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Musical Form and the Development of Schoenberg's "Twelve-Tone Method"

Martha M. Hyde; Schoenberg

Journal of Music Theory, Vol. 29, No. 1. (Spring, 1985), pp. 85-143.

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MUSICAL FORM AND THE DEVELOPMENT

OF SCHOENBERG S TWELVE-TONE METHOD

Martha

M Hyde

It is one of the oddities of scholarship that our regular response to

artistic inno vation is so much like a detective s response to a crime. The

more creative or outrageous the a ct, the mo re we tend t o avert our eyes

from the n ature of the act itself and inquire instead in to the facts th at

motivated or prod uced it . Ind eed , investigations of Schoenberg s twelve-

ton e me thod have comm only .begun with th e same questions th at our

favorite detectives ask: When exactly did the event take place? Who

knew or didn t know abou t it? What acts led up to it? What could have

been the motive? Was it fully preme ditated? Do the perpetrator s state-

men ts fit the othe r evidence? Or is he improving his alibi with the bene-

fit of hindsight? On the surface this scholarly detective work seems

misguided, for the deed in question is not a crime whose solution-

whose essential meaning-can be discovered by stockpiling extrinsic

facts.

Investigations of the development of Schoenberg s twelve-tone

method have, I think, gone astray because they have inadvertantly de-

fined th e problem as chronological-either establishing definite dates

of composition for the various serial movements that preceded the first

twelve-tone piece or inferring a chronological order from developments

in compositional technique. But chronology distorts the issue by



presupposing tha t either of these types of chronolog y, or bo th together,

will co ns titu te a teleological or evolutionary series of stages that will ex -

plain how and why Schoenberg developed th e twelve-tone m etho d.'

I d o no t m ean t o suggest tha t interest

in

how and why Schoenberg

developed the twelve-tone method is unjustified or th at identifiable

stages in its development are irrelevant. On the contrary, Schoenberg

was ofte n vague abou t the specifics of his me th od , but t hro ugh out his

life he affirmed that its strength could be understood through the

stages preceding its deployment in the

Suite

op. 25 , his first com plete

twelve-tone piece. And he cited particularly the two preceding works,

Five Piano Pieces, op. 23 , and

Serenade

op. 2 4 , in which he makes use

of wha t is usually called con textual serialism. But despite the intuitive

appeal of chronology , I want t o resist the urge t o play detective and for

th e mo m ent focus atte ntio n elsewhere-toward the issue Schoenberg

regards as crucial, namely, the ability of the twelve-tone method to

generate extended forms. As he asserts in his lectu re Com position

with Twelve Tones, no t until his discovery of the twelve-tone me thod

did h e solve the problem of brevity in atonal forms . Identifying form as

the crucial issue, he expressed confidence th at his new meth od seemed

fitted to replace those structural differentiations provided formerly by

ton al harmonies. * Indeed, one suspects that in 192 1 it was this as-yet

unproven potential that prompted Schoenberg's famous remark to

Rufer th at he had made a discovery which will ensure the supremacy

of German music for the n ext h und red years.

My suggestion that we concentrate on form rather than on chron-

ology results, ironically, from too much rather than too little chrono-

logical evidence. Schoenberg's numerous autograph manuscripts have

recently become available, bu t this new evidence has, I th ink , clarified

the chronology at the expense of the evolution or teleology which was

the goal of the chronological approach. In his elegant study in the

chronology of opp. 23-26, Jan Maegaard has recently shown that

Schoenberg worked simultaneously during 1920-23 on the various

movements of opp. 23-25, so that their chronological relationships are

exceedingly

complex.4 The chart shown in Example 1 in which Mae-

gaard outlines his findings should suggest how misleading is the notion

that even a precise chronology for op p. 2 3 and 2 4 will reveal a clear

evolution towards the twelve-tone meth od of op . 25. Notice, for in-

stance, t ha t the order of opus num bers reflects only the order in whlch

Schoenberg began each piece, but not the order in which he finished

them. Moreover, he often interrupted work on individual movements

and did not return to them until after working on new movements,

sometimes from different compositions. He completed the first two

movem ents of op. 23 in July 19 20 , for instance, bu t then shifted

attention to op. 2 4, tw o movem ents of which he had begun by August



CHART OF CHRONOLOGY

Op

23

A July 1920or U o r a

6

1920or b la re

o

1

o

2

No

4

vui ciom

A

6

Tmrrcam

Prbldum

k k r m a u o

o 5(Wahar]

altar

h b

13. 23

-

Apr. 4 L P . 23

i a

11 23 A , 14.

- 2 6

May10. 23

=@:::

baf.M ay 30. 233 7 t

ay 30. 2 3

July 15. 23

mfr July 15. 23

ug 17 24

A w . 26. 24

Example 1.Reproduced by courtesy of Jan Maegaard see note

4



6 , 1920. He then left these movements unfinished for two and a half

years, during which he had completed th e th ird, fourth , and fifth move-

ments of op. 23. Thus, although he began op. 23 as early as 1920,

almost three years passed before the composed the Waltz, its single

twelve-tone movement, and during these years he had already begun

(but n ot finished) th e single twelve-tone mov ement of op. 24 , the Son-

net. ' Maegaard's chronology also reveals anoth er surprising fact-tha t

before Schoenberg completed a single movement of op. 24, he had in

July 19 21 begun and finished the first twelve-tone mov ement of op. 25 .

Moreover, since he finished the rem aining five movements of op. 25 be-

tween F ebruary and March 19 23 , op. 25 was in fact completed before

mu ch of op. 24 , including its twelve-tone Sonnet

.

This tangled chronology thus thwarts our temptation to assume that

if we can discover the order of composition for Schoenberg's serial and

early twelve-tone compo sitions, then we will have understood the stages

through which the twelve-tone method developed. The composition of

so many movements overlaps th at , t o pursue such an approach, we

would be forced to decide whether our stages begin when Schoenberg

began a movement or when he finished it. Such a decision, even if

feasible, surely would have to consider specific com pos itiona l tech-

niques, and no t just dates of comp osition. Thus, Maegaard's ch rono-

logical findings should,

I

think, force us to reject the intuition that if

we can trace the steps that brought the criminal to the scene of the

crime, we will have solved the m ystery.

Jus t as the chronology of compo sition of opp . 23-25 does no t reveal

the etiology or evolution of the twelve-tone method, neither do the

procedures of com posit ion as revealed by his sketches-the variety an d

num ber of hls ex perimental techniqu es, bot h serial and twelve-tone. In

fac t, detective work on com positional technique, if it aims at unraveling

the origin of the twelve-tone meth od , will be as frustrated as the chrono -

logical-biographical approach. Let me support this claim with a brief

survey of some of the serial techniques Schoenberg uses in composing

these works, a survey which should suggest how difficult it would

be-

and perhaps impossible-to dem onstrate the evolution of the twelve-tone

me thod by studying compositional technique alone.

I want to focus on th e first four movements of o p. 23, movements in

which Schoenberg first uses serial techniques that he later revises and

ado pts in his twelve-tone me th od . These movements should fairly reflect

an evolutionary process-if one exists-since Schoenberg finished com-

posing the first tw o mo vem ents in July 1 92 0 , before he began either

op. 2 4 or op. 25 , but did no t complete the third and fo urth movements

until February 19 23 , after he had begun five movements of op. 2 4 and

two movements of op. 25. Among ll

the features of these m ovem ents, I

need t o consider here only three serial techniques-ordering, co nto ur,



and transpos ition an d inversion-in ord er to show the kind of difficulties

we encounter in trying to discover an evolutionary development of the

twelve-tone m etho d.

In the opening movement of o p. 2 3 , Schoenberg first uses what we

can loosely call a series. The mo vement op ens with three series, or sub-

jects, clearly delineated by separate voices and stemming in Example 2a

(mm. 1-6, labelled A,B,C). Although each contains only eight to ten

pitch-classes and man y repetitions-I call them series because they recur

untransposed thro ugh out the mo vement (though sometimes curtailed),

maintaining the original ordering of pitches and usually the original

co nto ur. Their fixed ordering and co nto ur makes these subjects more

like tonal motives than like the subjects of Schoenberg's earlier atonal

wo rks; for the first time the linear ordering of pitches is of primary im -

portance. That is, ordered repetitions of the series emphasize the suc-

cession of intervals, one at a time, as opposed to collections of intervals

in varied configurations that comprise what we usually call unordered

pitch-class sets.

Between entries of Series A B, and C, ho wever, passages occur tha t

simultaneously develop motivic fragments from all three series. But

unlike the entries of the series themselves, these developmental pas-

sages do not maintain the ordering and contour of the original series,

and they d o freely transpose and invert th e series' fragmen ts. In this

movement, then, a change

in

serial technique distinguishes between

wh at we can call motivic reprise a nd motivic dev elopm ent.

This important technique is well illustrated by the developmental

passage just after the first appearance of the three series (Ex. 2a, mm.

6-9). There is little correspon dence here between the linear motives of

the develop men tal lines and those of the three series; the ex act suc-

cession of pitches in the upper voice, for example, contains no

or ere

succession of three o r more pitches from Series A, B, or

C .

But if we

divide this up per voice in to two segments (mm. 6-8 and m . 9)-a group -

ing su pp or ted by surface changes in register and rhyth m-and iden tify

the

unor ere

pc set comprised by each , we discover one way th at this

passage relates t o the principal series. Taken as an unordered pc se t, the

first six pitches of the upper voice (mm. 6-8, 6-212: A

F

Bb E

D

G)

recur in an

inverte

form in Series B (mm. 3-4 6-212: B G

D D P

G A). The ordering and contour of successive pitches differ in both

forms of 6-212, but an ordering technique does nevertheless associate

their internal structures-both forms of 6-212 set forth the same suc-

cession of unord ered tetrachords and consequen tly share a com mo n

subset stru ctur e (Ex . 2b). In Series B, pc set 6-212 unfolds overlapping

forms of tetrachords 4 -1 8, 4- 6 , and 4-8 , and this same pattern-in retro-

grade-structures pc set 6-21 2 in mm . 6-8. Example 2b shows a similar

subset stru ctur e associating the final segment of the up per voice in m. 9



4 6

3

Series B: 9-18 series

C:

3 2 .

Example 2 Five Piano Pieces op

3

no



(F E G A G ) with a segment of Series C in mm. 4-6 (Bb A

C

C D).

Both segments represent forms of pc set 5-13 that are related by trans-

position, and, while their internal ordering differs, both share subset

structures t ha t overlap forms of trichords 3 2 and 3-1 and unfold pc se t

4-4

as their final t e t r a ~ h o r d . ~

These examples are typical of the first movement of op. 23 , through-

out which Schoenberg uses procedures of this sort to distinguish be-

tween passages that repeat the series and those that develop fragments

of the series. Repetitions almost always maintain the ordering and con-

tour of the original series and avoid both transposition and inversion.

Developmental passages, on the other hand, use unordered sets, which

often contain subset structures of the series and frequently use both

transposition and inversion. It appears, then, Schoenberg worked out

the form of his first serial movement by coordinating his new serial

techniques with those of his earlier atonal works. Such a procedure de-

parts radically from his earlier atonal techniques, bu t i t still stops sho rt

of the later twelve-tone m eth od .

The second m ovement of op. 2 3, composed immediately after the

first, combines the new serial techniques by a slightly differen t method .

As in th e first movem ent, this method w orks to delineate form by con-

trasting the function of discrete phrases. Here Schoenberg uses only

one series which again includes pitchclass repetitions and fewer than

twelve pitch classes (nine to be exact). Here he also allows transposi-

tions and inversions of the series; Example 3a shows those occurring

in just the opening five measures.' Perhaps more impo rtan t, in the

second movement Schoenberg deemphasizes con tour and orde r; each

repetition of the series presents a new contour, as well as a new order-

ing of pitches. As the movement progresses, however, the exact order-

ing and con tou r of the original series returns. Fo r instance, the beginning

of a canon towards the end of the movement (Ex. 3b) is structured by

inverted forms of the original series, form s tha t largely retain the origi-

nal ordering and contour (c.f., m.

1

[upper voice] and mrn.

18-19 .

To

summarize, the opening of the second movement deemphasizes the

linear ordering of pitches in the series, while the ending seems to

reestablish its importance. Repetitions of the series, both ordered and

unordered, freely use transposed and inverted forms, a technique that,

in the first movement, Schoenberg had used only to develop incomplete

or unordered segments of the series.

One technique strongly connects the first tw o mo vements, how-

ever. In both, repetitions of the series are often separated by passages

that develop short or incomplete portions of the original series. Exam-

ple 3a,

mm.

2-3 gives

an

instance. A developmental passage based on

several short motives contained in the original series immediately fol-

lows the first statement of the series. Example 4 shows that the first



a.

ehr rasch J S

Example 3 . Five Piano Pieces op . 2 3 n o . 2



short developmental phrase in m . 2 shares a number of subsets with the

origin4 series. Its upper voice unfolds a five-note segment whose pitch-

class con tent-b ut no t its order or contour-is identical t o a closing seg-

men t of the original series (pc se t 5-3). At the same time, a vertical

hexachord reproduces the first six-note segment of the original series

(pc s et 6-236). This techniqu e is n o t unlike the first mov ement, b u t the

remaining sets in m . 2 represent an impo rtan t departure. Relying on an

atonal tech niqu e, Schoenberg here develops the series by using the

omplements of sets tha t occur as linear segments of the original series,

some thing th at occurs infrequently in the first mov ement. PC set 8-1,

which represents the entire phrase in m . 2 , is the com plem ent of pc set

4-1 , a set tha t occurs twice as a linear segment of the series; and pc set

3-3 (the first and last vertical sonorities of m.

