different ways, but less so with P2. This inverse relation between temporal proximity and contextual association helps cast P2 as local and developmental, and E3 and P4 as events of more global significance. This associative organization within the set recommends El as a focal point; I call such a segment a contextual focus. (23) Within set B, it i s interesting that El, the contextual focus, is not only the first member of the set we hear, but the first clear segment in the piece as a whole. Starting the piece with E1 sets up the formal process that brings different facets of the opening idea into focus as the piece proceeds. E1's bland sonic profile contributes to its viability as a contextual focus: with its internal segmentation undefined, many relationships are latent therein.

Dispersed associative sets are common in Wolpe's music. Segments within dispersed sets often stand out from their surroundings in clear figure-ground relation (see Example 5). The four wiry chromatic springs in measures 14, 17, 20, and 39 form another dispersed associative set, F. Members of F are associated by pc content (their union gives the chromatic pc set from B[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] to E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] {e0123}), rising contours, an end on E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII], and short, even durations that stand out from their surrounds. Unlike set E, set Flacks a clear contextual focus. While the web of associations in set E has a clear profile, set F is more democratic. This is a qualitative difference between the two sets, a subtle distinction in their respective contributions to musical form. (24)

Like segments, associative sets need not unfold neatly one at a time. We define two basic ways associative sets (or subsets) may unfold with respect to one another, in monophony and polyphony. These are subject to infinite gradations, combinations, and refinements. Example 6a is a score for measures 43-56 (for now we focus on measures 48-56; measures 43-8 are included for later reference). Example 6b shows eleven segments from two associative sets H and I, from measures 48-56, in a cutaway score format. Each set, and chronological time, proceeds from left to right. Members of H share the pc set {9t0123}, or [T.sub.5] of the hexachord in measure 1; those of I, pcs from its complement (the pc set {45678e}). As the cutaway format makes clear, the two unfold in polyphony. Example 6c shows a contextual association graph for each set; vertical alignment indicates simultaneity. Pitch retrogrades recommend the associative subsets H1-H2 and I7-I8; invariance of a pitch dyad, the pairings I1-I2, I3-I4, and I5-I6. In th e music, the polyphony between sets H and I is modified by sonic continuities that interweave segments in one set with those in the other to produce unity between complements. Returning to the score in Example 6a, notice how the polyphony of sets H and I changes over time. The sets clearly contrast in measures 48-51, where six chords from the I set embellish individual notes within a long line that describes the single segment H2. Connections between the sets develop in measure 52 with the entrance of I7, which, with its high register and long notes, sounds like a continuation of H2. Pitch retrograde connects I7 and I8, but sonic continuity in note values, articulation, and dynamics reconnects I8 with the long, unaccompanied melody of H3. (25)

The theoretic concepts we have introduced support a view of Form based on the organization and disposition of segments and associative sets in time, and their interactions with the pairs of complementary pitch-class hexachords P = {45789t}/P' = {e01236} and [T.sub.5]P = {9t0123}/[T.sub.5]P' = {45678e}. The P/P' and [T.sub.5]P/[T.sub.5]P' pairs inform pitch-class structure in measures 1-27 and measures 59-62; and measures 30-58 and measure 63, respectively (Levy 1971, Brody 1977, Hasty 1978, Clarkson 1993). In some respects, associative organization and pc hexachord structure reinforce one another. Members of the prominent set B that opens the composition and returns at measures 11 and 59 outline the hexachord P; the wiry springs of set F derive from the complementary hexachord P' (in measures 14, 17 and 20), or from [T.sub.5]P (in measure 39). The tight associative organization of sets Hand I that partition the surface in measures 48-56 neatly articulate their [T.sub.5] counterparts [T.sub.5]P and [T.sub.5]P' , respectively. Sometimes studying associative organization refines an analysis based on pitch-class sets or recommends multiple interpretations of a single passage, as in measures 1-2 where strong sonic disjunctions and their interactions with pitch-class repetition suggest alternative segmentations of a single passage. It can also reveal contradictions between associative organization and pitch-class zones, as in measures 12-3 where the association between the melodic lines <E, F, F, G> and <D, C, C#, D>, each legato and embellished by a grace note, serves as a transition from the P to the P' hexachord.

Reinforcement, refinement, and contradiction are three basic ways associative organization and pc hexachord structure interact in Form; the realm of realized possibilities is far more subtle and complex. The transidon from the initial pair of hexachords P/P' to [T.sub.5]P/[T.sub.5]P' in measures 27-9 (Example 3a) is an interesting case. This passage exhibits what is arguably the clearest associative set organization in the piece: sets B, C, and D all involve strong contextual criteria (pitch repetition), their segments are connected by common tones and stepwise voice leading, and all three sets are in condensed disposition. If this tight associative organization seems at odds with the role of a transition, careful consideration of both local and long-range musical contexts suggests a more nuanced view of how this passage functions in Form as a whole. Locally, the chords in measures 27-9 subtly recall two pairs of chords in the low register in measures 23-4, each solidly within the realm of pc hexachord P = {1 45789t} and connected by semitone motions. They also foreshadow the pair of six-note chords in the high register in measure 30 (similarly connected by common tones and stepwise motion) that articulate the pc hexachord [T.sub.5]P' = {45678e}, which shares four tones with P. Based on these similarities in voice leading, pitch-class content, and continuity in register, the passage in measures 27-9 becomes the center of a longer transitional zone that extends from measure 23 at least through measure 30. Its tight associative organization focuses the process of transition precisely where the pitch-class structure of complementary hexachords breaks down--as the P/P' pair gives way to the emerging [T.sub.5]P/[T.sub.5]P' pair. Here the strengths of associative organization and hexachordal pitchclass structure stand in inverse relation, interacting as complementary--not contradictory--means for transition.

Recognition of a long-range association between measures 27-9 and the return to a homophonic texture at measure 43 provides yet another perspective on the passage and on relations between associative organization and pitch-class structure in Form (compare Examples 3a and 6a). Once again, the move to a homophonic texture involves strong voice leading and dynamic contrast between chords that define clear associative sets. The long-range association between measures 27-9 and measure 43 and the measures that follow transforms the earlier passage into a point of reference, a passage in its own right rather than only a transition between passages. As measures 43-8 unfold, they redefine the relationship between associative organization and pitch-class structure. Throughout this passage Wolpe uses only pcs of the [T.sub.5]P hexachord, for a sharp focus on pc structure enhanced by pockets of frozen register. Following the initial chord sequence (measures 43-4), Wolpe undoes the chords rhythmically, transforming simul taneities into successions (measure 45). He then projects the D into the highest register (measures 45-6), and transposes the chords' pitch material up an octave where not only the chords, but any possibility of clear segments to form associative sets, dissolve in a cloud of activity (measure 47). In contrast to measures 27-9, this passage begins with a solid correspondence between tight pitch-class structure and tight associative organization. Over measures 43-8, Wolpe gradually decouples the two until associative set organization vanishes in measure 47 (for lack of clear segments), while pc hexachord structure remains clear (in the frozen pitch field). The result is a reversal of the relationship between pc structure and associative set organization in measures 27-9--a reversal that is particularly interesting given the clarity of the initial association between measure 43 and measures 27-9, and the realignment between clear pc structure and tight associative organization that immediately follows in measure s 48-56. An analytical approach that considers both pc hexachord structure and associative set organization can get at these shifting and complex interactions, so that we may contemplate their intertwined and perhaps strategic contributions to musical form.


