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COPYRIGHT 2003 University of Washington
THIS ARTICLE INVESTIGATES a technique of pitch organization that is common in post-tonal music, involving static formations I call "pitch fields." After an introductory example and a brief, informal survey of pitch-field properties and their significance in works by Mel Powell, Elliott Carter, and Witold Lutoslawski, I develop a novel type of pitch field modeled on these precedents. Fields of this new type are generated by cyclic strings of intervals in a manner that I describe formally and relate to work in the theory of scales by Stephen Soderberg and others.
The second half of the article develops the idea of cyclically generated pitch fields, pursuing two paths. First, I examine chord formation and voice-leading in the context of this type of field. And second, I develop (in two stages) a concept of "pitch field systems," which give rise to families of related pitch fields and provide a principled means of modulating from field to field.
AN EXAMPLE FROM WEBERN
Example 1 presents the opening fourteen bars of the first movement of Webern's Symphony, op. 21. This movement proceeds as a double canon by inversion, organized in the form A :[parallel]: BA'. Its distinctive orchestration and other factors are omitted from the example, which is formatted according to the four-voice texture of the canon and runs through the first row statement of each voice: I9 answers P9 in canon 1, and P1 answers I5 in canon 2. (1)
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A well-known feature of this movement is its consistent placement of pitch classes in particular registers. This feature is evident in the excerpt given in Example 1, persists through the remainder of section A (measures 1-26), is largely absent from section B (measures 25-44), and returns--with different particulars--in section A' (measures 42-66). The registral disposition of pitch classes in the first and last of these three sections is given in Example 2 and represents in each case an instance of what I shall call a pitch field.
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The pitch fields depicted in Example 2 are structured in interesting ways that we shall soon explore. But in the general case, very little structure is required of a pitch field--so little, in fact, that a definition in structural terms hardly seems promising: a pitch field is an unordered collection of pitches (n.b. pitches, modeled by integers--not pitch classes--modeled by integers mod 12). What makes these constructs interesting, in the absence of special kinds of structure, is their usage: in a passage based on a pitch field, the pitches of the field are kept in circulation to the exclusion of other pitches. This usage places pitch fields at an intermediate position between the conventional notions of chord and scale.
Like a chord, a pitch field possesses a characteristic harmonic sonority to which all of its constituent pitches contribute, and we assume this sonority will color to some degree any music based on the field. We can hear our Webern example in this way as harmonically static, even though there is certainly more to its pitch organization than the elaborate "arpeggiation" of a few stationary chords. Unlike the usual notion of a chord, however, a pitch field may retain little of its identity when viewed as a collection of pitch classes. For instance, any distinctions between the two fields in Example 2 vanish when they are viewed in this way as twelve-pitch-class aggregates.
Like a scale, a pitch field provides a graded system of pitch levels, and melodies based on the field may be understood to move by step or by leap among these levels. Once we have become attuned to the underlying pitch field in our Webern example, we can hear figures such as the F#4-G2 descent (played by horn 2) in measures 2-3 as leaping over familiar intermediate pitch levels. Unlike the usual notion of a scale, however, the distinction between steps and leaps in a pitch field may not correlate with intervallic distance. In other words, some of the steps may involve larger distances than some of the leaps, producing what are known in the theory of scales as contradictions. (2) In Example 2a, for instance, the step from G2 to C3 is larger than the leap from G#3 to B[flat]3.
The pitch fields depicted in Example 2 possess a number of interesting properties. Each of these fields exhausts the twelve-tone aggregate, as it must on account of the serial structure of the music it supports. In addition, each of these fields is inversionally symmetrical. This, too, is a necessary consequence of other kinds of structure in the music, namely its canonic organization. In measures 1-26, each dux voice is inverted around A3 in the corresponding comes voice. This center pitch is present in the field--it is marked with an open notehead in Example 2a--and is the only instance of pitch class A. With one exception, the remaining pitch classes are also represented just once each and are positioned registrally so that each is balanced by its inversional partner an equal distance in the opposite direction from A3. The exceptional pitch class, of course, is E[flat], which is its own inversional partner and must therefore appear both above and below the pitch center A3--the duplicate E-fiats are stemmed in Example 2a.