2)

is the com plem ent of

pc set 9-3, the set represented by the entire series. ~ c h o e n b e r ~uilds

the form of the second movement of op. 23, then, by means of serial

techniques similar to those in the first movement, but combines them

in a new way and experiments also with complements. These experi-

ments, however, do not affect the formal similarity of the two move-

men ts-that the series do no t always recur intact as they would in a

twelve-tone piece and consequently do not structure all the musical

events in th e piece.

The third and most complex movem ent of op. 2 3 , composed almost

three years later, combinesthe serial techniques of the hrevious two

movements in still a new way. Some of these techniques come close to

the twelve-tone m eth od , while others , frustrating any simple, progressive

view of Schoenberg's develo pm ent, recall mo re strongly the earlier aton al

period. But more importantly, as in the first two movements, Schoen-

berg combines these techniques in such a way as to delineate form .

The opening of the third movement in Example 5 will illustrate

Schoenberg's new uses of the complementary relation, order, con-

tour, transposition and inversion. The principal series, labeled Plo,

unfolds in the upper voice (mm. 1-2) and now contains only five

pitches. ( Pl0 designates the o rdered form of the series th at begins on

pc 10 , that is ,

~

D E B

C . )

The complementary relation soon be-

comes p rominent: PI, (pc set 5-10) in the upper voice is immediately

followed by a transposed form of its complement,

F7

(pc set 7-10).

(Here

F7

designates the unordered complement of the original series,

Plo, transposed by nine half-steps.) Against this melody , the lower voice

sets transposed and inverted interlocking forms of the original series

(R IS , Ps, P8, P,, 19 ,etc .); and in mm. 3-4-as if responding to the newly

formed com plement in the upper voice-the lower voice begins to un-

fold interlocking forms of this same complementary set P4, 17, P8,

etc.). In this mov ement, then , the series' com plem ent, now used in bo th

transposed and inverted forms, assumes a more prominent role.



Example

4.

Five Piano Pieces op

3

no 2

Example

5


23

no

3



Since order and contour are, of course, harder to maintain when

forms of the series overlap and interlock, it should no t surprise us tha t

these features now generate more complexity. Nevertheless, contour

still seems linked, although more loosely, to the order of the original

series, for whenever the series appears in its original order-regardless of

its transp osition or inversion-it mo st o ften does maintain its original

co ntou r. (Compare, for instance, the co nto ur of PI,, [m . 11 with those

of P5 [m . 21,

I

[m. 41, an d P5 [m . 61

.

As in the first tw o mov emen ts, com plete forms of the series (and its

com plement) do n ot structure all the events in the piece. At first glance,

for instance, m.

5

appears to introduce wh at

I

have been calling a devel-

op me nta l passage (one th at develops linear segments extracted from th e

series), for it contains a succession of cho rds tha t d o no t fall within an y

transposed or inverted form of the original series or its com plem ent (pc

sets 3-5, 3-8, 3-12, 4-229). One trichord, 3-8, does represent a trans-

posed form of a segment of the original series, Plo (Bb D E), but the

sources of the remaining two trichords clearly depart from the tech-

niques Schoenberg used in the earlier movements. The trichord 3-5 is

n o t a linear segment of P,o, b u t rather of its complem ent, F7 (m. 3 :

A D ~ b ) , lacing further emphasis on the series complement. The

source of the last tricho rd, 3 -1 2, seems even furthe r removed; it derives

no t from the series or its complement, bu t from the initial intersection

of the series (Plo) with its transposed and inverted forms (R15, P5, Pa)

and with the first statement of its complement

p .

In Ex. 5 these

occurrences of 3 -12 are marked with do tted lines.)

The crucial development is th at th e trichords 3-5 and 3 -1 2, although

n o t derived from the series, gain importance as the movemen t progresses

and eventually delineate phrases and sections. That is, they come to

serve formal ends. At m.

8 ,

for instance, the trichord 3-5 controls the

three simultaneous transpositions of the series P7, P I, Pa). At m. 18

the trichord 3-12 initiates a new section, dictating three transpositions

of the series th at struc ture a canon-like figure (P5, P9, In oth er

words, in this movement Schoenberg derives new motivic material not

directly related t o the series itself in o rder t o generate form.

Th e techniques of th e third movem ent make a rather confusing con-

trast, then, with those of the previous two. Some techniques, such as

greater emphasis on the series com plem ent, seem to foreshadow the

twelve-tone method, but others, such as freer ordering (as well as its

association with fured contour) point in the opposite direction. But

more important than these techniques themselves, is that the modifi-

cation of these techn iques and their interaction seems governed by co n-

siderations of form. Schoenberg s need for new material, in order to

generate form , largely accounts for the growing role of the co mplem ent

an d the com plex overlapping of th e series and its transform ations. As I



argue late r, this need fo r new motivic material-that is, material n o t set

for th in the series itself-anticipates one of the m ost im port ant tech-

niques Schoenberg will invent t o generate

extended

twelve-tone form s.

The fou rth movement of op. 3 (begun before the th u d movem ent,

but finished after) works out yet another scheme for coordinating the

techniques of the first three movement^.'^ I must return t o this move-

ment later, but here it will suffice to point out that it is based upon

four hexachordal series, which for the first time can account for all the

motivic events in the m ovem ent; th at is, unlike the previous movem ents,

th e four th h as n o passages based o n incomplete forms of the series. But

as with the first three mov eme nts, any at te m pt here t o map a progressive

development toward the twelve-tone method is extremely difficult.

Some techniques again seem to approach the later me tho d; others back

off.

It might seem obvious, for example, that the texture of the fourth

movem ent is closer t o a twelve-tone tex ture, since it uses only com plete

forms of the series. But a closer look shows that Schoenberg achieves

the texture largely through overlapping forms of two or more of the

four series, thereby introducing ambiguities unusual in twelve-tone tex-

tures, bu t quite com mo n in the previous movem ents. Example

6

(based

on Schoenberg's own analysis of this passage) shows how the four hexa-

chordal series typically overlap. While a similar kind of overlapping

occurs in the third movement (Ex.

5 ,

here the context differs signifi-

cantly since it derives from four

different

hexachords, rather than from

a single pentachord and its complement. Schoenberg apparently ad-

dresed the problem of overlapping different hexachords in this move-

ment by choosing four that share many of the same trichords and

tetrachords. This similarity in subset structure allows the extensive

overlapping of the series that characterizes the movement's texture. But

at the same time this similarity causes major n lytic l problems, for

it ofte n makes it impossible to be sure which of the four series is being

unfolded. Consequently, all events in the texture may or may not be

controlled by complete forms of the series. Despite Schoenberg's ap-

parent intentions, we are closer here to atonal contextualism than to

the later twelve-tone m eth od .

To sum up my argument thus far: the complex chronology of opp.

23-25

is matched in the com plexity of Schoenberg's experimen tation

with serial techniques. The difficulties of using either chronology or

technique to chart the development of the twelve-tone method prove

daunting, and it is even less likely that the two approaches could be

made t o suppo rt one another. For that reason, think it better to con-

cede that chronological tangles and the vagaries of technical experi-

m enta tion work together here to make the detective's lo t a particularly

unh app y one-indeed t o call in to question the nature of our evidence,



6-244 6-219 complement)

B 6-24

C

6-14

6-210 D 6-239 complement)

Example Five Piano Pieces op 3 no



as well as our assumption that experiments will progress sensibly and

steadily toward the goal that we, with historical hindsight, know the

experimenter was seeking. Without the benefit of hindsight, that goal

may have appeared quite otherwise-a random miracle, an accidental

by-p rodu ct, the unlook ed-for end of exp eriments undertaken with

quite different aims. The teleological aura of that goal may derive pri-

marily from the fact that, however it may have looked to the com-

poser, for us it is the en d or goal of our q uest as historical detectives.

I1

Let me turn now to my alternative to the detective's approach to

the mystery of Schoenberg's invention of the twelve-tone method and

the reasons why he abandoned his earlier serial procedures in its favor.

I subm it th at focusing on extend ed forms and their generation-that is,

on the problem Schoenberg was trying to solve, rather than on the

meth od s he used or th e order in which he used them-will provide a

measure of his success bo th in these serial conte xtu al works and in the

twelve-tone works.

I

shall illustrate this approach wi th th e beginning of

the fourth movement of op. 23, but first

I

need t o com ment generally

on wh at we mean by contextual, whether contextual atonality or

contextual serialism.

When Schoenberg renounced tonality abo ut 19 08 , he opened exciting

new possibilities for motivic development, but also created new prob-

lems in the work s of wh at is com monly termed his contextual period

(from approximately 1908-1923). As David Lewin has argued, these

problems arose largely from freedom from the pre-compositional

assumptions or restrictions of ton ah ty , or according to Schoenberg,

from the lack of a unifying principle. In his conte xtu al music, the

events of a piece are heard only n relation to their specific contexts,

and not also in terms of a more general or abstract way of hearing (as,

for example, in tonal or twelve-tone music). Lack of a unifying prin-

ciple does no t mean lack of a central idea ; ma ny of these pieces do have

centra l ideas, b ut the y are self-defining. Th e central ideas of these pieces

do not display themselves in light of any precompositional system of

relations, bu t only in their particular conte xts . Hence the label con-

textual.

Lack of precompositional assumptions makes Schoenberg's contex-

tual works hard er to analyze in certain respects than tona l or twelve-tone

pieces. Analysis, especially of form, tends to be an optimistic report on

th e listening experience, since n o external standp oint exists from which

to criticize that experience. Once you have identified the various con-

textua l relations and associations, yo u have finished th e analysis; ex-

ploring these relations fur the r is extrem ely difficult.

Obviously, musical works are not written to satisfy analysts, but the



difficulty of analyzing Schoenberg's contextual works turns out to be

significant. These pieces are often very satisfactory as music, bu t most are

relatively short or else accompany a text, or both. This should not be

surprising. In a long piece, context so quickly diffuses into a welter of

motivic relations and associations that a listener all but requires a text

t o follow the piece. tex t helps t o solve one crucial problem in con-

textual works by providing a rationale for musical repetition. Repeti-

tion, in turn, is crucial because in a contextual piece we must always

relate even the most extended motivic variations t o the central idea,

and therefore we need to hear the essenti l features of that idea re-

peated throughout the work. But because repetition is so obviously

related to form, more than the listener's needs ought to justify it.

te xt , of course, provides exactly th e justification needed; repetition in

the tex t motivates repetition in th e music.

Som ewhat the same poin t can be made ab ou t the material preceding

a repetition or reprise. Because repetition in contextual music is so

closely concerned w ith form, repetitions need t o be prepared for so

that they will be heard as major points of articulation. That is, the

music preceding a repetition must justify or make compelling the repe-

tition itself. In tonal music a cadence or the end of a tonal prolonga-

tion may mark such a p oint of articulation, and analogous structures, I

will argue, operate in twelve-tone music. But

in

contextual pieces

motivic repetition often seems to replace elaboration simply because

the central idea is becoming to o ambiguous. This rationale is eminently

practical, b u t aesthetically troub ling, and the evidence suggests tha t

Schoenberg found it troubling.

Short pieces may avoid ambiguity and therefore evade the problem

of motivating or justifying motivic repetition, but longer works require

extending motivic material, and their success will therefore require the

problem t o be solved. In hisearlier atona l works, Schoenberg often side-

stepped the problem b y setting a te x t; in his serial contextual w orks, he

explores an intrinsic solution to wha t we call the reprise problem.

s I suggested earlier, the essential difference between Schoenberg's

contextual serialism and his earlier contextual music is that the linear

succession of intervals now claims primary importance. In o the r words,

as David Lewin pu ts it , later chords are heard in terms of earlier

lines. 12 T o show specifically ho w this linear hearing works,

as

well

as illustrate some inheren t problems of this style, let us return to the

fourth movement of op. 23, the movement that derives its structure

primarily from four hexachordal motives or series. This movement in

particular rewards s tud y because we have a partial analysis of the ope n-

ing by Schoenberg himself. We can therefo re do more than speculate

abo ut his intentions.

s

Example 1 shows, Schoenberg began this movement n 1920, but

did no t complete it until 1 92 3. It thus spans the period during which



he developed an d first used the twelve-tone me tho d. We have two au to -

graph drafts for the movem ent: one, dating from 19 20 , shows the first

fourteen measures; the seco nd, dating from 1 92 3, contains the re-

mainder of the movem ent. As Example 7 shows, th e first draft has been

heavily revised, but its final form is essentially identical t o the published

version.I3 The first draf t also con tains some cryptic markings-the let-

ters A, B,

C,

and D, occasionally accompanied by circles and by the

subscript num bers 1 and 2. These letters mark som e, bu t no t all, occur-

rences o f the four hexachordal motives in the first fourteen measures

(A: 6-24 4, B: 6-24,

C:

6-14, D: 6-z10).I4

On hearing this movement, one feature would probably attract our

atte ntio n first-the ordered or nearly-ordered repetitions of the princi-

pal hexachord (the one marked A b y Schoenberg in m.

1 ,

Ex. 7)

that

occur in m . 6 , m . 13 , and a t the end of the piece in mm. 34-35. (Ex. 8

indicates these repetitions with circles and order numbers correspond-

ing to the original ordering in m. 1.) These repetitions use different

rhythm s and one transposition, bu t Schoenberg insures th at we hear them

as repetitions and recognize their original form by maintaining nearly

identical ordering and contour. Moreover, these repetitions mark im -

por tan t divisions in the overall form of th e mov emen t. The mov ement

has a loose ternary form A B A'), in which the three parts are equal in

length and are themselves precisely divided into halves. (Ex. 8 indicates

these halves with do tte d lines in parts and 3.) The repetitions of hexa-

chord A, th en , mark the beginn kg of the second half of part (m. 6),

the end of part 1 (m. 13), and the end of part 3 (mm. 33-34). Though

not exact reprises, closely related forms of hexachord A mark the be-

ginnings and ends of the other halves. Their ordering and contour

change only slightly, so that we hear them as forms of hexachord A

when they occur in mm . 5, 23 , 24 , 28, and 29. In addition to these

motivic repetitions, Schoenberg uses surface features to reinforce the

movement's structure. For example, either a rest or an abrupt change

in rhythm or registral contour marks the beginning and end of each

half.

