A Theory of Pitch-Class-Set Extension in Atonal Music
Over years of teaching pitch-class-set theory and analysis as part of undergraduate twentieth-century theory courses, I have often reflected (and heard perceptive students remark) on an apparent shortcoming of the system. At that stage of their studies, students usually have come out of several semesters of tonal theory in which, especially in curricula based on Schenkerian principles, they have been taught the elements that provide large-scale coherence to tonal compositions. At the end of the theory sequence, however, students are suddenly presented with a system (pitch-class sets) which accounts for "surface" pitch relationships in atonal music, but which does not provide a satisfactory way to connect the resulting pitch-class collections among themselves. Various available techniques for the comparison of sets (similarity relations) are based on abstract set relationships, rather than on relationships immediately observable on the actual music.1
I intend to present in this article a technique of long-range pitch connection which I have called "pitch-class-set extension" (PCSE), and that may be found, as I will illustrate, in post-tonal works throughout the twentieth century, from the "atonal" compositions by Schoenberg, Berg, and Webern to serial or non-serial works by such present-day composers as Babbitt, Ligeti, or Crumb. Although a presentation of this sort will necessarily have to include technical explanations and formal definitions, I have strived to do so in a language and by means of a methodology which can be easily adapted for pedagogical purposes. This article may also help to dispel the idea, sometimes held by musicians who may not have examined this repertoire in any depth, that atonal music is made up of series of disconnected or discontinuous sounds. Quite to the contrary, the examples discussed below feature, in the light of PCSE, high levels of connection and continuity. Before presenting pitch-class-set extension, however, it will be useful to review some of the more general principles and questions underlying any study of long-range pitch coherence.2
1. Prolongation or Association?
Does the concept of prolongation apply to post-tonal music? Because any post-tonal coherence seeker is likely to face this question, numerous authors have investigated the issue from various perspectives. Some type of prolongation has been shown to exist in some particular repertoires, if only locally and at the foreground or middleground levels (see, for instance, Morrison 1991 and Wilson 1992). Attempts at formulating a theory of post-tonal prolongation, however, have not produced satisfactory results. (Some interesting studies dealing with this issue are Baker 1983, 1990, and 1993, Lester 1970, Morgan 1976, Pearsall 1991, Salzer 1962, Straus 1987, Travis 1959 and 1966, and Väisälä 1999).
Because I will be presenting a technique of long-range pitch coherence, and the question whether this technique is "prolongational" will necessarily arise, let us first define prolongation. As understood in Schenkerian theory, prolongation is the unfolding in time of the tonic triad. This element unfolding in time (a) is not actually present at every moment after its first appearance (although its influence remains in effect), and (b) constitutes an underlying structural framework which provides a direction towards a goal (see Salzer 1962, Schenker 1979, and Forte/Gilbert 1982).
In general, the definition in post-tonal music of a single element to be prolonged in time has eluded analysts. Moreover, the prolongational procedures from tonal music do not apply to post-tonal music (see Straus 1987), and as a result no sense of direction towards a predetermined goal can be clearly established. Even in cases in which the term "prolongation" has been used, what is often meant is "association." In his 1989 article, for instance, Lerdahl recognizes that while tonal space "has precise levels of elaboration and distinct paths in moving from one structure to another[,] an atonal space would look more like a free-for-all 'associational' space that plots similarity relations among motives" (p. 85). Lerdahl also points out that "listeners to atonal music . . . grab on to what they can: relative salience becomes structurally important, and within that framework the best linear connections are made" (p. 84). In Lerdahl's theory, tonal stability conditions are replaced by a set of "salience conditions."3 Lerdahl's theory, in spite of his article's title, is more associational than prolongational.4 Associational models have generally replaced prolongational models in post-tonal music.
Techniques of contextual association have also been studied by Straus (1987 and 1997a) and Boss (1994). Straus defines association as "the grouping together of notes according to similarities in register, metrical placement, duration, dynamics, instrumentation, and so forth," and points out that "these groupings may contain notes widely separated in time" (1987, 21, note 13). Associational models, however, do not attempt to relate these separated notes by means of the intervening notes, as prolongation does. As Straus puts it, "given three musical events X, Y, and Z, an associational model is content merely to assert some kind of connection between X and Z without commenting one way or another about Y" (1987, 13).5 Boss addresses the complex issue of trying to establish hierarchical distinctions between structural and ornamental pitches in atonal music by proposing nine contextual (that is, associational) factors which affect the likelihood that a pitch or a sonority will be structural. Although Boss's contextual criteria (having to do with beginning a phrase, length, repetition, loudness, accent, register, timbre, density, and parallelism with another event) are insufficient to determine the structural role of a pitch, he also proposes two additional criteria to distinguish structural from ornamental pitches in Schoenberg's atonal music: structural pitches must represent an unvaried or varied form of some motive in the piece, and ornamental pitches must satisfy either Lester's definitions of division tone or neighbor tone, or Boss's definition of motivic replication (p. 204).6 The first of these criteria points at the strongly motivic nature of Schoenberg's music, and at the fact that, as Boss writes, "structural levels are generated in Schoenberg's music through ornamentation [of motives] instead of prolongation" (p. 210).
Motives are indeed an essential constructive element not only in Schoenberg's music, but also in much of the post-tonal repertoire.7 Allen Forte has shown that the concept of "motive" in twentieth-century music does not necessarily imply registral ordering. Forte has developed the idea of a "motivic set," an unordered pitch-class set which functions as a motive in pitch-class space (Forte 1985, 475 and 478), and has argued not only that motivic sets can prolong a pitch class, but also that harmonic and linear projections of the same set can generate extended prolongational structures (Forte 1988a and 1992; see also 1973). The concept of motivic set is the point of departure for the present study of long-range connective techniques in post-tonal music.
2. Defining Pitch-Class-Set Extension
The purpose of this essay is to identify possible pitch cells which have the function and capability to generate extended coherent structures in post-tonal music, and to define the means by which this extension is effected. Existing studies have shown that in the post-tonal repertoire there is not a single type of pitch unit (such as the triad in tonal music) which we could generally define as being extended. If a pitch collection is extended in this repertoire, it would not be a predetermined collection, but rather a particular collection or set of collections for each particular piece. The motivic pc-set, ordered or unordered, often fulfills such a function. Let us examine several examples of motivic transformation of pitch cells abstracted from atonal compositions. In Example 1a, taken from Schoenberg's "Nacht" (Pierrot Lunaire, no.8), transposition of a motive preserves its contour and its ordered interval succession, <+3, -4>. While each three-note cell is of the same set-class, , we easily perceive the immediate relationship among the three cells as ordered sets. In Example 1b (from Webern's Concerto, op. 24, I) the initial cell is not only transposed, but moreover manipulated by means of serial operations. The motive is still preserved (but not its contour), and so is its interval content (but not the order or direction of intervals). The relationship among the four cells in Example 1c, from the same work, is the same as in 1b, although octave complementation has been allowed here. The motive is still preserved, but intervallic inversion (resulting from octave equivalence) is added to the possible reordering of intervals, providing a further element of variation (interval-class succession is preserved here, rather than interval succession). All six trichords in Example 1d (from Crumb's "Primeval Sounds," Makrokosmos, vol. 1, no. 1) also belong to the same set class, . The relative order of pitches, however, varies in each of the three pairs of adjacent trichords, and so does, hence, the intervallic configuration (, , and respectively). In other words, these cells are presented as unordered sets.
