A Study of Tripartite Forms in the Compositions of Anton Webern (With Original Composition by Author) Concerto for Orchestra

HOFFMANN, James Avery,

1929-A STUDY OF TRIP1929-ARTITE FORMS IN THE COMPOSITIONS OF ANTON WEBERN [with original composition by author] CONCERTO FOR ORCHESTRA.

University of Illinois, D.Mus.A., 1963 Music

University Microfilms, Inc., Ann Arbor, Michigan

A STUDY OF T R I P A R T I T E FORMS IN THE COMPOSITIONS OF ANTON WEBERN

BY

JAMES AVERY HOFFMANN

B . M u s . , New England C o n s e r v a t o r y of Music, 1951 B . M u s . , Yale University, 1952

M. M u s . , Yale University, 1953

THESIS

Submitted in p a r t i a l fulfillment of the r e q u i r e m e n t s for the degree of Doctor of Musical Arts

in the G r a d u a t e College of the University of Illinois , 1963

THE GRADUATE COLLEGE

A u g u s t , 1963

and COMPOSITION

I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY- JAMES AVERY HOFFMANN

ENTITLED. A STUDY OF TRIPARTITE FORMS IN THE COMPOSITIONS

OF ANTON WEBERN CONCERTO FOR ORCHESTRA

BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF MUSICAL ARTS

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PLEASE NOTE: Musical examples tend to "curl" due to glue used for mounted illustrations. Filmed in the best possible way.

I wish to acknowledge my g r a t i t u d e to D r . Hubert K e s s l e r for his encouragement given during the p r e p a r a t i o n of this study. F u r t h e r ' m o r e , I a m indebted to him for his extensive c r i t i c i s m , which was ' indispensible in the focusing and p r e s e n t a t i o n of m y i d e a s .

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T A B L E O F C O N T E N T S

P a g e

I N T R O D U C T I O N I C H A P T E R I - - THE ORGANIZATION O F ROWS - . . . . 5

F i r s t M o v e m e n t , Q u a r t e t for V i o l i n , C l a r i n e t , Tenor,':; * ' S a x o p h o n e , a n d P i a n o , Opus 22 .•'.*'.^{V*.'?* • • • 5 F i r s t M o v e m e n t , P i a n o V a r i a t i o n s , O p u s 27. .;.,:;.. i.;\;iS . . . 11 S e c o n d M o v e m e n t , S t r i n g Q u a r t e t , O p u s 2 8 . . '. . ,V .. V-r\ . . . 18 R e v i e w • 25 C H A P T E R II - - F I R S T M O V E M E N T , Q U A R T E T F O R VIOLIN, C L A R I N E T , T E N O R SAXOPHONE, AND P I A N O , O P U S 2 2 . . . . ,'" 29 • T h e F u n c t i o n of the M a t e r i a l in. M e a s u r e s 1-5 30 T h e C a n o n . . . 31 T h e R i t o r n e l l o , M e a s u r e s 1-5, 35 S e c t i o n A, M e a s u r e s 6-15 , . , , 40 S e c t i o n B , M e a s u r e s 16-27 51 S e c t i o n A", M e a s u r e s $8-37 _ 65 C o d a , M e a s u r e s 37b-41 ] 69 S u m m a r y and C o n c l u s i o n 73 C H A P T E R i n - - F I R S T M O V E M E N T , P I A N O VARIATIONS, O P U S 27 81 V a r i a t i o n s 81 S e c t i o n A, M e a s u r e s 1-18 . -, 84 S e c t i o n B , M e a s u r e s 19-36 93 S e c t i o n A», M e a s u r e s 3 7 - 5 4 ''. 104 C o n c l u s i o n . . . 11Q C H A P T E R IV - - SECOND M O V E M E N T , S T R I N G Q U A R T E T , , , O P U S 28 ^ . . . . 120 R h y t h m i c and Row O r g a n i z a t i o n 120 T h e m a t i c O r g a n i z a t i o n and P i t c h S h a p e . . . ' . 131 V e r t i c a l A g g r e g a t e s in S e c t i o n s A a n d A ' " . . 15,9. R e v i e w and C o n c l u s i o n 163 C H A P T E R V - - REVIEW. 165. F o r m a l D e s i g n 165 Row A p p l i c a t i o n a n d R h y t h m 169 P i t c h Shape ^ E71 B I B L I O G R A P H Y . 173 A P P E N D I X A 175 A P P E N D I X B . . . , . 177 V I T A . 180 I V