On first hearing this movement, then, most of us would assume

that order and contour are essential to the central idea, the principal

hex ach ord , since these properties delineate key parts of the form . For

the same reason, most of us would expect other occurrences of hexa-

chord A , those b etween these exact repetitions, also to involve these

two properties. By imposing this form of linear hearing, then, Schoen-

berg in one sense has arrived a t a solu tion t o th e reprise problem. Where

in earlier contextual works, repetition worked primarily to keep the

listener aware t o the piece's central idea, now , in a serial wo rk, motivic

repetition can be more constructive. Because order and contour now

define the cen tral idea, reprises can vary or develop it, as well as repeat



Used by p ermission of Be lmo nt Music Publishers, Los Angeles, California

9 49

Example

7a:

Draft of op. 3 no. 4

mm. 1-14;

Reproduced by courtesy

of the Arnold Schoenberg Ins titute Archive no . 17)



po o

rit..

Example

7b:

Diplomatic transcription of draft of op. 23 no. 4 , mm.

1-14, including Schoenberg s an alytic an not atio ns.



it. This flexibility in turn allows more flexibility in passages that elabo-

rate or develop the central idea, for now the y can employ motives only

rem otely suggested by the original series.

The same mo vem ent also uses the series t o prepare for major points

of articulation-in othe r words, to explain why each sect ion ends

when it does. Identical repetitions of hexachord A mark the begin-

nings of sections, but the most closely related repetitions mark their

endings. We can say, then, that the series has taken over the role in

articulating form that texts performed in earlier works. Schoenberg

would have preferred this solution to the reprise problem since it de-

rives musical form from musical relationships, instead of simply m

posing an extrinsic literary form.

This solution t o the reprise problem turns ou t not quite to be a

trium ph , however-the new use of the series poses a new problem , or

rather the old problem in a new guise. The problem is best stated in

general terms first, before I try to exemplify it in the fourth move-

ment's opening measures.

The general problem has to do with the relation between the series

and its contour or register. As the preceding discussion has shown,

Schoenberg usually associates the original series with a specific con-

tour, a feature that he experiments with variously in each of the first

four movements. But because the original series sometimes varies in

contour, the question becomes: Is the central idea of the piece an

ordered series of pitches whose registers are consequently fmed, or is

it merely an ordered series of pitch-classes whose registers are not

fmed? For example, in op. 23 no. 4 , must the second pitch of hexa-

chord A be a major third below th e first pitch (as it appears in m. I), or

can it just as well appear in any octave, above or below the first pitch?

If it appears in another register, does it represent a variation of the cen-

tral idea or remain an exact repe tition? Or, shou ld we define the cen-

tral idea as both a series of pitches and a series of pitch-classes, and if

so, how can these two ways of hearing be managed in the context of

a single piece? Will alternation between the two types of hearing suf-

fice?

Not only does confusion arise about the central idea itself, but also

about the techniques used to develop it. Here the crucial questions in-

volve the use of complementary relations and the occurrence of pitch-

class repetitions. When the com plem ent of the series seems to represent,

or at least refer to the series itself (as in op. 23 no. 3), how should we

interpret the ordering of its pitches? What relatio n, if an y , can exist be-

tween the con tou r of the original series and its complem ent? The use of

pitch-class repetitions raises many of the same questions. Throughout

op. 23, pc repetitions do frequently occur when the series is repeated.

When they do, should we view them as an essential feature of the



series, or merely a variation of it? But how can the order and contour

of the original series be main tained when new pitch-class repe titions are

introduced? These questions are essential, for they determine how we

hear and define form . An d these crucial issues, I think , motivate much

of Schoenberg's ex perimentation in his contextual serial works.

If we turn t o the opening section of op. 2 3 no. 4 and try t o describe

precisely how Schoenberg structures its form, we are confronted with

many of these same questions. The main difficulty arises, not in de-

scribing the compositional techniques in use or even their interaction,

but rather in determining what kind of form these techniques generate.

This example will suggest one reason th at Schoenberg may have become

dissatisfied with his serial techniques and consequently why it may be

wrong t o assume t ha t his serialism evolves progressively tow ard th e

twelve-tone m etho d.

Example 8 shows mo st occurrences of h exach ord A in the first section

of this movement (mm. 1-5). If we compare these with Schoenberg's

own analysis

in

Example

7,

appears that he has marked the more con-

cealed repetitions (for instance, those in m. 3), but not the most ob-

vious ones, such as the one in

m.

5 that ends the first section.''

An

analysis of form can begin by defming the central idea, the principal

hexachord labeled A in m . 1 and used to begin and end each sec-

tion.16 We will first need to specify the relation b etw een hex ach ord A

and its ordering and contour so as to distinguish between motivic

reprise and wha t we can call motivic development.

Hexachord A first occurs as the linear succession of pitches D#

B

~ D E

G

w h c h I have marked with order numbers

1

through 6 . The

corresponding succession of intervals describes its contour (preceded

+

y

a

or

to indicate ascent or descent): -4 -13, t 4 , t 2 , t 3 .

The first repetition of hexachord A (m. 2) represents an inverted form ,

16 , and with a new ordering of pitches, for order num ber 4 now pre-

cedes order num ber 3.' Most im po rtan t, the tw o contou rs differ

radically, since hexach ord A now has the interval succession -8, -3 , -8,

-6 -3. At this point, then, few listeners could guess that contour, or

even ordering, will be essential features of the central idea. These two

forms of hexachord A are connected, however; the

interva l classes

of

the three successive dyads that unfold both occurrences of hexachord

A are ic 4 , ic 4 , and ic 3 . We might conclude, then, that the central

idea essentially consists of three ordered dyads defined by interval

class, rather than of six ordered pitches. In that case, hexachord A in

m. 2 would be ordered with respect t o tha t in m .

1.

If we m ove o n t o m . 3-in this imaginary slow-motion hearing-we

find the exact complement of hexachord A structuring the melody

(A3) and three occurrences of hexachord A structuring the accompani-

men t (A8, A3 , IZ ), all marked by Schoenberg in Example

6 . (Here, as



in m. 1 , Schoenberg marks the last five pitches of the hexachord

A2. ) Th e com plement of hexachord A, p c set

6-219

preserves some

of the same properties of the first repetition of hexachord A in m. 2.

While the z-relation keeps it from identical ordering, its three dyads

do have the same interval classes (ic 4, ic 4, and ic 3). Our assumption

about the importance of dyad ordering seems confirmed, then, and we

have fu rther evidence that conto ur and order, in some form , may be

essential to the central idea.

But these assumptions are challenged by the three form s of hex a-

chord A in m. 3 that accompany the complement in the upper voice.

The first form (As) is a transposition of that in m. 1 and comprises a

vertical slice of all three voices (Eh, Gh,

~

h,

Ch

G . Here both

ordering and contour change, so that we hear neither the original con-

tou r nor the expec ted succe sion of dyads, ic 4 , ic 4 , ic 3 . At the same

time, the order numbers (2 4

5

suggest rota tion , with th e first pitch

rotated to last leaving the other pitches ordered. This notion contra-

dicts that suggested by m . 2, of course-that the central idea com -

prises three ordered dyads. But, again, a closer look shows Schoenberg

devising a new t e c h q u e to m aintain contour-one that he will use fre-

quently throughout the movement. The technique can be described as

a type of registral partitioning. If we place the pitches of hexach ord

A in m . in registrally ascending order-that is, from the lowest pi tch

in the motive t o the highest-we get a conto ur defined by the interval-

class patte rn: ic 4 , ic 2 , ic 3 , ic 4 , ic 4 . If we perform t h s same opera-

tion on A, in m . 3 , we o btain the id entical interval-class pa ttern . This

technique of registral partitioning creates a type of registral contour,

then , bu t does no t restrict the linear ordering of pitch es. We usually

think of a series with futed order and register either free or futed, but

here Schoenberg seems to define the series with register fixed, but

order free.

After only these few repetitions of hexachord A, then, we already

have several candidates for essence of the central idea. But because

these properties are not mutually compatible, we also have a consider-

able amount of ambiguity. The central idea may be represented by a

transposed or inverted form or by its exact complement, and its order-

ing may be rotated or treated as three ordered dyads. These ordered

dyads reproduce

interval class

but not

interval.

If its pitches are ro-

tated, then the central idea is defined by what we have called registral

partitioning, in which case contour

is

identical. Regardless of all of

these properties , con tour in the usual sense-at least a t this point-does

no t seem to be essential to the central idea.

As we by now might expect, the next occurrence of hexachord

A in m. 3 marked again A 2 , opens still other possibilities for the

central idea. As Example

8

shows, A2-the final five pitches of the



[ a r t

I

Example

8.


3

no 4



continued)



motive-comprises the p itches Bb, Dh , Bb, Eh , and Gh , exac tly the

same pitches that Schoenberg marked

A2

in m. 1. The first pitch of

the motive, Dtl, is missing, however, so that a repetition of hexachord

A can apparently occur with fewer than the original six pitches. Note

also that A2 in m. has the same ordering and con tour as in m. 1 . The

ambiguity of the cen tral idea includes, then , several new possibilities.

It may be repeated with fewer than the original six pitches, and this

incomplete segment will duplicate the ordering and contour of the

original series, or perhaps this duplication only occurs when the incorn-

plete segment contains the same pitches as the original.

Schoenberg's serialism does not really solve the problem of ambi-

guity in op .

23

then , an d in fact the radical ambiguity abo ut the central

idea continues to increase. Schoenberg marks the third and final occur-

rence in m . of hexachord A ( I2) to show how it structures the bassline

(marked by arrows). Here, for the first time, hexachord A contains

pitch repetitions (G and Bb), and therefore raises a new question: How

do we define order and patterns of contour in a series which can con-

tain repeated pitches? In m. 4, the nex t repetition of hexachord A turns

ambiguity into full-scale confusion because it works against the basic

no tion of the central idea as a series. Schoenberg has marked A be-

side six pitches identical t o those in m. 1 , but here the pitches are

neither successive, nor associated by register, rhyth m, voice, or dynam-

ics. This absence of any surface association prevents us from hearing

these pitches even as a discrete harm onic event, much less a series.

By now the essential features of hexachord A, the central idea of

this piece, seem hopelessly in d ou bt . We are only four measures in to

the piece, only halfway through half of the first section, and already

we find the number of possibilities, even plausibilities, confusing. The

possibilities increase in the next few measures, and I think it safe to

say that the section ends

in

the nick of time. When hexacho rd A recurs

in m.

6

with nearly its original ordering and contour, we are reassured

that, at least in some form, these two features are essential.

s

in his

earlier contextual works, Schoenberg here mus t resort t o repeating the

centra l idea in orde r to contr ol multiplying am biguities. The series does

enable listeners to follow more precisely how the motive is varied, but

does it control the ambiguity of the central idea enough to provide a

means of building an extend ed form ?

The difficulty in defining the movement's central idea in effect pre-

vents recognition of its form. ll the possible definitions seem too

vague to be useful. If we take the central idea as an unordered series

(whose contour may nevertheless be fixed), then how can we under-

stand the movement's principal sections as Schoenberg marked them?

If we take the central idea as an ordered series (whose contour may or

may not be fured), we wdl hear the points of articulation, but will be



I

unable to make sense of what comes between them. Or if we take it as

both ordered and unordered, then how do we distinguish between a

repetition an d a variation or dev elopm ent? We ough t t o conclude-as

think Schoenberg did-that wi tho ut definite criteria for motivic reprise,

variation, and development, musical forms can only be arbitrary or

derivative or very short. Contextual serialism left Schoenberg, then,

facing essentially the same difficulties as had his earlier contextual

idiom.

In addition to these difficulties, which we have grouped under the

rubric the reprise problem, serialism itself gave rise t o a second crucia l

problem. This problem arises when a new motive maintains the overall

order and contour of the original series while altering one or more

pitches. Such an alteration will not prevent us from recognizing the re-

lationship to the original series, bu t it may perceptibly change the pitch

an d intervallic con ten t o f the series. The problem then becom es: should

the motive be defined flexibly by its musical effec t-that is, by over-

all order and conto ur-or solely b y pitch and intervallic conte nt? If the

first, then changing a pitc h or tw o may affect the identity of the mo-

tive very l itt le; if the seco nd, changing a pitch will affec t it a great deal.

Or is a motive t o be defined in bo th ways? And if so, how can we relate

bo th within a description of form?

I will call this new difficulty the problem of motivic alteration

and discuss just one clear example amo ng the m any in o p. 2 3 no. 4.18

In m. 5 a closely related form o f hexachord A marks the en d of the first

section of part and at the corresponding place in part 3 , a very simi-

lar motive (marked b y dotted lines in Ex. 8 , m. 28) appears to m ark the

end of its first section. The contour and order of the two motives are,

in fac t, identical, exc ept for th e final pitch (in m. the final interval is

an ascending minor third , while in m. 28 it is an ascending major third).

But this slight change nevertheless changes completely the intervallic

content; the later motive (pc set 6-243) does not represent a repetition

of hexachord A (pc set 6-244). This new hexachorda l motive intersects

the two actual occurrences of hexachord A that mark the end of this

section and the beginning of the next, but its order and contour asso-

ciate it more strongly with the original occurrence of hexachord A

in

m . 5 than with either of th e actual intersecting forms of hexachord A

The implications of this problem come clearest, I thin k, if we imagine

trying to analyze this movement without the aid of Schoenberg's anno-

tations. Most llkely we would judge many more than four hexachords

to be structuring the piece, and therefore our analysis, at best, would

contradic t the composer's. We can con clude, then , tha t Schoenberg's

serial contextual works present two basic problems because they lack

specific premises to tell us,

n

a general w ay, how to hear basic kinds of

musical relations. Bo th problems-that of motivic reprise and of motivic



alteration-bear most directly on form. They make it difficult t o de-

scribe form s in these movements and work against the generation of ex -

tended forms. Schoenberg must have sensed these problems, I th ink. At

any rate, after 923 he never returned to the serial contextual idiom,

but used the new twelve-tone method to compose music in more ex-

tended forms.