Example 1a. Schoenberg, Pierrot lunaire, op. 21, no. 8, "Nacht"
Example 1b. Webern, Concerto, op. 24, I _||_ Example 1c. Webern, Concerto, op. 24, I
Example 1d. Crumb, Makrokosmos, vol. I, no. 1, "Primeval Sounds"
Example 1e. Ligeti, Ramifications
Example 1f. Webern, Five Movements for String Quartet, op. 5, no. 3
Example 1e (Ligeti, Ramifications) illustrates a progression of sonorities related by chromatic expansion, in which the initial  grows into  by a linear process involving contrary half-step motion in the outer voices. Each sonority is thus related to the previous one by a clearly perceptible half-step expansion in one of the voices. Chromatic expansion of a different type is also the connective bond in the passage from Webern's Five Movements for String Quartet, op. 5, no. 3 reduced in Example 1f, which presents a more abstract relationship among adjacent and overlapping trichordal cells. The unordered trichords are all of one of the following classes, listed in their order of appearance: , , , . Each of these trichords may be derived from another trichord in the collection by chromatic expansion or contraction of one or two pitch classes, a relationship among sets that I will refer to as "linear transformation of set classes." To be exact, a set class extends another one chromatically if the corresponding members of their respective adjacency interval series do not differ by more than 1. Thus,  <1,4> extends  <1,3>, and so does  <1,2>.
What do all of these fragments have in common? In all cases, a pitch cell is extended in time. The means used for this extension are repetition, set-equivalence operations (T or TI), serial operations (T, I, R, or RI), or chromatic expansion or contraction (whether it be applied linearly or to unordered sets). We will then define three basic principles of extension of a pitch collection: repetition, transformation, and voice-leading. Each of these can take place in pitch space or in pitch-class space. Table 1 clarifies the possibilities for repetition and transformation.8
literal motivic repetition
repetition of a set class (Ex. 1a)
In p-space: motivic transformation by T, I, R, or RI (Ex. 1b)
chromatic transformation (intervallic expansion or contraction) (Ex. 1e)
In pc-space: motivic transformation by T, I, R, or RI with octave complementation—IC succession is preserved (Ex. 1c)
pitch reordering (Ex. 1d)
linear transformation of set classes (Ex. 1f)
Voice leading as a technique of extension requires some attention. In the diatonic context of major/minor tonality, voice leading is regulated by the principle of proximity: the best voice leadings among registrally corresponding pitches in a harmonic connection are the common tone and the diatonic step. It has often been pointed out that, in the context of the total-chromatic pitch collection, the semitone is the quintessential voice-leading motion. Accounting for an associational "background" in Donald Martino's Pianississimo in which the only voice leading is by semitone, William Rothstein notes that "semitonal motion provides the only possibility for linear displacement within the total chromatic pitch collection" (Rothstein 1980, 141). In his important study of atonal voice leading, John Roeder follows Schoenberg's own intuition regarding common-tone and semitone relationships in atonal harmonic connections. In Harmonielehre Schoenberg remarks that what he calls "vagrant chords," which have no clear harmonic function, should be connected "by common tone or semitone in order to compensate for the lack of harmonic relations by 'compelling,' leading-tone-like motion between adjacent members of the chromatic scale" (Roeder 1989, 29. See Schoenberg's Harmonielehre, 134, 247, and 258-59). Jan Maegaard's study of voice leading in Schoenberg demonstrates the composer's pervasive use of common-tone and semitone voice leading among members of successive vertical sonorities (Maegaard 1972), and Roeder studies the harmonic properties of registrally ordered and unordered sets in the context of semitone voice leading (see Roeder 1987 and 1989).9
Following the intuition that common-tone and semitone voice leading provides the strongest connection among non-tonal adjacent pitch collections, I will define them as holding PCSE properties. Moreover, because a whole tone is the nearest possible motion to a semitone, I will also accept it occasionally, and under very specific circumstances which will be defined below, as having extension properties. In pitch class space, however, the proximity factor provided by the common tone and the half and whole steps loses its significance in favor of interval classes 0, 1, and 2. I will thus accept these three interval classes as holding pitch extension properties, and will abbreviate them henceforth as CT (common tone, or IC0, represented on the graphs by solid slurs), CHR (chromatic, or IC1, represented on the graphs by dotted slurs or lines) and WT (whole tone, or IC2, represented on the graphs by dotted slurs or lines with the indication WT). These same principles also account for the pitch-extension properties of "linear intervallic expansion or contraction" and "linear transformation of set classes" illustrated by Examples 1e and f above.10
The preceding discussion allows us to formulate the following definition:
Definition 1. Pitch-Class-Set Extension (PCSE). Pitch-class-set extension is the projection in time of a collection of pitches or pitch classes (an extension cell), ordered or unordered, dyads included, by means of repetition, transposition, inversion, retrogression, retrograde-inversion, or by linear intervallic expansion or contraction.
This definition suggests a number of questions. In the first place, are any transpositional levels better than others? Not necessarily as far as PCSE goes. However, for the same reasons formulated above with respect to voice leading, extension cells will be more closely connected if they have some pc in common (CT, or IC0) or if some of their pcs are related by CHR (IC1). And here we should stress that this is not a voice leading connection if by voice leading we understand a motion between registrally-corresponding pitches or order-corresponding pitch classes. Because we are defining a pitch connection between cells in pc space, and pitch reordering is a characteristic possibility in this context, exact registral or order correspondences lose their significance. If any pitch or pc in a cell is related to any pitch or pc of another cell by CT or CHR, then the two cells will be said to be connected.
Definition 2. Cell Connectedness. Extension cells are connected if they have at least one pitch class in common (common-tone connection; the degree of connectedness will be given by the number of common tones: the higher the degree, the stronger the connection), or if at least two of their respective pitch classes are related by IC1 (chromatic connection).11
Consider, for instance, the fragments from Schoenberg's "Nacht" (Pierrot Lunaire, no. 8) represented in Example 2. The movement is an example of extension of set , of which clear expressions can be found in the opening measures—with six embedded linear presentations of —, in the frequent statements of the , <+3, -4> head motive, and in the long chains of  cells in the piano part, measures 19-23.12 Especially interesting, however, are the various passages in which the trichord appears in different lines at different compositional levels. Example 2a reproduces the pitch content of the piano part's left hand, measures 14-15. The upper staff (which includes the upper two lines in the left-hand texture) features a chain of double  sets in which the exact intervallic contour <+3, -4> is preserved, connected by multiple common-tone relationships, and, in one case (between cells 3 and 4), by chromatic motion. The  cells in the lower staff do not preserve exactly the same interval succession as the original <+3, -4> cell. Pitches are reordered, resulting in two pairs of <-1, +4> and <-3, -1> sets.