The g e n e r a l goal of this paper is to gain an insight into Anton W e b e r n ' s

1 attitude toward form, arid in p a r t i c u l a r to investigate how and to what

extent his compositions r e m a i n within t r a d i t i o n a l f o r m a l c a t e g o r i e s while employing n o n - t r a d i t i o n a l , 1. e . , twelve-tone, t e c h n i q u e s .

One a n a l y t i c a l approach to twelve-tone m u s i c e m p h a s i z e s the r o l e of design in a r t i c u l a t i n g f o r m a l s t r u c t u r e s . Motives a r e delineated and t h e i r r e a p p e a r a n c e s , s t r i c t or v a r i e d , a r e t r a c e d throughout a m o v e m e n t . The r e s u l t s indicate that p i e c e s w r i t t e n m row-technique m a y often be d e s c r i b e d by t r a d i t i o n a l f o r m a l t e r m s .

In tonal m u s i c , however, f o r m a l p a t t e r n s a r e a l s o a r t i c u l a t e d by harmony through such a device a s c a d e n c e s indicating the e s t a b l i s h m e n t s of, d e p a r t u r e s from, and r e t u r n s to tonal c e n t e r s . This device p e r f o r m s

both a s e p a r a t i v e and, through the action of a c e n t r a l k e y c e n t e r , an i n t e g r a t i v e function. In view of this, the p r o b l e m a r i s e s a s to what c o u n t e r p a r t s to

traditional h a r m o n i c techniques would appear in twelve-tone m u s i c . An i m p o r t a n t a s p e c t of twelve-tone p r o c e d u r e s c o n c e r n s the a c t u a l compositional layout of row f o r m s , 1. e . , both the choices made from the forty-eight possible forms of a row (the set, its r e t r o g r a d e , i n v e r s i o n , and r e t r o g r a d e i n v e r s i o n , each at twelve t r a n s p o s i t i o n a l l e v e l s ) and the a r r a n g e -ment of t h e s e selected * r m s . It would s e e m that if a twelve-tone composition w e r e to be b a s e d on a t r a d i t i o n a l f o r m a l p a t t e r n such a p a t t e r n might find a p a r a l l e l in the layout of row f o r m s . Thus, for e x a m p l e , a group of row

f o r m s might r e c u r coincident with a r e c a p i t u l a t i o n . Since a r e c a p i t u l a t i o n m a y be l i t e r a l , a s the recapitulation of a minuet in a minuet and t r i o , or c o m

-p l e m e n t a r y , as in the sonata form, so likewise might a r e c a -p i t u l a t i o n of a 1

2

group of row forms be l i t e r a l or in some s e n s e c o m p l e m e n t a r y .

In tonal m u s i c the question of whether a l i t e r a l or c o m p l e m e n t a r y r e c a p i t u l a t i o n is called for is usually d e t e r m i n e d b y the s t r u c t u r e of the exposition which, in the f i r s t c a s e , i s r e l a t i v e l y complete in itself, i . e . , it does not modulate, and, in the second c a s e , i s incomplete in itself, i . e . , it does m o d u l a t e . Although it is not obvious on what b a s i s an initial grouping of row f o r m s could be t e r m e d complete or incomplete, n e v e r t h e l e s s if _ the row a r r a n g e m e n t i s to function in a m a n n e r analogous to that s e r v e d by t o n a l functions, this i s an i m p o r t a n t c o n s i d e r a t i o n .