I11

The twelve-tone system does solve the problem of generating ex-

tended forms, but in a fashion not yet completely recognized. At first

glance, Schoenberg s twelve-tone method seems formulated precisely in

response to the problems of motivic reprise and motivic alt era tion. The

new assumptions specify how to hear tonal relations

n general

and thus

clarify the relation betw een p itch and pitch-class which his serial con -

textual techniques left ambiguous. These new assumptions, particularly

the twelve-tone basic set, run counter to the basic tenet of contextual-

iam-that we hear material and relationships only in the c ontex t of the

other material in the piece. We now must hear material (including the

opening and ll central ideas), not only in co nt ex t, bu t also in reference

to the structure of a specific twelve-note set and to the operations of

the twelve-tone system (transposition, inversion, and retrogression).

It is imp orta nt t o understand t ha t Schoenberg s new precomposi-

tional assumptions did not solve the reprise problem-for instance, by

outlawing motivic reprise itself-but rathe r dete rmin ed only those fea-

tures that pertain to pitch and pitch-class relations. That is, the basic

set restricts how the twelve pitchclasses are used, but not features re-

lated only to pitch, such as register or motivic contour. These features

can still be developed contextually to accentuate motivic reprise, for

instance, bu t those pertaining t o pitch-class relations are predeterm ined

and thereby assume greater structural importance.

Adoption of a basic set not only controls ambiguity caused by the

problem of motivic a lterat ion ; the tech nique itself can now be used as

a structural determinant of form. Motivic alteration still occurs, but

now it derives from the specific operations of the twelve-tone system.

Now if Schoenberg wants to use motivic alteration to associate two

ideas o r suggest a th ird, for instance, he can use inversion, transposition,

or retrogression t o obta in th e desired motive. These alterations n o longer

thre aten the iden tity of the basic set itself or the structure of the piece.

In fact, manipulating the basic set (described in terms of specific row

forms and t he various operations) will now generate musical form .

Despite its apparent response to the problems of motivic reprise

and alteration, one would be wrong to assume that the twelve-tone

method solves these problems n itself, rather than merely allowing



their solution. Such an assumption invites two misconceptions on the

part of progressively minded historical detectives. First, it implies that

had Schoenberg arrived earlier at a twelve-tone basic set and the cor-

responding operation s, he would have had little reason to experimen t

with multiple series containing fewer than twelve pitches. Archival evi-

dence alone challenges this assumption, since Schoenberg's autographs

include sketches dating from March 19 20 (before he began op. 23) for

an orchestral Passacagha based on a twelve-tone row. Row tables for

this piece reveal that he had by then already formulated the various

twelve-tone operations. l This evidence suggests, the n, th at even thou gh

he had conceived of the twelve-tone me tho d as early as 19 20 , he had

no t yet worked ou t the techniques needed to implement it.

The second misconception is that Schoenberg's method explains

why a twelve-tone series is superior to one containing fewer pitches.

Schoenberg does assert in Composition w ith Twelve Tones th at only

a twelve-tone series will insure equal pitch-class distribution and there-

fore prevent the undesired tonal effect of pitches being unequally em-

phasized.'' But in fact the music composed withh is tw elv e-to nem eth od

seldom, if ever, distributes the pitch-classes equally. Schoenberg is

either mistaken in his lecture-manifesto or, more likely, he is trying to

dodge the issue.21 In either case, to avoid b oth m isconceptions is also

t o give up our initial noti on of the twelve-tone m eth od as the solu-

tion t o the formal problems of Schoenberg's con tex tua l music.

Schoenberg hm se lf recalls worries that confirm this view:

In the first works in which I employed this method, I was not yet

convinced that the exclusive use of one set would not result in

monotony. Would it allow the creation of a sufficient number of

characteristically differentiated themes, phrases, motives, and other

forms? At this time, I used complicated devices to assure variety.

But soon I discovered that my fear was unfounded.

In other words, when he began using the twelve-tone method he still

feared that it lacked techniques to compensate for the variety he had

earlier achieved by using multiple shorter series. His emphasis on form

is especially revealing. He remembers worrying not about the twelve-

tone method itself, but about the techniques it affords for generating

sufficiently distinct themes, phrses, motives, and other fonns. Testi-

mony like this further supports my initial argument that we can best

understand the development of the twelve-tone method by concen-

trating on th e main achievement of that method -that is, on its tech-

niques for generating large-scale form. To the two most important of

those techniques I now turn.

The Piano Suite, o p. 25 , will seem a transitional piece t o those rely-

ing on chronology.

On

chronological evidence and the compositional



techniques employed, for instance, Maegaard concludes that only

in the Trio of the Minuet (op. 25 does Schoenberg finally realize

the implications of the twelve-tone basic set.

23

If we grant his premise-

that detailed chronology will reveal the process of innovation-then

Maegaard's scrupulous scholarship and elegant argument will be per-

suasive. But if we focus instead on the problem that Schoenberg took

t o be crucial, the problem of form , we will find tha t op. 25 does first

exhibit his solution t o the problem of extended form.

Form in op. 25 depends primarily o n two techniques fo r structuring

discrete phrases, both of which respond directly to the problems of

motivic reprise and motivic alteration. In general, one technique defines

motivic reprise through harmonic structures derived from the linear seg-

ments of the basic set, while the other generates motivic alteration by

joining invariant pitch-classes th at connect specific transposed o r in-

verted forms of the basic set. Schoenberg uses these two techniques

together t o structure single mov emen ts, as well as to articula te form be-

tween and am ong movements.

Before going further,

I

should mention that

I

have only recently

und erstood the second of these techn iques. Therefo re, this essay revises,

as well as extends, earlier essays of mine on structure and form in

Schoenberg's twelve-tone com pos itions. sampling of Schoenberg's

later twelve-tone pieces persuades me that he continued to construct

forms through techniques similar t o those invented for o p. 2 5 , although

modifying them somewhat. While these tw o techniques d o clarify more

fully the relation between ha rmo nic stru cture , invariants, and specific

row forms, it will take more study of the later works before we can

draf t a general model for Schoenberg's twe lve-tone forms . Still, I ex-

pect that the two techniques Schoenberg first uses in op. 25 will form

the fo unda tion for such a model.

Because harmo nic structu re is the foun dation up on which Schoenberg

builds extended forms,

I

need t o begin by reviewing some principles of

twelve-tone harmonic structure which

I

have discussed elsewhere.24

Part way throu gh an early version of Com position with Twelve Tones,

Schoenberg advances an ambiguous redefinition of what constitutes a

harmony under his new m ethod

arrived at the concept whereby the vertical and horizontal,

harmonic and m elodic, th e simultaneous and the successive were all

in reality comprised within one unified space. It followed from this

th at whatever occurs at one po int in th e space, occurs not only there

bu t in every dimensional aspect of the spatial continuum , so tha t

any particular melodic m otio n will not only have its effect upon

the harmony, but on

v rything

subsequent that is comprised within

tha t spatial continuum.

25



Schoenberg does not conceive of a harmony, then, as merely a vertical

event with pitches sounding simultaneously, but asserts that melodic

events also have harmonic implications. He proposes that a legitimate

harmony comprises all pitches, either sim ultan eou s or successive, which

are tem porarily associated o r, as he describes i t , comprised within the

same spatial continuu m.

One of the examples from op . 25 cited in Composition w ith Twelve

Tones will clarify Schoenberg's rather obscure con cep t of multi-

dimensional musical space an d show precisely its imp ortan ce for har-

monic structure, and also for form. This example (Ex.

9)

contains the

opening two-measure phrase of the Minuet. Schoenberg points out the

irregular presenta tion of the basic se t, which is part itioned into three

tetrachords: The m elody begins wi th the fifth ton e, while the accom-

paniment, much later, begins with the first tone. 26 Schoenberg gives

tw o reasons t o justify t h s irregularity: first, the Minuet is the fifth

movem ent of the

uite

so th at th e basic set has become familiar; second,

each tetrachord maintains the correct ordering of its pitches and there-

fore can function as an independent small set. But Schoenberg stops

short of explaining what criteria control how these tetrachords are

used independently. Analysis of the example's harmonic structure re-

veals some of his criteria, however, by exposing at least two connected

dimensions of its musical space. The first, which can be term ed the

primary dimension, includes the first tw o measures and simply repro-

duces the twelve successive pitches of th e basic set. That is, the primary

harmonic dimension contains contiguous elements of the basic set and

occurs with each of it s ordered statements. The second, the second-

ary dimension, spans each individual measure and represents two hex a-

chordal harmonies (marked pc set 6-2) which contain pitches that are

non-adjacent in the basic set (for instance, order numbers 1,2,5,6,7,

8 ,

bu t which are equivalent t o its principal hex ach ord , pc set 6-2 .

Equivalent here means that they are no t identical, bu t rather are

unordered and related by transposition or inversion or both. While the

hexachords in both measures are equivalent to the principal hexachord

of th e basic se t, their ordering differs from tha t in the basic set. Thus,

even though this secondary h armo nic dimension contains hexachords

which are inversions or transpositions (or both) of the principal hexa-

chord of the basic set, they are not presented by the

s me

succession

of intervals.

This simple example shows how Schoenberg could claim th at a single

twelve-tone row can integrate all harmonic and melodic dimensions of a

composition, as well as showing the rudimentary techniques implied

by that assertion. It also illustrates how Schoenberg joins together pri-

mary and secondary harmonic dimensions to construct form, for here

the two simultan eous dimensions of harmonic structure clearly delineate



Basic

et P4:

E G D b G b E b , ,A b D B

C

A B b ,

6 - 2

6 - 2

Example

9 .

Piano Suite op . 25 Minuet



a two-bar phrase subdivided in to tw o measures of equal length-a phrase

structure typical of a tonal minuet form, but here recreated through

twelve-tone techniques alone.

Example 9 is wo rth a mo me nt more because it conveniently demon-

strates three principles of Schoenberg's method for organizing harmonic

structure. First, twelve-tone harmonies need no t be simultaneous, tha t is,

they need no t occur only n the vertical dimension. Ra the r, a harm ony is

defined by pitches occupying the same spatial continuum . Secon d, a

single harmonic event necessarily affects more than one dimension. (In

Example 9 , for instance, each pitch functions simultaneously in at least

two different harmonic dimensions.) This principle appears explicitly in

an early draft of wha t became Com position with Twelve Tones.

Th ird, an d m ost im por tan t, the ord er of the twelve pitch-classes defines

th e harm onies of th e basic set, bu t i t defines them primarily by

interval

lic content

rather than by pitch-class content. (In Ex. 9, for instance,

the harmonies marked 6-2 in the comp osition contain different pitches

from those marked 6-2 in the basic set.) Moreover, these harmonies

need not be presented by the same succession of intervals; the internal

ordering of their pitches need n o t be the same. Thus, multiple harm onic

dimensions are connected through unordered pc sets that associate by

identical intewallic content and that represent linear segments of the

basic se t.

Schoenberg's me tho d of building form in o p. 25 derives from his use

of multi-dimensional harmo nic structures t o define or delineate discrete

phrases. He relies mainly on tw o related, b u t nonetheless distinct tech-

niques. Because evidence for both techniques appear in his composi-

tional sketches for op. 25, I shall briefly review a few sketches that

seem crucial.

8

Schoenberg experimented with four different orderings of the basic

set for op. 25 before selecting the fifth an d final ordering.29As the five

orderings in his preliminary sketch show (Ex. lo), he began by having

already decided on the pitch-class content of the three principal tetra-

chords (order num bers 1- 4,5 -8 , 9-12), as well as the ordering of pitches

in the first. He then experiments, searching for the best ordering of

pitches in the final two tetrachords. In the second ordering he arrives

at the order of the final tetrachord and in the fifth the order of the

middle tetrachord. To understand Schoenberg's reasons for selecting

the fifth ordering, we need to return to the relations that structured

the harmonies of Example 9. There Schoenberg grouped nonadjacent

pitches in the basic se t together t o form harmonies equivalent to ordered

linear segments of the basic set itself. Th e basic set for op. 25 proves to

have remarkable propensities for this type of relation, which clarify

Schoenberg's experim ents with the ordering of the basic set, as well as

his reasons fo r settling on th e final ordering.



Ordering6 of

the

B a r i c S e t

f i n a l )

Cl:

CZ:

(3:

4:

(5:

E

E

E

E

E

F

F

F

F

P

C

G

D b E b G b D A b C

D b D E b A b G b B

D bD E b G b A b B

D b G b D A l E b B

D b C b E b A b D B

B

C

C

C

C

BbA

A Bb

A Bb

A Bb

Bb

I n v a r i a n t ~ y a d s *

B y p h r n s m a r k i n v a r i a n t d y a d s ; a r r o v a m a r k i m p li e d p a r t i ti o n s

xample

1



The chart in Example 1 a shows th at if one divides the basic set in to

six successive dyads, and then associates

ll

pairs of nonad jacent d yads,

all b u t one such association produces a tetrachord equivalent t o a trans-

posed or inverted linear segment of the row. (For example, the nonad-

jacent dyads 1 and form a chromatic tetrach ord, pc set 4-1, which

represents a transposition of the final tetrachord -that is, the adjacen t

dyads 5 and 6-of the basic set.) None of the rejected orderings of the

basic set produces as many correspondences of this type. This same

kind of property also helps to explain why, as the sketches reveal, be-

fore he experimented with the basic set, Schoenberg had already de-

cided to base the enti re piece on only fo ur transpositions and inversions

of th e basic se t: P4, Plo , 14, I AS shown by the chart in Example 1 b ,

if one tetrachord from the prime form of the basic set (P4) is joined

with a tetrachord from one of the oth er three row forms, harmonies

arise th at are o ften equivalent to linear segments of the basic se t. In

fact, these equivalent harmonies frequently represent a form of the

principal hexach ord of the basic se t, pc set 6-2.30

Thro ugho ut o p. 2 5 these properties consistently determine bo th the

rhythmical and registral shapes of individual row forms, as well as the

succession of row forms and the ways they are joined. In fact, they rep-

resent the means by which Schoenberg implem ents wha t I will now refer

to as hls first techniqu e for structuring discrete phrases. Two examples

from op . 25 mu st suffice to illustrate this techniq ue.