Example 2. Schoenberg, Pierrot lunaire, op. 21, no. 8, "Nacht"
a. Mm. 14-15
b. Mm. 11-13
Example 2b reproduces the complete content of the piano part, measures 11-13. The left hand is a further example of PCSE involving repetition and transposition. All the cells in the passage—also  sets presented motivically as a <+3, -4> interval succession—are connected by CT with a single case of CHR connection. It will also be noticed that the common tones themselves are connected in chromatic pairs, both in Ex. 2b (G-, E-) and in 2a (G-, E- in the upper voice; E-, -C in the lower voice), thus establishing a CHR connection which involves groups of three cells.
Both passages feature, moreover, several layers of extension. In Ex. 2a the three lines of the piano part's left hand present  cells moving at different rates: the two upper layers (upper staff in the example) move in eighth-notes, while the lower layer (lower staff in the example) moves in quarter notes. In 3b, a string of eighth-note  cells at the foreground is connected by underlying  cells at the middleground, formed by the first note of each eighth-note trichord. At the same time, the right hand outlines melodic  cells in the upper and lower voices of a series of  chords.
The passages from "Nacht" illustrate cases of PCSE and cell connectedness in which a single set is clearly dominant. A different type of PCSE is present in the opening of Webern's Five Movements for String Quartet, op. 5, no. 5 (Example 3a). Segmentation is facilitated in this passage by clear textural functions: an almost-ostinato independent bass line in the cello, an upper voice in the first violin which is related to the cello line by complementary rhythm, and an inner-voice accompaniment made up of harmonic tetrachords. The cello line begins with overlapping  and  trichords, followed by  dyads in mm. 3-4, and a repeated  in mm. 5-9. The upper voice consists of two tetrachords,  in mm. 3-4, and  in mm. 5-9 (which breaks up into  and ). The  tetrachord is accompanied in the inner voices by two vertical tetrachords,  and , each repeated twice. This inner voice accompaniment may also be considered as four registrally-ordered horizontal lines, and the resulting tetrachords are then , , , and . The upper-voice tetrachord  is in turn accompanied by only two vertical trichords,  and , while the resulting lines yield three  and one  dyads. Although we can perceive immediate abstract CHR voice-leading connections among many of these sets which may indicate PCSE, we have not yet established the criteria to compare sets of different cardinalities (that is, sets that include different numbers of pitch classes) to determine their PCSE relationship. We need then to determine these criteria, and we will do so following some basic intuitions which involve voice leading once again, along with inclusion properties.
Example 3a. Webern, Five Movements for String Quartet, op. 5, no. 5, mm. 1-9
3. Comparing Sets for PCSE: Similarity Criteria
1. If two sets are of the same class, extension results (Example 4a)
2. Two sets may be of the same cardinality and different classes. The principles of CT or CHR voice leading will determine the existence of PCSE. We need to differentiate, for this purpose, between two types of voice leading: voice leading between actual pitch classes (C to or to B for CHR), or voice leading between set classes (abstract CHR voice leading involving order-corresponding members of the set, as between  and ). Moreover, we should allow for possible cases of WT voice leading (as in between C to D or , or between set classes  and ; both CHR and WT are present between  and ).
PCSE between sets of the same cardinality and different classes results if:
a. The corresponding members of their respective adjacency-interval series (AIS) do not differ by more than 1. Example:  extends  (AIS <1,4> and <1,3>) (Example 4b). Or,
b. The corresponding members of their respective AIS differ by 2 in no more than one entry, and at least one actual pc in the first set is related to one pc in the second set by CT or CHR. Example:  extends  (AIS <1,5> and <1,3>) if at least one of the actual pcs in the first set is related by CT or CHR to at least one of the actual pcs in the second set, but not otherwise. We will refer to this type of PCSE as WT expansion or contraction (Example 4c). We should note that this type of PCSE allows for all three of the pcs to be transformed by CHR, as between <C, , E>  and <B,E,F> .13
Connection between PCSE-related sets of the same cardinality and different classes:
c. If the set classes are related by CHR (such as  and ), connection results if at least one actual pc in the first set is related to one pc in the second set by CT or CHR (Example 4d). If all pcs between the sets are related by CT, CHR, or WT, the sets are maximally connected (Example 4e).
d. If the set classes are related by WT expansion or contraction (such as  and ), connection results only if all the actual pcs in the two sets are related by CT or CHR (Example 4f).
Note: The PCSE and connection criteria outlined above allow for WT voice leading between either pcs or set classes, as long as the other type of voice leading uses only CT or CHR.
3. Sets of different cardinalities.
a. A set B of cardinality n+1 extends a set A of cardinality n if and only if A is a subset of B. Example:  is extended by any of the tetrachords which contain it as a subset, that is, , , , , , , , , and  (Example 4g).
b. A set B(n+1) which extends set A(n) will be connected to A if at least one actual pc of A is related to at least one pc of B by CT or CHR (Example 4g).
c. A set B(n-1) extends a set A(n) if and only if B is a subset of A (Example 4g, reverse arrows).
d. Set B(n-1) which extends set A(n) will be connected to A if at least one pc of A is related to at least one pc of B by CT or CHR (Example 4g, reverse arrows).
Going back to Webern's op. 5, no. 5, Example 3b shows all the set-class PCSE relationships between the cello and the first violin lines. Single lines (with double arrows, because PCSE works both ways) indicate PCSE by CHR, double lines indicate PCSE by WT. A comparison of this chart with Example 3c will allow us to place it in the context of Webern's music. In this case, all musically adjacent or simultaneous sets display PCSE, with the only exception of  with either  or . The connection, in both cases, is made through , a subset of both tetrachords, and a musically significant trichord (it begins both  and  in their actual musical ordering). Another apparent conflict appears between  and , as well as  and , in Example 3b. These two pairs of trichords do not display abstract CHR voice leading. The actual musical voice leading as shown in Example 3c, however, does feature two cases of CHR between  and , and two cases of CT between  and . Both pairs are then related by PCSE according to Similarity Criterion 2.b.
Set-class PCSE charts for lines and chordal sonorities in mm. 3-9 appear in Example 3d. The first three charts compare sets from lines and sonorities in mm. 3-4, then in mm. 5-9, and finally all the harmonic trichords in mm. 3-9. Only two sets in all three charts do not display PCSE relationships,  and , although they are related by means of . The final chart in Example 3d shows PCSE among all the sonorities in mm. 3-9 resulting from combining the upper line with the accompanying tetrachords. This illustrates relationships among sets of different cardinalities. In almost all cases, adjacent sets are related by PCSE.  is not extended by , but the latter can be represented by its rotation (01378), which makes the relationship apparent. The last three sets, ,  = (01378), and , do not extend each other. However, each of them is related by PCSE to the preceding superset, the pentachord .