We now turn to Form IV (1969) for a detailed study of musical form in the opening passage based on segments, associative sets, and their temporal disposition. This analysis focuses on associative set organization in measures 1-11, and some longer-range connections with moments later in the piece. Example 7 provides a score for measures 1-17. (26)

Several features mark measures 1-11 off as a distinct passage. The half-rest in measure 11 is the longest silence to that point and, ultimately, in the composition overall. Changes in tempo, dynamics (piano to mezzo forte), and note values (running sixteenths to lyrical eighths) flank the rest, creating a strong boundary in the sonic domain. Contextual criteria also contribute. While two contextual criteria bridge this boundary--the falling minor third repeated from left hand to right in the eighth-note figures E-C# and E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]-C ([C.sub.ip-3]), and the pc ordering <E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII], C> within the right hand ([C.sub.pc <30>])--these are weak compared to those active only on one side or the other, such as the pitch-class transposition [T.sub.1] that relates the first four notes of the sixteenth-note gesture in measure 10 to that in measure 11, or the pitch transposition [T.sub.2] (with order reversal in the opening dyad) between the first two phrases in the right hand, measures 11-3. These stronger contextual criteria set up fields of attraction that pull away from the sonic boundary in measure 11, reinforcing it with indirect opposition in the contextual domain.

Typical of Wolpe's late music, this opening passage is full of rapid changes in texture, dynamics, register, and durations. Its fickle surface belies the underlying coherence in pitch-class vocabulary limited to the eight-note chromatic set from B [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] to F [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] ([te012345]), but for two exceptions: measure 8, where G and A are introduced in a prominent left hand move, and measure 9, where A returns in the left hand. Example 8 is an analytical segmentation of measures 1-11, again in cutaway score format. Its 49 segments reflect my interpretation of the most musically significant units. Based on recurrence of pitch and pitch-class sets, intervals, rhythms, and behaviors (e.g., ordered sets, vs. partially or unordered sets, long or short segments, etc.), I organize these into five associative sets A through E, plus the move from G to A in measure 8, which stands outside all of these (shown in box).

Some points on notation. Segments are shown one of two ways: most, as before, by a new staff, clef, segment name, and measure number; or, for larger segments that embed one or more smaller ones, with a box around two or more such notations. (Segments are named in chronological order, first by starting note, then by ending note. A segment that includes several smaller segments is numbered chronologically after the last of its subsegments.) For each segment I indicate the one or two strongest contextual criteria active within measures 1-11. In all cases, sonic criteria contribute to segment definition, but are not shown.

Some points about segments, segmentation, and associative sets both here and in general. Notes shown as adjacent within a segment need not be temporally adjacent in the score (see C1, C#-D-E). A segmentation is not (necessarily) a partitioning of the score-notes may contribute to more than one segment (within a set, as in C1 and C2, or between sets, as in C1 and A1), or may be omitted (although here none are). While not so here, sets can even share segments if the reach of similarity relations among segments within two sets overlaps. (27) Associative sets, then, can be either fuzzy or crisp; investigating their qualities in particular cases is part of the analytic process and its findings.

A quick scan of the segments in each set and their supporting criteria suggests each set's distinguishing characteristics. With only two members, set E has clear criteria for membership. El and E2 are related by pitch inversion around the three shared notes C, D, and E. Criteria for membership in the more extended sets C and D are also clear; for each, I indicate primary contextual criteria or other distinguishing features. Members of set C generally exhibit one of two pitch-class orderings, <C#, D, E, F> or <B, C, E, F>, each realized in close spacing. These two orderings define distinct associative subsets. Members of set D tend to involve the pcs {D, Eb, B, F} with prominent semitone motion and symmetrical construction as an organic mode. For sets A and B it is much harder to name criteria for each set as a whole, due in part to their greater number of associative subsets. This suggests one kind of qualitative difference between associative sets: some, like sets C, D, and B, are tightly organized with one or two clear criteria for membership, while others, like A and B, are loose, with many criteria supporting segments within the set but no criteria presiding overall. This difference is one of many that shape musical form.

The cutaway score format makes it easy to see which sets are active at a given time. A survey of set activity in measures 1--11 proves revealing. Sets A and B are never active at the same time, but together account for most of the score. There are a few prominent exceptions: the end of measure 2 and measures 6--7, where set D is active; and the start of measure 10, where set B appears. Throughout the passage, sets A and B tend to alternate, and set A gradually gives way to set B. Each set has a typical mode of behavior: set A tends to move in clusters of short segments; set B, longer segments, almost always in pairs. A look at the criteria involved suggests segments that are temporally adjacent or proximate tend to be more closely related to one another, than to set members farther away. Each cluster of temporally adjacent segments in set A, and each pair in set B, comprises a condensed associative subset, and alternation between sets A and B articulates associative organization within each set.

Complementing the alternation between sets A and B in measures 1--9 is set D, which emerges in the two points of hiatus, measure 2 and measures 6--7. The cluster of segments D2--D7 in measures 6-7 partitions into pairs in two dimensions described by temporal adjacency (e.g., D2--D5) and by simultaneous pitch inversion (e.g., D2-D3). These pairings, particularly the pitch inversion, are distinctive organic habits of the set, evident even in the isolated D1 which includes the pitch inversion E-Eb, E--F, and the temporal succession <[D, E}, {Eb, F}> via pitch transposition [T.sub.1].

Drawing notes from segments in other sets and active only intermittently, set C is a melodic overlay. Its segments often come in pairs, but this is not consistent. Segments of set C often duplicate notes found in segments of other sets; notes adjacent in segments of C are not always temporally adjacent in the score (see, e.g., C1, C4, and C6). These segments can be harder to hear, particularly at the start of the piece (or analysis) before the orderings that define set membership are established. But therein lies their interest: what is initially vague becomes clarified; threads of an idea become cloth.

