andrew mead tonal forms in schoenberg s twelve tone music

Society for Music Theory

'Tonal' Forms in Arnold Schoenberg's Twelve-Tone MusicAuthor(s): Andrew MeadSource: Music Theory Spectrum, Vol. 9 (Spring, 1987), pp. 67-92Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/746119 .

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'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music

Andrew Mead


Andrew Mead


Andrew Mead


Andrew Mead


Andrew Mead


Andrew Mead

Many of Arnold Schoenberg's twelve-tone compositions display surface features that strikingly invoke large-scale tonal forms. This has led some to conclude that Schoenberg, like Moses, having brought his followers to the edge of the prom- ised land, could not bring himself to enter-that having discovered twelve-tone composition, he used it as a means of filling the shells of tonal forms rather than pursuing its own particular formal and form-generating properties.1 Analysis reveals, however, that even Schoenberg's most ostensibly "tonal" forms are not some sort of musical taxidermy-rondo and sonata-allegro skins stuffed and mounted with chromatic saw- dust. On the contrary, the surfaces of his music reveal compositional strategies animated by relations provided by the twelve-

'The following is typical: "[T]he Wind Quintet is composed of the four movements of the Beethovian [sic] sonata. That simple enumeration permits me to assert that Schoenberg was employing the nascent serial technique to enclose preclassic and classic forms in the elaboration of a world ruled by functions antagonistic to those very forms; the articulation of an architecture not flowing entirely from serial functions, the hiatus between the structural edifice of the work and the determination of its material is clear." (Pierre Boulez, Notes of an Apprenticeship [New York: Alfred A. Knopf, 1968]: 255-256, translated from the French by Herbert Weinstock.)

This sort of criticism is also used to motivate the discussion in Martha

Hyde, "The Roots of Form in Schoenberg's Sketches," Journal of Music The-

ory 24/1 (1980): 1-36.





ory 24/1 (1980): 1-36.





ory 24/1 (1980): 1-36.





ory 24/1 (1980): 1-36.





ory 24/1 (1980): 1-36.





ory 24/1 (1980): 1-36.

tone system. Despite surface similarities to tonal idioms, Schoenberg's twelve-tone music represents a distinctly different form of musical life. I will attempt to demonstrate this point in the present paper through analysis of two of the more dra- matically imitative movements in Schoenberg's work, the first and last movements of the Wind Quintet, opus 26, and will suggest some extensions by means of a few brief examples from some of his other twelve-tone compositions.2

Before proceeding, it is necessary to define briefly the nature of relations provided by the twelve-tone system. Relations in a twelve-tone composition depend on the variety of relations

2The interaction of form and twelve-tone structure in Schoenberg's music has been a central question in twelve-tone analysis from the outset. Among the historically significant writings are Milton Babbitt, "Set Structure as a Compo- sitional Determinant," Journal of Music Theory 5/1 (1961): 72-94; David Le- win, "A Theory of Segmental Association in Twelve-Tone Music," Perspec- tives of New Music 1/1 (1962): 89-116; George Perle, Serial Composition and Atonality (London: Faber and Faber, 1962); and Schoenberg's own essay, "Composition with Twelve Tones," in Style and Idea (New York: Philosophi- cal Library, 1950).

Recently there has been renewed interest in this area, yielding a number of articles. An analysis of the entire Wind Quintet is found in Langdon Corson, Arnold Schoenberg's Woodwind Quintet Op. 26 Background and Analysis (Nashville: Gasparo Company, 1984). Corson describes some of the row rela-























68 Music Theory Spectrum 68 Music Theory Spectrum 68 Music Theory Spectrum 68 Music Theory Spectrum 68 Music Theory Spectrum 68 Music Theory Spectrum

tions found below, and makes a number of general observations about the large-scale forms of the work's movements. His discussion, however, attempts to bridge the gap between large-scale form and twelve-tone structure by creating analogies between rows used and tonal functions.

Ethan Haimo and Paul Johnson, in "Isomorphic Partitioning and Schoen- berg's Fourth String Quartet, " Journal of Music Theory 28/1 (1984: 47-72, de-

velop various invariance relations among rows based on non-segmental partitioning, and illustrate their consequences with examples from the first movement of the Fourth Quartet

Martha Hyde has explored the interaction of twelve-tone structure and form in a number of articles, starting with "The Roots of Form," and most recently in "Musical Form and the Development of Schoenberg's Twelve-Tone Method," Journal of Music Theory 29/1 (1985): 85-143. In the latter Hyde outlines two basic techniques she has discovered in the Suite, Opus 25, and suggests their significance for later twelve-tone works. The first, initially developed in her earlier essays, identifies collections projected in the musical surface derived from non-adjacent row elements, or elements from more than one row, that belong to the same collection classes as the various collections found segmentally within the members of the row class. These she calls secondary harmonies. For the most part, Hyde concentrates on collection class member- ship, rather than actual pitch class content. In order to deal with collections arising in the musical surface that do not belong to collection classes represented segmentally, Hyde has identified a second technique, which she calls invariant harmonies. These consist of collections belonging to the same collection classes as collections assembled from various segments held invariant

among specific members of the row class used in the movement, or the composition as a whole. As with the secondary harmonies, collection class member-

ship is emphasized over pitch class content. Hyde restricts herself to segmental invariants, and does not distinguish between those invariant segments arising as properties of pitch class collections at a fixed set of order numbers in two members of a row class, and those arising from the presence of more than one member of a pitch class collection class associated with more than one member of an order number collection class in the row. The two techniques serve to articulate and define formal sections in Hyde's analyses within and among movements of Opus 25, based on the collection classes represented in the musical surface.

Steven Mackey, in The Thirteenth Note (Ph.D. Dissertation, Brandeis Uni- versity, Spring 1985), considers the question of registral placement of tones in the unfolding of material at the outset of the first movement of the Third Quar- tet, and relates the resulting observations to the way the material returns at the

recapitulation. From this analysis he develops a discussion concerning the









































among rows of the work's row class.3 A row is a specific order-

ing of the twelve pitch-classes, or in other words a specific as-

signment of the twelve pitch classes to the twelve order num-

function of pitch class registral placement in local and long-range unfolding of

aggregate structures. While largely concerned with a detailed account of local progression,

Stphen Peles, in "Interpretations of Sets in Multiple Dimensions: Notes on the Second Movement of Arnold Schoenberg's String Quartet no. 3," Perspectives of New Music 22, nos. 1 and 2 (Fall-Winter 1983, Spring-Summer 1984): 303- 352, makes a number of points crucial to the understanding of the movement's large-scale formal procedures. First, he notes the projection of multiple simul- taneous collections in the surface based on different criteria of grouping, and further notes their partitional derivations. His analysis reveals the recurrence of specific collections, derived in a variety of ways from the underlying rows, and suggests the complex network of interdependent strategies based on different collectional invariance and order-relations among rows that forms the structure of the movement

Bruce Samet, in HearingAggrgates (Forthcoming, Pennsylvania State Uni- versity Press), traces in minute detail the use of row relations through the opening sections of the third movement of the String Quartet no. 4. Samet is rigor- ous in his perceptual criteria, and is able to reveal the local and global structural significance of the way each of the aggregates of the passage unfolds, individu- ally and as parts of larger structures. His accounting takes in details of articula-

tion, dynamics, register, note repetition and so forth to reveal the consequences of Schoenberg's compositional choices to a profound degree.

My "Large-Scale Strategy in Arnold Schoenberg's Twelve-Tone Music," Perspectives of New Music 24/1 (1985): 120-157, traces long-range connections among sections of movements of the Violin Concerto, Wind Quintet and Piano Concerto based on the use of recurring pitch class collections held invariant in various ways, both segmentally and non-segmentally among rows and collections of rows, and examines the use of interdependent strategies to articulate large-scale form.

3Throughout the paper I shall make the distinction between an instance of an entity and the equivalence class to which it may belong. Notationally, a collection, whether of order numbers or pitch classes, will be indicated with curly brackets: 0, and a collection class with parentheses: (). This reflects the distinction made between sets and set-types outlined in Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973). Another outline of pitch class set theory is found in John Rahn, Basic Atonal Theory (New York: Longman, 1980).





















































'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music 69 'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music 69 'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music 69 'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music 69 'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music 69 'Tonal' Forms in Arnold Schoenberg's Twelve-Tone Music 69

bers.4 A row class is the equivalence class of rows generated from some row together with a specification of operators and their ranges.5 Row classes are closed systems: any operator applied to any member of the row class will yield another member of the row class.6 Obviously, a composition employing a given row class need not exhaust the row class.

In classical descriptions of twelve-tone theory, the operators are Tx and Iy applied to the pitch classes, with x and y ranging from 0 to e (10 = t, 11 = e) mod 12 (where x is the constant of

transposition, and y the constant of inversion, or index number),7 and TO and Ie applied to the order numbers, yielding the identity and retrogression.8 (In this paper, pitch classes and their operations will be notated in roman: Tx, Iy, 0, 1, 2, .. .;

4This manner of describing twelve-tone rows originates in Milton Babbitt, "Twelve-Tone Invariants as Compositional Determinants," Musical Quarterly 46 (1960): 246-259, reprinted in Problems in Modern Music, ed. Paul Henry Lang (New York: Norton, 1962).

5This matches the approach used in Daniel Starr, "Sets, Invariance, and Partitions," Journal of Music Theory 22/1 (1978): 1-42. However, I define my operators in a slightly different way.

6The group theoretical implications here implied by the twelve-tone system are indicated in Babbitt, "Twelve-Tone Invariants," and are examined in

depth by Robert Morris in his forthcoming book on group theory and twelve- tone music.

7Index number is first defined in Milton Babbitt, "Twelve-Tone Rhythmic Structure and the Electronic Medium," Perspectives of New Music 1/1 (1962): 49-79. It is the constant from which an element of a collection is subtracted in order to determine its inversion. Thus, for the operators described above, Tx{a, b} = {a + x, b + x} and Iy{a, b} = {y-a, y-b}.

8The implied isomorphism of pitch classes and order numbers has been ex-

plored in a number of articles, including David Lewin, "On Certain Tech-

niques of Re-ordering in Serial Music," Journal of Music Theory 10/2 (1966): 276-282; Andrew Mead, "Some Implications of the Pitch Class/Order Num- ber Isomorphism Inherent in the Twelve-Tone System," Perspectives of New Music forthcoming; Robert Morris, "On the Generation of Multiple-Order- Function Twelve-Tone Rows," Journal of Music Theory 21/1 (1977): 238-263; Walter O'Connell, "Tone Spaces," Die Reihe 8 (1968): 35-67; John Rahn,
























































order positions and their operations will be notated in italics

Tx, Iy, O, 1, 2,....) In practice, Schoenberg used both smaller and larger row classes.9

Relations among rows of a row class are basically of two

types: those involving unordered pitch class collections held invariant in some way, and those based on ordered pitch class collections. Each of these basic distinctions contains a number of

sub-types, and the two general types are highly interdependent. The concept of the mosaic is useful for discussing all types of row relations.10 A mosaic, notated W or W, is a parsing of the twelve pitch classes or order numbers into discrete collections. The collections of a mosaic are not ordered with regard to each

other, nor are the elements of each constituent collection. Mo- saic classes are equivalence classes of mosaics under Tx and Iy (or Tx and Iy).11 A mosaic may be mapped onto itself if for some operation at some value there exists in the resulting mo-

"On Pitch or Rhythm: Interpretations of Ordering Of and In Pitch and Time," Perspectives of New Music 13/2 (1975): 182-203; Larry Solomon, "New Sym- metric Transformations," Perspectives of New Music 11/2 (1973): 257-264; Mi- chael Stanfield, "Some Exchange Operations in Twelve-Tone Theory: Part

One," Perspectives of New Music 23/1 (1984): 258-277, and "... Part Two," Perspectives of New Music 24/1 (1985): 72-95.

9The rows of the Suite Opus 25 are a subset of the classical row class of that

piece, but form a closed system under the operators. This piece is discussed in Martha Hyde, "The Roots of Form." The Wind Quintet employs the full range of operators on the order numbers as well as the pitch classes, yielding a grand row class of 576 members. This is detailed in my "Large-Scale Strategies." Confirmation from the sketches is offered in Fusakao Hamao, "The Historical

Origin of Schoenberg's Combinatorial Hexachord," presented at the 1986 So-

ciety for Music Theory annual conference, Bloomington, Indiana. "The term "mosaic" is introduced in Donald Martino, "The Source Set

and Its Aggregate Formations," Journal of Music Theory 5/2 (1961): 224-273. "An interesting feature of mosaic equivalence classes is that while two

members of an equivalence class must have collections of the same classes, their having collections of the same classes is not sufficient to guarantee mem-

bership in a mosaic class. Consider the following mosaics:















































































saic a collection of identical content for each collection of the

original mosaic.12 Note that this means that individual collections may map onto themselves or onto each other, in some combination. Figure 1 is an example of a mixture of collectional

mappings.

Figure 1

W: {3,7,9, e} {l, , t, 2} {4,6,8,5}

I6(W): {3, e, 9,7} {5,6,8,4} {2,0, t, l}



mappings.

Figure 1

W: {3,7,9, e} {l, , t, 2} {4,6,8,5}

I6(W): {3, e, 9,7} {5,6,8,4} {2,0, t, l}



mappings.