The pitch-field symmetry in measures 42-66 is similar; here the pitch center E[flat]5 is not itself present in the field, so two pitch classes, A and E[flat], occur twice, as shown with stems in Example 2b. The centers of symmetry involved in measures 1-26 and 42-66 represent the same case of pitch-class inversion ([T.sub.6]I, assuming C = 0), which enables these fields to support the same succession of row forms in the same canonic voices; this is the main sense in which the latter section "recapitulates" the material of the former one.
One consequence of the great majority of pitch classes being fixed in single registral positions in the Symphony fields is that striking unisons occur whenever multiple twelve-tone rows-in-progress reach the same one of these pitch classes at the same time. For instance, the whole-note dyads G2-A[flat]3 and B4 to B[flat]3, which the horns contribute to canon I in measures 3-6, occur against the same dyads in the same measures played by the pizzicato strings as part of canon II. Meanwhile, the arco violin contributes an additional B[flat]3 (in the dux voice of canon II), also in measure 6.
Just as striking as these frequent unisons are the numerous occurrences of pitch interval 5 in measures 1-26 of the Symphony. Interval class 5 is unavailable between adjacent pitch classes in the row class of the Symphony, but it occurs reasonably often when elements of different rows coincide. When this interval class does occur, the underlying pitch field ensures that it nearly always appears as an interval of five semitones rather than some octave-equivalent alternative--for, as Example 3a demonstrates, the structure of this pitch field is heavily weighted in favor of pitch interval 5. Similarly, Example 3b reveals a marked emphasis on pitch interval 3 and certain spacings of set class [036] in the field underlying measures 42-66. In both cases, the intervallic makeup of the pitch field lends a particular coloration to the music based on it.
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A FIELD SURVEY
Webern's use of pitch fields in the Symphony is an early example of a technique that has become important to many posttonal composers? Before constructing and investigating the distinctive type of pitch field that is the main concern of this article, we consider here a few additional precedents, focusing on the special pitch-field properties that various composers have favored in their work.
We begin with the case of Mel Powell, whose music from the late 1950s onwards reflects in many ways his careful study of Webern. Stylistically, there is the same miniaturization, economy of materials, and preference for quiet, transparent textures. And procedurally, there is a frequent use of pitch field techniques for which the Webern Symphony provides a plausible model. Especially in his music of the 1980s and 1990s, Powell based his pitch syntax on a particular type of pitch field, which he called a tableau.
In Powell's formulation, a tableau fixes each of its constituent pitch classes in a single register; it need not consume the entire twelve-tone aggregate, but it often does. When it does not, "harmonic friction" remains characteristic, with each of its pitch classes normally participating in at least one instance of interval class 1 or 6. (4) This sort of "friction" is evident in each of the first four tableaus occurring in Powell's 1985 Woodwind Quintet, which are labeled F1-F4 in Example 4.
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A striking feature of each of the illustrated Quintet fields is the replication of a simple pitch-interval pattern at multiple pitch levels. For instance, the figures p and q in Example 4 show how the pattern , formed at two pitch levels 11 semitones apart, constitutes tableau F1. It is characteristic of Powell's compositional practice that the identities of p and q are not projected motivically in the musical foreground (compare the music based on tableau F1 in Example 5)--more often than not, musical gestures cut across the boundaries separating these two subsets.