A sample of the actual voice leading among adjacent or simultaneous sets in this passage should suffice to illustrate the multiplicity of CT and CHR relationships featured in the music. Example 3e shows the CT (solid lines) and CHR (dotted lines) relationships among the combined sonorities of the two violins and viola. We have seen that all adjacent sets in this group are related by set-class PCSE by CHR. The condition for connection is, then, that at least one actual pc in a set be related by CT or CHR to at least one pc in the next set. Example 3e shows, at a glance, the multiple connections between all pairs of adjacent sets in the fragment. In conclusion, PCSE and connection provide a very strong element of coherence and organic growth in Webern's passage, in such a way that every pitch, musical segment, and section in the fragment can be referred back to the original - pair by means of a unified scheme of progressive musical unfolding.
4. PCSE Regions
If in PCSE an initial cell is extended by means of repetition, transformation, or linear processes, does this mean that the specific pitches or pcs of the initial cell are sustanied in time? PCSE and cell connectedness unquestionably create consistent regions of influence generated by an initial pitch event. They do not necessarily, however, sustain a focus on the actual pitches which constitute the initial event. PCSE based on repetition certainly sustains pitches in time, although in a trivial way. Linear connections and set-class expansion of the type we have discussed in Webern's op. 5 no. 5 create very strong pitch or set-class associations which provide cohesion to a musical region. But in the absence of a consistent hierarchical pitch system, we cannot argue that they sustain or prolong (in the Schenkerian sense) the initial pitch event. Neither do motivic transformations such as Tn, In, Rn, or RIn sustain the original pitch cell which they transform, although, as in the case of linear association of sets, they do create a region in which the initial cell maintains a "unifying influence" by means of motivic transformation. We can then define the relationship between the initial cell and the subsequent elements in PCSE space by means of the following two concepts:
Definition 3. PCSE Region. A PCSE region is a musical area in which PCSE of a set or group of sets can be traced.
Definition 4. PCSE String. A PCSE string is a collection of PCSE-related sets which extends an initial set or group of sets to create a PCSE region.
The opening of Webern's "Five Movements for String Quartet," op. 5, no. 3, (mm. 1-8, reduced in Example 5), may be viewed as a PCSE region extending a collection of two set classes,  and .14 Each pair of adjacent cells throughout the passage (all of them trichordal) is connected by either CT, CHR, or both. The piece opens with a set of chordal statements of . Cells 1 and 2 are connected by a common B and a chromatic B-, 2 and 3 by three CHR motions (-A, G-, and B-C), 3 and 4 by a common C. M. 4 features two lines, in first violin and viola, which do not contain any  as a trichord of adjacent pitch-classes as they appear on the score ( is, however, a subset of hexachord , resulting from considering either the violin or the viola pitches as unordered sets). Rather, the totality of adjacent trichords—, , ,  in both violin and viola—presents an intervallic trichordal expansion from  to  by means of  and the "missing"  which they actually extend (the complete PCSE string is then    ; see also Ex. 1f above). The return of  chords at m. 5 is related with the last  in the viola by two CHR connections (G- and F-E), while the two  trichords at m. 5 are themselves related by a CT and three CHR connections. The  chords are followed by a series of  melodic statements with multiple connections, and by three more  chords at m. 6, in which chords 1 and 2 are connected by a common and a chromatic E-, and chords 2 and 3 by a common G and three CHR relationships. It will also be noticed that the three lines which form these chords (violin 1, violin 2, and viola) are also made up of melodic  cells. This passage of total  saturation precedes the two converging lines (violin 1 and cello) at m. 7, in which  saturation is featured: each of the trichords formed by adjacent pitches in each of the lines is a  cell. The section closes with a return, at m. 8, of the  chords (related by a common C and three CHR connections), and a final, cadential statement of  in the cello. The unifying principle of this section (besides the pedal in the cello) is thus the PCSE of two sets,  and , with multiple CT and CHR connections, and a brief passage of trichordal intervallic expansion.
Example5. Webern, Five Movements for String Quartet, op. 5, no. 3. mm. 1-8
Many other fragments or complete pieces by Schoenberg, Berg, and Webern feature PCSE. Among those which come easily to mind and which have been analyzed elsewhere, I will briefly mention Schoenberg's "Farben" (Five Pieces for Orchestra, op. 16, no. 3) and Berg's Wozzeck, op. 7, Act II, mm. 465-74. "Farben" features "middleground" extension based on transposition of a single set, , used as a chord which appears in the same position every time it is transposed, and in such a way that each T of the chord has at least one common tone connection with the previous chord. At the surface level, each T is effected by means of a canon (a <+1, -2> motion in each voice). Within each canon, the pitch class originally in voice 2 from the bottom is present in every simultaneity of the canon, thus connecting by CT the foreground span between "middleground" chords.15 Berg's passage, on the other hand, includes a harmonic accompaniment in the strings in which a tetrachord constantly in motion creates a strictly connected PCSE space, as evidenced in Roeder's analysis of these measures (which includes a reduced score with annotated set classes; see Roeder 1989, 51-53). The voice leading in the tetrachord is effected mostly by semitone in only one voice (occasionally in more than one voice, and with one single instance of WT motion between mm. 465 and 470; larger voice-leading intervals are introduced in mm. 472-73). Besides this strict actual CHR voice leading among the chords, all adjacent set classes in the passage are systematically related by abstract CHR. I will list, as an illustration, the set classes in mm. 465-66: ------------------. The only exception to CHR voice leading in this PCSE string can be found between set classes 2 and 3, which feature WT voice-leading. The rotation of set 3 into (0458), however, makes the CHR connection apparent.
Techniques of PCSE are not exclusive to the composers of the Schoenberg circle. Joseph Straus's motivic analyses of the music of Ruth Crawford Seeger reveal numerous instances of highly connected PCSE. Mm. 20-32 from Crawford's Diaphonic Suite, for instance (reproduced in Straus 1995, 30), are unified motivically by two PCSE strings involving trichords closely connected by CT (overlapping pitches) and CHR. The main string features only the principal cell, , and its extension, . An overlapping string includes set classes , , , and . All of the set classes in the passage are linked by the PCSE string       (Example 6a). Straus also notes instances of embedded statements of the same motive, in which structures similar to those represented in Example 2 above create two levels of PCSE (Example 6b; see Straus 1995, 28, 32, and 138).