Consider first the opening phrase of the Intermezzo (Ex. 12), one of

the earliest sections of op. 25 to be composed. Here a repeating tetra-

chorda l drone spans and defines the phrase by joining tetrachord al

segments from two row forms. The drone, which appears in the right

hand, presents the first tetrachord from P4 followed by the second

tetrachord from Ilo (notate d 'LP4(1) 110(2) in E x. 1 b) . Together

these two tetrachords form a secondary harm ony, pc set 8- 8, which is

equivalent to the linear segment of the basic set comprised of order

numbers 3-1 0. Precisely this harmon y joins the two row forms into a

single phrase. In this opening phrase the left hand too depends largely

upon properties worked out in the sketches, those determining the as-

sociation of nonadjacent dyads within a single row form. For instance,

the first left-hand harm ony, marked pc set 4-18, groups together the

third and fifth dyads from the basic set (order numbers 5,6,9,10);

and the second, marked pc set 4-5, similarly groups the fourth and

sixth dyads (order numb ers 7,8 ,11 ,12 ).31 As Examples 11 and 1 2

show,

ll

the harmonies that structure this phrase-with the single

excep tion of pc set 4-9 t o which I will return later-contain nonad -

jacent pitches, but are nonetheless equivalent to linear segments of

the basic set. In both its internal harmonies and the harmony that

defines its duration, then, the structure of this phrase exemplifies



E G D ~ G ~ E ~ A ~ D

A ~

(a) BS P4

: \

B

Dyads: 1 3 4 5 6

yads Tet. yads Tet.

Te t: (1) (2) (3

(b)

BS P4 : E F G Db Gb Eb ~ D B C A ~

P l 0 : s b B D ~ G

C A D A b F

G b E b E

1 4 : E E b D b G

D

F C G ~ A

A b B ~h

I , , : B b A G Db

A b B

G b C ~ D F E

Example 11 . Basic S et, op .

5 ;

Derivation of secondary harmonies



Example

12.

Piano Suite op. 25 Intermezzo



the first of the two principal techniques that Schoenberg devises for

joining successive row form s int o a single phrase.

The opening phrase of the first movement (Prelude) offers a second

example of this techniqu e, now used to join segments from three row

forms (Ex. 13).

As

in Exam ple 12 , Schoenberg defines the length of the

phrase by using textural and rhythmic features to create a secondary

harmonic dimension, which joins together a single tetrachord from each

of three row forms, P4(3) 110(1)

P1O(l).Especially appropriate for the

opening phrase of the Suite this secondary dimension sets fo rth pc set

6- 2, the principal hexachord of t he basic set. The exact pitch classes

that make up this secondary harmony here seem important, for they

repre sent those of the principal hex achord of Pi,-the only row form

that occurs twice and that serves to conclude the phrase. As in the

previous example, Schoenberg also uses additional secondary har-

monies (that is, harmonies equivalent to linear segments of the basic

set) to provide harmonic structure within as well as between discrete

row

forms.32 (The pc sets marked with do tted lines do not represent

secondary harmonies and will be discussed later.) Examples 12 and 1 3

both illustrate the first of the two principal techniques that Schoen-

berg uses to generate discrete phrases. While in both the secondary har-

mon ic dimension th at spans the phrase joins ordered segments of equal

length from more than one row form , this need no t be the case. In fac t,

Schoenberg frequ ently varies th e nu mber and size of the segments tha t

comprise such secondary dimensions, and he uses this variation to

create the addition al connections required for an extended form.

Unlike the first, Schoenberg s second techn ique for structuring dis-

crete phrases derives not from the internal structure of the basic set,

but from invariant segments that occur among

specific

inversions and

transpositions of the basic set. If the first technique develops the inter-

vallic struc ture of a specific basic set, then the second develops a group

of specific transformations of that same basic set. This second tech-

nique, I submit, finally solved the problem of generating extended

twelve-tone form s.

I shall return to the sketches for op. 2 5, which do exemplify this

second technique, but it will be useful to explain the technique first

through on e brief exam ple. Until recently one of the puzzles of op . 25

was the harmonic structure of phrases like the one shown in Example

14a. This is the opening phrase of the last movement (Gigue), and

here the first technique clearly delineates the length of the phrase, but

does not organize its internal secondary harmonies. For instance, the

two row forms tha t make up the phrase (P4, 110) are joined tog ether

by two forms of pc set 8-18 (right and left hands), which represents

the complement of a linear segment of the basic set (4-215: order

numbers 5-8). Each form of pc 8-18 , in turn, comprises two hexachords



4-12 4-215

P4:

F G D b G b E b A b D

B C

A B b

6-2

$5-2

4-5 4-18

Example

13

Piano Suite op. 25 Prelude



(6-21716-z43), each conta ining pcs nonadjacent in the basic set. Unlike

the examples discussed above, however, these extracted hexachords are

not equivalent to linear segments of the basic set and consequently do

not represent what I have defined as secondary harmonies.

I

began to suspect these structures to represent a second technique

for organizing harmony (and not merely the inconsistent use of the

first) when I noticed a pattern in their occurrence. They occur more

oft en in the second half of every movem ent, bu t occurrences in the first

half increase markedly in the final movements. When I recognized

the connection between the opening phrase of the final movement

(Gigue) and that of the second movement (Gavotte), the nature of the

second technique finally came clear, as well as its importance in estab-

lishing large-scale formal connections amo ng movem ents.

The opening phrase of the Gavotte (Ex. 14b) shows Schoenberg

vertically grouping together pitches nonadjacent in the basic set (order

numbers 1 ,2 ,3 , 4 ,9 ,1 0 and order numbers 5 ,6 ,7 ,8 ,11 ,12) to form pc

sets 6-24316-217, which are the same complementary hexachords that

struc ture the opening of the Gigue. Unlike those in the Gigue, however,

these hexachords in the Gavotte are comprised largely of dyads occur-

ring as invariant segments among the four row forms that structure the

piece. The first occurrence of

6-217, for instance, is formed by the

dyads (9,10), (6,3), and (8,2); (9,lO) is an invariant segment between

P and lo while (6, 3) and (8,2) are invariant segments between and

PIo (see Ex . 1 4 ) ~ ~ ~n this phrase, Schoenberg seems to derive new har-

mon ies-beyond those defined by linear segments of the basic set-by

joining together invariant segments that connect specific forms of the

basic set. This technique is conceptually attractive, since it provides a

means of developing the group of row form s th at structu re the piece,

as well as a way of deriving new motivic material from the intemallic

structure o f the basic set.

My analysis of op. 25 confirms the source of these previously unex-

plained harmonies; they are all junctures of invariant segments. Like

Schoenberg's secondary harm onies , these invariant harmonies relate

by transposition and inversion; for example, the pitches that form

6-21716-243 in the Gigue are differen t from those tha t make up the

invariant segments among the four row forms. Schoenberg uses this

second technique also to generate contrasting sections within move-

ments and to associate the overall forms of movements themselves. The

concealed connection between the openings of the Gavotte and Gigue

points toward this second technique-the innovation th at ,

I

think, con-

vinced Schoenberg th at the new twelve-tone m etho d could generate the

extended forms lacking in his earlier serial works.

If from the outset of op. 25 Schoenberg planned t o use bo th tech -

niques for structuring harm ony , then we should expect to find evidence



in his sketches fo r t he second techniqu e as well as for the first. Clearly

the second technique would require precise control of the number and

placem ent of invariant segments arising am ong th e piece s fou r struc-

tural row forms. In fact, the five orderings Schoenberg sketched for h s

basic set exhibit several features that throw light upon his choice of the

final ordering.

Concern for the occurrence and placement of invariant segments

would explain decisions made prior to the experiments with ordering.

Example 10 shows tha t in addition t o determining the pc conten t of

each tetracho rd (as well as the ordering of the first), Schoenberg ha d

already decided 1) th at fo ur tritone-related row forms would structure

the piece and (2) that the basic set would unfold primarily in tetra-

chordal partitions. The choice of tritone-related row forms must, in

part, have been determined by invariant properties, since it guarantees

that the invariant relations between any row form and the other three

would be the same whichever row form (P4,

Plo

14, II 0) were singled

o u t. The choice of a single (tetrachorda l) partitioning is more com plex,

bu t can be inferred from the kind of concealed association between the

Gavotte and Gigue. At first glance the hexachords in the Gavotte may

seem structurally insignificant, merely an incidental feature of the

ordered presentation of a tetrachorda lly partitioned row form. We have

little reason t o revise this assum ptio nu ntil they reappear in the Gigue.

Al-

though even here their derivation from invariant dyads is not explicit,

since some internal segments contain only single pitch classes, we will

still hear the recurrence and grasp its formal significance. The use of a

single partitioning thro ug ho ut the piece makes m uch easier this kind o f

extended connection because it allows association of invariant seg-

ments between row forms before they are used to derive new motivic

material.

In addition to these initial decisions, Schoenberg still had to solve

one rem ainingpro blem t o insure th e effectiveness of this new techniqu e.

He had to construct the basic set so that the length and placement of

invariant segments would complement a tetrachordal partitioning

of the basic set. The five orderings of the basic set sketched in Example

10 show a revealing pat tern in th e placement of invariant dyads among

the four row forms generated by each ordering. Only in the fifth and

preferred ordering of the basic set do the invariant dyads (marked with

hyphens) partition the row exactly into three tetrachords and-with

only one excep tion- into six discrete dyads. Th at is, the final ordering

creates invariant segments between order numbers 1 -2 , 3-4, 5-6 , 7-8,

10-1 1 , and 11 12 ; this pa ttern intersects with bo th a tetrachordal par-

titioning (order numbers 1 -4 ,5 -8 ,9 -1 2) and, except for order numbers

10-1 1 , a dyadic partitioning. The fou r orderings tha t Schoenberg con-

sidered but rejected, have invariant segments that never delineate all



three tetrachordal partitions. The first three or de ri ng create asyrnmet-

rical partitions that delineate not more than one tetrachord: 51413,

4

2121612, a n d m / 8 . The fourth ordering comes closer with more dyad

partitions, but still thwarts the tetrachord partition between order num-

bers 8 and

9 :

2/2/3/2 /1/2. The fifth ordering-the one employed in the

finished piece-improves on this partitionin g by rearranging the invariant

dyads so that they intersect exactly with the three tetrachordal parti-

4 4 4

1

ions: 2 /2 /2 /2 /1 /3 .34

. . .

Schoenberg s sketches suggest, th en, tha t h e settled on the final

ordering of the basic set at least in part because it best accommodates

both techniques for structuring harmony. The association of nonadja-

cent dyads in one row form a nd of tetrachords amon g all row forms

creates the greatest number of secondary harmonies, while the place-

ment of invariant dyads implies a partitioning that best complements

both tetrachordal and dyadic partitioning of the row, thus allowing

explicit derivation of invariant harmonies. Knowledge that Schoenberg

had devised bo th tech niques before beginning op. 2 5 proves im po rtan t

because it prevents us from misinterpreting the greater frequency of

invariant harmonies as the movements progress. Without the evidence

of the sketches, we might speculate tha t, as in opp. 23 and 24 , Schoen-

berg tinkered in op. 25 with various compositional techniques not yet

recognizing the potential of his new twelve-tone method. Instead,

as

I

shall con clud e by sho win g, the prevalence of invariant harmonies results

primarily from Schoenberg s te chn ique of deriving new motivic m ateria l,

the technique that, in turn, produces extended forms both within and

among movements.

IV

Two analyses will show how Schoenberg s two techniques of struc-

turing phrases work in op. 25 to generate form for a single movement

and t o create an exten ded form spanning several movements. The first

explores the two-part form of the Intermezzo to reveal further aspects

of the solution t o the problem of motivic reprise. The second examines

the formal connection between the Prelude and the Trio of the Minuet

to clarify further Schoenberg s solu tion t o the problem of motivic

alteration. This latter exam ple will draw in to question Maegaard s con-

clusion th at the change of techn ique i n th e Trio betrays an inconsistency

in th e twelve-tone m eth od of op . 25 itself (see p. 1 1 2 above). R ath er, I

shall argue that this apparent change results from using invariant har-

monies t o derive the new motivic material essential t o extend ed forms.

Returning to the opening phrase of the Intermezzo (Ex. 12), I want



t o propose th at Schoenberg's me tho d of struc turing phrases solves one

aspect of the reprise pro blem by establishing criteria bo th for ending

motivic elab oration and for preparing a formal point of articu lation. As

I suggested earlier, a listener to contextual pieces often cannot under-

stand why motivic elaboration breaks off at a certai? point; often the

only reason is apparently tha t the piece, or its central idea, is becoming

to o ambiguous. Schoenberg's phrase structures respond directly to this

problem , for defining the length of the phrase through secondary or

invariant harmonies allows the com pletion of either to convey why the

motivic elaboration has ended and also that we have reached a formal

point of articulation. In the opening phrase of the Intermezzo, for

instance, we know that we have reached a formal point of articulation

wh en the drone figure-in its entirety-reproduces a harm ony equiva-

len t to a linear segment of the basic set ; com pletion of this har m ony

signals the beginning of a new phrase. In one sense, then, Schoenberg

has devised a twelve-tone technique analogous to a tonal cadence or

prolongation.