Figure 1

W: {3,7,9, e} {l, , t, 2} {4,6,8,5}

I6(W): {3, e, 9,7} {5,6,8,4} {2,0, t, l}



mappings.

Figure 1

W: {3,7,9, e} {l, , t, 2} {4,6,8,5}

I6(W): {3, e, 9,7} {5,6,8,4} {2,0, t, l}



mappings.

Figure 1

W: {3,7,9, e} {l, , t, 2} {4,6,8,5}

I6(W): {3, e, 9,7} {5,6,8,4} {2,0, t, l}



mappings.

Figure 1

W: {3,7,9, e} {l, , t, 2} {4,6,8,5}

I6(W): {3, e, 9,7} {5,6,8,4} {2,0, t, l}

A mosaic of one domain applied to a row yields a mosaic of the other domain: W(P) is thus an order number mosaic, and

W(P) is a pitch class mosaic. This is illustrated in Figure 2.

Figure 2

P: 0123456 789te

379e10t24685

W:{3,7,9,e} {l,0,t,2} {4,6,8,5}

W(P):{0, 1, 2, 3} {4, 5, 6, 7} {8, 9, t, e}

W: {0, 1, 2, 3, 4} {5, 6} 7, 8, 9, t, e

W(P):{3,7,9,e, l} {0,t} {2 4 6 8 5}

W1: {0,1,2} {e,3,4} {6,9,t} {5,7,8} W2: {0,1,2} {5,9,t} {7,8,e} {3,4,6}

While both mosaics each contain a member of each of the collection classes (0,1,2), (0,1,3), (0,1,4), and (0,1,5), there is no simple operation that will map W1 onto W2. A study of trichordal mosaics and their interaction with hexachordal mosaics is found in Steve Rouse, "Hexachords and their Trichordal Gener- ators: An Introduction," In Theory Only 8/8 (1985): 19-43.

12Studies of collectional invariance properties are found in Forte, The Structure of Atonal Music, Herbert Howe, Jr. "Some Combinational Proper- ties of Pitch Structures," Perspectives of New Music 4/1 (1965): 45-61, and Rahn, Basic Atonal Theory.



Figure 2

P: 0123456 789te

379e10t24685

W:{3,7,9,e} {l,0,t,2} {4,6,8,5}

W(P):{0, 1, 2, 3} {4, 5, 6, 7} {8, 9, t, e}

W: {0, 1, 2, 3, 4} {5, 6} 7, 8, 9, t, e

W(P):{3,7,9,e, l} {0,t} {2 4 6 8 5}

W1: {0,1,2} {e,3,4} {6,9,t} {5,7,8} W2: {0,1,2} {5,9,t} {7,8,e} {3,4,6}





Figure 2

P: 0123456 789te

379e10t24685

W:{3,7,9,e} {l,0,t,2} {4,6,8,5}

W(P):{0, 1, 2, 3} {4, 5, 6, 7} {8, 9, t, e}

W: {0, 1, 2, 3, 4} {5, 6} 7, 8, 9, t, e

W(P):{3,7,9,e, l} {0,t} {2 4 6 8 5}

W1: {0,1,2} {e,3,4} {6,9,t} {5,7,8} W2: {0,1,2} {5,9,t} {7,8,e} {3,4,6}





Figure 2

P: 0123456 789te

379e10t24685

W:{3,7,9,e} {l,0,t,2} {4,6,8,5}

W(P):{0, 1, 2, 3} {4, 5, 6, 7} {8, 9, t, e}

W: {0, 1, 2, 3, 4} {5, 6} 7, 8, 9, t, e

W(P):{3,7,9,e, l} {0,t} {2 4 6 8 5}

W1: {0,1,2} {e,3,4} {6,9,t} {5,7,8} W2: {0,1,2} {5,9,t} {7,8,e} {3,4,6}





Figure 2

P: 0123456 789te

379e10t24685

W:{3,7,9,e} {l,0,t,2} {4,6,8,5}

W(P):{0, 1, 2, 3} {4, 5, 6, 7} {8, 9, t, e}

W: {0, 1, 2, 3, 4} {5, 6} 7, 8, 9, t, e

W(P):{3,7,9,e, l} {0,t} {2 4 6 8 5}

W1: {0,1,2} {e,3,4} {6,9,t} {5,7,8} W2: {0,1,2} {5,9,t} {7,8,e} {3,4,6}





Figure 2

P: 0123456 789te

379e10t24685

W:{3,7,9,e} {l,0,t,2} {4,6,8,5}

W(P):{0, 1, 2, 3} {4, 5, 6, 7} {8, 9, t, e}

W: {0, 1, 2, 3, 4} {5, 6} 7, 8, 9, t, e

W(P):{3,7,9,e, l} {0,t} {2 4 6 8 5}

W1: {0,1,2} {e,3,4} {6,9,t} {5,7,8} W2: {0,1,2} {5,9,t} {7,8,e} {3,4,6}



Figure 3

A: P: 379e|l 0t214685

I6(P): 3e9715 684120t invariant at W: {3, e, 9,7} { ,0, t, 2} {4,6,8,5}

Figure 3

A: P: 379e|l 0t214685


Figure 3

A: P: 379e|l 0t214685


Figure 3

A: P: 379e|l 0t214685


Figure 3

A: P: 379e|l 0t214685


Figure 3

A: P: 379e|l 0t214685


B: P: 3 B: P: 3 B: P: 3 B: P: 3 B: P: 3 B: P: 3

T3(P): T3(P): T3(P): T3(P): T3(P): T3(P):

6 6 6 6 6 6

t 5 {0, 5, 6, e} 7 1 2 8 {,4,7, t}

9e 46 {2,3,8,9}

53 Ot {2,3,8,9} 8 7 1 2 {1,4, 7, t}

9e 4 {0, 5, 6, e}

t 5 {0, 5, 6, e} 7 1 2 8 {,4,7, t}

9e 46 {2,3,8,9}

53 Ot {2,3,8,9} 8 7 1 2 {1,4, 7, t}

9e 4 {0, 5, 6, e}

t 5 {0, 5, 6, e} 7 1 2 8 {,4,7, t}

9e 46 {2,3,8,9}

53 Ot {2,3,8,9} 8 7 1 2 {1,4, 7, t}

9e 4 {0, 5, 6, e}

t 5 {0, 5, 6, e} 7 1 2 8 {,4,7, t}

9e 46 {2,3,8,9}

53 Ot {2,3,8,9} 8 7 1 2 {1,4, 7, t}

9e 4 {0, 5, 6, e}

t 5 {0, 5, 6, e} 7 1 2 8 {,4,7, t}

9e 46 {2,3,8,9}

53 Ot {2,3,8,9} 8 7 1 2 {1,4, 7, t}

9e 4 {0, 5, 6, e}

t 5 {0, 5, 6, e} 7 1 2 8 {,4,7, t}

9e 46 {2,3,8,9}

53 Ot {2,3,8,9} 8 7 1 2 {1,4, 7, t}

9e 4 {0, 5, 6, e}

invariant with P at W. {0, 5, 6, e} {1, 4, 7, t} {2, 3, 8, 9} invariant with P at W. {0, 5, 6, e} {1, 4, 7, t} {2, 3, 8, 9} invariant with P at W. {0, 5, 6, e} {1, 4, 7, t} {2, 3, 8, 9} invariant with P at W. {0, 5, 6, e} {1, 4, 7, t} {2, 3, 8, 9} invariant with P at W. {0, 5, 6, e} {1, 4, 7, t} {2, 3, 8, 9} invariant with P at W. {0, 5, 6, e} {1, 4, 7, t} {2, 3, 8, 9}

C: T6, 19(P): C: T6, 19(P): C: T6, 19(P): C: T6, 19(P): C: T6, 19(P): C: T6, 19(P): 53 53 53 53 53 53 Ot Ot Ot Ot Ot Ot

7 1 2 8 e 46 9

invariant with P at the same W as in B above

7 1 2 8 e 46 9


7 1 2 8 e 46 9


7 1 2 8 e 46 9


7 1 2 8 e 46 9


7 1 2 8 e 46 9


D: P: 379e 10t24685

Ie,T2(P): 7 t 8 6 4 0 2 3 1 e95

invariant with P at W. {0, 1, 6, 7} {2, 3, 4, 5, 8, 9, t, e}

Using mosaics, we can describe the various conditions of col-

lectional invariance between two rows, P and H,H(P), where

H,H is some combination of operators and values in the row

class. Two rows are collectionally invariant for some mosaic, W

(or some mosaic W) if W(P) equals W(H,H(P)). Four distinct

patterns of this sort of relationship are illustrated in Figure 3.

Each example in Figure 3 arises from a different set of condi-

tions. In 3A, the two rows are related solely by pitch class operation, and their invariance is the result solely of the property of

the pitch class mosaic that allows it to be mapped onto itself at

D: P: 379e 10t24685

Ie,T2(P): 7 t 8 6 4 0 2 3 1 e95











D: P: 379e 10t24685

Ie,T2(P): 7 t 8 6 4 0 2 3 1 e95











D: P: 379e 10t24685

Ie,T2(P): 7 t 8 6 4 0 2 3 1 e95











D: P: 379e 10t24685

Ie,T2(P): 7 t 8 6 4 0 2 3 1 e95











D: P: 379e 10t24685

Ie,T2(P): 7 t 8 6 4 0 2 3 1 e95














the appropriate operator value. Similarly, in 3B, the two rows are related solely by an order number operation, so that their invariance relation depends on the fact that the constituent collections of the order number mosaic in question map onto themselves or each other at the operator value. In other words, the relationship in 3A is not dependent on the order number collections at which the collections of the pitch class mosaic appear, and in 3B, the relationship does not depend on the nature of the pitch class collections arising at the order number mosaic's collections. Thus, relations of these two types do not

depend on a row's ordering. However, the relationships in 3C and D are dependent on

the specific association of order number collection with pitch class collection in a row and its row class. In both 3C and 3D, the two rows are related by operations in both the pitch class and order number domains. In 3C, the invariance relationship depends both on the properties of the order number mosaic and the pitch class mosaic, such that each may map onto itself under the appropriate operation. This, in effect, combines the restrictions of 3A with 3B, and in all three cases invariance de-

pends on the invariance properties of the mosaics in question.13 Figure 3D, on the other hand, depends on a completely different aspect of the relations of order number mosaics with pitch class mosaics in a row. In this case, there must be two instances of a member of some pitch class mosaic class associated with a member of some order number mosaic class in a row for there to be an invariance relationship with a transformation of the row.14 Thus, in 3D, we find that the row P contains two in-

13The preceding discussion is a way of formalizing some of the kinds of rela-

tionships outlined in Haimo and Johnson, "Isomorphic Partitioning," and can be used as the basis of generalizing from segmental to all order number collections the ideas in Lewin, "A Theory of Segmental Association."

A4This idea is extended with segmental collections in Robert Morris, "Set-

type Saturation Among Twelve-tone Rows," Perspectives of New Music 22, nos. 1 and 2 (Fall-Winter 1983, Spring-Summer 1984): 187-217.









































stances of the association of a member of the pitch class mosaic class (0,1,5,8) (2,3,4,6,7,9,t,e) with a member of the order number mosaic class (0,1,6,7) (2,3,4,5,8,9,t,e). Note that the sort of relationship found in 3D is not dependent on invariance properties of the mosaics. The fact that the mosaics used in this example have particular invariance properties merely means that a wide variety of rows may be related to P based on all of the relational techniques outlined in Figure 3, at this particular pair of mosaics.

Figure 3D is a particular case of another form of collectional relationship between rows of a row class, based on the various order number collections associated with a pitch class collection class, and the various pitch class collections associated with an order number collection class. (Collections may be thought of as de facto two-part mosaics, the collection and its complement. All of the following discussion may of course be extended to multi-part mosaics as well.) Figure 4 illustrates the catalogue of pitch class collections associated with a particular order number collection class, and the catalogue of order number collections associated with a particular pitch class collection.15

In Figure 4A, we find two instances of the pitch class collection classes (0,1,2,3,4,6) and (0,2,3,4,6,8) at members of the same order number collection class: that which yields segmental hexachords. In 4B, we find a number of instances of members of the pitch class collection class (0,1,5,8) appearing at two members of some order number collection class. As one might imagine, it is possible to set up a wide variety of relational strategies involving pitch class collections found at similar order number collections, or dissimilar pitch class collections found

'5A technique for generating catalogues of pitch class collections over an order number collection class and vice versa is found in the author's "Some

Implications." It entails the construction of a matrix of order number rows ar-

rayed against normalized pitch classes.


