Example 6a. Crawford, Diaphonic Suite, mm. 20-32
Example 6b. Crawford, Diaphonic Suite, mm. 1-3; Piano Prelude no. 1, mm. 1-3; "Rat Riddles," mm. 13-19
A remarkable example of connected PCSE by a composer closer to us chronologically is provided by Crumb's "Primeval Sounds," Makrokosmos, vol. 1, no. 1.16 The piece opens with two consecutive events, each of which contains seven pairs of chordal sonorities (Example 7a reproduces the pitch content of the first event). Each pair is made up of two unordered  trichords related by T6. The resulting seven hexachords, all of the class , are arranged in a network of T5 (or T11) transformations, as indicated over the first pair of chords in Example 7a. The pairs of T6-related trichords in the example have two CHR connections, while the pairs of T5-related trichords are always connected by a CT. Pitch extension in the passage is further stressed by the trichordal content of the upper voice in the texture: adjacent  trichords overlap with adjacent  trichords in a line with multiple CHR connections, as indicated in Example 7b.17
Example 7. Crumb, Makrokosmos, vol. 1, no.1, "Primeval Sounds"
a. Opening event
b. Top voice, opening event
While pitch extension is mostly a "foreground" technique (that is, operating at the surface level), some of the examples we have mentioned feature connective "middleground" relationships which establish a level of pitch extension deeper than the surface (see, for instance, Schoenberg's "Farben," and "Nacht," and Crawford's Diaphonic Suite, all discussed above). Similar "middleground" connective devices (in this case covering a much larger formal span) are used by Schoenberg in Klavierstück, op. 33a, and in his Piano Concerto. In the first of these works, the three long-range combinatorial areas found throughout the complete piece (that is, the families of four rows each formed by a row, its inversional complement, and their retrogrades), A0, A2, and A7, mirror the pitch-class set of the three initial pitches of the row, -F-C, (027). In the Piano Concerto, all twelve combinatorial areas may be found in the beginning section (mm. 1-333, up to the return of the opening melody). As Andrew Mead has pointed out (1985, 130), the order of transpositions used in this complete succession of combinatorial families follows the order of pitch classes of the row itself, as first stated in the opening melody. Thus, the row 07e21935t684 produces the succession of combinatorial areas T0-T7-Te-T2-T1-T9-T3-T5-Tt-T6-T8-T4.
5. Pitch-Class-Set Extension and Musical Space
While the definition of background structures in the realm of pitch extension does not seem plausible, musical space is made uniform by such techniques of "foreground" and "middleground" extension as the ones we have discussed so far. We may then define such uniformity of musical space, based on our observations and musical examples in the preceding pages.
Definition 5. Extension of Musical Space. Musical space is extended if it contains pitch extension as characterized in Def. 1. "Foreground" extension of musical space results from the literal surface-level occurrence of associations described in Def. 1. "Middleground" spatial extension results from the occurrence of associations described in Def. 1 at a level deeper than the immediate surface.
Definition 6. Connectedness of Musical Space. Musical space is connected if a) it contains pitch extension and b) the pitch-extension cells are connected among themselves.
Definition 7. Uniformity of Musical Space. Musical space is uniform if (a) it is saturated with a single set and/or (b) its two coordinates display the same pitch-class collection or collections.
Musical space is extended and connected in all of the above examples after (and including) Example 2. Moreover, musical space is uniform in the fragments analyzed in Examples 2a (although the complete musical space of the passage, including all the parts other than the left hand of the piano, is not uniform) and 6.
A brief reference to an example of a highly uniform musical space will help clarify the concept. Webern's musical space is often closely uniform because of his use of derived rows. Among many possible examples which would illustrate this point, let us examine two passages from the first movement of his Concerto, op. 24. In the reductions of mm. 1-10 and 13-17 reproduced in Example 8 I have applied the principle of octave equivalence for ease of reference. The presentation of the same sets in the actual music includes compound intervals (as in Ex. 1b) and octave complementation (as in 1c).
Example 8. Webern, Concerto, op.24
a. Mm. 1-10
b. Mm. 13-17
The row is divided into four ordered  trichords, themselves related by P, RI, R, and I operations, and all following the basic motivic contour <-1, +4> (also presented as <+4, -1>, <-4, +1>, or <+1, -4> depending on whether the trichord considered is a P, RI, R, or I form; and varied in the actual music by the application of octave equivalence, into forms such as <-13, +4>, <+11, -8>, etc.). The division of the row into four discrete  trichords is systematically preserved in the music, where trichords are actually differentiated in timbre by being assigned to different instruments. Pitch extension thus occurs at two levels: the complete twelve-tone row is a set which is extended, and so is the  trichord.18 Mm. 1-10 contain five forms of the row (P0, RI1, RI0, P1, and I1). The graph shows the web of CT connections among these row forms, considered as trichordal connections. The degree of connectedness is indicated by a numeral next to each arrow. Thus, P0 and RI1 feature the maximum degree of connectedness, 3, between their corresponding trichords. Trichordal relationships between RI1 and RI0 are as follows (with 3 representing a trichord, and the superscript letter indicating the position of the trichord in the row): 3a-3d = 1, 3b-3c = 2, 3c-3b = 1, 3d-3a = 2. A similar relationship of trichords between RI0 and P1 (first with last, second with second to last, etc.) yields a degree of connectedness = 2 in all four cases, while corresponding trichords between P1 and I1 are related by the degrees 1, 2, 1, and 2.
Example 8b shows a further instance of extension and connection. In the passage, Webern exploited the hexachordal and trichordal invariance between R7 and I2, and I7 and R0. Both pairs of rows are presented horizontally, with the respective common hexachords overlapping, while the remaining two pairs of trichords feature maximum degree of connectedness (grouped R73a-I23c, R73b-I23d, and similarly for I7 and R0, as indicated by the arrows). At the same time, vertical connections between trichords presented simultaneously (see R73b-I73a, R73c-I73b, etc.) produce the degrees 1, 2, 1, 2, 1. In short, both passages (the complete composition, as a mater of fact) feature pitch extension with a high degree of horizontal and vertical interconnections. Musical space is not only extended and connected, it is also totally uniformed by the exclusive use of two sets: the twelve-tone row and the  trichord.
We have examined so far PCSE, cell connectedness, and uniformity of musical space in a variety of examples, regardless of the compositional method applied in the particular pieces. We will now focus our discussion on the specific PCSE characteristics of the twelve-tone row.
6. The Twelve-Tone Row, PCSE, and Uniformity of Musical Space
In itself, the twelve-tone row and its derivations extend and uniform musical space, according to Definitions 5 and 7. Because serial music is based on the projection in time of a collection of pitch classes (the row), pitch extension is an intrinsic characteristic of serialism. All serial music, moreover, produces uniformity of musical space. All twelve-tone music fulfills condition (a) in Def. 7 (it is saturated with a single set), and music which exploits the combinatorial properties of a row and the formation of aggregates fulfills both conditions (a) and (b) (its two coordinates display the same pitch-class collection or collections).19
A brief reference to a celebrated passage by Schoenberg will further illustrate the uniforming properties of combinatorial rows.20 Mm. 27-28 of Schoenberg's Fourth String Quartet, I, feature the simultaneous presentation of P0 and I5, two combinatorial forms of the row. The complementary hexachords (P06a-I56a and P06b-I56b) are stated simultaneously, so that in the two measures there are four horizontal statements of the row (P0-I5 in the violins, I5-P0 in the viola-cello) and four vertical statements of the aggregate (P06a-I56a, P06b-I56b, I56a-P06 a, I56b-P06b). Considering that the aggregate is an unordered twelve-tone collection, we see that the two coordinates in the passage display the same pitch-class collection, and hence musical space is uniform (see Example 9). Furthermore, all of these twelve-tone collections are interconnected, both vertically and horizontally, by two invariant trichordal segments between P0 and I5 (D--A and G--C), as indicated by the arrows on the reduction. Space is thus extended, connected, and uniformed by the row, its combinatorial forms, and their invariant segments.