Studying associative organization within sets proves as interesting as relations between sets. Sets A and B generally unfold in clusters or pairs of segments that suggest associative subsets, but these subsets vary considerably in clarity and strength. The perceived strength of an associative set or subset derives from four factors: the strength of its segments (which involves sonic as well as contextual criteria), strength of supporting contextual criteria, number of segments in the set, and extent to which sets can be represented by a few primary contextual criteria rather than many criteria with no clear priority among them. In measures 1--11, set A has five clusters of segments; associative subset organization is clearest within and between the third and fourth (in measures 4 and 5), less so in the second, and weakest in the first and fifth (measures 1 and 7). Example 9a is a contextual association graph for six of the eight segments in the third and fourth clusters (measures 4 and 5); each cluster appea rs as a subgraph. Different weights of line and dotted lines indicate relative strengths of connection as shown in the key at upper right. The two subgraphs are isographic; that is, they exhibit the same pattern of nodes and arcs. Moreover, comparable arcs are labeled with contextual criteria of same subtype. The strongest associations involve pitch orderings in the right hand (shown with horizontal arrows at the top of each subgraph that indicate repetition in measure 4 and retrograde in measure 5). The relation of right hand to left by pitch inversion (diagonal arrows) is also strong, but not as strong. The two subgraphs are connected, by still weaker criteria--two pitches shared in the right hand (ES, D#6) and two pcs in the left (e, 0). It is musically obvious that the material in measures 4 and 5 is similar; segmentation and contextual association graphs clarify the nature and extent of this similarity.

Contrast this situation with the more fuzzy associations among Al, A2, and A3 in the opening cluster (Example 8). Register distinguishes Al from A2; the two segments share the pc ordering <C#, D, Eb>, but the more audible relation may well be the semblance of contour inversion starting on C#. A3 replicates the pc content of Al in what sounds like a varied contour repetition of A2, but is hard to grasp with a contextual criterion. (28) The result is much like the bland tune that opens Form: once again, hazy relations at the outset constitute potential to refocus later; as in Form, the strongest associations come only much later, in measures 47, 48, 56, and 66, which all clearly recall A2. (29)

Associative organization within set B is similar to that in set A in that it involves a series of many small subsets. As noted above these are generally pairs of segments rather than clusters, but the segments are longer, and criteria that associate segments within pairs, stronger. Example 9b is a contextual association graph that organizes nine of the ten segments of set B into four associative subsets. Supporting criteria involve varied repetition of ordered segments with four to six elements, to within retrograde. The dotted curved arrow from the first subset to the second indicates an association between subsets. These subsets, B1-B2 in measure 3, and B3-B4 in measure 5, come before and between the two subsets of A shown with isographic networks above. The result is an alternation not only between different musical ideas identified with two sets A and B, but also different strengths of criteria. Changes in the pattern of the associative fabric are intensified by changes in the tightness of weave, another important aspect of musical form.

The pairing of segments in set B, and alternation between subsets of A and B, exemplifies a propensity for duple associative organization in the moment-to-moment flow of Wolpe's music. The Trio in Two Parts for Flute, Cello and Piano (1963-4) and Piece for Trumpet and 7 Instruments (1971) are full of this kind of thing. If this recalls binary structures in tonal music or some twelve-tone works of Schoenberg (e.g., the Suite, op. 25), its sources are different. Duple organization in antecedent- consequent phrase construction and larger binary forms of the Classical and early Romantic periods derives from tonal opposition between tonic and dominant; in Schoenberg it often reflects twelve-tone combinatoriality, or the operations of inversion or retrograde. (30) In Wolpe's music, duple organization arises more directly from the action of individual and diverse contextual criteria through repetition or varied repetition, of which pitch inversion and temporal retrograde as means for symmetrical construction are sp ecial cases; it is the minimum to required define a "now," a facet of a musical idea. Only when these facets have been formed with sufficient clarity can they be successfully juxtaposed, or even opposed, as in the alternating action of associative sets A and B.

Example 9c is a contextual association graph for the eight segments of set C. Boxes indicate two associative subsets, defined according to the primary criteria on Example 8. Reading across the segment names in the top row points up a disparity between the associative organization into two subsets and the segments' chronology implied by the names C1, C2, etc. The graph shows that C1 is most closely related to segments C2 and C8. The association between C1 (measure 1) and C8 (measure 6) through the ordered pc set <C#, D, B, F> repeated one octave higher flanks the web of associations among C3, C4, C6, and C7, via the ordering <B, C, E, F>. In the music, this misfit between associative and temporal proximity is subtle rather than obvious. When C1 enters in measure 1, no context yet provides an incentive to recognize its supporting contextual criterion as such; and, with its notes embedded in other segments, C1 has only weak support in the sonic domain. Further, none of the segments C3, C4, C6, C7 is particularly strong in the sonic domain, so even the web of associations among them is implicit rather than explicit. But rather than deem these reasons to dismiss the misfit as a curious analytical artifact, I think they help account for the peculiar flavor of how I hear C8 in measure 6--a strong melodic line that is at once strangely familiar and right, yet also unprecedented. The misfit grounds this curious perception in musical part iculars: C8 does have a precedent in C1, but that precedent is obscure as precedent until C8 arrives; further, C8 associates with the segments C3, C4, C6, and C7, but less clearly than with C1 and C2, which are associatively closer, but temporally more distant. When sets A and B pause in measure 6 and C8 and the nexus of relations in set D emerge, an associative layer peels off, revealing a dimension previously hidden, but immediately engaging.

The complexity of associative organization we have explored in the opening passage suggests the individuality and multidimensionality of Wolpe's musical forms, and both the detail and scope of analysis required to weave their various associative threads into a tapestry. We now consider some long-range connections for segments of sets A and D, and how local and long-range relationships can work together in the context of the whole.

Example 10a shows four core segments of an associative set F, with the primary contextual criterion the pitch-class dyad {F, E}, spaced as an F with an E eleven semitones higher. A full extension of set F by this criterion would contain many more segments; these are simply the strongest. Similarly, Example 10b shows three core segments of an associative set G, with primary contextual criterion the pc ordering <EF> realized as a rising semitone. The two primary criteria for sets F and G share the pc content ({E, F}), but differ in its realization--moving the E up or down by one octave turns each idea inside out and into the other, which suggests that sets F and G might be seen as two parts of an associative superset. The two sets have distinct organic habits: members of F generally involve arpeggiation; those of G, a persistent oscillation or trill. Four of the seven core segments of sets F and G appear in measures 50-65, making these bars an important focus for set activity.

The primary criteria for sets F and G gain prominence only well into the composition, but they are implicit from the start. The (F, E) dyad identified with set F defines the extremes of the bassline in measure 1 that is Al. The <EF> semitone of set G links the two associative subsets of set C and shapes the prominent melodic cadence of D5 (measure 7). Additional connections among sets A, D, F, and G play out in measures 66-70 (see Example 11), a passage just after the focus of activity in sets Band G. We look first at the web of relations among members of sets A and D, then proceed to relations between members of these sets and those of B and G.