The phrase structure of the Inte rmezzo also exemplifies how Schoen-

berg's tw o techniques can serve to generate a two-part form . Example

12 shows tha t the opening phrase of the In termezzo derivesits harmo nic

structure from secondary harmonies, and thus represents what I am

calling Schoenberg's first technique for structuring phrases. Analysis of

the harmonic structure of later phrases shows that this same technique

prevails up to and including what sounds like a reprise of the opening

phrase in m . 20 (Ex . 15a). (Here phrase means all passages th at join

tw o or mo re row forms.) As Example 16a shows, Schoenberg uses four

secondary harmonies to delineate the length of the six phrases that

make up th e Intermezzo's first part: p c sets 8-8, 6- 2, 8- 12 ,8 -2 15 . Ex.

16 b shows where these secondary harm onies occur in th e basic set.)

After the apparent reprise in m. 20, the phrase shown in Example

17 marks the demise of the first technique. Analysis of this phrase

(which joins ll four row form s) shows tha t only what I term in-

variant harmonies organize all harm onic dimensions tha t associate

nonadjacent pitches or segments. (In this and the following examples,

parentheses mark invariant harmonies; all sets not marked by paren-

theses are secondary harmonies.) PC set 5-8 structures each row fo rm

internally, pc sets 5-1, 5-8, and 6-223 link successive row forms, and

pc set 7-8 and 8 -25 join ll four row forms into a single phrase. None

of these sets are equivalent t o linear segments of the basic set, bu t all

can be derived by connecting invariant dyads among the four row

form s. (Ex. 16b shows the derivation of these harmonies by invariant

segments.) Because some invariant harmonies here are not formed by

the invariant segments from which they derive, the phrase itself does

not make their derivation explicit. (For example, pc set 5-1 is formed



HARMONIC PHRASE STRUCTUR E IN THE INTERME ZZO

(a)

Phrases Secondacv and Joined Segnzn s;

Invariant Hartnorlies

Tets Order No s

mm. 1-3

mm . 5-7

mm . 7-9

rnm. 11-15

mm. 15-18

mm. 18-20

mm . 20-23

mm. 25-26

mm . 29-30

mm . 3 1-33

mm. 35-37

mm . 37-43

mm . 43-45

pc set 8- 8

pc set 6-2

pc set 6-2

pc set 8-1 2

pc set 8-21 5

pc set 8-12

pc set 8-8

pc set (7-8)

pc set (8-25)

pc set (5-1)

pc set (5-8 )

pc set (8-25)

pc set (8-28)

pc set (5-8)

(b) Secotzdary Hart7zonies:

Invariant Harntonies:

Linear Segn-t.,?tso f BS

Derivation frorn Azv. Dvads

pc set (5-1)

=

(2 5 (3 4) (3 6)

pc set (5-8)

=

(1 7 (3 4) (4 5)

8-2 15

pc set (7-8) = (0 6) (2 8 ) (9 10)

- - -

10 11)

8-8

pc set (8-25)

=

(1 7)

3

4) (4 5)

5 7 1 6 3 8 2 1 1 0 9 1 0 (9 10) (10 11)

pc se t (8-28)= (0 6) (0 9)

(2

5)

8 12

Example 16



by joining the final tetrachord from two row forms, but no invariant

segment structures the initial dyad of these tetrachords, i.e., orde r num-

bers 9-10.) To understand the source of these new harmonies, then, the

listener must recognize that the invariant relations implicit in the first

part have in the second part been un folded in to harmonic structure.

The notion of invariant harmonies also explains several oddities in

set usage that occur in this phrase. For example, two complementary

sets, 5 - 8 /7 4 , structure the outer voices. But to form pc set 5-8, one

pitchclass in each row form must be repeated, in different registers:

pc 5 in I 4 and P4; pc 11 in Il o and Plo. If we did n ot understand the

role of invariant harmonies in this phrase, we would be hard pressed to

explain why Schoenberg chooses t o ignore his method s p rohibition

on repeating pitch classes (from a single row form) in different registers

and row partitions.

To return to the Intermezzo, the second part , U e the first, again

uses fo ur harmonies t o determine phrase length: 5-817 -8,5 -1,8 -25 ,and

8-25 (E x. 16a).35 Unlike the first part s harmonies, however, the sec-

ond s are all invariant harm onies, indeed t he same four harmonies th at

structured the phrase shown in Example 17 . The first phrase in the

movem ent to derive from invariant harmonies thus marks the two-part

form with a change in harmonic structure. The Intermezzo s two-part

form emerges, the n, fro m t wo different harmonic techniques for struc-

turing phrases.

Like the Intermezzo, most of Schoenberg s forms develop from phrase

structure and therefore from the harmonic techniques that produce

phrase structure. Twelve-tone reprise, then, may be supplemented by

textural, rhythmic, or other echoes, but essentially consists of phrases

structured- that is, spanned-by equivalent secondary or invariant har-

monies. If we do not recognize that form derives principally from the

harmonic dimensions that delineate phrase length, we encounter the

same ambiguities as in Schoenberg s serial con tex tua l forms. Three

phrases from the Intermezzo can quickly illustrate this point.

The opening phrase of th e Intermezzo (Ex. 1 2) never recurs exactly,

bu t surface similarities in tex ture and motivic conto ur suggest two later

phrases as possible reprises (Ex. 15a,b). In each of these phrases the

drone figure in one hand accompanies what I have called the motivic

tetrachords in the oth er. If our definition of reprise rests o n the recur-

rence of texture, contour, specific row forms, or intern l secondary

harmonic dimensions, we could defend several plausible, albeit contra-

dictory , conclusions abou t how motivic reprise works in this m ovement.

For instance, if we take tex ture as primary, th en Example 15b is closer

to t he opening tha n Example 15 a; if we choose motivic conto ur, then

the opposite wiU be true. Trying to resolve the issue by considering

additional surface features merely deepens the ambiguity. Example 15 b



a a a a a



uses one of the same row forms (Ilo), but structures its motivic tetra-

chords with different secondary harmonies, while Example 15a uses

neither of the same row forms, bu t does structure its motivic tetrachords

with equivalent secondary harmonies. Unless we ta ke longer-range ha r-

monic structures into account, we will be unable to formulate useful

criteria for motivic reprise, variation, or development. If motivic re-

prise is defined as the recurrence of harmonies that determine phrase

length, then only Exam ple 15a constitutes a reprise of th e opening

phrase, since both are structured by equivalent forms of the secondary

harmony 8-8 . As ofte n in Baroque binary forms, this kind of trans-

posed or inverted reprise begins th e second par t, thereby enhancing

the underlying two-part harmon ic form.

What, the n, should we say about Example 15b ? Perhaps Schoenberg

has here devised a twelve-tone analogue for the tension between an

incipient three-part motivic form and a two-part harmonic form that

so often arises in Baroque binary forms. But just as in these ton al form s,

tension is not th e same thing as irresolvable ambiguity or formal contra-

diction. Example 15b clearly recalls the opening , bu t nevertheless repre-

sents a h rmonic variation since its structure derives from the invariant

harmony 5-8. This harmony , which occurs only in the second part of

the Intermezzo, becomes increasingly important as the movement pro-

gresses. It first functions internally, where it gives rise to a short-term

secondary h armonic dimension; it later structures the complete phrase

shown in Example 15 b; finally an d most significantly, it structures the

motivic tetrachords that span the final phrase of the movement rnrn.

43-45), a phrase tha t so unds much like a m ot to or short coda. In a

sense, then, the motivic tension introduced by the pseudo-reprise of

Example 15 b is finally and decisively resolved by the movement's

final phrase.

The notion of motivic reprise that I have been illustrating, coupled

with the concept of harmonic form, thus helps to reveal how melody

and harmony interact in Schoenberg's twelve-tone music and begin to

provide a means of describing how a twelve-tone form as whole de-

velops. In other words, these ideas together can reveal why sections, as

well as discrete phrases, occur in a specific orde r. The discussion above

of motivic reprise and the emergence of the invariant harmony 5-8

move in this direction, but the Intermezzo offers a more important

example in the way it relates the harmonic structures that generate

each of its two parts.

In the first part, four row forms explore or develop the internal

structure of the basic set. The harmonies that structure its phrases,

however, in effect partition the basic set into the sets 6-2, 8- 8, 8-12 ,

and 8-215 , in fact emphasizing its principal hexachords and tetrachords

(see Ex . 16b). But while t he first part develops a specific partitioning of



the basic set, it does not explore the relations etween the four row

forms t ha t generate thi s partitioning. (Invariants between row forms do

receive local emphasis, of course, but are no t given a structura l or form -

defining role-that is, they do not wo rk as what I have termed invariant

harmonies.) This exploration occurs only in the second part. Just as in

th e first part harmo nic s tructure in effect partitions a single basic set, so

in the second part it partitions the group of row forms that generate the

movem ent. Fo r instance, the most im portan t invariant harm ony , pc set

5- 8, derives from joining th e invariant dyads

(1 7), (3 4) an d (4 5), as

shown by Example 16b. This harmony thus uses the on e dyad that

occurs in ll fou r row form s, (1 7), to explore invariant relations that

associate either the beginning or end of each of the four row forms,

dyads (3 4) and (4

5).

A second derivation of 5-8 from invariant d yads

[(I 7)

9

10) (10 l l ) ] yelds th e identical partition, but with the op-

posite ends of th e row forms.) Moreover, the harm on y 5-8 has implica-

tions that suggest why it particularly makes a fitting continuation of

the Intermezzo's first par t. PC set 5- 8 also represents the set formed by

the invariants occurring between the first hexachords of P4 and I4 and

between those of P lo and IlO. (For exam ple, P,: 4

5

7 1 6 3 and

14:4 3 1 7 2 5 share pcs 1,3 ,4,5 ,7, forming pc set 5-8.) Since this

hexachord (6-2) represents one of the principal secondary harmonies

used to struc ture phrases in pa rt one, it turns out t ha t part two not o nly

develops part one's generative row fo rms, but also the relations between

the specific partitions that generate the movements two parts. I offer

these kinds of connections between the Intermezzo's two parts only as

illustrative. They are by no means th e only connec tions on e could d raw,

bu t the y do begin to reveal how th e form as a whole develops and why

phrases and pa rts occur

in

the order they do .

I want to argue against a view of op. 25 as a transitional piece in

which Schoenberg has not yet perfected the techniques of his twelve-

tone method, but such a view does claim a certain plausibility. Even if

we consider only form, and disregard both chronology and composi-

tional techn ique, the n op . 2 5 still seems t o lack the extended forms of

th e later twelve-tone work s. It may seem paradoxical, t he n, to claim that

op. 25 fully represents the twelve-tone method, although not fully

embodying its main formal achievement. I hope that my last examples

will suggest an answer by illustrating, in miniature, the intricate net-

work of relations among mov ements tha t co nstitutes the Suite's ex-

ten ded form . These relations arise from the same principles tha t

genera te th e forms of single movements. More study of the later works

is needed, b ut their form s appear to depend upon techniques similar to

those of op . 25 . My last analysis illustrates a few of these techniq ues by

showing ho w the Trio of th e Minuet (Mvt. 5) derives fr om , as well as

develops, the opening phrase of th e Prelude (Mvt.

1).



I have suggested above how in op. 23 Schoenberg used material

derived from a movement's opening to structure progessively larger

parts of the form. This material sometimes seemed closely related to

the series and sometimes only distantly or ambiguously related. Much

the same happens in op. 25, except that now the new material derives

ultimately from the structu re of the basic s et, even in the Trio where a

unique texture has persuaded many that it marks a departure or dis-

continuity. Row forms n the Trio unfold linearly for the most part,

and imitation emphasizes hexachordal rather than tetrachordal parti-

tions. The results are motives and harmonies that sound realtively new

or unfamiliar. This texture seems simpler than n earlier movements,

where tetrachords from one or more row forms usually overlap or un-

fold simultaneously. In the Trio, most row forms remain registrally

intact. Why, then, does this apparently simpler texture not appear

earlier, when the listener needs t o learn the internal s truc ture of th e

basic set in order to follow the more complicated textures and har-

monic structures to come? To some, this anomaly bears witness to a

change in th e twelve-tone me thod within the piece, and th us also to

op. 25's transitional character. Maegaard, for instance, argues that not

until the Trio does Schoenberg recognize compositionally that th e

three tetrachro ds of th e basic set con stitu te a twelve-tone continuum

(see note 23). In my view, the Trio does indeed represent a unique

moment in op. 25, but not because its new texture marks a discontin-

uity in technique. Instea d, here Schoenberg develops formally a group

of harmonies that first appear, though unemphasized, in the opening

phrase of the S uite. These harmonies in turn provide essential motivic

material for the Suite's final movements. Thus, as I will show, what

Schoenberg mean t t o sound new in the Trio also associates th e first

and fifth movements of the Suite.

Since the basic set itself structures horizontally each of the Trio's

two voices, the effect of newness arises largely from th e vertical ha r-

monies created by their simultaneous presentation. Example 1 8 shows

how these vertical harmonies, though formed by only two voices, often

contain segments from three different row forms. (The harmony

6-238

n m m. 35-36, for example, is formed by dyads from

Ilo,

P and Id.)

The Trio's texture (with its repeating patterns of articulations) suggests

this segmentation, as does the distance between imitative entrances.

Moreover, these harmonies form a rhythmic pattern ]

)

which is interrupted only at the end of each part to provide rhythmic

closure ). This pattern strongly implies a 618 meter

which, against the clearly audible 314 meter of the preceding Minuet,



Trio

-12

ord)

4 - 1 8 6 - 2 3 8 ( 6 - 5 )

J

( - )

J J

Example

18

Piano Suite op .