Example 9. Schoenberg, Fourth String Quartet, mm. 27-28
Reference to such unifying characteristics of the row can be found in the writings of both Schoenberg and Babbitt. In Style and Idea, Schoenberg makes the following often-quoted remark:
The unity of musical space demands an absolute and unitary perception. In this space, there is no absolute down, no right or left, forward or backward. . . A musical creator's mind can operate subconsciously with a row of tones, regardless of their direction, regardless of the way in which a mirror might show the mutual relations. . . The employment of these mirror forms (that is I, R, and RI forms) corresponds to the principle of the absolute and unitary perception of musical space (Schoenberg 1950, 113-115).
Babbitt points out that the most immediately powerful cohesive property of a row is the preservation of the interval number succession under T.
The totality of twelve transposed sets associated with a given row constitutes a permutation group of order 12; as such it is closed, disjunct with regard to any other collection of sets T-derived from a set whose intervallic succession differs from that of any member of this totality (Babbitt 1960, 249).
Under I, the interval succession of P is substituted by a succession of its complementary intervals. All T levels of R present the intervallic succession of I in reverse order, while the RI forms present the intervallic succession of P in reverse order. In other words, the intervallic succession of a row, whether reversed or presented as complementary intervals, is always preserved throughout all of the serial operations which may be applied to the original row. The properties of the row as a pitch-extension structure are firmly based on this intervallic element of cohesiveness.
Besides the immediate and sufficient PCSE properties of the twelve-tone set, however, composers have often taken advantage of the unique connective properties of specific row relationships. Such is the case, for instance, of Webern in his first Cantata, op 29, I. The piece has been thoroughly studied by such authors as Rochberg (1962), Kramer (1988), and Mead (1993). Because it displays numerous PCSE properties, however, it is worth reviewing some of its characteristics in the context of our present discussion. All adjacent trichords in the row, reproduced in Example 10a, are of the  class. The row is RI-symmetrical, and hence each I form is also an R form. Rochberg and Kramer use the R labels and omit the I forms. Mead, however, has shown that the preserved repertoires of dyads between pairs of voices and of tetrachords between all four voices result from the fixed inversional relations among the rows, rather than from any ordering property of the row class (Mead 1993, 175-77). I will thus use I forms instead of R forms, although I will keep the conventional integer labels for this row, which assign integer 0 to pitch A (rather than Mead's labels based on C = 0).
Example 10. Webern, Cantata, op. 29, I
Because the opening and closing intervals are both , the row offers the possibility of dyadic invariance between the end of one row and the beginning of another as a means of row connection. The transformations which produce such invariance (T9 for P forms, or T3 for I forms) generate a cycle of four rows followed by a return to the row of departure. Example 10b illustrates a cycle connected by ending/beginning dyads and covering the forms P0-P9-P6-P3-P0. Webern uses four simultaneous strands of such transpositional cycle throughout the movement, thus connecting each pair of adjacent row forms by their invariant dyad. The chain is broken only at mm. 13-14, where the end of the orchestral introduction (covering four simultaneous half cycles) is not linked to the beginning of the choral section by invariant dyads. Mm. 14-47 (the choral section and the orchestral conclusion) contain four strands of a cycle and a half each, as indicated in Example 10c (a complete cycle for each of the choral parts, and four half cycles for the conclusion).
In summary, the following are the techniques of PCSE, cell connection, and uniformity of musical space found in this movement: (1) All adjacent horizontal trichords are of the  class. (2) All adjacent row forms (with one single exception) are connected by their invariant ending/beginning dyads. (3) Vertical dyads are preserved throughout the chain transpositions, between the pairs soprano/alto and tenor/bass. (This property results from the simultaneous presentation of any P and I forms of the same row in which the sum of the respective transpositional operators remains the same; in this case, the sum of transpositional operators in the choral section, for instance, is always 11 for the soprano/alto pair, and 7 for the tenor/bass). (4) Resulting harmonic tetrachords are also preserved throughout the entire cycle of transpositions: there are only three tetrachord forms (, , and ), and only six different chords (two groups of three related by T6). The Cantata thus provides examples of extension of a single trichord—the initial melodic —, extension of a collection of three harmonic tetrachords, connection of twelve-tone row sets by invariant melodic dyads, connection of pairs of twelve-tone sets by invariant harmonic dyads, and uniformity of musical space resulting from space saturation with a single set, .
We will now refer to a final well-known example from the serial repertoire, Schoenberg's Klavierstück, op 33a, and I will point out the most salient techniques of pitch extension in the opening measures (the first-theme area, mm. 1-11).21 As we have already established, because of being serial, op. 33a features PCSE by definition. Because of the combinatorial properties of the row, the musical space of the piece is uniformed. Extension and connection techniques in this composition, however, go beyond these general characteristics built into the serial method. Two types of tetrachords are significant in the piece. In the first place, the three discrete tetrachords of the row are frequently presented as musical units (, , ). In the second place, the simultaneous presentation of combinatorially-related row forms (such as P0 and I5) results in six tetrachords which include corresponding discrete dyads of both rows (see Example 11a). These six tetrachords form a symmetrical collection of sets:      . The actual symmetrical presentation of collections of tetrachords in the music is a further element of PCSE (see Example 11b). Thus, mm. 1-2 feature two symmetrical statements of the discrete tetrachords of the single row; mm. 3-5, displaying simultaneous statements of RI5 and R0, present the symmetrical succession of tetrachords from paired combinatorial row forms; in mm. 6-7 we see a return to two symmetrical statements of discrete tetrachords of single rows; mm. 8-9 present two simultaneous statements of the three discrete tetrachords, thus creating a vertical (rather than horizontal) symmetry, while mm. 10-11 present two simultaneous and symmetrical double statements of the same discrete tetrachords, creating both vertical and horizontal symmetry. Collections of tetrachords are thus not only extended by serial operations, but moreover are presented as closed symmetrical cycles which have extending properties of their own.