Example 12 shows seven segments of set A (A21--A28) and eleven more of set D (D8--D19), that together account for all notes in the passage except those in the left hand of measure 69. (Of course, one might recognize additional set members in the intervening bars; I name only those I discuss.) A21 in measure 66 stands out as one of the three strong returns of the opening alluded to earlier; strong contrasts in register, dynamics, and associative pull versus the end of the <EF> trill from G3 highlight its entry. A22 and A23 interact with A21 much as Al, A2, and A3 did in measure 1. D8--D10 strongly recalls D5--D7 as a literal repetition concatenated with its retrograde; D11--D18 follow from this local reference point. Example 13 is a contextual association graph for segments D1--D10. The layout of segment names indicates absolute consistency between ordering and strength of association in two dimensions, time and register. The process of accretion from associations within a single segment (Dl), to a pair of seg ments (D2--D3), to two pairs of segments (D5--D10) suggests a kind of teleology realized with the arrival of D8--D10 in measure 67. The force of the long-range connection derives from the pitch inversion that binds D2 and D3 into a single unit, D4, defining a "now" so distinct from its surrounds that its return sixty bars later is easily recognized in D5--D7. The quality of recognition is markedly different from that between C8 and C1 discussed earlier; here, a clear referent grounds the long-range association; there, an obscure referent leads to a sense of fuzzy recollection.

Returning to Example 12, many of the eighteen segments of sets A and D in measures 66--70 embed the primary criteria of sets F and G; these associative echoes between sets are shown with dotted rectangles and ovals as per the key at top right. Heard in context, the (F, E) dyad in A22 in measure 66 resonates with the recent arpeggiations from F up to E in F2 and F3 (measures 50 and 59, respectively), and <EF> oscillations in G2 and G3 (measures 60 and 64). The repeated (F, E) dyad in A23 has no counterpart in the opening cluster of A1--A3, but here adds to the texture of local associations; it is soon answered by the same two pitches in reverse order in A26 (measure 68). Similarly, the prominence and richness of D8--D10 (measure 67) secured by the long-range association with D5--D7 (measure 7) is further enhanced by connections between these segments and sets F and G. The melodic line <EFE> of D8 (and D10) recalls recent action in segments of set G, while the (F, E} extremes of the opening and closing vertical ities of D10 instantiate the primary criterion of set F. D10, then, becomes an important point of conflux among three associative sets, D, F, and G. In the passage as a whole, note that segments of set A mainly recall the (F, E) dyad of set F and secondarily the <EF> oscillation of set G, while the reverse is true for the segments of set D. These connections between segments of sets A and D, on one hand, and F and G, on the other, open a channel of communication between sets A and D. The result is a complex web of similarity relations among the segments of set A, through those of sets F and G, to members of D, that defies any single view of musical form. The form can be heard and studied from many perspectives, through the various strengths of connection among segments, and how these are arranged in time.


I close by considering three issues raised by this approach to the analysis of musical form, first in Wolpe's music, then in general. These are: how the theory models some of Wolpe's ideas about musical form; how it can inform comparative and style analysis of Form and Form IV; and the role of interpretation in my analysis.

Simultaneity, symmetry/asymmetry, and opposition are three primary components of Wolpe's form-sense; the theory provides means to model and illuminate all three. The idea of simultaneities connects with that of associative sets, and in two respects. An associative set is a set of segments, but more fundamentally it is a set of associations among segments: every segment in the set must associate with at least one other by some contextual criterion. Simultaneities might be seen as patterns of associative activation set up by each member of an associative set, relative to all others. Situations such as the conflux among sets D, F, and G in measure 67 of Form IV suggest a second interpretation of simultaneity: it can also be seen as a tangent between two or more associative sets.

Symmetry is easily modeled by contextual criteria that indicate the grounds for equivalence under transformation. To model asymmetry, contextual criteria for a referential symmetry are modified by inclusion relations to handle the misfit between their associative power and segment boundaries. But while this is an expedient solution, it is not complete: since contextual criteria are predicated on repetition, in principle they cannot grasp asymmetry as such.

The theory also sheds light on our three views of opposition. In the first view, opposition defines a polarity and therein a dimension of sonic space as it contrasts high and low, soft and loud, etc. Opposition in this sense is simply a sonic disjunction great enough to activate that sonic dimension in perception, and establish clear boundaries between segments, and even sections, as elements of form. (31) The second view conceives opposition as complementation that confers unity. Here the connection with aggregate completion and intervallic relationships between complementary set-classes suggests the action of contextual criteria, such as between the Z-related hexachords in Form. The third view is the one I find most interesting: it suggests that we simply do not perceive opposition as such; rather, the mind so seeks to connect, that it may forge a connection. This may occur in the sonic domain, cast as a limited range of values in some dimension, much as one might connect Dorothy's slippers with a fire eng ine and say "well, they're both red." More subtle, though, is the question of what this means in the contextual domain. The "will to connect" indicates that the mind must search for an association; none is readily apparent. This suggests that the contextual criteria available are very weak, as in "both gestures go up." Just how persuasive these criteria turn out to be depends on musical context. If they relate two "opposing" segments with no stronger ties to other segments, they may actually define a weak associative set. But if the opposing segments do belong to strong and disjunct associative sets; these sap the weak criterion of binding force. The sets create opposing associative fields that pull the segments in question away from one another. Opposition in the contextual domain, then, is not itself a force, but rather the residue of attraction. This has interesting implications for the perception of continuity in musical form, in that if one seeks a connection under these conditions and in some small way still succeeds, once through the point of opposition the stream of musical associations flows easily through less-resistant and more constructive channels.

Wolpe's ideas about musical form, and the theoretic framework I have proposed, provide a starting point for comparative stylistic analysis of Form and Form IV. In contrast to the clear hexachordal pc structure of Form, Form IV is characterized by gradual chromatic saturation that comes with the growth and accretion of gestures (and associative sets) defined in part by pitch-class chroma. The opening eight-note chromatic set {et012345} accounts for all but a handful of notes in measures 1-11: it yields the characteristic pc material of set A, set B (whose members tend to emphasize pcs e, 0, 1, 2, and 3), and set D (emphasis on pcs 2, 3, 4, and 5). As the piece goes on, sets Fand G develop, focusing on pcs 4 and 5 in different configurations. Where in Form pc structure based on pairs of complementary hexachords ultimately confers large-scale balance among pcs in the aggregate, in Form IV the aggregate remains lopsided--pcs {et012345} tend to preside through their association with key gestures, while pcs 6, 7, 8, and 9 are subordinate in the piece overall. Some prominent exceptions prove the rule, most notably the passage marked "crudely" (measure 79 ff.) and its residual influence that extends through the emphasis on F#, G#, and A in the final bars (measures 90-5).

A passage from "Thinking Twice" (1959) suggests some grounds for stylistic comparison between Form and Form IV with respect to pc balance in the aggregate and the idea of focusing on particular tones. Wolpe writes:

being a series means a relentless mobility of cutting, folding, stratifying serial aspects, layers, levels. Each tone within the set is equally aroused to do it, equally capable, an equal spatial component. Each serial tone's faculty is of the same order, of the same serial fate: to be focused or unfocused. Because of each tone's serial omnipresence, there are perhaps no focuses. Yet the organizational mode, the organizational assignment of serial tones or serial segments could be radically different.