25

Trio

34



creates through hemiola a contrasting syncopated interlude that post-

pones, as well as prepares f or , th e Minuet s retu rn.

Despite its brevity an d elusiveness, the Trio has a miniature two -part

form itself, but a more complex two-part form than the Intermezzo,

where secondary harmonies dominated one part an d invariant harmonies

the other. Here both kinds intermingle freely. Except for the initial

4-12, all the harm oniesin Example 18 , for instance, are either secondary

or invariant. (As before, parentheses mark all invariant harmonies.) The

com plexity of t he Trio s form arises in par t from the secondary har -

monies that provide its structure. Not only do they all represent linear

segments of the basic set, but all can also be derived by joining invari-

ant segments occurring among the four row forms. In other words, all

of these secondary harmonies simultaneously represent what I have de-

fined as invariant harmonies. (Ab out half of the basic set s harmonies

have this property, which arises from the large number of invariant

dyads amo ng the fo ur row forms.) Othe r movements of o p . 2 5 use these

same secondary harmonies, but only the Trio uses them exclusively.

Their predominance helps to unify its form. So does the fact that ex-

cept for pc sets 5-16 and 6 -2 in mm. 40-41 , all of these secondary har -

monies result from joining only segments that are actually invariant

among th e four row forms. None of the invariant harmonies, however,

are so constructed. For example, bo th dyads tha t form the secondary

harmony 4-18 (m. 35) represent invariant segments [(6 3) (10 9)],

whereas on ly tw o o f th e three dyads t ha t form the following invariant

harmony (6-5) represent invariant segments [(2

8

(1 7 ] . By using

only invariant dyads to form secondary harmonies, Schoenberg em-

phasizes their dual nature. The manner of constructing both invariant

an d secondarv harmonies. then . furthe r unifies the movement as a wh ole.

If these com plex relations help to ensure the Trio s u ni ty , equally

complex ones determine its two parts. The two parts are superficially

distinct in using different groups of harmonies (with the important ex-

ception o f th e invariant harmony 7-8 to w hich I shall return). Recurring

harmonies always do so within the part where they first appear-for

exam ple, the repetitions of 5-1 9, 6-5, and 6-238 in th e first part. But

these harmonies delineate a two-p art for m in oth er , subtler ways, which

depen d upon structural differences between each part s invariant har-

monies. The first way has to d o with subset relations. The first part has

tw o invariant h armon ies, 5-19 and 6 -5, whe re 5-1 9 is a subset of 6 -5;

th e second part also has tw o invariant harm onies, 4-6 and 6-2 41, which

are sirnilarlv related. Subset relations do not. however. associate har-

monies f rom different parts; pc set 5-19 is no t a subset of 6-2 41, nor is

4-6 a subset of 5 -19.

Second and more important, these groups of invariant harmonies

distinguish the Trio s tw o parts according to their derivation fro m



specific invariant dyads. As th e fou r row form s th at structure t he piece

(Ex. 14) show, two invariant dyads associate all four row form s; the

first, (1 7) occupies order numb ers 3 and 4 in all fou r form s, and th e

second, (4 l o ) , order numbers 1 and 12 , tha t is, the first and last

pitches of the row. While each of these invariant harmonies can be de-

rived from invariant segments in at least two different ways, all deriva-

tions of th e invariant harm ony 6-5 include the dyad (1 7), bu t exclude

the dyad (4 lo ), whereas th e reverse is true for th e invariant ha rm ony

6-241. Sim ilarly, th e dyad (1 7) can be used to derive the invariant

harmony 5-19 , bu t no t the dyad (4 lo ), whereas the reverse is true for

the invariant harmon y 4 -6. The invariant harmonies that structure the

Trio thus fall into tw o groups corresponding t o its two parts.

Two harmonic structures support th e form of t he Trio, then. A

special group of secondary harmonies and uniform methods of con-

structing

secondary and invariant harmonies unify the movement,

while different invariant harmonies, and modes of deriving them, ful-

fill its two-p art form. This latter techniqu e also explains wh y one in-

variant har mo ny , 7- 8, occurs in bo th parts-in the second ending of

part one and in the first ending of part two. This harmony can be

derived in only two ways from invariant segments-(0 6)(2 8)(9 10 )

(10 11) and (0 6)(2 8)(3 4)( 4 5)-both of which ex lude th e dyads

(1 7) and (4 lo), the dyads respectively unique to each part. But

b o th

in lude

other invariant dyads common to all possible derivations

of the four remaining invariant harmonies. The appearance of pc set

7-8 at t he end of p art on e, then serves as a kind of mod ulatory har-

mo ny; it drops the dyad unique to part one and adds several comm on

to part two. To confirm this view, when 7-8 reappears just before the

repeat of part two, it serves to introduce the repetition as if it were

again following directly from part on e.36

The harmonic structure of the Trio derives in part from that of the

Minuet, for b ot h feature prominently many of the same secondary and

invariant harmonies (for example, 5-4 , 5-16, 5-19, 6-2 ,

6-238, 6-241).

But its harm onic struc ture is not entirely derivative, for several har-

monies important in the Trio do not appear in the Minuet (invariant

harmonies 6-5, 7-8, and 4-6, for instance); moreover, while various

prop erties d o relate th e Trio s harmonies (as discussed above), a number

of other harmonies share the same properties and could have served

equally well. Why, the n, did Schoenberg choose these specific harmonies

t o play a structural role in th e Trio?

To answer that question,

I

must first backtrack to a few general ob-

servations. My analysis of the Suite shows that while all its movements

use the same four row forms (usually partitioned tetrachordally), each

derives its form from a unique group of harmonies. While individual

harmonies may appear in several movements, each movement has its



own group of harmonies. For example, the invariant harmony 7-8 ap-

pears in bo th the Intermezzo and Trio, bu t an equally imp ortant invar-

iant harm ony 6-5 in the Trio never occurs in th e Intermezzo. Furth er-

more, after harmonies have occurred in one movement, they tend to

reappear as structural harmonies in following movements, and vice

versa. (7-8 is an example; it plays a structural role in the Intermezzo

an d th en reappears in the Trio. Likewise, 6-5, which appears in th e Trio,

becomes crucial in the Gigue.) Harmonies may appear in passing in one

mov ement, th en , b u t echo or recall structural harmonies of earlier move-

ments. The reverse is also true; harmonies may appear in passing in one

mo vem en t, bu t play a crucial role in a later mov ement. Such proves the

case with th e Trio s invariant harmonies th at do no t occur in the

Minuet-6-5,4-6, and 7-8.

Perhaps th e most im por tan t of these is 6-5, which makes its first

appearance, albeit concealed, in the opening phrase of the Prelude. As

we have seen, this phrase is structured primarily b y secondary harmonies

(see Ex. 13). Invariant harmonies (marked by dotted lines) do occur

vertically (linking right- and left-hand parts), bu t the c omp lex textu re

and rh ythm make these harmonies less audible th an t he more prominent

secondary harmonies (marked by solid lines). Indeed, although 6-5

occurs four times in succession (always formed by associating dyads

1,3,5 and 2,4,6) the contra pun tal independence of th e voices make its

occurrences sound incidental, as if an undesigned consequence of the

more important secondary harmonies. Later in the Prelude 6-5 emerges

from the background, however, and becomes increasingly important.

(The association of the same dyads becomes progressively more audible,

first in

mrn.

13-14 and then in mm. 20-21

.

When 6-5 reappears in th e

Trio, the n, it recalls the first mov emen t of t he Suite an d creates th e

potential for an extended form .

This form will seem more than potential if we compare the remain-

ing invariant harmonies th at structure th e Trio with those of th e Pre-

lude s opening phrase. With the exception of 7-8 , which recalls the

Intermezzo, all appear as concealed harmonies, like 6-5 , th at link to-

gether various elements of the textu re. PC set 6-241 , for exam ple, occurs

in part t wo of th e Trio (m. 42) an d twice in Example 1 3 ; t represents

the first six pitches of PIo in m . 1 (no t marked) and the upper voice of

RP n

mm. 5-6, both times formed by associating order numbers

1,2 ,3,4,5 ,9. PC set 4-6, which also structu res part two of the Trio, also

makes two appearances in Example 13. Formed by associating order

num bers 1,2,5 ,9, it first occurs at the end of

RPIo

in m. 5, and then

more prominently in the following

RP

(neither occurrence is marked

in Ex. 13). (Like 6- 5, 4-6 becomes increasingly im por tan t as th e move-

ment progresses. In mm. 20-21, for example, it introduces and con-

cludes th e phrase.) Like its invariant harmonie s, th e Trio s secondary



harmonies

4-5,4-12,4-18,6-2,6-2616-238

also derive largely from t he

opening of the Prelude and are for the most part those most audible in

Example 13 . (The Trio's tw o remaining secondary harmonies, 5- 4 and

5-16, become prominent in m m. 6-9, th e phrase tha t follows Example

13.)

The group of harmonies that structure th e Trio derive from , the n, as

well as develo p, those tha t appear a t t h e opening of the Suite, thereby

creating an extend ed form tha t associates widely separated movements.

This ex tended fo rm , llke many in Schoenberg's later works, does no t

merely repea t or vary slightly what has been presented before, but

works out a specific kind of harmonic development. This development

seems often to depend on a kind of interplay between concealed and

emphasized harmonic structures and between secondary and invariant

harmonies. The Trio, for instance, brings to the foreground an earlier

group of nonstructural invariant harmonies and develops properties

within the group so as to sustain th e two-part form. Through this kind

of interplay , whose variables derive u ltimately from the basic set itself,

Schoenberg derives the new motivic material needed to generate ex-

tended forms. It represents, then, one solution t o the problem of

motivic alteration. Where motivic alteration seemed sometimes to

threate n th e identity of the central idea in th e serial works, now it serves

to develop further the internal structure of the central idea or the basic

set.

v

Schoenberg's forms, as I have described them in op. 25, may seem

based on methods too complex to be audible, perhaps too complex

even to be theoretically satisfying. In the abstrac t, the techniques of o p.

2 5 are neither arcane nor com plicated, however. Generating new har -

monies by joining invariant segments between row forms, for instance,

elegantly solves the problem of relating compositionally the form of a

piece t o its generative row forms. Both tech niques can work in conjunc -

tion with the combinatorial property, but d o not depend upon it , and

more importantly, both are consistent with Schoenberg's requirement

that a twelve-tone piece derive entirely from a single basic set. More-

over, th e need to make its complexities audible seems to have induced

Schoenberg t o permit serious theoretical ambiguities in the S uite. Once

he had set up the Suite's four row forms to guarantee invariants for

constructing form s, he confro nted, as I read his acco unt, an unexpected

difficulty-an excess of invariants. With characteristically subt le self-

criticism, he wrote: In the first works in which I employed this m eth od ,

I was no t y et convinced th at the exclusive use of one set would no t

result in m ono ton y. At this time,

I

used complicated devices to

assure variety. But soo n I discovered that my fear was unfo und ed (see



note

22 .

The four row forms, in other words, produce too many pos-

sibilities; th at is, to o man y invariant harmonies result from joining the

invariant dyads, especially in view of the limited number of secondary

harm onies. To reduce the n um ber of invariant harm onies, he restricted

invariant dyads t o on ly three interval classes (ics

1,3,6 ,

but th at remedy

creates new difficulties.

For on e, partitions generated by invariant harm onies become a m-

biguous because invariant harmonies often derive from invariant seg-

ments in more than one way. Multiple derivations generate multiple

partitions, th at is, equivocal partitions. Y et, as I have argued, partitions

are crucial in relating parts of a form. Seco nd, the com plem ents of

invariant harm onies are also ambiguous. Complements of th e basic set's

linear segments must be considered secondary harm onies, bu t mu st th e

com plem ents of invariant harmonies be considered invariant harmonies?

Op. 25 leaves this question unanswe red, because its basic set allows not

onl y m ultiple derivations of invariant ha rmo nies, bu t also derivation of

most of their complements. Schoenberg uses these complements

throughout op. 2 5 , bu t their rationale remains unclear. Th ird, harmonies

that are simultaneously secondary and invariant may be both fewer in

number and more audible than othe rs, bu t they, too , blur the deriva-

tions of harmonies, as well as the form s tha t depend on them . These

difficulties are specific to op.

2 5 .

In later works, Schoenberg did not

need such complicated devices because he usually began with row

forms less abunda nt in invariants. In composing op . 2 5 , he properly

sacrificed theoretical elegance and simplicity to the demands of the

piece at hand .

In investigating op. 25's complexities, I may have played detective

myse lf, merely sub stituting intrinsic clues for extrinsic. Similarly, in

noting how Schoenberg's later twelve-tone works avoid the theoretical

difficulties of op.

2 5 ,

I ma y seem to embrace the very evolutionary a nd

teleological assumptions that I began by challenging. But it is an uneasy

hug and,

I

think, inevitable for anyone who attempts to account in-

trinsically for artistic change. To trace the inner logic impelling what

Schoenberg himself clearly thought of as a progression is not to set

atonal below twelve-tone works in a scale of value or to disinherit the

many modern descendants of Schoenberg's atonalism as illegitimate.

Neither the reprise problem nor the problem of motivic alteration are

defe ts in an y single wo rk , bu t rath er areas of irresolution, challenges t o

th e composer's ingenu ity. No art or science has made a simple ma rch of

progress, and often difficulties or anomalies have proven as productive

as successes.