Figure 4 Figure 4 Figure 4 Figure 4 Figure 4 Figure 4

A: P: 379e10t24685

I II I

_ji I I

A: P: 379e10t24685

I II I

_ji I I

A: P: 379e10t24685

I II I

_ji I I

A: P: 379e10t24685

I II I

_ji I I

A: P: 379e10t24685

I II I

_ji I I

A: P: 379e10t24685

I II I

_ji I I

I I IL I I IL I I IL I I IL I I IL I I IL

W: (0, 1, 2, 3, 4, 5)

(6, 7, 8, 9, t, e)

(0,2,3,4,6,8)

(0, 1,2,3,4,6)

(0, 1,2,3,4,5)

(0, 1,2,3,4,6)

(0, 2, 3, 4, 6, 8)

(0, 2, 4, 6, 8, t)

W: (0, 1, 2, 3, 4, 5)

(6, 7, 8, 9, t, e)

(0,2,3,4,6,8)

(0, 1,2,3,4,6)

(0, 1,2,3,4,5)

(0, 1,2,3,4,6)

(0, 2, 3, 4, 6, 8)

(0, 2, 4, 6, 8, t)

W: (0, 1, 2, 3, 4, 5)

(6, 7, 8, 9, t, e)

(0,2,3,4,6,8)

(0, 1,2,3,4,6)

(0, 1,2,3,4,5)

(0, 1,2,3,4,6)

(0, 2, 3, 4, 6, 8)

(0, 2, 4, 6, 8, t)

W: (0, 1, 2, 3, 4, 5)

(6, 7, 8, 9, t, e)

(0,2,3,4,6,8)

(0, 1,2,3,4,6)

(0, 1,2,3,4,5)

(0, 1,2,3,4,6)

(0, 2, 3, 4, 6, 8)

(0, 2, 4, 6, 8, t)

W: (0, 1, 2, 3, 4, 5)

(6, 7, 8, 9, t, e)

(0,2,3,4,6,8)

(0, 1,2,3,4,6)

(0, 1,2,3,4,5)

(0, 1,2,3,4,6)

(0, 2, 3, 4, 6, 8)

(0, 2, 4, 6, 8, t)

W: (0, 1, 2, 3, 4, 5)

(6, 7, 8, 9, t, e)

(0,2,3,4,6,8)

(0, 1,2,3,4,6)

(0, 1,2,3,4,5)

(0, 1,2,3,4,6)

(0, 2, 3, 4, 6, 8)

(0, 2, 4, 6, 8, t)

at similar order number collections, or dissimilar order number collections carrying similar pitch class collections.

Order relations occur between two rows when one or more collections of some pc mosaic are ordered the same way in each row.16 This may occur when a given collection may be mapped onto itself in order under some non-zero operation,'7 or because two members of a given pitch class collection class in the row are ordered the same way to within the prescribed operations. These two situations are illustrated in Figure 5.

Figure 5

A: P: 379e 0t24685

Ie,5I(P): 09e1375468t2



Figure 5

A: P: 379e 0t24685

Ie,5I(P): 09e1375468t2



Figure 5

A: P: 379e 0t24685

Ie,5I(P): 09e1375468t2



Figure 5

A: P: 379e 0t24685

Ie,5I(P): 09e1375468t2



Figure 5

A: P: 379e 0t24685

Ie,5I(P): 09e1375468t2



Figure 5

A: P: 379e 0t24685

Ie,5I(P): 09e1375468t2

B: P: 379e10tt2 468

Ie,T4(P): 90t862 4531 e7

B: P: 379e10tt2 468

Ie,T4(P): 90t862 4531 e7

B: P: 379e10tt2 468

Ie,T4(P): 90t862 4531 e7

B: P: 379e10tt2 468

Ie,T4(P): 90t862 4531 e7

B: P: 379e10tt2 468

Ie,T4(P): 90t862 4531 e7

B: P: 379e10tt2 468

Ie,T4(P): 90t862 4531 e7 B: P: 379 e10t24 685 B: P: 379 e10t24 685 B: P: 379 e10t24 685 B: P: 379 e10t24 685 B: P: 379 e10t24 685 B: P: 379 e10t24 685 C: (0, 1,5,8)

(0, 1, 6, 7)

(0, 2, 5, 7)

(0, 1, 6, 7)

(0, 2, 4, 7)

(0, 3, 6, 9)

(0, 2, 5, 7)

(0, 2, 6, 8)

(0, 2, 3, 7)

(0, 2, 3, 6)

(0, 1, 5, 8)

(0, 3, 6, 9)

(0, 2, 4, 7)

C: (0, 1,5,8)

(0, 1, 6, 7)

(0, 2, 5, 7)

(0, 1, 6, 7)

(0, 2, 4, 7)

(0, 3, 6, 9)

(0, 2, 5, 7)

(0, 2, 6, 8)

(0, 2, 3, 7)

(0, 2, 3, 6)

(0, 1, 5, 8)

(0, 3, 6, 9)

(0, 2, 4, 7)

C: (0, 1,5,8)

(0, 1, 6, 7)

(0, 2, 5, 7)

(0, 1, 6, 7)

(0, 2, 4, 7)

(0, 3, 6, 9)

(0, 2, 5, 7)

(0, 2, 6, 8)

(0, 2, 3, 7)

(0, 2, 3, 6)

(0, 1, 5, 8)

(0, 3, 6, 9)

(0, 2, 4, 7)

C: (0, 1,5,8)

(0, 1, 6, 7)

(0, 2, 5, 7)

(0, 1, 6, 7)

(0, 2, 4, 7)

(0, 3, 6, 9)

(0, 2, 5, 7)

(0, 2, 6, 8)

(0, 2, 3, 7)

(0, 2, 3, 6)

(0, 1, 5, 8)

(0, 3, 6, 9)

(0, 2, 4, 7)

C: (0, 1,5,8)

(0, 1, 6, 7)

(0, 2, 5, 7)

(0, 1, 6, 7)

(0, 2, 4, 7)

(0, 3, 6, 9)

(0, 2, 5, 7)

(0, 2, 6, 8)

(0, 2, 3, 7)

(0, 2, 3, 6)

(0, 1, 5, 8)

(0, 3, 6, 9)

(0, 2, 4, 7)

C: (0, 1,5,8)

(0, 1, 6, 7)

(0, 2, 5, 7)

(0, 1, 6, 7)

(0, 2, 4, 7)

(0, 3, 6, 9)

(0, 2, 5, 7)

(0, 2, 6, 8)

(0, 2, 3, 7)

(0, 2, 3, 6)

(0, 1, 5, 8)

(0, 3, 6, 9)

(0, 2, 4, 7)

The foregoing gives us a basis for understanding relations in a twelve-tone composition. As any row may be sliced into any mosaic, either of order numbers or pitch classes, the criteria by which we decide on the mosaic interpretations of rows become

16Order relations are discussed in the following articles, among others:

Philip N. Batstone, "Multiple Order Functions in Twelve-Tone Music," Per-

spectives of New Music 10/2 (1972): 60-71, and 11/1 (1972): 92-111; David Kowalski, "Construction and Use of Self-Deriving Arrays," Perspectives of New Music 25, nos. 1 and 2 (Fall-Winter 1986, Spring-Summer 1987); Morris, "Multiple-Order-Function Rows"; Daniel Starr, "Derivation and Polyph- ony," Perspectives of New Music 23/1 (1984): 180-257; Peter Westergaard, "Toward a Twelve-Tone Polyphony," Perspectives of New Music 4/2 (1966): 90-112.

17This is discussed in Fusakao Hamao, "On the Origin of the Twelve-Tone Method: Schoenberg's Sketches for the Unfinished Symphony (1914-1915)," presented at the conference of the Society for Music Theory, Vancouver, B.C., November 1985.


























0 0

0 0 0

0 0 0

0

0 0 0

0 0 0

0 0 a 0

0 0 0

? ? ?

0 0 0

0 0 0

0 0

0 0 0

0 0 0

0

0 0 0

0 0 0

0 0 a 0

0 0 0

? ? ?

0 0 0

0 0 0

0 0

0 0 0

0 0 0

0

0 0 0

0 0 0

0 0 a 0

0 0 0

? ? ?

0 0 0

0 0 0

0 0

0 0 0

0 0 0

0

0 0 0

0 0 0

0 0 a 0

0 0 0

? ? ?

0 0 0

0 0 0

0 0

0 0 0

0 0 0

0

0 0 0

0 0 0

0 0 a 0

0 0 0

? ? ?

0 0 0

0 0 0

0 0

0 0 0

0 0 0

0

0 0 0

0 0 0

0 0 a 0

0 0 0

? ? ?

0 0 0

0 0 0

* * * * * * * * * * * *




critical to our analyses. If grouping strategies are not based on the way the surface of the music is perceived, then the analysis ceases to reflect the music as it is heard, and becomes purely self-referential. I shall rely on common sense for grouping for the purposes of this paper, but clearly we need a formalized and systematic investigation of grouping strategies, based on the perceived surface of the music.18

In Schoenberg's practice, rows appear in the musical surface projecting several different mosaic interpretations simultaneously, based on different grouping criteria.19 This allows a given musical passage to participate in several different relational tra- jectories at once, both long-range and local. Furthermore, significant pitch class mosaics and collections frequently arise from combinations of several rows. This occurs within aggregates formed from segments of more than one row, or by means of surface grouping across spans of two or more aggregates.20 We must therefore extend our use of pitch class mosaics to in- clude multiple-row groupings. As we shall observe, pitch class mosaics derived from more extended groupings are usually members of pitch class mosaic classes that have received specific treatment within single rows. By tracing relations through a movement, we can gain an insight into its compositional strategy.

18Such a need has been addressed by C. F. Hasty, most recently in "Mate- rial and Form in Webern's Twelve-Tone Music," presented at the annual conference of the Society for Music Theory, Bloomington, Indiana, November, 1986. An outline of a formal theory of segmentation has been recently offered by Fred Lerdahl and John Covach in "A Generative Approach to Set Theory and Analysis," presented to the Michigan Music Theory Society, Ann Arbor, Michigan, January 1987

'9This feature, among others, is addressed in detail in Peles," Interpreta- tions of Sets," and is demonstrated to a profound degree in Samet, Hearing Aggregates.

200ne instance of grouping across spans of aggregates entails the notion of secondary sets, as outlined in Babbitt, "Twelve-Tone Invariants."


























Wind Quintet, Opus 26, First Movement, Schwungvoll

The first movement of Schoenberg's Wind Quintet, op. 26, is perhaps the most notorious example of a twelve-tone movement imitating a tonal form. It appears to be a text-book sonata-allegro, with a repeated exposition complete with "first theme," "second theme," and a transitional passage connect- ing them; it also contains a development section, and, most damning of all, a recapitulation in which the "second theme" is transposed up a perfect fourth, the appropriate interval had this indeed been a tonal work with the second key area the dominant. All this looks mighty suspicious, but we shall see that each feature, even the transposition in the recapitulation, reflects relations within the piece's row class working in an overriding compositional strategy for the whole movement.

When we examine the rows used in the exposition, we find they exhibit a number of collectional and order relationships involving row segments. The principal criterion for projecting rows in the exposition is the assignment of rows or row segments to individual instruments. Virtually all of the haup- stimme are either complete rows, or discrete hexachords from the row. Row use changes with the sections; the three sections of the exposition are distinguished in the musical surface not only by the change of rows used, but also by the use of ritar- dandi and change of motive. Figure 6 is a list of rows found in the exposition, and a chart of their segmental relations.

As may be seen, all of the rows found in the exposition are closely related by row segment, both by collection and by order. Each of the two primary sections uses a principal row: P for the first section, and It(P) for the second section. These two rows are conjoined only in the transitional section. They also have the greatest degree of connectedness with the other rows used in the exposition. The subsidiary rows used are frequently connected to the principal rows by means of an order relation, suggesting that they act as motivic extensions of the principal rows.
























Figure 6 Figure 6 Figure 6 Figure 6 Figure 6 Figure 6 Figure 6 continued Figure 6 continued Figure 6 continued Figure 6 continued Figure 6 continued Figure 6 continued

P: 379e10t24685

First main section (bars 1-28): P, I6(P), also Ie(P), plus their retrogrades (Ie) and some rotations (T)

Transitional section (bars 29-41): P, It(P), and retrogrades

Second main section (bars 42-73a): principally It(P) and its retrograde; also I5(P), I6(P), I3(P), T5(P) and some of their retrogrades.