Instead of hexachordal, tetrachordal, or other divisions that only satisfy claims for symmetrical correspondences or for reductions in the number of intervallic cells within the series, the reproductive, combinatorial activity is applied to zones--that is, half of the serial area is, for example, dynamic and progressive, the other half is stagnant, repetitive, and fixed. One zone is in motion, the other is motionless, one mobile, the other dead, one warm, the other cold. ("Thinking Twice," 295) [italics in original]

The passage suggests two (related) grounds for stylistic comparison: the focusing/unfocusing of individual tones; and the use of pitch-class zones as a basis for binary opposition.

In Form, pitch-class zones are largely systematic due to hexachordal complementation. Within zones, changes in pc order, registral placement, dynamics, articulation, and duration continually reconfigure relations among pcs. Passages limited to a single pc zone are often highly mobile in these respects, e.g., Wolpe's treatment of tones within the P hexachord in measures 1-3, and those within its complement P in measures 19-23. In both passages, one might well say that "Each serial tone's faculty is of the same order ... to be focused or unfocused. ... there are perhaps no focuses" (or, every note is a potential focus). Form also has moments of relative stasis created by repetition or symmetry. The passage from measures 43-8 based on [T.sub.5]P is a good example; here repetition and the development of frozen register constrain activity in a way that might be described as "stagnant, repetitive, and fixed" or "motionless," "cold." In Form, hexachordal complementation guarantees a certain balance among pcs in the aggregate at the large level; within individual pitch-class zones, high mobility and continual "refocusing" often achieves a similar effect locally.

Ten years later in Form IV, Wolpe forsakes the large-scale balance between complementary hexachordal pc zones for progressive imbalance and new ways of focusing and refocusing individual tones. Within the set of tones favored at the outset of the composition and to a significant extent throughout ({et012345}) there is further prioritization based on the formation and return of characteristic gestures and how each treats the tones it includes. Pitch-classes 4 and 5 (E and F) figure prominently in segments of associative sets D, F, and G (they also contribute to segments in sets A and C), but each set focuses on these tones in its own way: in set D, pcs 4 and 5 are registral edges or linear goals; in set F, pc 5 is a ground tone for activity; in set G, the "trill" confers prominence by repetition. In the composition as a whole, the focus on pcs 4 and 5 is itself refocused as new associative sets form and members of different sets succeed one another.

Another significant stylistic difference between the two pieces resides in the moment-to-moment flow of events. Form proceeds largely by juxtaposition. Wolpe's famous statement "The form must be ripped endlessly open and self-renewed by interacting extremes of opposites" ("Thinking Twice," 302-3) aptly describes its remarkable contrasts in the sonic domain--oppositions in register, dynamics, articulations, densities, and durations. Proportions and relations in Form tend toward asymmetry (including hidden symmetries masked by temporal order or superimposition) rather than literal repetitions or symmetries by inversion or retrograde that combine content with order relations in the strongest contextual associations among segments. (32) As a result, segmentation in Form is often multivalent or ambiguous, and the strong sonic disjunctions so prevalent throughout the piece often supercede contextual criteria as motivation for segmentation. Where associative sets do form, they tend to be either dispersed sets with a few prominent segments (e.g., sets E and F) or short-lived (e.g., sets B, C, and D). Other than succession, there is little clear higher-level organization among associative sets.

In Form IV literal symmetries in pitch and time (often subject to embedding) are much more common, with implications for musical flow and associative organization at every level. One might cast the stylistic differences between the two pieces in terms of pattern and weave. With its far greater use of clear pitch inversion and retrograde, Form IC involves stronger contextual criteria that create more vivid and precise patterns; pairs of segments related by inversion or retrograde often succeed one another directly or are superimposed, suggesting tighter weave. Vivid pattern and tight weave: these conjoin to define a "now." Moment to moment the music proceeds as a series of nows; recall Wolpe's words: "It has its presence, its now situation, / and then the now situation is joined with another, / with the next now-- / an unfoldment of nows!" ("Lecture on Dada," 214). The clarity of each now raises the possibility of connection between nows that are temporally distant; for example, the crystallization of set D i n D2 through D7 in measures 6-7 helps create the flashback quality of D8 through D10 in measure 67. The clarity of each now also supports opposition in the contextual domain: the appearance of D8-D10 in measure 67 not only refers back to measures 6-7; it also interrupts and "opposes" the recall of material from set A in measure 66. The clearer the now, the greater the complexity of relation it can support and survive. Thus, in a further significant difference with Form, the associative sets in Form IV tend to be much larger and long-lived: vividly patterned nows are pockets of activity within larger associative sets, some active throughout the composition (e.g., set B; and, to a lesser extent, sets A, D, F, and G). Framing this contrast between Form and Form IV in another way, we might say that they exhibit different "tunings" for the roles of sonic and contextual criteria. In Form, opposition is associated with the sonic domain; in Form IV, by fine-tuning the symmetries that yield stronger contextual criteri a and stronger and larger associative sets, Wolpe transfers some of the resources for opposition from the sonic to the contextual domain. Paradoxically, as Wolpe strengthens contextual criteria and their potential for association and unity, he also enhances their potential to define oppositions.

While progress moment-to-moment is clearer in Form IV than in Form, large-scale organization is not reductive and so is no easier to discern. Wolpe's phrase "multiple instances knotted together" is a good description for the webs of relationships one might explore among segments and associative sets. Among the possibilities realized in Form IV are: segments of one set that extract notes from segments in other sets (as in set C, vs. sets A, B, and D in measures 1-7); alternation between patterns of two different associative sets that render both active at the next level of consideration (e.g., sets A and B in measures 1-11); and tangents between associative sets, where aspects of segments in one set are shared by those in another (in some cases, even actual segments may be shared), blurring distinctions between them (e.g., pitch repetition of {C4, E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]4, E4, B4} between members of set B in measures 10-1 and the start of a new associative set governing the tight-knit passage beginning in measure 12). While the stream of now formation makes it easy enough to define passages and even sections, a complex network of associations across sections precludes their dear prioritization. In the end, Wolpe's fine-tuning the nows yields a piece rich in sonic and contextual oppositions, but also remarkably continuous; it also spawns a wealth of formal processes and relationships, a music that "yields the view from all sides."

Finally, a few words on the role of interpretation in my analysis: throughout, I have made innumerable choices--which passages? which criteria? which segments? how to group these into associative sets? how to understand relations among sets? I cannot emphasize this enough. The theory supports many interesting discoveries about Form and Form IV through its particular view of musical form as associative organization. It also clarifies how Wolpe's ideas on musical form translate, in practical terms, into notes and relationships. But it does not predict or prescribe even the range of choices, much less which choices we should make. This is much in keeping with the lively, idiosyncratic spirit of Wolpe's music and his musical sensibility. He would not have told us what to hear; he would only have asked us to listen.