Example 11. Schoenberg, Klavierstück, op. 33a
a. Row forms and tetrachords
b. Tetrachordal content, mm. 1-11
c. Mm. 1-11, the stacked-IC5 sets
d. The connective function of set C
As shown in Example 11a, the row begins with two fifths (or fourths) stacked up, -F-C, , while the first and last tetrachords of paired combinatorial row forms can also be organized as three stacked fifths (or fourths) each, such as ---F , and D-A-E-B . The recurrence of stacked-IC5 sets  and  at the same pitch level creates an element of direct pitch connection among the row forms in the piece. Because combinatorial areas are extended throughout large spans of the piece (there are only three areas in the complete composition, A0, A2, and A7, and of these A0 is greatly predominant), the connections relating stacked-IC5 sets at identical pitch levels are of very strong structural significance, as shown in Example 11c. The graph is a spatial representation of all the stacked-IC5 sets in the passage, with indication of their function within row forms. Considering the absolute pitch content of all these sets, there are only three different pitch collections, and each has a definite function: set A, C-F- , begins P0 or ends R0; set B, -- , begins I5 or ends RI5; and set C, D-E-A-B , connects the end of P0 with the beginning of RI5 (or the beginning of RI5 with the simultaneous beginning of R0; or the simultaneous endings of P0 and I5). The double function of set C is further outlined in Example 11d. The set connects the P0/RI5 complex in one of two ways: 1) if the P0/RI5 complex is presented linearly, as in mm. 1-2 and 6-7, then set C acts as a link between the two rows; 2) if the P0/RI5 complex is presented simultaneously, then set C functions as a common departure set (mm. 3-5), or as a common goal set (mm. 8-9), or as a common connective element between the end of one complex and the beginning of the next (mm. 10-11).22
7. Conclusions and Pedagogical Applications
PCSE provides the means to relate pitch collections in a passage or a composition according to similarity criteria which take mostly into consideration set cardinality, inclusion relationships, pc invariance (CT), and pc voice-leading proximity (CHR or WT). In summary, two sets are related by PCSE if (1) they belong to the same set class (identity), or (2) they do not belong to the same set class but they have the same cardinality and their pcs and adjacency-interval series fulfill some conditions of closeness determined by voice leading (involving CT, and CHR or WT expansion or contraction), or (3) their cardinalities are not the same but their difference is not bigger than 1, and the two set classes are related by inclusion. Passages in which pitch collections are thus related by PCSE display a high level of coherence provided by close relationships among the collections, as defined by a very limited set of similarity properties. Cell connection and uniformity of musical space imply even higher levels of pitch coherence.
As demonstrated in this article, PCSE constitutes a theoretical and analytical tool which contributes to our understanding of extended pitch coherence in post-tonal music. As such, the theory has immediate pedagogical implications. After teaching the basics of pitch-class-set theory, instructors can easily introduce PCSE, and illustrate its applicability to a very broad repertoire. The short fragments in Example 1 in the present article can be used to introduce the concepts and lead to the basic definitions (Definitions 1 and 2 above). The discussions of Schoenberg's "Nacht" and Webern's op. 5, no. 5 (Examples 2 and 3 above) provide instances of PCSE in larger musical contexts. If serialism has not yet been covered at this point, the concept of PCSE may be revisited later in the course, after serialism is introduced. The above discussions of Webern's Concerto, op. 24, Cantata, op. 29, and Schoenberg's op. 33a can be used for this purpose, and will provide an appropriate pedagogical link between pitch-class-set theory, serialism, and the principle of long-range pitch coherence in post-tonal music. Finally, although the above section on similarity criteria to compare sets for PCSE may be somewhat demanding due to its more formalized language, its study is perfectly appropriate for advanced undergraduate students or in courses with sufficient time to explore this type of technique in depth.
In summary, PCSE can enrich the students' appreciation of twentieth-century musical structure by providing a means to (a) investigate and understand aspects of long-range coherence associated to (and complementing) their study of pitch-class-set theory; (b) apply the same analytical techniques to the study of serial compositions; and (c) further apply the techniques to the study of more recent, post-serial repertoires (represented in our discussions by composers such as Crumb and Ligeti), thus stressing the existence of underlying structural currents in twentieth-century post-tonal music, regardless of style.
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1In numerous studies in recent years, scholars have attempted to define the elements that provide long-range pitch coherence in post-tonal music. Although authors such as Mead (1985), Straus (1995), Wilson (1992) and others have succeeded in demonstrating extended pitch coherence in specific repertoires, no general methodology has emerged which might be applicable to multiple stylistic repertoires.
2Most of the technical principles that I will discuss in this article have been individually studied by other scholars, in some cases at great length: motivic association and transformation (Boss 1992 and 1994, Forte 1985 and 1992, Lewin 1987 and 1993, and Santa 1999), connections between pitch collections by means of invariance relationships (Babbitt 1960; Haimo/Johnson 1984; Hyde 1985; Lewin 1962 and 1967; Mead 1985, 1987, and 1993; Peles 1983-84; and Starr 1978), motivic correspondences between the horizontal and vertical dimensions (Dean 1973-74, Forte 1988a), large-scale projection of surface pitch collections (Mead 1985), and common-tone and semitonal voice leading (Morris 1998, Roeder 1987, 1989). My contribution in this article is to bring together these and other intuitions regarding long-range pitch coherence in post-tonal music, and to illustrate the applicability of the resulting theory to a broad spectrum of twentieth-century compositions.
3Lerdahl clarifies the concept as follows: "Given that two events connect, the more stable is the one that is more consonant or spatially closer to the (local) tonic; the more salient is the one that is in a strong metrical position, at a registral extreme, or more significant motivically" (1989, 73).
4Lerdahl has recently stressed that his use of the term "prolongation" is non-Schenkerian: "My disagreement with Straus and Larson on this point reduces to the semantics of the term 'prolongation.' I employ it as an ordinary English word that resonates with centuries of music theory and define its specific usage in terms of the atonal theory I am building; they restrict it to a specifically Schenkerian usage" (Lerdahl 1997, 153). See also Lerdahl 1999.
5For a recent critique of Straus 1987, see Larson 1997, to whom Straus responds in Straus 1997b. See also Väisälä 1999.
6Lester's "division tone" is the atonal equivalent of a passing tone, although in a division tone no limit is placed on the dividing intervals's size—they need not be seconds. Boss defines "motivic replication" as the replication of the ordered pitch-interval succession of a motive defined as structural in a piece.
7For a compelling study of motive and motivic transformation in the music of Ruth Crawford Seeger, for instance, see Straus 1995. Much of what Straus uncovers in this music in the form of motivic coherence is pertinent to the present study.
8In his article "Schoenberg's Op. 22 Radio Talk and Developing Variation," Jack Boss shows in much detail that in his 1932 radio talk on the Four Orchestral Songs, op. 22, Schoenberg discussed each of the categories which I include in Table 1 as types of developing variation (see also Schoenberg's Fundamentals of Musical Composition, pp. 3-9). Boss rewords Schoenberg's kinds of motivic variation into the following categories: (A) every collection having the same succession of unordered pitch intervals between its adjacent pitches (in Boss's example, all the set forms including the two-interval successions generated by combining ordered pitch intervals +1 or -1 with +3 or -3, such as <-1, +3>, <-1, -3>, <-3, +1> etc.); (B) allows for octave complementation; (C) allows for pitch reordering; and (D) allows for semitone expansion of one or more of the ordered pitch intervals (Boss 1992, 132-133). In this article Boss proposes a model of long-range connection similar to my PCSE, but rather than basing it on pitch and pc-set connections, he bases it on intervallic and pitch-interval successions treated as motives. For a study of linear coherence resulting from motivic permutation in Schoenberg's Das Buch der hängenden Gärten, see Forte 1992. In his article "A Principle of Voice Leading in the Music of Stravinsky," Joseph Straus discusses the principle of "pattern completion," according to which "a certain unordered collection or set of notes is established as a structural norm for the composition, pervading the surface of the music (both melodic and harmonic) and governing the tonal motion at all levels of structure" (Straus 1982, 106). Straus's examples in this article illustrate some of the same concepts as my PCSE does. A striking case of this is presented by Straus's Ex. 14, mm. 549-53 from Agon, totally permeated by overlapping forms of set  in all voices.