Segmental relations:

Tx(P), I(2x+5) (P) (a):

P: 379e 10t24685

I5(P): 2t86457 3 1 e90

Tx(P), T(x+5) (P) (b):

P: ,379e 0t24685

T5(P): 80 2 4 6 53 7 9e 1, t

P: 379e10t24685





Tx(P), I(2x+5) (P) (a):

P: 379e 10t24685

I5(P): 2t86457 3 1 e90

Tx(P), T(x+5) (P) (b):

P: ,379e 0t24685

T5(P): 80 2 4 6 53 7 9e 1, t

P: 379e10t24685





Tx(P), I(2x+5) (P) (a):

P: 379e 10t24685

I5(P): 2t86457 3 1 e90

Tx(P), T(x+5) (P) (b):

P: ,379e 0t24685

T5(P): 80 2 4 6 53 7 9e 1, t

P: 379e10t24685





Tx(P), I(2x+5) (P) (a):

P: 379e 10t24685

I5(P): 2t86457 3 1 e90

Tx(P), T(x+5) (P) (b):

P: ,379e 0t24685

T5(P): 80 2 4 6 53 7 9e 1, t

P: 379e10t24685





Tx(P), I(2x+5) (P) (a):

P: 379e 10t24685

I5(P): 2t86457 3 1 e90

Tx(P), T(x+5) (P) (b):

P: ,379e 0t24685

T5(P): 80 2 4 6 53 7 9e 1, t

P: 379e10t24685





Tx(P), I(2x+5) (P) (a):

P: 379e 10t24685

I5(P): 2t86457 3 1 e90

Tx(P), T(x+5) (P) (b):

P: ,379e 0t24685

T5(P): 80 2 4 6 53 7 9e 1, t

Tx(P), I(2x-2) (P) (c):

P: 379e1 0t24685 I 1t 11 I

It(P): 7 31e9t08 6425 1 It .. ,

Tx(P), I(2x-2) (P) (c):

P: 379e1 0t24685 I 1t 11 I

It(P): 7 31e9t08 6425 1 It .. ,

Tx(P), I(2x-2) (P) (c):

P: 379e1 0t24685 I 1t 11 I

It(P): 7 31e9t08 6425 1 It .. ,

Tx(P), I(2x-2) (P) (c):

P: 379e1 0t24685 I 1t 11 I

It(P): 7 31e9t08 6425 1 It .. ,

Tx(P), I(2x-2) (P) (c):

P: 379e1 0t24685 I 1t 11 I

It(P): 7 31e9t08 6425 1 It .. ,

Tx(P), I(2x-2) (P) (c):

P: 379e1 0t24685 I 1t 11 I

It(P): 7 31e9t08 6425 1 It .. ,

d b d b d b d b d b d b

I6(P)

\b

I6(P)

\b

I6(P)

\b

I6(P)

\b

I6(P)

\b

I6(P)

\b

\e(P Ie(P) \e(P Ie(P) \e(P Ie(P) \e(P Ie(P) \e(P Ie(P) \e(P Ie(P)

I5(P)

a/ b /

_, P -c - It(P)

I5(P)

a/ b /

_, P -c - It(P)

I5(P)

a/ b /

_, P -c - It(P)

I5(P)

a/ b /

_, P -c - It(P)

I5(P)

a/ b /

_, P -c - It(P)

I5(P)

a/ b /

_, P -c - It(P)

T5(P) - c I3(P)

d /

d

T5(P) - c I3(P)

d /

d

T5(P) - c I3(P)

d /

d

T5(P) - c I3(P)

d /

d

T5(P) - c I3(P)

d /

d

T5(P) - c I3(P)

d /

d

The prevalence of projected discrete segmental hexachords in the musical surface suggests a particular aspect of the relationship between the two principal rows of the exposition. The greatest intersection between two inversionally related hexachords of the segmental collection class (0,2,3,4,6,8) is five pitch classes. This occurs in two ways, as exhibited by the two different inversionally symmetrical five element collections found in Figure 6c. Thus there will be two rows with maximal hexachordal intersection with P: It(P) and IO(P). It(P), used here, not only has maximal intersection, but also displays the high degree of segmental invariance found in Figure 6.

The musical surface at the opening of the first and second sections emphasizes the relationship between P and It(P). This is illustrated in Example 1.











Example 1 Example 1 Example 1 Example 1 Example 1 Example 1

-Y I r Ir i 1- L. I -Y I r Ir i 1- L. I -Y I r Ir i 1- L. I -Y I r Ir i 1- L. I -Y I r Ir i 1- L. I -Y I r Ir i 1- L. I

Tx(P), I(2x+6) (P) (d):

P: 379el 0t214685

I6(P): 3e975684120tl

Tx(P), I(2x+6) (P) (d):

P: 379el 0t214685

I6(P): 3e975684120tl

Tx(P), I(2x+6) (P) (d):

P: 379el 0t214685

I6(P): 3e975684120tl

Tx(P), I(2x+6) (P) (d):

P: 379el 0t214685

I6(P): 3e975684120tl

Tx(P), I(2x+6) (P) (d):

P: 379el 0t214685

I6(P): 3e975684120tl

Tx(P), I(2x+6) (P) (d):

P: 379el 0t214685

I6(P): 3e975684120tl

42 , ob b

^-j7r I r fb I A r J' I 42 ,

ob b

^-j7r I r fb I A r J' I 42 ,

ob b

^-j7r I r fb I A r J' I 42 ,

ob b

^-j7r I r fb I A r J' I 42 ,

ob b

^-j7r I r fb I A r J' I 42 ,

ob b

^-j7r I r fb I A r J' I




In both passages, the shared collection is projected in a nar- row span, with the differing pitch class separated by a large skip. As will be seen at the end of the exposition, the resulting emphasis of the final pitch class of the hexachord will itself become motivically significant. In addition, each passage opens with the same unordered dyad, {3,7}, presented in a similar manner.

While the two main sections of the exposition project row

segments based on the discrete hexachords, the transitional section, in contrast, opens with a segmental hexachord spanning the discrete hexachordal boundary, across order numbers 3-8. The role of this new hexachord, projected in the same manner as the discrete hexachords of the main sections (by in-
















strumental line) will become clear in the summary of the exposition. The passage is illustrated in Example 2.

Example 2


Example 2


Example 2


Example 2


Example 2


Example 2

29

The criterion of instrumental line has yielded rows and hexachordal row segments, for the most part. Within this criterion, we have seen registral shift articulate collectionally invariant

29


29


29


29


29


Example 3 Example 3 Example 3 Example 3 Example 3 Example 3 --

- --- L_ -

-- - ---

L_ -

-- - ---

L_ -

-- - ---

L_ -

-- - ---

L_ -

-- - ---

L_ -

Fl.

Ob.

C1.

Hrn.

Bsn.

Fl.

Ob.

C1.

Hrn.

Bsn.

Fl.

Ob.

C1.

Hrn.

Bsn.

Fl.

Ob.

C1.

Hrn.

Bsn.

Fl.

Ob.

C1.

Hrn.

Bsn.

Fl.

Ob.

C1.

Hrn.

Bsn.

Hi- Hi- Hi- Hi- Hi- Hi-

2

2

i - - ~ ;

2

2

i - - ~ ;

2

2

i - - ~ ;

2

2

i - - ~ ;

2

2

i - - ~ ;

2

2

i - - ~ ;

! I ! I ! I ! I ! I ! I ) ) ) ) ) )




segments. When we shift to additional criteria, we shall see additional mosaic-forming groupings that operate in the music.

Example 3 contains the opening six bars of the movement. As with many of Schoenberg's works, the opening reveals a number of clues to the way the music will progress. The opening passage consists of a statement of P in the flute, accompanied by T6(P) in the other instruments, and finally by its own trichords in bar 6. In addition to the hexachordally and trichor- dally partitioned presentation of P in the flute, we can observe a number of interesting pitch class mosaics and collections. The opening aggregate, through the third beat of bar 3, can be parsed by discrete timespans into a pair of hexachords belonging to the (0,1,2,5,7,8) collection class. This is equivalent to combining alternate discrete trichords of P. If we consider the accompanimental presentation of T6(P) as a trichordal mosaic parsed by individual instrumental part, we find that the resulting hexachordal mosaics formed by taking the various pairings of trichords are either segmental, or belong to the same hexachordal collection class as the hexachord found temporally in the passage (see Figure 7).












{2, 5, 3} {4, 6, 7} (0, 1, 2, 3, 4, 5)

{t, 8,9} {e, 1, 0} (0, 1,2,3,4,6)

(0, 1,2,5,7,8)

Each pairing may be perceptually argued. The oboe may be conjoined with the bassoon because of their shared timbre; likewise the clarinet with the bassoon as bottom edge of the en- semble; the oboe with the clarinet as lower half (note that the horn does not cross the oboe part until the pairing of oboe and clarinet is completed); and the horn with bassoon by the moti-

{2, 5, 3} {4, 6, 7} (0, 1, 2, 3, 4, 5)

{t, 8,9} {e, 1, 0} (0, 1,2,3,4,6)

(0, 1,2,5,7,8)


{2, 5, 3} {4, 6, 7} (0, 1, 2, 3, 4, 5)

{t, 8,9} {e, 1, 0} (0, 1,2,3,4,6)

(0, 1,2,5,7,8)


{2, 5, 3} {4, 6, 7} (0, 1, 2, 3, 4, 5)

{t, 8,9} {e, 1, 0} (0, 1,2,3,4,6)

(0, 1,2,5,7,8)


{2, 5, 3} {4, 6, 7} (0, 1, 2, 3, 4, 5)

{t, 8,9} {e, 1, 0} (0, 1,2,3,4,6)

(0, 1,2,5,7,8)


{2, 5, 3} {4, 6, 7} (0, 1, 2, 3, 4, 5)

{t, 8,9} {e, 1, 0} (0, 1,2,3,4,6)

(0, 1,2,5,7,8)


vic juxtaposition of the last note of the horn with the first note of the bassoon; and so on.

We may also note in the passage the delay of the final note of the bassoon, allowing us to group the pitch class t with the preceding five pitch classes, 3, 7, 9, e, 1; this yields the collection of the first hexachord of It(P), in the collectional partition found at the opening of the second main section.

Still another detail is worth noting. The downbeat of bar 6 is the first place in the piece where more than two instruments at- tack simultaneously. The resulting collection is a member of the collection class (0,1,5,8), found at order numbers (0,3,6,9).

One more significant collection is derived from among row statements at the end of the exposition. The passage consists of statements of the hexachords of T5(P) with a significant alteration of the last pitch class, It(P), and I3(P), all projected so as to accent the final notes of the hexachord, either by dynamic in- flection or by registral separation in the manner of the opening of each of the main sections. The passage is illustrated in Exam- ple 4, and the row parsing is found in Figure 8.





















Figure 8 T5(P): 802465 379e10*

I3(P): 086423 51e97t

It(P): 731e9t 086425

* note the alteration from the expected pitch class t

This passage produces an emphasis on the collection {0,3,5,t}. Note the collectional associations produced by the re- maining pitches, and the resulting quotation of P's first hexachord through the alteration of the second hexachord of T5(P).

Spanning the entire exposition is a series of high points illustrated in Example 5. As may be seen, the resulting collection is a member of the (0,1,2,3,4,6) collection class, the class of the hexachord found twice segmentally as noted in Figure 4.

Figure 8 T5(P): 802465 379e10*

I3(P): 086423 51e97t

It(P): 731e9t 086425




Figure 8 T5(P): 802465 379e10*

I3(P): 086423 51e97t

It(P): 731e9t 086425




Figure 8 T5(P): 802465 379e10*

I3(P): 086423 51e97t

It(P): 731e9t 086425




Figure 8 T5(P): 802465 379e10*

I3(P): 086423 51e97t

It(P): 731e9t 086425




Figure 8 T5(P): 802465 379e10*

I3(P): 086423 51e97t

It(P): 731e9t 086425








66 A

66 A

66 A

66 A

66 A

66 A

ob. 1' fl> d ob. 1' fl> d ob. 1' fl> d ob. 1' fl> d ob. 1' fl> d ob. 1' fl> d

h r. h

hm tISLJ FiL V~~~~~~~~~~m

h r. h


h r. h


h r. h


h r. h


h r. h



6 7 23 24 45 57 60 61 68

E - bi ^ v - -- - #~~_ & -0 6 7 23 24 45 57 60 61 68

E - bi ^ v - -- - #~~_ & -0 6 7 23 24 45 57 60 61 68

E - bi ^ v - -- - #~~_ & -0 6 7 23 24 45 57 60 61 68

E - bi ^ v - -- - #~~_ & -0 6 7 23 24 45 57 60 61 68

E - bi ^ v - -- - #~~_ & -0 6 7 23 24 45 57 60 61 68

E - bi ^ v - -- - #~~_ & -0

Before examining the ways the collections and mosaics discussed above behave in the rest of the movement, I will summarize the way the exposition unfolds. In the first section, primary material articulated by individual instrument projects a series of row and discrete hexachordal statements from a limited number of rows. These rows are related by segmental invariance. At the outset of the transitional section, not only do we have a new row, It(P), but the customary mode of projection, by individual instrument, yields a parsing of the row heretofore not found as primary material. It is true that during the opening section we have had examples of the hexachord found as the conjunction of the first and last trichords, or the middle two trichords, but it has arisen as subsidiary material, rather than as primary material. However, with the advent of the second main section, the role of the primary material of the transitional section is made clear (see Ex. 6).

As may be seen, in the passage characterized by the close melodic connection with the opening, the primary material of











42

V 6_

C. r r

If 1

42

V 6_

C. r r

If 1

42

V 6_

C. r r

If 1

42

V 6_

C. r r

If 1

42

V 6_

C. r r

If 1

42

V 6_

C. r r

If 1 bsn. I bsn. I bsn. I bsn. I bsn. I bsn. I r r r r r r

the transitional section is repeated almost literally, within the accompaniment, demonstrating the subsumption of the transitional section within a larger argument spanning the exposition as a whole. In other words, our first impression of the transitional section, both in its use of a new row and in its promotion of formerly subsidiary material to primary material, is recon- textualized by the emergence in the second major section of a specific association of the new row with the opening, employing the same manner of projection, and including the same subsidiary role for the material found at the surface at the outset of the transitional section.