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The design of tableaus F2-F4 is related to that of F1. The relationship is particularly close in the case of F3, which can be constructed from F1 by transposing q down an octave (with the result q'). In tableau F4, the segments labeled [w.sub.1] and [w.sub.3] together constitute a transposition by six semitones of tableau F1, while [w.sub.2] is an additional instance of the pattern and an octave transposition of p; furthermore, F4 partitions into the segments labeled x and y, which are related to one another, as pitch-class collections, by transposition ([T.sub.6]). Tableau F2, finally, is assembled similarly to F1; but its replicated interval pattern is ; the two instances of this pattern, labeled r and s, stand at a distance of thirteen semitones from one another; and s is altered by the displacement of E[flat] up one octave.
Replicated interval patterns of the sort just mentioned are observable in nearly every tableau used in the Quintet. In a discussion that draws examples from Powell's Setting for Guitar (1986), Anthony Kroyt Brandt calls pitch fields organized in this way Powell's "favored tableau type" and suggests that "the resultant structural 'resonance' is a nontonal analog of the resonance of octaves in tonal music." (5) We shall incorporate a similar feature into the pitch fields we construct further on in this article.
Elliott Carter's use of pitch fields, normally in the form of "twelve-note chords," is well documented--most broadly so in David Schiff's The Music of Elliott Carter, which summarizes the role of these formations in each major work beginning with the Double Concerto (1961), where they first appear. (6) At times, Carter's twelve-note chords emerge complete as harmonic events--a well-known instance is the transition from the first to the second movement of the Piano Concerto (1965). And Andrew Mead has demonstrated that in some cases these chords serve more specialized purposes. In String Quartet No. 3 (1971), for instance, Mead has shown how Carter extracts all of the tetrachords formed by registrally adjacent pitches in that work's governing twelve-note chord and uses transformations of them in passages in which the twelve-note chord is not otherwise present. (7)
But the primary function of Carter's twelve-note chords is to serve, in Mead's words, "as a background grid against which collections of smaller cardinality are projected." (8) Jonathan Bernard relates this function to the composer's "deliberately global" use of smaller pitch sets--for instance, the systematic inclusion of all possible trichord types in the Piano Concerto, all possible tetrachord types in String Quartet No. 3, and so on. With so many distinct pitch formations in play, twelve-note chords provide a "means of restriction," determining the most characteristic spacing of each smaller set. (9)
A favored pitch field type in Carter's music from String Quartet No. 3 onwards locates each pitch interval from 1 to 11 semitones in a unique position between adjacent pitches; in other words, it corresponds to an all-interval twelve-tone row whose order is expressed in register. But Carter's music of the 1960s and 1970s demonstrates an interest in other types of pitch fields as well. For instance, the field depicted in Example 6, from the Duo for Violin and Piano (1974), unfolds the interval string repeatedly as it ascends. (10) This field is organized very much like the ones we shall construct further on, and its interval structure makes it possible (as we shall later see) to predict very efficiently how various smaller sets embed into it.
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Twelve-note chords similar to Carter's form the foundation of Witold Lutoslawski's harmonic language. (11) But whereas Carter tends to keep these pitch fields in the background, scanning them for smaller embedded formations, Lutoslawski, as a rule, treats them as integral chords with distinctive harmonic qualities. Lutoslawski's interest in these chords principally for their sonority influences the kind of structure he favors, as the following remarks, from a conversation with Charles Bodman Rae, indicate:
One rule which it is possible to formulate about my experiments with twelve-note chords is that the fewer different intervals between neighbor[ing] notes the chord contains, the more characteristic the result is. If, for instance, you use all possible intervals in one chord, the final result is, in a way, faceless, something which has no character, which in color is grey. (12)
Lutoslawski's tendency to limit the intervals between adjacent chord members to a few different types has led Steven Stucky and Charles Bodman Rae to propose similar classification systems for these pitch fields. Stucky's simpler system involves three categories: chords emphasizing interval classes 1, 5, and 6 (such as Example 7a); those emphasizing interval class 2 (such as Example 7b and c); and those in which a presumed emphasis on interval classes 3 and 4 yields a structure containing various types of triads and seventh chords as adjacencies (such as Example 7d, where the relevant adjacencies are delineated by beaming). (13) In relatively complex cases like Example 7d, Rae proceeds by parsing a twelve-note chord into three tetrachordal segments, which he finds to be often types based on various arrangements of pitch class intervals 3, 4, and 5. (14) The segments involved in Example 7d--and demarcated there by dotted barlines--represent two of the possible types: the upper and lower tetrachords are equivalent to dominant seventh chords (which stack the intervals ), and the middle one can be formed by stacking the intervals .