1. Disagreement a bo ut th e essent ial nature of Schoenberg's twelve-tone metho d

explains in pa rt wh y earl ier scholars such as Rufer an d Leibowitz, even though

they wrongly assume a correspondence be tween op us number and chronology

of composition, nonetheless differ in identifying Schoenberg's first "real"

twelve-tone piece. Leibowitz selects op. 23 n o. 5 , while Rufer op ts for th e

Piano Sui te , op . 25, the fi rst extended piece that uses the twelve-tone m etho d

thr ou gh ou t. Most recen tly M aegaard, using mo re accurate chronological evi-

de nc e, has assigned this status to the Wind Q uin tet, op. 26. All of these pieces

do , o f course, use th e twelve-tone metho d, b ut these disagreements arise from

wh at are argued to be "primitive" or "mature" uses of the method-essen-

t ial ly, that is , from interpretat ions of the m ethod's main achievement . See

Josef Rufer, Composition with Notes t rans. Humphrey Searle (New York :

MacmiUan, 1 954 ), pp.

55

ff and Rene Leibowitz,

Schoenbergan d His Scho ol

t rans. Dika Newlin (New Y ork : Da Capo Press, 194 9), pp. 9 8 ff ; for Maegaard,

see n . 4 below.

2. "Com posi t ion with Twelve-Tones (1)" in Style and Idea ed. Leonard Stein,

trans. Leo Black (New Y or k: S t. Martin's Press, 19 75 ), p. 218.

3. H. H. Stuckenschmidt , Arn old Schoenberg: His Li fe World and Work t rans.

Hu mp hrey Searle (New York : Macmillan, 197 8), p. 277.

4. "A Stu dy in the Chronology of op. 23-26 by Arn old Schoenberg," Dansk

aarbog for musicforskning (1 962 ) 9 3-1 15 .

5. Also am ong th e sk etches for o p. 24 is one undated loose sketch sh eet , prob-

ably wri t ten between August 3-5, 19 20 , which con tains an interesting l is t of

main ideas for four mo vem ents: "Marsch," "Menuet," "Sonet," an d "Tanz."

Because the succession of tones for these them es is ident ical to the theme of

the "Variat ions," Schoenb erg apparent ly planned to derive the thematic

material of these four movements from the same fourteen-note series. Al-

thou gh h e eventually aba nd on ed this plan, this early sketch suggests tha t be-

fore August 6 , 19 20 , he was already considering the idea of a mult i -movem ent

work based on a single series. (For a more complete discussion see Maegaard,

p . 101 . )

The three series share a number of segments that represent equivalent forms

of the same unordered pc sets . Schoenberg seems to use similar internal sub-

set s t ructures in t he developm ental l ines to ident i fy w hich of several possible

segments in the series are being developed. Accordingly, when no shared sub-

set structu re b etwee n ehuivalent sets structur es a segment of the series an d a

developm ental l ine, usual ly that set occurs only o nce am on g the three series.

7. Analysis of the entire mo vem ent suggests that the o rdered series S 2 initially

appears in the right hand of m. 1 : D F Ab F # G A B b B D b ; the label "Sz"

designates the ordered form of the series that begins on D or pc 2. Labels fo r

successive repet i t ions o f the series-both ordered and unordered-derive from

this init ial ordering and follow the same notational form.

Some published analyses of this

movement exclude from the series the

pi tch-class repet i t ions that occur in the left hand of m. 1 I have included

these repetit ions since Scho enbe rg usually repeats these same pitches wh en

the series recurs a t i ts original tra nspo sition (e.g., m m . 1 0, 14 , 22).



8. The derivation and function of the all-interval tetrachord 4-229 is less clear. It

does n ot occur as a l inear segment of th e ser ies or i ts complement, al though i t

does represent a subse t of the com plemen t 7-10. It may derive fro m the initial

tetrachord of the series

Plo,

4-215, its only all-interval pair. This seem sub-

stant iated by two occurrences of 4-215 (pcs 1 ,4 ,5 ,1 1) in the f i g two mea-

sures where the y link

Plo

with its initial transformation s, R S and

P7

9. For a fuller discussion o f this passage, as well as an alterna te derivation of

trichord 3-12, see Milton Babbitt, "Since Schoenberg,"

Perspectives

of

N e w

usic

1211 (1 97 3) : 3-9.

The canon in the second movement (Ex. 3b) also is structured by 3-12; bu t

here it neither represents a linear segment of the series, nor receives emphasis

in the opening section.

10 . Several new techniques d o appear in this movement, howeverTmost impor-

tantly, a type of registral partitioning that reproduces the series by preserving

its contou r , bu t no t i ts order . This technique al lows use of conto ur and order

separately to define the hexachords as ordered pc sets, i .e., series.

I

discuss

this technique more fully below (Ex. 8).

11 . Th e following exposition of the problem s of Schoenberg's contextu alism is

heavily indebted to the argument and terminology of an unpublished manu-

script by David Lewin, "An Ap proa ch to Classical Twelve-T one Music," esp.

pp . 7-9, 18-19, 27-34. Schoen berg, "Composition w ith Twelve Tones" ( I) ,

pp . 216-219.

12 . Lewin, p. 28.

13. The d iplomatic t ranscript ion of the d raf t show n in E xample 7a was prepared

from the original by Bryan Simms. (Arno ld Schoe nberg Ins titu te, archive no.

17) .

14 . I t remains unclear w hether Schoenberg made these markings in 19 20 or in

19 23 , b u t some evidence-repeated revisions of m. 14-suggests tha t the anal-

ysis implied by the markings may have helped to resolve the problem s of con-

t inuing the piece past m. 14 .

15. His selective markings suggest that Schoen berg probably prepared the analysis

for his own use rather than for teaching, since he would not need to mark the

mo st obvious repetitions of the series.

1 6. In m . 1 "A3" designates the form of the series tha t begins on DX or pc 3. La-

bels for successive repetitions o f hexach ord A-both ordered and unordered-

derive from this initial ordering and follow the same notat ion .

17. The o rder numb ers marked for all repeti t ions of hexachord A correspond to

the ordering of i ts init ial form in m . 1 .

18 . Lewin discusses a similar prob lem und er the term "motivic inflection" (p p.

31-32).

19. Arnold Schoenberg Inst i tute, archive no . U272-U290. Schoenberg's row

tables also indicate that he was aware of the semi-combinatorial property of

his basic set, deriving from the all-combinatorial hexachord 6-20.

20. "Comp osition w ith Twelve-Tones," p. 219.

21. See my "The Telltale Sketches: Harmonic Structure in Schoenberg's Twelve-

Tone Method,"

usical Quarterly

66 (19 80) : 578-80.

22. Composit ion with Twelve-Tones," p . 224.

23. See Maegaard, "A Stud y in the Chron ology of op. 23-26":



Th e change in the application of the four-tone sets of op. 25 takes place

at the transit ion from Minuet to Trio. What happens is that the three four-

ton e sets are transformed into tw o six-tone sets , which is technically

significant since it presupposes the recognition of th e four-to ne sets as

cons t i tu t ing a 12- tone cont inuum . The char t of chronology [see Ex. 11 tells

us that this transit ion was mad e through the Gigue (p. 113).

24. See "The Tell tale Sketches."

25. Schoenb erg, "V ortra g/l2 T K/Prince ton," ed . Claudio Spies,

Perspectives of

Ne w Music

1311 (1974) : 83 .

26. "Composition with Twelve Ton es (I) ," p. 23 5. In this examp le I have added

row labell ings, pc set nam es, and t he basic set below.

2 7 . " V o r t r a g / l 2

T

K/Pr inceton ," p . 88:

Any part icular melodic motion-for instance, a chrom atic s tep-will no t

only have i t s ef fect upon the ha rmo ny, b ut on everything subsequent that

is comprised with in that spatial conti nuum . This circumstance

enables the composer to ass ign one par t of his thinking to the vertical,

and ano ther in th e hor izonta l .

28.1 discuss composit ional sketches in greater detail in "The Format and Func-

tion of Schoenberg's Twelve-Tone Sketches,"

Journal o f the American Musf

cologial S ociety

36 (1 98 3) : 453-80.

29. Arnold Sch oenberg Insti tute, archive no. 27G and 27N.

30. Because the f irs t a nd las t pitches of all four row-forms intersect at the tr i tone

E-Bb, th e rela tions betw een P4 and Plo, 14, and Ilo , are t he same as those be-

tween an y s ingle row form an d th e oth er three.

31. Th e third harm ony , p c set 8-215, spanning measures 2 and 3 is more co m-

plex, bu t non etheless derives from the sa me kind of s truc ture as the f irs t tw o

harmonies . This eight-note set is the complement of a l inear segment of the

row (pc set 4-215) and comprises two tetrachordal harm onies tha t join nonad-

jacent pitches , b ut n ot th e pitches of the paired dy ads discussed above. The

firs t tetrachord (m. 2), pc se t 4-9 (order numbers 1 , 2 , 9 , 12 ) , is equivalent to

the harm ony formed by jo in ing the second and four th dyads . Th e second te t-

rachord (m . 3) , p c set 4-215 (order num bers 3,4 ,1 0,1 1) , is equivalent to a

l inear segment of the basic set (order numb ers 3 ,4 ,5 ,6 ) . T he reasons for in-

cluding as harmon ies o f the basic set the comp lemen ts of i t s l inear segments

should be obvious, bu t fo r a discussion see m y Schoenberg s Twelve-Ton e Har-

mony: The Suite Op.

29

and the Compositional Sketches,

Stud ies in Musi-

cology, ed. George Buelow (A nn A rbor: UMI Research Press, 19 82) , pp. 9-11.

32. The f irs t , marked pc set 4-12, joins the second dyad of P4 with the f irst dyad

of Plo and form s a tetrachordal h arm ony equivalent to the f irs t tetrachord o f

th e basic set. In m . 2 a similar kind of dyad pairing betw een P4 and Plo creates

pc set 4-215, the second principal tetrach ord of the basic set . As Exam ple 1 3

shows, Schoenberg continues to create secondary dimensions by joining to-

gether non-adjacent dyads from single row forms, forming three occurrences

each of the secondary h armon ies 4-1 8 and 4-5.

33 . The repet i tion of pcs 0 , l l in P4 (order numbers 9 , l O) funct ions wi th the f i rs t

tetrachord of I lo to create pc set 6-2 ( the principal hexachord of the basic

set) , a secondary harmon ic dimension serving to l ink together the two row

forms. As in the opening phrase of the Gigue, the first technique again defines

the length of the phrase; the f inal p i tch of the r ight ha nd , pc 0 , creates a 12-

ton e aggregate-one th at contains p c repeti t ions-to span the entire length of



thephrase. Thisaggregate, too, formsasecondaryharmonicdimension, s ince

i t is equivalent to a linearsegmentof th ebasic set-namely thebasicset itself .

Th e fac t tha t Schoenbergwi thholds the f ina lpi tch o i the aggregate unt il the

endof thephrasepersuadesmethat heintendedthest ructure.

34. Tw o addi tiona l fea tures tha t make the second technique more e ffec tive a l so

c la r ify Schoenberg's preference for the f ina l ordering. A mong the five order-

ings , the four th and f i f th have the fewestnumber of invar iantdyads-exac tly

1 1 ( the fus t has 1 2 and the secondand thi rd 1 5) . At the same time, the num-

ber of uniq ue interval classesprojected by the invariantdyad sdecl inesin the

course of t he five orde rings. In t he f ir st t h ree t he i nva ri an t dyads repre sen t

four inte rva l classes , whi le in the four th and f i fth they represent only three .

Byreducingboth the n umb er of invariant dyads and the interval classes they

projec t , Schoenberggre tly reduced the n umb er of unique se t -types ( tha t is,

invar iant harmonies) tha t the invariant segments couldgenera te . Perhaps he

f ea re d t ha t t he m a ny inva ria nt s egm en ts ge ne ra te d by h is bas ic s et w ou l d

create t o o largea vocabulary of invariant harmon ies, a likelyconcernin light

of thesmall numberofsets that canrepresent secondaryharmonies.

35 . At f ir st g lance, i t may seem incons is t en t i n mm. 29-30 t o de li neat e phrase

lengththroughthemotivic f igurein theleft -hand,rather thanthedronein the

r ight -hand. But in fac t , bycompari sonwi th the openingphrase andmm.

20-

23, the te trachords compris ing the two f igures have been switched. In other

words, where Schoenbe rg swit ches t he reg is te rs o f t he two figures be tween

the openingphrase and tha t of mm . 20-23, he now keeps the regis te rsf ixed,

b u t switchesth e tet rachords that m akeupth efigures.Onlyby designat ingpc

s et 5 -1 a s t he ha r m ony t ha t de li ne at es ph ra se l eng th i n m m . 29 -30 , do w e

keep consistent ourcriteria fordeterminingphrasest ructure.

36. Several char ts could i llus tra te ho w th e Tr io's invariant harmonies supp ort i t s

two-part form.Oneappearsbelow, l is t inginvariant harmonies in theorderof

appearance.BecauseintheTrioinvariantharmoniesarenotformedexclusively

from invariant segmentsa nd because they canbederivedinseveralways, this

cha rt r ema ins hypothe ti cal . I o f fe r i t on ly t o showho w one might compare

di f ferent par t i tions a nd thereby describe more prec ise lyho w harmonic s truc-

t u re d iffe rs in v ario us p ar ts o f a f o rm .

Here , fo r i nst ance , each pa r ti ti on

associates two different t r i ton e invariants [(4 10) (0 6)and

( 1 7 )( 2 8 ) ) w i th

dya d s egm en ts t ha t occ ur a t t he beg inning o r e nd o f e ac h o f t he fou r r ow

forms.

P A R T 1 P A R T 2

I

6-5: (9 1 0 ) ( 1 7 ) ( 2 8 )

5-19: (1 7 ) (2 8 ) ( 8 11 )

7-8 : ( 2 8) (4 5 ) ( 3 4 ) (0 6 )

4-6: (4 5 ) (3 4 ) ( 4 10 )

6-241

:

(4 5 )( 3 4 ) (4 1 0 )( 0 6 )

Such a chart a lsohelpsto clari fy theuniquefu nct io nof theinvariantharm ony

hyde schoenberg

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