Throughout the exposition, and spanning it, are projections of material not associated with the primary material drawn from discrete hexachords. This additional material is grouped in the musical surface by different criteria as well, creating a si-












4 4 4 4 4 4




multaneous series of different collectional and mosaic projections. Because this additional material is drawn either from non-segmental sources crossing the discrete hexachordal boundary, or segments crossing the same boundary, it has a very different intervallic quality from the discrete hexachords. Each discrete hexachord consists of five pitch classes from one whole-tone scale, and one pitch class from the other whole- tone scale. Thus, all odd intervals within discrete hexachords must employ the single pitch class from one whole-tone scale. Such a condition rules out a wide variety of collections, and it is not surprising that the secondary material drawn from crossing the hexachordal boundary abounds in discrete dyads with odd intervals. The distinction between the hexachordal and secondary materials is of course dependent on the hexachord theorem,21 but the preponderance of even intervals within the hexachords and odd intervals between the hexachords makes the distinction in this case extremely vivid.

In contrast with the exposition, the development section makes use of non-segmental row parsings for its primary material. While row segments do appear, they tend to be subsidiary to the non-segmental materials. The change of emphasis is ele- gantly made at the outset (see Ex. 7).

The statement of P is unmistakeable, but its mode of projection underlines the change of instrumental roles.

The first collection class to receive attention in the development section is the one whose members were found segmen-

21The theorem states in part that two complementary hexachords will contain the same index of interval classes; a straightforward corollary of this says that intervals not found within the hexachords must therefore be found between them. Given the propensity for the discrete hexachords of P to contain even (whole-tone) intervals, it is not surprising that the collections formed between hexachords will tend to have odd intervals. A clear presentation of the hexachord theorem is found in Rahn, Basic Atonal Theory. It is additionally found in Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music 13/1 (1974): 191-212, among other places.



























Oh.

CI.

Hrn.

Bsn.

Oh.

CI.

Hrn.

Bsn.

Oh.

CI.

Hrn.

Bsn.

Oh.

CI.

Hrn.

Bsn.

Oh.

CI.

Hrn.

Bsn.

Oh.

CI.

Hrn.

Bsn.

)^ 76 )^ 76 )^ 76 )^ 76 )^ 76 )^ 76

-tzrp I | /ht= -tzrp I | /ht= -tzrp I | /ht= -tzrp I | /ht= -tzrp I | /ht= -tzrp I | /ht=

eJ- If

w -r - _ I . r .

eJ- If

w -r - _ I . r .

eJ- If

w -r - _ I . r .

eJ- If

w -r - _ I . r .

eJ- If

w -r - _ I . r .

eJ- If

w -r - _ I . r .

1. AI- A v 1. AI- A v 1. AI- A v 1. AI- A v 1. AI- A v 1. AI- A v

tally crossing the hexachordal boundary in the exposition. The

passage illustrated in Example 8 is saturated on the surface with members of this collection class, derived both segmentally and

non-segmentally from the rows present. Figure 9 illustrates the means of derivation.

















Is(P): .4 t0 1 c 9 7 3 5 8 6

10,16(P): 31 tO2486579e I ,Lj, ,L ,I

Is(P): .4 t0 1 c 9 7 3 5 8 6

10,16(P): 31 tO2486579e I ,Lj, ,L ,I

Is(P): .4 t0 1 c 9 7 3 5 8 6

10,16(P): 31 tO2486579e I ,Lj, ,L ,I

Is(P): .4 t0 1 c 9 7 3 5 8 6

10,16(P): 31 tO2486579e I ,Lj, ,L ,I

Is(P): .4 t0 1 c 9 7 3 5 8 6

10,16(P): 31 tO2486579e I ,Lj, ,L ,I

Is(P): .4 t0 1 c 9 7 3 5 8 6

10,16(P): 31 tO2486579e I ,Lj, ,L ,I

Note that in contrast with the passage in Example 7, the passage in Example 8 is composed of the other set of discrete dyads in the row. A later pair of passages contrasting the two discrete

dyad parsings of the row brings to the surface collections we observed at the opening and close of the exposition. The first of these two passages pairs the three discrete dyads of each discrete hexachord of T8(P). The passage is illustrated in Example 9, and the resultant mosaic revealed in Figure 10.











I I I I I I





Example 9

93

, */ / \ etc.

: $7121 4) r^_ - r I

Example 9

93

, */ / \ etc.

: $7121 4) r^_ - r I

Example 9

93

, */ / \ etc.

: $7121 4) r^_ - r I

Example 9

93

, */ / \ etc.

: $7121 4) r^_ - r I

Example 9

93

, */ / \ etc.

: $7121 4) r^_ - r I

Example 9

93

, */ / \ etc.

: $7121 4) r^_ - r I

Example 10 Example 10 Example 10 Example 10 Example 10 Example 10 fl.

(m- rg ?J I .. r banSS

fl.


fl.


fl.


fl.


fl.



T8(P): c3 57 98

6t 02 4 1

(0, 1,5,8)(0,2,5,7)(0, 1,5,8)

T8(P): c3 57 98

6t 02 4 1

(0, 1,5,8)(0,2,5,7)(0, 1,5,8)

T8(P): c3 57 98

6t 02 4 1

(0, 1,5,8)(0,2,5,7)(0, 1,5,8)

T8(P): c3 57 98

6t 02 4 1

(0, 1,5,8)(0,2,5,7)(0, 1,5,8)

T8(P): c3 57 98

6t 02 4 1

(0, 1,5,8)(0,2,5,7)(0, 1,5,8)

T8(P): c3 57 98

6t 02 4 1

(0, 1,5,8)(0,2,5,7)(0, 1,5,8)

The resultant mosaic conjoins members of the two four- element collection classes represented at the beginning and end of the exposition. Furthermore, the first member of (0,1,5,8) is the same collection as that found at the downbeat of bar 6. Note that the order number collection associated with the pitch class







P: 3 7 0 e 0 t 2 4 6 8 5 P: 3 7 0 e 0 t 2 4 6 8 5 P: 3 7 0 e 0 t 2 4 6 8 5 P: 3 7 0 e 0 t 2 4 6 8 5 P: 3 7 0 e 0 t 2 4 6 8 5 P: 3 7 0 e 0 t 2 4 6 8 5

collection at bar 6 is different from that associated with it here. The following passage features a member of the (0,2,5,7) collection class, in fact the same one as found at the end of the exposition. This is illustrated in Example 10, and the mosaic is offered in Figure 11.








80 Music Theory Spectrum

Example 11


Example 11


Example 11


Example 11


Example 11


Example 11

Ob.

Cl.

Hrn.

Bsn.

Ob.

Cl.

Hrn.

Bsn.

Ob.

Cl.

Hrn.

Bsn.

Ob.

Cl.

Hrn.

Bsn.

Ob.

Cl.

Hrn.

Bsn.

Ob.

Cl.

Hrn.

Bsn.

115

3 3

9.r 1 V) 1 " I 1 r3 r3

i> -(^^r^^-

? 'r j? 7^ 'i^ P"3

115

3 3

9.r 1 V) 1 " I 1 r3 r3

i> -(^^r^^-

? 'r j? 7^ 'i^ P"3

115

3 3

9.r 1 V) 1 " I 1 r3 r3

i> -(^^r^^-

? 'r j? 7^ 'i^ P"3

115

3 3

9.r 1 V) 1 " I 1 r3 r3

i> -(^^r^^-

? 'r j? 7^ 'i^ P"3

115

3 3

9.r 1 V) 1 " I 1 r3 r3

i> -(^^r^^-

? 'r j? 7^ 'i^ P"3

115

3 3

9.r 1 V) 1 " I 1 r3 r3

i> -(^^r^^-

? 'r j? 7^ 'i^ P"3

Thus, the specific transpositions of non-segmental collections found in the exposition appear featured in the development, arising from different order number sources.

Members of the (0,1,5,8) collection class continue to appear in the development section, as illustrated in Example 11.

In the exposition, the second main section produced a rein-

terpretation of the relationship between the first main section and the transition, by providing a context in which the two dis-

parate passages were subsumed. In a similar manner, the reca-

pitulation reveals a higher strategy within which the differing strategies of the exposition and the development are subsumed. This is done both in detail and in the large. The strategy of the exposition is to present primarily segmental materials related by different degrees of collectional invariance. The strat-

egy of the development is to draw a variety of non-segmental materials from the rows used, with certain transpositional ref- erences to the exposition's secondary material. In the recapitulation and coda, the primary invariance link between principal




































Fl.

Ob.

Cl.

Hrn.

Fl.

Ob.

Cl.

Hrn.

Fl.

Ob.

Cl.

Hrn.

Fl.

Ob.

Cl.

Hrn.

Fl.

Ob.

Cl.

Hrn.

Fl.

Ob.

Cl.

Hrn.

128

_-- -_ . ,i --f r- J

8 f^J J ,

128

_-- -_ . ,i --f r- J

8 f^J J ,

128

_-- -_ . ,i --f r- J

8 f^J J ,

128

_-- -_ . ,i --f r- J

8 f^J J ,

128

_-- -_ . ,i --f r- J

8 f^J J ,

128

_-- -_ . ,i --f r- J

8 f^J J ,

transpositional areas is based on instances of collectional invariance whose order number mosaics do not represent row segments. Thus the strategy of invariance in the exposition is meshed with the strategy of drawing out non-segmental materials in the development section to produce the recapitulation.

The initial return of the opening of the work at the beginning of the recapitulation helps to underline the unification of strate-

gies by presenting the recognizably segmental material divided

among the instruments. In contrast to the opening of the devel-

opment section, the passage is rhythmically profiled to suggest the return of the opening, rather than a move away from the

procedures in use (see Ex. 12). Of far greater significance is the relationship between P, the

principal row of the first half of both the exposition and the re-

capitulation, and I3(P), the row that replaces It(P) as the principal row of the second major section in the recapitulation. Here at last we arrive at that most suspiciously "tonal" move, the transposition of the second half of the recapitulation up a










































) ) ) ) ) ) - sc 7t 4 - sc 7t 4 - sc 7t 4 - sc 7t 4 - sc 7t 4 - sc 7t 4





perfect fourth. But consider the two mosaic relations between P and I3(P) as shown in Figure 12.22

These two mosaics represent the four element collection found at the end of the development and in the recapitulation, and the hexachordal mosaic found temporally in the very opening of the movements. Lest this seem just a theoretical conceit, consider the passage in Example 13. Clearly the pitch class mosaics used to relate P with I3(P) are strongly projected in the surface of the music at the close of the movement.

To summarize, we can see that an underlying strategy of this movement is the gradual extension of modes of extracting material from the rows combined with a gradual extension of modes of establishing invariance relationships among rows, articulated in ways such that elements that seem initially highly contrasted (the first section of the exposition and the transition section; the exposition as a whole and the development section) are revealed retrospectively to be parts of larger overarching strategies. Even the repeat of the exposition works within this

22The first of these mosaics is related to the pattern that underlies the third movement of the Wind Quintet. This pattern has been noted in Arnold

Schoenberg, Style and Idea (New York: Philosophical Library, 1950) and discussed in Martha Hyde, "The Roots of Form."


























scheme. By allowing the version of P found divided among the instruments at the outset of the development to replace the return of P heard in the flute at the repetition of the exposition, we can see the expected row transformed in its mode of projection as a departure; and the eventual return of the same row similarly projected at the beginning of the recapitulation reveals all the more strongly the synthesis of the diverse strategies of exposition and development.






Figure 12 Figure 12 Figure 12 Figure 12 Figure 12 Figure 12 P: 3 7 9 e t 4 6 8 5 P: 3 7 9 e t 4 6 8 5 P: 3 7 9 e t 4 6 8 5 P: 3 7 9 e t 4 6 8 5 P: 3 7 9 e t 4 6 8 5 P: 3 7 9 e t 4 6 8 5

13(P): 0864235 e97t

P: e 26 8 5

13(P): 0 8 6( )51

Wind Quintet, Opus 26, Fourth Movement, Rondo

Many of the parsing techniques used in the first movement are also used in the finale, but they are used to animate a mark-

13(P): 0864235 e97t

P: e 26 8 5

13(P): 0 8 6( )51



13(P): 0864235 e97t

P: e 26 8 5

13(P): 0 8 6( )51



13(P): 0864235 e97t

P: e 26 8 5

13(P): 0 8 6( )51



13(P): 0864235 e97t

P: e 26 8 5

13(P): 0 8 6( )51



13(P): 0864235 e97t

P: e 26 8 5

13(P): 0 8 6( )51







edly different overall strategy. The finale lives up to its name in that it contains a pattern of major sections distinguished motivically that fall comfortably into the rondo scheme of A, B, A', C, A", B', A-Coda. Closer inspection reveals that not only are these sections distinguished motivically, but they are also distinguished both by the way rows are projected in the surface, and by what rows are used. As we shall see, there is also a long- range series of collectional invariance relationships at work over the span of the movement, with implications that extend into the earlier movements of the quintet as well.

The first major section, running to bar 39, employs the familiar P and I6(P) almost exclusively, with infiltrations of I8(P) and I1(P) at the end. Ie and T6 are also employed on some of the rows. The primary strategy of row projection in the first major section and its subsequent varied returns is analogous to that found in the exposition of the first movement, with the addition of rotation. Instruments project complete rows of segmental hexachords. Thus the primary material of these sections is segmental. As with the first movement, secondary grouping strategies reveal additional material.

A statement of P in the clarinet is followed by an identically parsed statement of T6(P) in the oboe in bars 1-10, as illustrated in Example 14.
