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Repeating interval patterns like those evident in the first three instances in Example 7--namely, in (a), in (b), and in (c)--are common in Lutostawski (although equally restricted interval schemes without precise repetition are also plentiful). The structure of these pitch fields makes them akin to the ones depicted in Example 4 (Powell) and Example 6 (Carter) and to the special class of pitch fields we shall now begin to construct and investigate.
In addition to the idea of repeating interval patterns, we shall consider other issues raised in the above "survey" as we proceed. We shall examine some of the ways in which distinct fields may share structure, as they did in our example from the Powell Quintet. And we shall explore the ways in which various small formations embed within a larger pitch field--an issue that is central, as I have suggested, to Carter's use of these materials. In addition, we shall consider how successive pitch fields may be linked by the pitches that they hold in common, an important process in the work of each composer we have considered, but one whose details exceed the limited scope of our survey. (15)
INTERVAL STRINGS AS PITCH FIELD GENERATORS
We begin by recalling that in the most general case, a pitch field p is simply an unordered collection of pitches. Indexing these pitches in ascending order gives us p = {..., [p.sub.-2], [p.sub.-1], [p.sub.0], [p.sub.1], [p.sub.2], ...} = [{[p.sub.i]}.sup.[infinity].sub.-[infinity]] when the field is infinite, or p = {[p.sub.1], [p.sub.2], ... [p.sub.n]} = [{[p.sub.i]}.sup.n.sub.1] when it is finite. We consider the pitches [p.sub.i] to be situated in a referential pitch space P of the type Robert Morris calls p-space. (16) We write P = {..., -2, -1, 0, 1, 2, ...} and assume by default that the minimal interval is the semitone [sup.12] [square root of 2] : 1 and that the pitch corresponds to middle C.
Next we construct a model for the repeating interval patterns we have encountered in the examples presented above. An interval string is an ordered n-tuple of positive integers s = that is taken to be circular, so that [i.sub.1] is the successor of [i.sub.n] in the same sense that [i.sub.2] is the successor of [i.sub.1].
An interval string x is reducible if there is a non-zero rotation of x, [r.sub.j]x, such that [r.sub.j]x = x. If x is reducible and j is the smallest nonzero integer for which [r.sub.j]x = x, then the string w consisting of the first j elements of x is said to be the reduction of x. For example, the string is reducible to , the string is reducible to , and the string is not reducible.
Given an interval string x = , we define the cardinality of x (length of x, number of elements in x) #x = n, and the modulus of x (sum of the elements in x) SUM(x) = [x.sub.1] + [x.sub.2] + ... + [x.sub.n]. If x is reducible, we normally prefer to assess the cardinality and modulus of its reduction.
With our intervallic building blocks in place, we now consider the relation between a repeating interval pattern and the pitch field in which we find it. Given an initial pitch [P.sub.0], the interval string x = generates the pitch field where p = ([p.sub.0],x) = [{[p.sub.i]}.sup.[infinity].sub.-[infinity]],
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For example, if [p.sub.0] = 0, then the interval string x = generates the pitch field ([p.sub.0],x) = {...,-9,-7,-6,-2,0,1,5,7,8,...}. If the interval string s generates the pitch field p, we say conversely that p unfolds s. It should be evident that a reducible string and its reduction will generate identical pitch fields.