(1.) Quotations from Wolpe's writings are referenced by lecture title and page number; full citations appear in the references list. In a prefatory note, Austin Clarkson indicates that this lecture probably originated as a talk at Black Mountain College in the summer of 1953.

(2.) In the 1962 interview with Eric Salzman, for instance, Wolpe challenges Salzman on his reference to "literal recapitulation" in one of Wolpe's compositions (Wolpe 1999, 405). In a footnote, Austin Clarkson suggests the work in question is Piece in Two Parts for Six Players, first performed in April of 1962. He writes: "Wolpe appears not to recall this case of 'recapitulation,' possibly because he thought of the reprise not as recapitulation, but rather as the null case of variation" (412, note 32).

(3.) "Thus, to a pitch set is given an additional fundamental function to provide for the organic conditions (I call them organic modes) of musical-matter-making shapes, events, and a course of action. Sets are no longer merely generators of distributive, combinational, and formally related pitch material. Each of the transpositions, transformations, constellations, and autonomous fragments is appointed for specific organic tasks or organic habits" ("Thinking Twice," 304).

(4.) William Rothstein represents Schenker's view of form as "the disposition of the fundamental structure, the Ursatz, over the course of [a] composition, along with those levels of middieground closest to the Ursatz" (Rothstein 1989, 103). For a survey of relations between fundamental structure and form in Schenker's thought see Smith 1996.

(5.) The pertinent passage from the "Lecture on Dada" extends from "To hear a piece of music at the same instant" on page 211 to "involving multiple aspects of the same thing" on page 212.

(6.) In this connection, consider Wolpe's words: "To move in all directions, because all aspects are exposed to each other in that space of omnipresent notions, will yield the view from all sides. Everything becomes connectable, and, because there is no hierarchic order of succession, everything is equally prophetic, and all-inclusively discovered and revealed" ("Thinking Twice," 302).

(7.) In his preface to Eric Salzman's 1962 interview with Wolpe, Austin Clarkson notes that many of Wolpe's ideas about musical form recall or develop formal principles associated with visual arts at the Bauhaus: "texts that outline the basic courses at the Bauhaus by Itten, Klee, Kandinsky, and Schlemmer reveal many links to Wolpe's thinking: the principles of opposites, proportions, and multidimensional space; the nature of intervals and motions as elements of music; the inclusion of nonpitched elements as musical material" (Wolpe 1999, 381).

(8.) References to symmetry and asymmetry abound in "On Proportions": Examples 1-16 illustrate symmetries; Examples 17-43, asymmetries. Oppositions are explored most extensively in the "Lecture on Dada."

(9.) Matthew Greenbaum writes: "What precisely is meant by 'proportion'? ... [Wolpe] is trying to describe something underlying pitch, a perceptual level beneath that of specific events: the realm of general categories ('larger,' 'smaller,' 'interpenetrating,' et cetera). And, as he is at pains to point out, these categories apply in any perceptual field (the 'proportions of the audible and visible are twins of one and the same totality'). His long association with visual artists, from the Bauhaus days on, certainly form the ground of this approach, which betrays an unusually 'painterly' attitude toward musical phenomena" (Greenbaum in translator's introductory note to Wolpe 1996, 134).

(10.) In his review of Lewin 1993, Robert Morris points out that David Lewin's analytical applications of musical transformations and transformational networks suggest similar associations not only between symmetry and stability, and asymmetry and tension, but also between asymmetry and teleology as a kinetic desire for symmetric (that is, transformational) completion. Morris identifies this teleological view of musical transformations as one of two theoretic themes in Lewin's book and in this regard he writes: "Lewin often describes musical transformations as phenomenologically teleological" (Morris 1995, 343), and "Lewin often considers his transformations to exert pressures and resolve tensions that seem to operate teleologically" (Morris 1995, 344), and "Asymmetry produces tension; symmetry releases it" (Morris 1995, 375). This connection between musical transformations and teleology is established early in Lewin's work on transformations. Of differences between posited literal transposition or inversion relations (i.e., symmetries in Wolpe's sense), and the actual relations between pairs of chords in Schoenberg's Klavierstuck op. 19 no. 6, Lewin says: "I find it suggestive to think of these generative lusts as musical tensions and/or potentialities which later events of the piece will resolve and/or realize to greater or lesser extents" (Lewin 1982-3, 341). For a more detailed account of a teleological view of musical transformations and its implications, see Morris 1995, 374-6.

(11.) Noted by Clarkson in his preface to Eric Salzman's 1962 interview with Wolpe (Wolpe 1999, 380).

(12.) My use of the term "domains" differs from Christopher Hasty's (Hasty 1978 and 1981b). My "dimensions" and "criterion subtypes," or "individual criteria" are roughly comparable to Hasty's "domains."

(13.) Details of this process follow those presented by Tenney and Polansky 1980, but differ in that I apply this only at a single surface (note-to-note) level, rather than hierarchically through recursion as Tenney and Polansky do. Most often adjacency is reckoned in time, but pitch, duration, or other linear sonic dimensions may also be used to determine adjacency as when proximity of pitch spacing is used as a criterion for segmenting a chord into two or more parts. In Hanninen 1996 and 2001, [S.sub.1] and [S.sub.2] criteria distinguish the cases where time is the basis for reckoning adjacency, versus when it is gauged in some other linear dimension (e.g., pitch or dynamics).

(14.) To focus on the matters at hand--contextual associations among groupings--sonic criteria will not be shown in subsequent musical examples, but are ever-present as forces in segmentation.

(15.) Hasty 1978, 37-44 and Examples 2-12 through 2-14; also 206-11 and Examples 4-45 through 4-49.

(16.) That is, considering notes within each of the trichordal pc sets (F, Ab, Bb) and (E, G, A) in measure 1 to be unordered in time, we may consider how they are ordered in register; this yields the pitch interval ordering <32> from low to high in the first case, and its inversion, <23> in the second.

(17.) I part with Tenney, Polansky, and Hasty, though, to say that the perceived relative strength of segments is such a complicated subject that it resists quantification. While perceived strength often correlates positively with the number of supporting criteria, it does not reduce to the number of criteria. It involves a host of considerations from simple to complex, including the relative magnitudes of sonic disjunction within each active sonic dimension (how big are the intervals?); questions of weighting among sonic dimensions (how many sixteenth notes is a semitone worth?); relative strength among contextual criteria of same and different subtypes, this involving both the criteria per se and their frequency of activity in a composition; relative strengths of sonic and contextual criteria (an apples and oranges question--which is "stronger," a grouping by contextual association or a strong sonic disjunction that might divide that grouping?), and even the interests and dispositions of the analyst. Questi ons of perceived relative strength, then, reference individual acts of interpretation.

(18.) When one segment is literally included in another, "consecutive order" is determined by noting which starts first. If they begin simultaneously, consecutive order is decided by which ends first-that is, the embedded segment precedes.