An answer to this question m a y lie m the possibility of placing a c e r t a i n row form s t r a t e g i c a l l y . A specific f o r m , for example, appearing both at the beginning and end of a group of row f o r m s might be considered a " t o n i c " r o w , thus suggesting that the group itself is r e l a t i v e l y c o m p l e t e .

Yet the fact is that a p a t t e r n shaped by a p a r t i c u l a r a r r a n g e m e n t of row f o r m s m u s t be c o n s i d e r e d a b s t r a c t in the sense that by itself it does not r e v e a l what concrete pitch r e l a t i o n s h i p s e m e r g e from a molding of the row m a t e r i a l into thematic m o t i v e s , a c c o m p a n i m e n t s , p h r a s e s , e t c . Thus the u s e of rhythm and the choice of octave r e g i s t e r for pitches would seem to be i m p o r t a n t e l e m e n t s m the shaping of these c o n c r e t e r e l a t i o n s h i p s . It is

p o s s i b l e , for e x a m p l e , to emphasize v e r t i c a l s o n o r i t i e s that m a y suggest pitch groupings of formal significance, to r e p e a t the same pitch o c c u r r i n g in different row f o r m s in the s a m e octave to make p h r a s e a r t i c u l a t i o n s or to bind sections t o g e t h e r , and to a r r a n g e the highest and lowest n o t e s of successive p h r a s e s to

This idea, a s well a s o t h e r s in the above p a r a g r a p h s , h a s been v a r i o u s l y s u g g e s t e d . Cf., for i n s t a n c e , George P e r l e , S e r i a l Composition and Atonality; An Introduction to the Music of Schoenberg, B e r g , and W e b e r n . (Berkeley: University of California P r e s s , 1962), Chap, VI, and especially pp. 126 and 129.

form s m a l l - and l a r g e - s c a l e stepwise p r o g r e s s i o n s or neighbor tone motions implying "tonal" m o v e m e n t or l a c k thereof. Such concrete r e l a t i o n s h i p s a s these m a y or m a y not r e i n f o r c e the formal implications of an a b s t r a c t row a r r a n g e m e n t . T h e r e f o r e , in determining how far twelve-tone p r o c e d u r e s

a r e able to provide a p a r a l l e l to traditional h a r m o n i c techniques it i s n e c e s s a r y to examine both a b s t r a c t row a r r a n g e m e n t s and the o v e r - a l l shape of concrete pitch r e l a t i o n s h i p s .

In o r d e r to d i s c u s s in some detail within the scope of this paper W e b e r n ' s use of twelve-tone p r o c e d u r e s in constructing movements a p p a r e n t l y organized along t r a d i t i o n a l f o r m a l lines it seemed a p p r o p r i a t e to limit the a n a l y s i s

to c e r t a i n m o v e m e n t s of Webern*s compositions in which the formal design i s obvious, clear cut, and r e l a t i v e l y s i m p l e . Guided by this c o n s i d e r a t i o n , the ABA form wherein after a contrasting middle section a r e t u r n to the beginning a p p r o x i m a t e s a lateral recapitulation did seem to offer a valid b a s i s for the selection of m o v e m e n t s .

T h e r e f o r e , the m o v e m e n t s that have been selected for a n a l y s i s a r e the f i r s t m o v e m e n t s of the Quartet, Opus 22 (1930), and the Piano V a r i a t i o n s , Opus 27 (1936), and the second movement of the String Quartet, Opus 28

(1938), these being the outstanding r e p r e s e n t a t i v e s of the chosen form amongst the tone row compositions of W e b e r n .