The two rows are parsed in such a way as to group the opening dyads, the subsequent hexachords, and the final tetra- chords. The grouped hexachords are members of the (0,1,2,3,4,5) collection class. When we associate the two initial dyads with each other, we get a member of the (0,1,5,8) collection class, familiar from the first movement.

Bars 15-17 yield a different segmental hexachord. Here, the segmental member of the (0,1,2,3,4,6) collection class found at order number mosaic {1,2,3,4,5,6} {7,8,9,t,e,0} is played in the horn, as shown in Figure 13.

Figure 13



Figure 13



Figure 13



Figure 13



Figure 13



Figure 13

FI.,C.: e9 756 8

Hrn.: I6(P): 3

FI.,C.: e9 756 8

Hrn.: I6(P): 3

FI.,C.: e9 756 8

Hrn.: I6(P): 3

FI.,C.: e9 756 8

Hrn.: I6(P): 3

FI.,C.: e9 756 8

Hrn.: I6(P): 3

FI.,C.: e9 756 8

Hrn.: I6(P): 3 420t l 420t l 420t l 420t l 420t l 420t l

The following passage at bar 18 resumes the parsing scheme of the opening bars, but applied to Ie(P) and I6(P), presented in canon. The resulting mosaics are presented in Figure 14.






A I 1, A I 1, A I 1, A I 1, A I 1, A I 1,

Cl. Cl. Cl. Cl. Cl. Cl.

A hb A 6 1b 7. , A hb A 6 1b 7. , A hb A 6 1b 7. , A hb A 6 1b 7. , A hb A 6 1b 7. , A hb A 6 1b 7. ,

I I , II ' 1 ' L_ , I i ^ I i ' -

I I , II ' 1 ' L_ , I i ^ I i ' -

I I , II ' 1 ' L_ , I i ^ I i ' -

I I , II ' 1 ' L_ , I i ^ I i ' -

I I , II ' 1 ' L_ , I i ^ I i ' -

I I , II ' 1 ' L_ , I i ^ I i ' -

ob.^ t Lr "F I ' .I 1 !L I I ..I t I ob.^ t Lr "F I ' .I 1 !L I I ..I t I ob.^ t Lr "F I ' .I 1 !L I I ..I t I ob.^ t Lr "F I ' .I 1 !L I I ..I t I ob.^ t Lr "F I ' .I 1 !L I I ..I t I ob.^ t Lr "F I ' .I 1 !L I I ..I t I

, I , e I .V I , I , e I .V I , I , e I .V I , I , e I .V I , I , e I .V I , I , e I .V I

> > > > > >





92 A A

j_ e -IV n F19 , 92 A A

j_ e -IV n F19 , 92 A A

j_ e -IV n F19 , 92 A A

j_ e -IV n F19 , 92 A A

j_ e -IV n F19 , 92 A A

j_ e -IV n F19 , Ie(P): 58 642t01 e973

16(P): 01 e97358 642t

Ie(P): 58 642t01 e973

16(P): 01 e97358 642t

Ie(P): 58 642t01 e973

16(P): 01 e97358 642t

Ie(P): 58 642t01 e973

16(P): 01 e97358 642t

Ie(P): 58 642t01 e973

16(P): 01 e97358 642t

Ie(P): 58 642t01 e973

16(P): 01 e97358 642t

Once again, the pair of dyads at the head of each row, now musically juxtaposed by the canon, form a member of the (0,1,5,8) collection class. Note that the availability of members of this collection class from the same parsing scheme applied to two transformations of the row is in this case dependent on there being two pairings of members of this collection class with two members of a particular order number collection class in the row.

The segmental hexachords now exposed in Figure 14 are members of the same collection class as the discrete hexachords of P, and therefore may themselves be found as discrete hexachords of two inversionally related transformations of P. One of these is I1(P), found in the bassoon at bar 34, the last portion of the opening section (see Ex. 15).

Example 15



Example 15



Example 15



Example 15



Example 15



Example 15

The remainder of the first section deals with materials outlined above, in similar ways. In this section four of the six possible segmental hexachordal pairs derived through order number transposition are exposed. One of the two omitted pairs, the whole-tone collection found at order number mosaic {5,6,7,8,9,t} {e,0,1,2,3,4}, is eventually derived in the same manner as the passage in bars 15-17 during the A' section (see Ex. 16).






The other missing segmental hexachordal pair is that found at order number mosaic {3,4,5,6,7,8} {0,1,2,3,4,5}, which

played a role in the first movement. This rotation forms the basis for the middle section of the finale, and is also used in a brief transitional passage into the next section.

The next section, labeled B in the rondo scheme, runs from bar 39 through bar 77, and is distinguished from the initial section both by the principal row used, I0(P), and its principal mode of projection, the extraction of two pitch class mosaics involving pairs of members of the (0,1,5,8) collection class. This is illustrated in Example 17, and explicated in Figure 15.

As may be seen, I0(P) has the greatest possible hexachordal intersection under inversion with P, as did It(P) in the first movement, but the change in mode of projection does not em- phasize this.

Despite the high degree of differentiation between the first and second sections, there are a number of details that effect a smooth transition. As part of the accompanimental figuration of bars 30-31, Ie,I6(P) is parsed between the clarinet and the oboe so that the oboe has order numbers {2,5,8,e}. By inspect- ing the constituent collections of W2 and W2(I0(P)) in Figure 15, we can see that each mosaic will be preserved under the two operations that transform I0(P) into the retrograde of I6(P). This collectional invariance, which will play a large role later, integrating diverse sections of the movement, is here intimated in the accompaniment.

A second local linkage may be observed in the passage con- necting the two sections, in bars 39-42. Here, rather than I0(P), we find T9,I3(P) parsed as illustrated in Figure 16.
































usn. _ usn. _ usn. _ usn. _ usn. _ usn. _

~o.L _. ,/r'. 35., ttifb~o o (.1 ~o.L _. ,/r'. 35., ttifb~o o (.1 ~o.L _. ,/r'. 35., ttifb~o o (.1 ~o.L _. ,/r'. 35., ttifb~o o (.1 ~o.L _. ,/r'. 35., ttifb~o o (.1 ~o.L _. ,/r'. 35., ttifb~o o (.1





Fl. Ob.

Cl.

Hrn. Bsn.

Fl. Ob.

Cl.

Hrn. Bsn.

Fl. Ob.

Cl.

Hrn. Bsn.

Fl. Ob.

Cl.

Hrn. Bsn.

Fl. Ob.

Cl.

Hrn. Bsn.

Fl. Ob.

Cl.

Hrn. Bsn.

43

) A

43

) A

43

) A

43

) A

43

) A

43

) A

yJ1 AI vL I ' i yJ1 AI vL I ' i yJ1 AI vL I ' i yJ1 AI vL I ' i yJ1 AI vL I ' i yJ1 AI vL I ' i r , J 4J?' I " I -" I r , J 4J?' I " I -" I r , J 4J?' I " I -" I r , J 4J?' I " I -" I r , J 4J?' I " I -" I r , J 4J?' I " I -" I

379e10t24685

9531e02t8647

379e10t24685

9531e02t8647

379e10t24685

9531e02t8647

379e10t24685

9531e02t8647

379e10t24685

9531e02t8647

379e10t24685

9531e02t8647

WI:

Wi (IO(P)):

WI:

Wi (IO(P)):

WI:

Wi (IO(P)):

WI:

Wi (IO(P)):

WI:

Wi (IO(P)):

WI:

Wi (IO(P)):

{9, 5, 2, t} {3, 1, 8, 6} {e, 0, 4, 7}

{0, 1, 6, 7} {2, 3, 8, 9} {4, 5, t, e}

{9, 5, 2, t} {3, 1, 8, 6} {e, 0, 4, 7}

{0, 1, 6, 7} {2, 3, 8, 9} {4, 5, t, e}

{9, 5, 2, t} {3, 1, 8, 6} {e, 0, 4, 7}

{0, 1, 6, 7} {2, 3, 8, 9} {4, 5, t, e}

{9, 5, 2, t} {3, 1, 8, 6} {e, 0, 4, 7}

{0, 1, 6, 7} {2, 3, 8, 9} {4, 5, t, e}

{9, 5, 2, t} {3, 1, 8, 6} {e, 0, 4, 7}

{0, 1, 6, 7} {2, 3, 8, 9} {4, 5, t, e}

{9, 5, 2, t} {3, 1, 8, 6} {e, 0, 4, 7}

{0, 1, 6, 7} {2, 3, 8, 9} {4, 5, t, e}

W2: {9, 1, 2, 6} {5, e, t, 4} {3, 0, 8, 7}

W2(IO(P)): {0, 3, 6, 9} {1, 4, 7, t} {2, 5, 8, e}

W2: {9, 1, 2, 6} {5, e, t, 4} {3, 0, 8, 7}

W2(IO(P)): {0, 3, 6, 9} {1, 4, 7, t} {2, 5, 8, e}

W2: {9, 1, 2, 6} {5, e, t, 4} {3, 0, 8, 7}

W2(IO(P)): {0, 3, 6, 9} {1, 4, 7, t} {2, 5, 8, e}

W2: {9, 1, 2, 6} {5, e, t, 4} {3, 0, 8, 7}

W2(IO(P)): {0, 3, 6, 9} {1, 4, 7, t} {2, 5, 8, e}

W2: {9, 1, 2, 6} {5, e, t, 4} {3, 0, 8, 7}

W2(IO(P)): {0, 3, 6, 9} {1, 4, 7, t} {2, 5, 8, e}

W2: {9, 1, 2, 6} {5, e, t, 4} {3, 0, 8, 7}

W2(IO(P)): {0, 3, 6, 9} {1, 4, 7, t} {2, 5, 8, e}

Mosaic Collection Classes:

(0, 1,5,8) (0,2,5,7) (0, 1,5,8)

(0, 1, 6, 7)x3

(0, 1,5,8) (0, 1,6,7) (0, 1,5,8)

(0, 3, 6, 9) x 3


(0, 1,5,8) (0,2,5,7) (0, 1,5,8)

(0, 1, 6, 7)x3

(0, 1,5,8) (0, 1,6,7) (0, 1,5,8)

(0, 3, 6, 9) x 3


(0, 1,5,8) (0,2,5,7) (0, 1,5,8)

(0, 1, 6, 7)x3

(0, 1,5,8) (0, 1,6,7) (0, 1,5,8)

(0, 3, 6, 9) x 3


(0, 1,5,8) (0,2,5,7) (0, 1,5,8)

(0, 1, 6, 7)x3

(0, 1,5,8) (0, 1,6,7) (0, 1,5,8)

(0, 3, 6, 9) x 3


(0, 1,5,8) (0,2,5,7) (0, 1,5,8)

(0, 1, 6, 7)x3

(0, 1,5,8) (0, 1,6,7) (0, 1,5,8)

(0, 3, 6, 9) x 3


(0, 1,5,8) (0,2,5,7) (0, 1,5,8)

(0, 1, 6, 7)x3

(0, 1,5,8) (0, 1,6,7) (0, 1,5,8)

(0, 3, 6, 9) x 3

,.. I rf vI

-ti. Ir r

,.. I rf vI

-ti. Ir r

,.. I rf vI

-ti. Ir r

,.. I rf vI

-ti. Ir r

,.. I rf vI

-ti. Ir r

,.. I rf vI

-ti. Ir r

52 I A

52 I A

52 I A

52 I A

52 I A

52 I A

Fl.

Ob. Cl.

Fl.

Ob. Cl.

Fl.

Ob. Cl.

Fl.

Ob. Cl.

Fl.

Ob. Cl.

Fl.

Ob. Cl.

Bsn. Bsn. Bsn. Bsn. Bsn. Bsn.


P:

IO(P):

P:

IO(P):

P:

IO(P):

P:

IO(P):

P:

IO(P):

P:

IO(P):

vY

r

r

`S ~ ~~ -9I ,, ? p:P

vY

r

r

`S ~ ~~ -9I ,, ? p:P

vY

r

r

`S ~ ~~ -9I ,, ? p:P

vY

r

r

`S ~ ~~ -9I ,, ? p:P

vY

r

r

`S ~ ~~ -9I ,, ? p:P

vY

r

r

`S ~ ~~ -9I ,, ? p:P

r r r r r r

ll ll ll ll ll ll I I I I I I

tt t*4tAr- - tt t*4tAr- - tt t*4tAr- - tt t*4tAr- - tt t*4tAr- - tt t*4tAr- - I ,r I ,r I ,r I ,r I ,r I ,r

)v --- I- - -t - - MF )v --- I- - -t - - MF )v --- I- - -t - - MF )v --- I- - -t - - MF )v --- I- - -t - - MF )v --- I- - -t - - MF





Fl: T9,I3(P): 42 51 Fl: T9,I3(P): 42 51 Fl: T9,I3(P): 42 51 Fl: T9,I3(P): 42 51 Fl: T9,I3(P): 42 51 Fl: T9,I3(P): 42 51

C1: C1: C1: C1: C1: C1: 3 e97 08 3 e97 08 3 e97 08 3 e97 08 3 e97 08 3 e97 08

Hrn: Hrn: Hrn: Hrn: Hrn: Hrn: t 6 t 6 t 6 t 6 t 6 t 6

The change from bold to plain type in the clarinet indicates a change of motivic projection allowing us to associate its first dyad with the horn's dyad. However, the instrumental continu- ation of the clarinet part yields the first four elements of I6(P), found in the preceding section. The resulting tetrachord between the clarinet and horn, however, brings to the fore a latent collectional invariance relation between P and I3(P) not composed out in their interaction in the first movement. Note that the collection is the same as that found at the down beat at bar 6 in the first movement. The relationship, dependent on the parsing scheme of W2 in Figure 15, is illustrated in Figure 17. As shown there, the operation maps the contents of each collection onto itself or each other preserving the mosaic.