9Allan Chapman (1978 and 1981) has studied voice leading among registrally ordered sets in terms of intervallic relationships between adjacent pitches in a vertical. Several authors have recently studied voice leading from a transformational perspective (that is, following the recent theoretical work by David Lewin). See Klumpenhouwer 1991, Roeder 1995, and Straus 1987. The most complete and general study of voice leading that I am aware of can be found in Morris 1998.
10Similar voice leading principles underlie Richard Cohn's concepts of maximally smooth voice leading, which he defines as "only one voice moves, and that motion is by semitone" (Cohn 1996, 15), and parsimonious voice leading connecting two triads, in which two common tones are retained while the third voice proceeds by semitone or whole tone (1997, 1-2). Referring to the stepwise motion in the third voice, Cohn notes that "this feature is not without significance to the development of a musical culture where conjuct voice-leading in general, and semitonal voice-leading in particular, are enduring norms through an impressive range of chronological eras and musical styles" (p. 2). Cohn's following observation is also very pertinent to our present discussion of voice leading: "Beginning with (at least) the eighteenth century, the normative status of common-tone retention and stepwise motion is not only statistical but cognitive: one conceives of them as occuring even when the actual leading of the 'voices' violates them, e.g. when instantiations of the common or step-related pitch-classes are realized in different regions" (p. 62, note 5).
11George Perle has noted the significance of what he calls the "basic cell" in free atonal music, as well as of invariant pitches to connect adjacent cells: "It may operate as a kind of microcosmic set of fixed intervallic content, statable either as a chord or as a melodic figure or as a combination of both. Its components may be fixed with regard to order, in which event it may be employed, like the twelve-tone set, in its literal transformations: prime, inversion, retrograde, and retrograde inversion. (. . .) Individual notes may function as pivotal elements, to permit overlapping statements of a basic cell or the linking of two or more basic cells" (Perle 1991, 9-10).
12For an analysis of transformational networks in this piece following David Lewin's methodology, see Gillespie 1992. See also Straus 1990, 23-25. Lewin's transformational theory addresses similar concerns as those addressed in this paper (Lewin 1987 and 1993). The theory points towards a type of unifying structure (the motive or pitch collection to be transformed) and towards prolongational processes (the transformational networks) more appropriate for post-tonal music than linear models derived from the tonal repertoire. Lewin's methodology seems indeed to be closer to a true theory of atonal prolongation than any post-Schenkerian methodology has proved to be. Although Lewin does not refer to his methodology as prolongational, he has used the concept of prolongation to refer to different hierarchical levels of Klumpenhouwer Networks (Lewin 1990, 94 and 115). Lewin notices that a higher-level network (interpreting a chord progression) "prolongs" a lower-level one (interpreting a chord). Lewin's analysis of the opening of Dallapiccola's "Simbolo," from Quaderno musicale di Annalibera, for instance, illustrates the parallelisms between PCSE and transformational analysis. See Lewin 1993, chapter 1, Example 1.2.
13In his unpublished article, "New Modes of Linear Analysis," Allen Forte defines a set-class transformation similar to my linear transformation of set classes, and which he calls "unary voice leading transformations" (Forte 1988b). Forte defines unary transform as "the mutation of one pitch-class set into another by a change of a single element." Forte's unary transform and my linear transform, however, differ in several aspects. Forte's unary transform in based on actual pcs, it allows for only one element to change, and the change can be by any interval. My linear transform, on the other hand, applies to set classes, allows for more than one element to change, and the change can be effected only by CHR or WT.
14For a study of transformational structures in this piece, see Lai 1989.
15Analyses of "Farben" in which the principles of PCSE are made apparent can be found in Burkhart 1973-74 and Rahn 1980, 59-72. See also Coppock 1975 and Forte 1973, 166-77.
16See Bass 1991, 8-14.
17Two fragments from Ligeti's Ramifications which I have analyzed elsewhere, on the other hand, illustrate PCSE based on linear chromatic expansion and contraction similar to the Wozzeck example we have discussed above. See Exx. 2 (mm. 1-10) and 3 (mm. 10-26) in Roig-Francolí 1995.
18Perle has discussed the role of both the twelve-tone row in general, and trichord  (although Perle does not use Forte's set-theoretical terminology) in Webern's Concerto in particular, as "basic cells" with a function similar to his basic cell in free atonal music. See Perle 1991, 79-80, and note 11 above.
19For an example of a serial piece featuring total uniformity of extended musical space due to the use of aggregates, see Milton Babbitt's Music for Twelve Instruments. The combinatorial properties of the row used in this piece have been discussed by the composer in Babbitt 1961, 81-82. See also Hush 1982-83 and Westergaard 1965.
20There are numerous references to this passage in the published literature. See, for instance, Haimo/Johnson 1984, 48-51; Lewin 1962, 92; Westergaard 1966, 100-101; and Wittlich 1975, 410-11.
21Analyses of op. 33a can be found in Cook 1987, 322-33; Graebner 1973-74; Perle 1991, 111-16; and Straus 1990, 173-79.
22Because of the extension of combinatorial areas in this piece, any other segments of the row could be traced (and found at the same pitch level) throughout large spans of music. However, sets  and  have special significance because (a) they begin and end row forms or row pairs, (b) they include the stacked-IC5 motive which has structural significance at both the foreground and middleground levels, and (c) they occur at both the horizontal levelas  in a single row—and the vertical level—as  encompassing the opening dyads of a pair of rows, and hence they are an element of spatial uniformity.
Miguel A. Roig-Francolí (Ibiza, Spain, 1953) is a Distinguished Teaching Professor of Music Theory and Composition at the College-Conservatory of Music, University of Cincinnati. His work as a music theorist, composer, musicologist, and pedagogue has been recognized internationally. His compositions have been widely performed in Spain, England, Germany, Mexico, Italy, Switzerland, and the United States. He has published over twenty articles in leading scholarly journals and collections of essays in the U.S.A., England, Italy, and Spain, on topics ranging from sixteenth-century compositional theory and the music of Spanish Renaissance composers Antonio de Cabezón and Tomás Luis de Victoria to the music of György Ligeti. Roig-Francolí is the author of two textbooks published by McGraw-Hill and widely used at universities in the U.S. and Canada. The recently issued Chinese translation of Understanding Post-Tonal Music is quickly becoming the standard post-tonal textbook at major Chinese conservatories. Among his many honors are first prize at the National Composition Competition of the Spanish Jeunesses Musicales (1981) and second prize at the UNESCO International Rostrum of Composers (Paris, 1982), both for Five Pieces for Orchestra; the Medal of Honor from the Superior Conservatory of Music of the Balearic Islands (Spain, 2004); the University of Cincinnati’s A.B. "Dolly" Cohen Award for Excellence in Teaching (2007) and George Rieveschl Jr. Award for Creative and/or Scholarly Works (2009), and the Ramón Llull Prize from the Goverment of the Balearic Islands (2010).