In the f i r s t c h a p t e r of this paper the a b s t r a c t row o r g a n i z a t i o n s of the — chosen movements will be examined and compared, and it will be shown how

row presentations in t h e m s e l v e s a r e organized with specific r e g a r d to the forms of the m o v e m e n t s . Succeeding chapters will be devoted to the individual

m o v e m e n t s , analyzing both t h e i r individual designs (the r h y t h m i c , m e t r i c ,

4

These c h a p t e r s will show that both the formal implications of the row a r r a n g e -m e n t s and the designs of the -move-ments a r e supported by c o n c r e t e pitch r e l a t i o n s h i p s . In addition, attention will be given to p r o b l e m s concerning the a c t u a l f o r m s of the m o v e m e n t s , such a s , for example, to the problem of combining a t r i p a r t i t e form with v a r i a t i o n p r o c e d u r e in the f i r s t movement of Opus 27. Analogues from traditional l i t e r a t u r e will be cited. A conclusion will c o n s i d e r the r e l a t i o n of row manipulation, t h e m a t i c design, and pitch

shape to the t r i p a r t i t e f o r m s of these examples and to what extent traditional f o r m a l analogues a r e r e l e v a n t to the twelve-tone p r o c e d u r e s exhibited in these c o m p o s i t i o n s .

This chapter will be devoted to the row o r g a n i z a t i o n s of the t h r e e m o v e m e n t s under c o n s i d e r a t i o n . Both the i n t e r n a l s t r u c t u r e s of the r o w s t h e m -s e l v e -s and the a r r a n g e m e n t -s of t h e i r f o r m -s w i l l be examined. Manife-st f o r m a l i m p l i c a t i o n s will be d e t e r m i n e d . T h e s e i m p l i c a t i o n s will then be c o m p a r e d b r i e f l y to the g e n e r a l designs of the m o v e m e n t s in o r d e r to judge whether these implications b r o a d l y p a r a l l e l the a c t u a l f o r m s of the m u s i c . A c o m p a r a t i v e s u m m a r y will conclude t h i s c h a p t e r .

F i r s t Movement, Quartet for Violin, C l a r i n e t , Tenor Saxophone, and Piano, Opus 22

The row

The row (SQ) of Opus 22 is i l l u s t r a t e d in E x a m p l e 1. It m a y be divided

ExMtfU I .

into two h e x a c h o r d s , of which the f i r s t m a y be subdivided two w a y s , i. e . , into t h r e e s e g m e n t s , each containing a t h i r d , or into two s e g m e n t s , each

spanning a m a j o r third and containing a m i n o r s e c o n d . The second h e x a c h o r d m a y b e subdivided into two s e g m e n t s , the f i r s t containing m i n o r seconds and the second containing a tritone and a perfect f o u r t h . The two hexachords a r e c o n t r a s t e d in i n t e r vallic content and together contain a l l i n t e r v a l s . The f i r s t and l a s t notes of the row, its f r a m e , form a t r i t o n e .

\

Layout of row f o r m s

The a r r a n g e m e n t of the forms of this row used in the f i r s t movement is laid out in Example 2 in the following m a n n e r : (1) each form is place'd within a single " b a r " or box; (2) a whole-note indicates the f i r s t note of the row (or the last note of a r e t r o g r a d e ) and a filled-in note-head indicates the l a s t note of the row (or the first note of a r e t r o g r a d e ) ; (3) the original form of the row (S) is shown by an ascending i n t e r v a l , the r e t r o g r a d e (R) by a descending i n t e r v a l , and, correspondingly, the inversion (I) is shown by a descending interval, the r e t r o g r a d e of the inversion (RI) by an ascending i n t e r v a l . (Cf. boxes 6 and 7. ) Since r e g u l a r l y m o r e forms than one a p p e a r together, the row form that a p p e a r s f i r s t in the m u s i c is placed on the top staff, the row form that a p p e a r s next i s placed below on the second staff, e t c .