Figure 17

P: 3 e t 6

7 1 2 8

9 0 4 5

I3(P): 0 4 5 9 8 2 1 7

6 3 e t

I omit the analysis of section A', except to say that it resumes the projectional strategies and rows of A, with certain interesting changes of detail, intimated by Example 16 above.

Of all the sections of the finale, the middle section is at the greatest remove from the others both by ways material is pro-


Figure 17

P: 3 e t 6

7 1 2 8

9 0 4 5

I3(P): 0 4 5 9 8 2 1 7

6 3 e t




Figure 17

P: 3 e t 6

7 1 2 8

9 0 4 5

I3(P): 0 4 5 9 8 2 1 7

6 3 e t




Figure 17

P: 3 e t 6

7 1 2 8

9 0 4 5

I3(P): 0 4 5 9 8 2 1 7

6 3 e t




Figure 17

P: 3 e t 6

7 1 2 8

9 0 4 5

I3(P): 0 4 5 9 8 2 1 7

6 3 e t




Figure 17

P: 3 e t 6

7 1 2 8

9 0 4 5

I3(P): 0 4 5 9 8 2 1 7

6 3 e t



jected and by the rows used. There are two basic modes of projection found in this section. The first, found in bars 116-124, extracts every third segmental trichord in a concatenation of statements of Ie, Tt(P) in the bassoon to construct an aggregate equivalent by trichordal and hexachordal mosaic to T9,Tt(P). This is illustrated in Figure 18.23

It is interesting to note that this manipulation of the local row forms an invariance relation with material from the third movement. The central section of the third movement employs T3 and T9 transformations of I5(P), which yields the same hexachordal mosaic as that found in the passage under inspection.24 The second principal mode of projection in the central section consists of canonic statements of the discrete segmental hexachords of the row in use, accompanied by its own dyads.

This central section alternates modes of projection, first employing Tt(P) and its retrograde, and then employing I8(P) and its retrograde. Note that Tt(P) and I8(P) are in the same relationship as are P and It(P) in the first movement. During I8(P)'s presentation of the first mode of projection, the scheme is al- tered slightly through canonic presentation to juxtapose alternate discrete trichords of the row in the music's surface. Not surprisingly, this passage is immediately followed by the use of T5(P), which is related to I8(P) through the preservation of the hexachordal mosaic produced by alternate trichords, as illustrated in Figure 19. This section employs in miniature and with a different batch of rows the relationships found spanning the entire first movement, linking P first with It(P) and then with I3(P).

The transition back to the A material at the end of the central section contains many interesting details, but in the interest of pursuing the overview of the movement, I will leave its in-

23This is discussed in Schoenberg, Style and Idea. 24An analysis of the third movement of the Wind Quintet is found in the

author's "Large-Scale Strategy."



































Ie, Tt(P): 364208te9751 T9,Tt(P): 4631579et802 Ie, Tt(P): 364208te9751 T9,Tt(P): 4631579et802 Ie, Tt(P): 364208te9751 T9,Tt(P): 4631579et802 Ie, Tt(P): 364208te9751 T9,Tt(P): 4631579et802 Ie, Tt(P): 364208te9751 T9,Tt(P): 4631579et802 Ie, Tt(P): 364208te9751 T9,Tt(P): 4631579et802

Bsn.: 364

Fl., Ob.: 208te9

Bsn.: 364

Fl., Ob.: 208te9

Bsn.: 364

Fl., Ob.: 208te9

Bsn.: 364

Fl., Ob.: 208te9

Bsn.: 364

Fl., Ob.: 208te9

Bsn.: 364

Fl., Ob.: 208te9

751 751 751 751 751 751 te9

364208 751364

Example 18

te9

364208 751364

Example 18

te9

364208 751364

Example 18

te9

364208 751364

Example 18

te9

364208 751364

Example 18

te9

364208 751364

Example 18

208 208 208 208 208 208

etc. etc. etc. etc. etc. etc.

I8(P): 51e978t64203

T5(P): 802465379e t

I8(P): 51e978t64203

T5(P): 802465379e t

I8(P): 51e978t64203

T5(P): 802465379e t

I8(P): 51e978t64203

T5(P): 802465379e t

I8(P): 51e978t64203

T5(P): 802465379e t

I8(P): 51e978t64203

T5(P): 802465379e t

spection to the reader. Similarly, I shall gloss over the return of the A material except to point out that it once again returns to the rows of the opening, and similar modes of projection. One point worth noting, however, is the postponement of the presentation of P in the manner of the opening to the very end of the passage, in bars 218-225. This passage marks the sole appearance in this movement of the piccolo (used throughout the second movement, and nowhere else in the work), which adds extra emphasis to the first reappearance in the entire movement of the presentation of P found at the outset.

There follows now a return of the B section, initiating a series of events revealing invariance relationships that tie diverse sections of the movement into an overarching strategy. The initial row used in the return of B is I5(P), the row that exchanges discrete hexachordal contents with P. This row also preserves the mosaic with P that parses out initial dyads of each discrete hexachord, so that its first appearance not only invokes the motivic surface of the initial appearance of B, but also the initial dyads of the two opening phrases of the movement. This is illustrated in Example 18.











The other primary row of the return of B is I6(P), whose relationship with P has been discussed above. I6(P) is also T6 of I0(P), as mentioned earlier, and thus preserves the tetrachordal mosaic labeled W2 in Figure 15. This connection is made explicit in the music, as shown in Example 19.

Example 19

A^ -661 , b.f L f -t- f.


Example 19

A^ -661 , b.f L f -t- f.


Example 19

A^ -661 , b.f L f -t- f.


Example 19

A^ -661 , b.f L f -t- f.


Example 19

A^ -661 , b.f L f -t- f.


Example 19

A^ -661 , b.f L f -t- f. Fl. Fl. Fl. Fl. Fl. Fl.

Fl. Fl. Fl. Fl. Fl. Fl.

243

n A ) 243

n A ) 243

n A ) 243

n A ) 243

n A ) 243

n A )

From the return of B to the end, a number of passages serve to link disparate portions of the movement and the quintet as a whole. Beginning at bar 259 we find a resumption of the motivic shape of the A material used to project a special relationship







Hrn. Hrn. Hrn. Hrn. Hrn. Hrn.

A k LU- I

j W

W ^ P ol1 A k LU- I

j W

W ^ P ol1 A k LU- I

j W

W ^ P ol1 A k LU- I

j W

W ^ P ol1 A k LU- I

j W

W ^ P ol1 A k LU- I

j W

W ^ P ol1

I"? -' I I- : ia,g ^ I"? -' I I- : ia,g ^ I"? -' I I- : ia,g ^ I"? -' I I- : ia,g ^ I"? -' I I- : ia,g ^ I"? -' I I- : ia,g ^ 4f 0

M rb " ( i score) 4f 0 M rb " ( i score) 4f 0 M rb " ( i score) 4f 0 M rb " ( i score) 4f 0 M rb " ( i score) 4f 0 M rb " ( i score)





259 A 259 A 259 A 259 A 259 A 259 A

Fl. Fl. Fl. Fl. Fl. Fl.

= Ie (P) _IV

= Ie (P) _IV

= Ie (P) _IV

= Ie (P) _IV

= Ie (P) _IV

= Ie (P) _IV

Ob. Ob. Ob. Ob. Ob. Ob.

4)/ \ / \/ \\ etc.

->: t

4)/ \ / \/ \\ etc.

->: t

4)/ \ / \/ \\ etc.

->: t

4)/ \ / \/ \\ etc.

->: t

4)/ \ / \/ \\ etc.

->: t

4)/ \ / \/ \\ etc.

->: t Bsn. Bsn. Bsn. Bsn. Bsn. Bsn. Bsn. Bsn. Bsn. Bsn. Bsn. Bsn.

between P and T6(P), a row found in the third movement. The passage in question depends on the order number collections of two pitch class mosaics that are related by T6 to one another. The hexachordal collection class is (0,1,2,5,7,8), a member of which was found at the very opening of the first movement, and served to link P and I3(P) in the recapitulation of that movement. The mosaics are depicted in Figure 20.







P: 3 7 9 t 24

e 0 685

P: 3 7 9 t 24

e 0 685

P: 3 7 9 t 24

e 0 685

P: 3 7 9 t 24

e 0 685

P: 3 7 9 t 24

e 0 685

P: 3 7 9 t 24

e 0 685

P: 3 9 1 t 4 8

7 e 0 2 6 5

P: 3 9 1 t 4 8

7 e 0 2 6 5

P: 3 9 1 t 4 8

7 e 0 2 6 5

P: 3 9 1 t 4 8

7 e 0 2 6 5

P: 3 9 1 t 4 8

7 e 0 2 6 5

P: 3 9 1 t 4 8

7 e 0 2 6 5

T6(P): 9 3 7 4 t 2

1 5 6 8 0 c

T6(P): 9 3 7 4 t 2

1 5 6 8 0 c

T6(P): 9 3 7 4 t 2

1 5 6 8 0 c

T6(P): 9 3 7 4 t 2

1 5 6 8 0 c

T6(P): 9 3 7 4 t 2

1 5 6 8 0 c

T6(P): 9 3 7 4 t 2

1 5 6 8 0 c

T6(P): 9 1 3 T6(P): 9 1 3 T6(P): 9 1 3 T6(P): 9 1 3 T6(P): 9 1 3 T6(P): 9 1 3 48t 48t 48t 48t 48t 48t

576 02e 576 02e 576 02e 576 02e 576 02e 576 02e

The tritone relationship maps the two mosaics into each other's order number collections in the two rows. The relationship is spelled out in the surface of the music, and illustrated in Ex- ample 20. Lest there be any mistaking the relationship, there is present shortly after this passage a quotation of the opening of the first movement (see Ex. 21).






Beginning in bar 282, a double reference to the procedures of the B material and the central section takes place, using the retrogrades of P and I6(P). Concatenations of the retrograde of P are treated in the manner found at the outset of the central section, yielding a line in the clarinet equivalent by both trichordal and hexachordal mosaics to T3(P), one of the rotations yielding discrete hexachords of the (0,1,2,3,4,6) collection class. The retrograde of I6(P) is registrally parsed in its projection by the bassoon so as to repeat the mosaic employed to link the two B sections together. The coda continues with P and I6(P), projecting in various ways collections that have partici- pated in the relations across the span of the composition, nota-







=, 299 -

\

b

- 4 - I

( ̂: ,, ^ -,

=, 299 -

\

b

- 4 - I

( ̂: ,, ^ -,

=, 299 -

\

b

- 4 - I

( ̂: ,, ^ -,

=, 299 -

\

b

- 4 - I

( ̂: ,, ^ -,

=, 299 -

\

b

- 4 - I

( ̂: ,, ^ -,

=, 299 -

\

b

- 4 - I

( ̂: ,, ^ -,

* _ f7A

Z" , I h b~ ^J

^ J

^'J~~~~~~~#01

"'^j IgJ =i

* _ f7A

Z" , I h b~ ^J

^ J

^'J~~~~~~~#01

"'^j IgJ =i

* _ f7A

Z" , I h b~ ^J

^ J

^'J~~~~~~~#01

"'^j IgJ =i

* _ f7A

Z" , I h b~ ^J

^ J

^'J~~~~~~~#01

"'^j IgJ =i

* _ f7A

Z" , I h b~ ^J

^ J

^'J~~~~~~~#01

"'^j IgJ =i

* _ f7A

Z" , I h b~ ^J

^ J

^'J~~~~~~~#01

"'^j IgJ =i





bly members of both the (0,2,5,7) and (0,1,5,8) collection classes, derived in the various manners discussed above and illustrated in Example 22.

We can now summarize the overall strategy found in the finale of Opus 26. The first B section represents a departure both in terms of rows and projection from the opening A section. This is composed to be a smooth departure, but B's connection with A is tenuous, and dependent on secondary details rather than primary details. The return to A's material in A' does not seem an absorption of B, as did such returns in the first movement; on the contrary, despite the smoothness of the connections, we get the impression that B is left hanging. The central section, while elaborate and self-contained, gives initially a similar impression, despite its integration into the composition as a whole by means of collectional invariance with the central section of the third movement. Once again, the return of the A material seems to be the resumption of an earlier activity, but











the central section is better integrated with the resumption of A

through the interpenetration of its end with the beginning of the A material. Compare bars 187-193 (the close of the central section) with bars 204-208 (within the return of A, but replac- ing motivic material formerly found in analogous places earlier). The integration of the central section with this passage is further enhanced by the postponement of the clearest reference to the opening of the movement until the very end of the section, in the passage employing the piccolo. The return of the B section, with its local connection to the A material allows us locally to integrate the material of B and A; the preserved mosaic between I6(P) and I0(P), articulated in the surface of the music, allows us also to integrate retrospectively the initial appearance of B into the overriding scheme. The coda provides us with still more means of integrating rows and modes of projection found not only within the finale, but also across the span of the composition as a whole.