There is, we shall soon demonstrate, an advantage to defining cyclically generated pitch fields so that they extend infinitely low and high. But in compositional and analytical contexts, a pitch field will necessarily be truncated to some portion between a lower and an upper bound, and where the context is clear, we shall continue to call this finite result a cyclically generated pitch field. We should note, however, that any finite pitch collection may be viewed as a truncated, cyclically generated pitch field if we assume a sufficiently long generating interval string. Normally, we expect the cardinality of a truncated, cyclically generated pitch field to significantly exceed the cardinality of the interval string that it unfolds, so that its circular features will be apparent.
MODULAR STRUCTURE OF CYCLICALLY GENERATED PITCH FIELDS
Readers familiar with Stephen Soderberg's treatment of "white note systems" will recognize that my interval string model corresponds closely to his. (17) But the two models do not entirely coincide. Soderberg defines interval strings in relation to modular spaces, requiring that the elements of a string in a mod-m space sum to m, whereas I define interval strings in relation to an unchanging p-space and associate a modulus with each string. Soderberg's strings generate collections of pitch classes, whereas mine generate collections of pitches: the cyclically generated pitch fields described above.
A modular interpretation of these pitch fields is both straightforward and useful, however. Consider the field p illustrated in Example 8a, which unfolds the interval string with initial pitch 0. Assume p extends infinitely low and high, even though only a finite portion is illustrated. Now consider what happens when p is transposed upwards one semitone at a time. The first six results are given in Example 8b, while one additional transposition produces the result given in Example 8c--a field identical to p at its original pitch level. (The truncations of 8a and 8c differ, but they stand for exactly the same infinite field.) Thus there are only seven distinct transpositions of p. This result is quite general: any pitch field that unfolds an interval string x has m distinct transpositions, where m = SUM(x), the modulus of x.
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Furthermore, when an interval string with modulus m generates pitch fields, it organizes the underlying p-space into a modular space of m pitch classes. The mod-7 space corresponding to the fields of Example 8 is illustrated in Example 9a, and the original field p = (0, )--from Example 8a--is written as a two-element pitch-class set in Example 9b. In Example 9, pitches that are vertically aligned are equivalent mod 7. This modular structure derives from the sum of the interval pattern , which generates the fields depicted in Example 8 and Example 9b.
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In general, the cyclically generated pitch field q = ([q.sub.0],y) = [{[q.sub.i]}.sup.[infinity].sub.i=-[infinity]] corresponds to the set of mod-m pitch classes [{[q.sub.i]}.sup.n-1.sub.i=0], where m = SUM(y), n = #y, and all the [q.sub.i] are defined as before but reduced mod m. In case [q.sub.0] = 0, no modular reduction is necessary: all of the stipulated values [q.sub.i] are already valid in the mod-m space.
With this shift from pitches to pitch classes, the gap between my interval string model and Soderberg's has essentially closed. However, I shall continue to grant significance to the underlying p-space in ways that Soderberg does not. He begins with a referential space of pitch classes determined by the psychoacoustical phenomenon of "octave equivalence," (18) and his interval strings are made to conform to the modular structure of this underlying space. In contrast, the modular structure in my model is "induced" by the unfolding of an interval string across p-space; the cyclic nature of this process, and of the resulting structure, creates, rather than conforms to, a system of equivalence classes.
Pitch-class spaces in which the modular interval differs from the octave 2:1 have been explored by a number of composer-theorists, notably John Rogers and William E. Benjamin. (19) Rogers calls these variable intervals of modularity "contextual octaves," regarding them as "contractions or expansions" of the true octave. For this contraction or expansion process to yield a mod-m organization of the usual chromatic, it would have to be applied to a space with m pitch classes per true octave. It would have to expand or contract that space so that twelve pitch classes comprise a true octave in the result, which would consist of m pitch classes per contextual octave, with a contextual octave smaller than the true octave when m 12. As the inelegance of this account may hint, Rogers's main interest is not in organizing the usual chromatic into m [not equal to] 12 equivalence classes, but...
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