(19.) One might indicate associative subset membership with one or more lower case letters appended to the segment name: for instance, Ala and A2a are members of an associative subset a within a larger set A, while Ala and A3b belong to different subsets.

(20.) Note that A2 and A3 are not delimited by any sonic criterion other than temporal adjacency. Within measure 1, only the contextual criterion [C.sub.SC 3-7[025]] recognizes these segments by relating them to one another; as measure 2 unfolds, repetition of the two pitch-class tinchords (with added notes) lends additional support, but only in retrospect. The fact that the segments paired (A2-A4, A3-AS) are not temporally adjacent in the music is another reason this pairing is harder to hear.

(21.) Sonic disjunctions in dynamics are remarkable for their magnitude and unpredictability: the shift from ppp to fortissimo coincident with the change in note values and tempo at the upbeat to measure 30 clearly separates the end of this passage from the start of the next, but within the passage associative sets are more often fragmented than articulated by changes in dynamics.

(22.) Identity between A1 and E1 implies that the segments of E are actually an associative subset of A. I recast this subset as a distinct associative set for clarity in illustration.

(23.) An analyst may identify a segment as a contextual focus for an associative set based on two kinds of considerations: (1) the typicality or frequency of its supporting contextual criteria; and (2) their salience, strength, or significance (Hanninen 1996,90-4). In his work on categorization and musical motives, Lawrence Zbikowsi makes a similar distinction between "prototype effects" and "conceptual models" (Zbikowski 1999, 16 and 20).

(24.) For more on dispositions of associative sets (condensed, dispersed, and tangents between associative sets), and associative sets and aspects of form, see Hanninen 1996 and 1997. For another author's perspective on how varying strengths of relation map onto time, see Zbikowski's discussion of instances of the motive that opens the Megro of the first movement of Mozart's String Quartet in C Major, K 465 (Zbilcowski 1999, 25-37).

(25.) There is another connection: a loose contour inversion between 18 and 113 through Michael Friedmann's contour adjacency series or CAS, that records the sequence of ups (+) and downs (-) in a contour: the CAS of I8 is <+, -, -, +, ->; that of H3 is <-, +, +, -, +> (Friedmann 1985).

(26.) Peter Serkin has recorded Form IV on his CD "Works by Igor Stravinsky, Stefan Wolpe, Peter Lieberson" (New World Records CD NW 344-2). Austin Clarkson indicates a number of corrections to the published score for Form IV. In measures 1-17 most involve notation or placement; some, dynamics. Two are particularly significant: (1) measure 9, last A[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] in the right hand should be A[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]; E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII] quarter note below should be an eighth note; (2) tempo change in measure 16 should occur at measure 17; add "poco rit." in measure 16.

(27.) In principle, individual contextual criteria are equivalence relations: they indicate repetition of the aspect named in the criterion, so segments related by that criterion are equivalent in that respect. In practice, though, contextual criteria usually only indicate similarity between segments, for additional contextual criteria may involve one segment but not the other, and any salient differences between the segments simply aren't modeled by contextual criteria at all. Associative set membership, then, indicates a similarity relation, not an equivalence class. As such, it is non-transitive, which implies that segments in one set may approach those in another set by degrees, or even merge with them.

(28.) The inchoate quality suggests the first two measures might be an introduction rather than part of set A proper, but if one hears a reinitiation in measure 3, then A1 and A2 are part of set A.

(29.) In the context of the piece as a whole, then, the potential for a weak condensed associative set in the opening cluster is later superseded when A2 is absorbed within a stronger dispersed associative set.

(30.) Wolpe makes a similar connection between duality in tonal and twelve-tone contexts: "the concept that the coherence of the whole-of-the-unity-of-pitches has to be continually restored--such a concept of an equilibrium of unified pitches, of a pitch-totality, is classical thinking. As such, it is, at least at this moment, an undissolved remainder of the contextual properties of tonality and of that obsessive preoccupation (in an early phase of atonality) with decomposing historically-encrusted formations of sound" ("Thinking Twice," 277)

(31.) Robert Cogan has developed a theory of oppositions as the basis for sonic design, a particular view of musical form. See Cogan 1984 and 1998.

(32.) As examples of clear symmetries, consider the retrograde relation between right and left hand in measure 17 (right hand dyads <F#5/ E[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]6, D5/D[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]6, B4/C6>) or the inversion relation between right and left hand in measure 18 (pitch inversion around the dyad Db4/E1~4). Examples of asymmetrical proportions include the two segments in measure 2 identified earlier as A4 and A5; the six-note melody in measure 1 provides an example of hidden symmetry, with the pitch inversion between the first and the last three notes (slightiy) obscured by pitch order.


Bell, Adrian D. 1991. Plant Form: An Illustrated Guide to Flowering Plant Morphology New York: Oxford University Press.

Brody, Martin. 1977. "Sensibility Defined: Set Projection in Stefan Wolpe's Form for Piano." Perspectives of New Music 15, no. 2 (Spring--Summer): 3-22.

Clarkson, Austin. 1984. "'On New (and Not-So-New) Music in America' by Stefan Wolpe: A Translation." Journal of Music Theory 28, no. 1 (Spring): 1-45.

-----. 1985. "The Works of Stefan Wolpe: A Brief Catalogue." Music Library Association Notes 41, no. 4 (June): 667-82.

-----. 1993. "'The Fantasy Can Be Critically Examined': Composition and Theory in the Thought of Stefan Wolpe." In Music Theory and the Exploration of the Past. Edited by Christopher Hatch and David W. Bernstein. Chicago: University of Chicago Press, 505-24.

-----, editor. In press. The Music of Stefan Wolpe: Essays and Recollections. Stuyvesant, N.Y.: Pendragon.

Cogan, Robert. 1984. New Images of Musical Sound. Cambridge, Mass.: Harvard University Press.

-----. 1998. Music Seen, Music Heard: A Picture Book of Musical Design. Cambridge, Mass.: Publication Contact International.

Friedmann, Michael L. 1985. "A Methodology for the Discussion of Contour: Its Application to Schoenberg's Music." Journal of Music Theory 29, no. 2 (Fall): 223-48.

Hanninen, Dora A. 1996. "A General Theory for Context-Sensitive Music Analysis: Applications to Four Works for Piano by Contemporary American Composers." Ph.D. dissertation, University of Rochester.

-----. 1997. "On Association, Realization, and Form in Richard Swift's Things of August." Perspectives of New Music 35, no. 1 (Winter): 61-114.

-----. 2001. "Orientations, Criteria, Segments: A General Theory of Musical Segmentation." Journal of Music Theory 45, no. 2 (Fall). (In press)

Hasty, Christopher Francis. 1978. "A Theory of Segmentation Developed from Late Works of Stefan Wolpe." Ph.D. dissertation, Yale University.