I$i f "r '

Al 1 'b - I rif -!!iz M 9- 1

vU ~Jv; 6 |4 t t b: b:~~~~~

I$i f "r '

Al 1 'b - I rif -!!iz M 9- 1

vU ~Jv; 6 |4 t t b: b:~~~~~

I$i f "r '

Al 1 'b - I rif -!!iz M 9- 1

vU ~Jv; 6 |4 t t b: b:~~~~~

I$i f "r '

Al 1 'b - I rif -!!iz M 9- 1

vU ~Jv; 6 |4 t t b: b:~~~~~

I$i f "r '

Al 1 'b - I rif -!!iz M 9- 1

vU ~Jv; 6 |4 t t b: b:~~~~~

I$i f "r '

Al 1 'b - I rif -!!iz M 9- 1

vU ~Jv; 6 |4 t t b: b:~~~~~




Conclusions

The analysis of the first and last movements of the Wind Quintet suggests a number of conclusions. First, examination of the surface reveals that the surface signals of change that have been associated with the pejorative descriptions of these movements do in fact signal changes, both of the rows used and their modes of projection. Examination of the resulting rows and their projected mosaic interpretations reveals ways that rows are related by various means, including collectional invariance of different kinds, and order invariance to various degrees. It is important to note that the various ways two rows are related in the preceding discussion have been dependent upon the ways the rows are projected in the surface of the music. Thus, for instance, the potential collectional invariance between P and I3(P) based on a preserved mosaic containing a pair of members of the (0,1,5,8) collection class at a pair of members of the (0,3,6,9) collection class was not invoked in the discussion of the recapitulation of the first movement, as it was not manifested there; discussion of the relationship was with- held until the analysis of the finale, where at last it appears in the surface of the music.

Examination of the surface has also revealed that rows participate in a variety of different relationships simultaneously, depending on the various mosaics resulting from multiple inter-

pretations of the row's projection. Multiple interpretations are based on various common-sense means of grouping events in the musical surface. Assembling the network of row relation-

ships provides us with a means for describing the compositional strategy of a section or a movement, such as the ever-expanding strategy of the first movement, or the retrospectively-absorbing strategy in the finale.

From this we can conclude that the musical surfaces of these movements are not just superficial imitations of tonal forms filled out with padding generated from a row class, but that they reveal networks of relationships fundamentally based in the

Conclusions






Conclusions






Conclusions






Conclusions






Conclusions






twelve-tone system from which we may come to understand the

compositional strategies of these works on their own terms. Although we have dealt with only one of Schoenberg's

twelve-one compositions, a look at a few examples from other works helps to confirm the impression that the large-scale forms are articulated by twelve-tone relations. The first and second movements of the String Quartet no. 3, opus 30, have been discussed by Steven Mackey and Stephen Peles, respec- tively, in ways that confirm the articulative significance of the relations in that work's row class.25 An example from the rondo finale will suffice here. The row class of the quartet displays a variety of mosaic-preserving interpretations. Figure 21 illustrates a familiar assortment.26

















Q: 7 43 90 5 6et 8 2

5(Q): t 1 e 6 7 49 3

T6(Q): 1 9 3 6 e 5 0"7 4"2 8

Ie(Q): 4 7 8 2 e 6 5 0 1 t 3 9

Q: 7 43 90 5 6et 8 2

5(Q): t 1 e 6 7 49 3

T6(Q): 1 9 3 6 e 5 0"7 4"2 8

Ie(Q): 4 7 8 2 e 6 5 0 1 t 3 9

Q: 7 43 90 5 6et 8 2

5(Q): t 1 e 6 7 49 3

T6(Q): 1 9 3 6 e 5 0"7 4"2 8

Ie(Q): 4 7 8 2 e 6 5 0 1 t 3 9

Q: 7 43 90 5 6et 8 2

5(Q): t 1 e 6 7 49 3

T6(Q): 1 9 3 6 e 5 0"7 4"2 8

Ie(Q): 4 7 8 2 e 6 5 0 1 t 3 9

Q: 7 43 90 5 6et 8 2

5(Q): t 1 e 6 7 49 3

T6(Q): 1 9 3 6 e 5 0"7 4"2 8

Ie(Q): 4 7 8 2 e 6 5 0 1 t 3 9

Q: 7 43 90 5 6et 8 2

5(Q): t 1 e 6 7 49 3

T6(Q): 1 9 3 6 e 5 0"7 4"2 8

Ie(Q): 4 7 8 2 e 6 5 0 1 t 3 9

Like the finale of the Wind Quintet, the last movement of the String Quartet no. 3 is marked "Rondo," and displays those surface changes of motive and texture associated with the form. Analysis reveals that, like the finale of the Quintet the changes of surface arise as changes of row use and mode of projection. The first major section runs through bar 12 and employs four rows; Q, I7(Q), and their retrogrades. I shall trace only one

25Mackey, The Thirteenth Note, and Peles, "Interpretations of Sets." 26These relations are described in Babbitt, "Twelve-Tone Invariants," and

George Perle, Serial Composition and Atonality: An Introduction to the Music

of Schoenberg, Berg, and Webern, (Berkeley: University of California Press, 1962).
























chain of relations, dealing with Q and its transformations as listed in Figure 21. In doing so, I ignore a number of relations that operate across this and subsequent passages, and the movement as a whole, in order to make a specific point. It must be noted that this is a far more complex movement than the following discussion will suggest.

The second major section, starting with the pickup to bar 23, is preceded by a transitional passage. Both the transitional passage and the second major section employ modes of row projection different from each other and from those in the initial section. However, among the rows they employ are Ie,T6(Q) in the transition, and Ie(Q) in the second major section. These two rows are related to each other by a shared tetrachordal mosaic, and their mode of projection enables one to make the association with ease. They are part of the dyad-sharing complex with Q, and so one may make an additional association of these rows with the first section. The first section is articulated over its span to reveal gradually the dyads of Q, along with a number of other important mosaics. Another transitional passage leads to the return of the opening material in a slightly varied form at bar 41. The first row to appear at this point is I5(Q), the final member of the complex of rows in Figure 21. However, the im-

portant point is that Q and I5(Q) are related across this span by a different mosaic association projected in the musical surface, as illustrated in Figure 22.

Figure 22




Figure 22




Figure 22




Figure 22




Figure 22




Figure 22

and the ways they are projected in the surface of the music. Se- lected passages are illustrated in Example 23.

Later works reveal even more complex series of interlocked strategies to delineate and relate formal sections. Passages of return in the Piano Concerto and Violin Concerto reveal rich complexes of completed row relations.27 Bruce Samet has demonstrated to a profound degree the way details on the surface and larger spans of music arise from row use in the first portion of the third movement of Schoenberg's String Quartet no. 4.28 The return of the opening melody in the first movement of that work, at T6, reflects the tritone relationship of the long dyads of its initial appearance,29 as well as the preserved double mosaic formed by pairs of discrete dyads at like order numbers between a member of that work's row class and its inversional combinatorial complement.

The preceding are but a few of the examples we may draw from Schoenberg's music to confirm the significant interaction among row relations, the perceived musical surface, and the compositional strategies thus created.

Post Script

It is interesting to note in the preceding discussion of the Wind Quintet that while both movements we examined use es- sentially the same sorts of row relationships, each movement manifests a markedly different strategy, and produces a markedly different effect when heard. It is here that perhaps we can




Post Script





Post Script





Post Script





Post Script





Post Script


Q: 7439056et 1 82

I5(Q): t 12850e67493

Q: 7439056et 1 82

I5(Q): t 12850e67493

Q: 7439056et 1 82

I5(Q): t 12850e67493

Q: 7439056et 1 82

I5(Q): t 12850e67493

Q: 7439056et 1 82

I5(Q): t 12850e67493

Q: 7439056et 1 82

I5(Q): t 12850e67493

Thus, the various sections are made distinct from each other

by row and row use, but different degrees of association are created by the relations among the various rows of this complex











27Mead, "Large-Scale Strategies." 28Samet, Hearing Aggregates. 29This is described in William Lake, "Structural Functions of Segmental

Interval-Class I Dyads in Schoenberg's Fourth Quartet, 1st Movement," In

Theory Only 8/2 (1984): 21-29.



Theory Only 8/2 (1984): 21-29.



Theory Only 8/2 (1984): 21-29.



Theory Only 8/2 (1984): 21-29.



Theory Only 8/2 (1984): 21-29.



Theory Only 8/2 (1984): 21-29.





Vn. 1

Vn. 2

Va.

Vc.

Vn. 1

Vn. 2

Va.

Vc.

Vn. 1

Vn. 2

Va.

Vc.

Vn. 1

Vn. 2

Va.

Vc.

Vn. 1

Vn. 2

Va.

Vc.

Vn. 1

Vn. 2

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begin to understand the source of the surface similarities to tonal music that initiated the investigation.

While the forms of tonal music are dependent upon tonal relationships, the variety of formal strategies, as well as the variety of strategies manifested in movements of like form, attests to the flexibility of the tonal system to articulate compositional strategies. Tonality provides a system of measurement and differentiation along with a hierarchy of relationships that allows one to create a variety of different strategies for making music. These strategies in particular are the pieces themselves; their shared characteristics tend to get abstracted as "forms."

Properly understood, the twelve-tone system can also pro- vide a system of measurement and differentiation along with a hierarchy of relationships that allows one to project a variety of different strategies for making music. In the case of the twelve- tone system, that which is differentiated is not the triad or the scale degree, but the aggregate-the totality of twelve pitch classes parsed multiply by the various modes of projection in
















the musical surface.30 Different aggregates are related by their mosaic interpretations, and may themselves be grouped into larger spans based both on shared mosaic interpretation, and the formation of mosaics from surface groupings that extend over several aggregates. The row class of a composition provides the means of controlling relationships among aggregates. It contains the potential for a wealth of relationships among rows and groups of rows, and provides the mechanism whereby one may construct hierarchies among rows. Such a hierarchy is suggested in the preceding discussion of the Wind Quintet in the way the rows have been chosen so that the greatest number of invariance relationships are manifested between the variety of rows and P.

30A related discussion may be found in the opening pages of Milton Bab- bitt, "Since Schoenberg," Perspectives of New Music 12, nos. 1 and 2 (1973- 74): 3-28.











1I 1I 1I 1I 1I 1I

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As with tonal music, forms in twelve-tone music are dependent on twelve-tone relationships, but are not inherent in those relationships. Just as in tonal music, the twelve-tone system may manifest a variety of formal strategies in general, and a variety of strategies within compositions of similar form.

I do not wish to suggest, however, that the twelve-tone system is simply a replacement for tonality. That would be far too mechanical, and depressingly simplistic. Clearly, tonal forms as they have developed reveal an extraordinary sensitivity to the possibilities of the tonal system, to the point where certain as- pects of tonal form are inextricable from tonality. Similarly, the twelve-tone system possesses its own particular form- generating tendencies, based on the sorts of relationships available within it. However, given the wide range of strategies available in each system, it is not inconceivable that there may be an intersection of the two systems' strategies which might lead to a degree of similarity that would not bely the integrity of either tonality or the twelve-tone system.

For example, the development section of the first movement of Beethoven's Piano Sonata, opus 31, no. 1, opens with the same music at the same transposition level as the opening of the exposition. At the development, the passage functions as a V of IV, as opposed to a true tonic. Part of the compositional strategy of the piece is the reinterpretation of this music, first heard as a true tonic in this same post-exposition context with the repeat of the exposition. The reinterpretation becomes part of the formal strategy of the piece, and a striking part of our hearing of the piece.31 In the Schoenberg Wind Quintet, as noted

31Jack Adrian is currently working on a dissertation at the Eastman School of Music investigating sonata movements whose development sections open in the tonic.





















above, the development section of the first movement opens with the row heard at the opening of the exposition projected in the mode of the development section. Part of the compositional strategy of the Schoenberg is the reinterpretation of this row, previously heard in this same post-exposition context in the repeat of the exposition projected in the manner of the exposition. In both pieces, the reinterpretation of principal material becomes part of the compositional strategy, and both pieces proceed to offer larger contexts in which the two interpretations are reconciled. While each piece depends on its re- spective musical system for creating both reinterpretation and reconciliation, in the abstract, both pieces are doing very similar sorts of things.

In this light, we can begin to understand that those surface features of Schoenberg's twelve-tone music which resemble the surface of tonal forms might indeed arise from similarities between the compositional strategies of twelve-tone and tonal musics, but that the means whereby they do so can still main- tain the integrity of the twelve-tone system. Just as in nature, two different life forms of radically different physiology might nevertheless display on some level remarkably similar patterns of behavior, so may a twelve-tone composition display compositional strategies similar to tonal music, despite the fundamentally different forces animating tonal and twelve-tone music. Schoenberg's genius was not only in seeing the twelve-tone system as a living musical system, but also, in an act of radical con- servatism, in perceiving the possibilities of its sharing strategic potentials with tonality.











Excerpts from Schoenberg's Wind Quintet and Third String Quartet are used by permission of Belmont Music Publishers.








andrew mead tonal forms in schoenberg s twelve tone music

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