-----. 1981a. "Rhythm in Post-Tonal Music: Preliminary Questions of Duration and Motion." Journal of Music Theory 25, no. 2 (Fall): 183-216.

-----. 1981b. "Segmentation and Process in Post-Tonal Music." Music Theory Spectrum 3: 54-73.

Itten, Johannes. 1961. The Art of Color: The Subjective Experience and Objective Rationale of Color. Translated by Ernst van Haagen. New York: Reinhold Publishing.

-----. 1964. Design and Form: The Basic Course at the Bauhaus. Translated by John Maass. New York: Reinhold Publishing.

-----. 1970. The Elements of Color: A Treatise on the Color System of Johannes Itten, Based on his Book The Art of Color. Edited by Faber Birren. Translated by Ernst van Hagen. New York: Van Nostrand Reinhold.

Kandinsky, Wassily. 1977. Concerning the Spiritual in Art. Translated and with an introduction by M.T.H. Sadler. New York: Dover. Reprint of 1914 edition, London: Constable.

Levy, Edward. 1971. "Stefan Wolpe: For His Sixtieth Birthday." In Perspectives on American Composers, edited by Benjamin Boretz and Edward T. Cone, 184-98. New York: Norton.

Lewin, David. 1982-3. "Transformational Techniques in Atonal and Other Music Theories." Perspectives of New Music 21: 312-71.

-----. 1986. "Music Theory, Phenomenology, and Modes of Perception." Music Perception 3, no. 4 (Summer): 327-92.

-----. 1993. Musical Form and Transformation: Four Analytic Essays. New Haven: Yale University Press.

Morris, Robert D. 1995. Review of Musical Form and Transformation: Four Analytic Essays by David Lewin. Journal of Music Theory 39, no. 2 (Fall): 342-83.

New Harvard Dictionary of Music. 1986. Edited by Don Michael Randel. Cambridge, Mass.: Harvard University Press.

Patterson, David. 1996. "Appraising the Catchwords c. 1942-1959: John Cage's Asian-Derived Rhetoric and the Historical Reference of Black Mountain College." Ph.D. dissertation, Columbia University.

Rothstein, William. 1989. Phrase Rhythm in Tonal Music. New York: Schirmer.

Smith, Charles. 1996. "Musical Form and Fundamental Structure: An Investigation of Schenker's Formenlehre." Music Analysis 15, no. 2-3 (July-October): 191-297.

Tenney, James. 1986. Meta + Hodos. Oakland: Frog Peak Music.

Tenney, James, and Larry Polansky. 1980. "Temporal Gestalt Perception in Music." Journal of Music Theory 24, no. 2 (November): 205-41.

Whitford, Frank. 1984. Bauhaus. World of Art Series. London: Thames and Hudson.

-----. 1993. The Bauhaus: Masters & Students by Themselves. Edited by Frank Whitford with additional research by Julia Engelhardt. Woodstock, New York: Overlook.

Wolpe, Stefan. 1978. "Thinking Twice." In Contemporary Composers on Contemporary Music. Edited by Elliott Schwartz and Barney Childs. 2d edition, 274-307. New York: Da Capo. Reprint of 1967 edition, New York: Holt, Rinehart & Winston.

-----. 1979. "Thoughts on Pitch and Some Considerations Connected with It." Perspectives of New Music 17, no. 2 (Spring-Summer) : 28-55.

-----. 1982. "To Understand Music." With an introduction by Austin Clarkson. Sonus 3, no. 1 (Fall): 4-17.

-----. 1982-3. "'Any Bunch of Notes': A Lecture." Edited by Austin Clarkson. Perspectives of New Music 21: 295-310.

-----. 1984. "'On New (and Not-So-New) Music in America' by Stefan Wolpe: A Translation." Translated, with an introduction and notes by Austin Clarkson. Journal of Music Theory 28, no. 1 (Spring): 1-45.

-----. 1986. "'Lecture on Dada' by Stefan Wolpe." Edited by Austin Clarkson. Musical Quarterly 72 , no. 2: 202-15.

-----. 1996. "on Proportions." Translated and with an introductory commentary by Matthew Greenbaum. Perspectives of New Music 34, no. 2 (Summer): 132-84.

-----. 1999. "Stefan Wolpe in Conversation with Eric Salzman." Edited and with a foreword by Austin E. Clarkson. Musical Quarterly 83, no. 3 (Fall): 378-412.

Zibkowski, Lawrence M. 1999. "Musical Coherence, Motive, and Categorization." Music Perception 17, no. 1 (Fall): 5-42.


Associative set: An unordered collection of two or more segments interrelated by contextual criteria. Each segment in an associative set relates to at least one other by one or more contextual criteria; conversely, every contextual criterion shown to contribute to the set must support at least two of its segments. We name associative sets with letters, e.g., associative set A; and individual segments with the set name followed by a number that reflects chronology in the score, e.g., Al, A2.

Associative subset: A set of two or more segments embedded in a larger associative set and distinguished from its total membership by the action of additional, often stronger, contextual criteria.

Condensed disposition: A (temporal) disposition in which most segments of an associative set are temporally adjacent (perhaps overlapping) or proximate.

Contextual association graph: A system of named nodes representing segments, connected by arcs that indicate contextual associations among segments in the associative set.

Contextual criterion: A criterion that identifies the means for association between two groups of notes. For example, "repetition of the ordered pitch-class set <C#, D, E, F>" is a contextual criterion. Contextual criteria are notated in shorthand: C for contextual, followed by a subscript that indicates first the type of association (pitch-class, set-class, rhythm, etc.), then the specific means for association. For example, [C.sub.pc] C#, D, E, F> denotes the contextual criterion described above.

Contextual domain: That domain of music discourse concerned with equivalence or similarity between groups of notes such as "same pitch-classes," "same rhythm," "same set class," etc. These relations arise only at the group level within a musical context that contains both groups.

Criterion: A rationale for cognitive grouping of musical events. Criteria support segments.

Dispersed disposition: A (temporal) disposition of segments in an associative set in which all or most of the segments are temporally nonadjacent or relatively distant, such that the span of time over which the set is active well exceeds the sum of durations of its member segments.

Disposition: An arrangement of two or more segments in a musical dimension, usually time or pitch.

Monophonic disposition: A disposition in which associative sets unfold successively, one at a time.

Polyphonic disposition: A disposition in which two or more associative sets unfold simultaneously.

Segment: A grouping of tones recognized by an analyst as a readily audible unit.

Sonic criterion: A criterion that responds to changing attribute values within a sonic dimension. Large changes in values delineate groupings.

Sonic domain: That domain of music discourse concerned with psychoacoustic attributes of individual notes such as their pitch, duration, loudness, or articulation. The full range of values a sonic attribute might assume constitutes a sonic dimension. For example, pitch, duration, and loudness are three sonic dimensions.

DORA A. HANNINEN is an Assistant Professor of Music Theory at the University of Maryland.
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Date:Jun 22, 2002
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