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The o v e r - a l l f o r m a l schema of the m o v e m e n t

In Example 2 it can be seen that in two p l a c e s , n a m e l y , n e a r the beginning (boxes 2-3) and n e a r the end of the m o v e m e n t (boxes 9-10), three row f o r m s . a r e p r e s e n t e d together, and that in both c a s e s the s a m e row f o r m s a r e -used. The f i r s t instance is p r e c e d e d by a box containing two row f o r m s (box 1),

and the second instance is followed by a box containing the r e t r o g r a d e of these s a m e two row forms (box 11). The middle s e r i e s of boxes p r e s e n t two row f o r m s together (boxes 4 - 8 ) . These p r e l i m i n a r y o b s e r v a t i o n s suggest an ABA form with an introduction and c o d a .

The "canon" and theme

In those boxes where only two forms a r e p r e s e n t e d together the original form (or r e t r o g r a d e ) i s always accompanied by an inversion (or r e t r o g r a d e -i n v e r s -i o n ) . The t r a n s p o s -i t -i o n a l l e v e l of the -invers-ion -is chosen so that when the number r e p r e s e n t i n g its f i r s t pitch i s added to the number r e p r e s e n t i n g the first pitch of the accompanying row the sum will always be ten (or ten plus twelve). In those boxes where t h r e e rows a r e p r e s e n t e d together this coupling applies to the p a i r s on the second and third s t a v e s . These facts clarify how Webern was able to w r i t e a continuous quasi-canon by inversion about a pitch a x i s , in this c a s e F , a s will be explained in Chapter two. The third row f o r m s superimposed on this "cannn" in boxes 2 and 3 and again in boxes 9 and 10 turn out to be the pitches of an independent theme that does not d i r e c t l y r e l a t e to the "canon. "

The n a t u r e of this introduction will be examined in Chapter, two wher.ei. it will be found that the m a t e r i a l involved h e r e s e r v e s for m o r e than a m e r e i n t r o d u c t i o n , that it functions a l l through the movement as a r i t o r n e l l o .

8

Introduction and section A, b o x e s 1-3

The a r r a n g e m e n t of t r a n s p o s i t i o n s can be seen by examining the

o r i g i n a l form of the row a s it a p p e a r s throughout the " c a n o n . " SQ I S followed by S^ (box 2, second staff). This is a t r i t o n e t r a n s p o s i t i o n ; and, b e c a u s e the t r i t o n e f r a m e s the r o w , t h e r e i s a common tone, G, between SQ and S&. This t r a n s p o s i t i o n a l s c h e m e c o n t i n u e s . The next row of the s a m e form i s another t r i t o n e t r a n s p o s i t i o n with a common tone connection, i . e . , SQ again (box 3, second staff). This i s a closed o r c i r c u l a r s c h e m e of t r a n s p o s i t i o n s c h a r a c t e r i z e d b y (1) a specific p a t t e r n of selection of row f o r m s or an o r d e r of f o r m s (in this c a s e a l l S f o r m s ) , (2) a r e p e a t e d t r a n s p o s i t i o n a l i n t e r v a l (in this c a s e the t r i t o n e ) , (3) common tone connections, and (4) a r e t u r n to the initial r o w . This c i r c u l a r s c h e m e i s m i r r o r e d by the i n v e r t e d f o r m s in box 1, second staff, and boxes 2 and 3, t h i r d staff.

The third row p a i r which i s s u p e r i m p o s e d on the " c a n o n " in boxes 2

and 3 , top staff, i s a t r a n s p o s i t i o n of the i n v e r t e d p a i r in boxes 2 and 3, bottom staff, and u s e s the s t a r t i n g pitches of the pair of S f o r m s in boxes 2 and 3 ,

middle staff, in the r e v e r s e o r d e r . In t h i s m a n n e r it is i n d i r e c t l y r e l a t e d to the "canon" and l i k e w i s e a d h e r e s to a closed t r a n s p o s i t i o n a l s c h e m e (including the r e p e a t ) .

If one of the two r o w s of the " c a n o n , " S~ or S , , w e r e to be d e s i g n a t e d a s the " t o n i c " r o w , the choice would fall upon SQ (combined with IJQ) not only b e c a u s e it begins the m o v e m e n t but b e c a u s e it i s the second of the succeeding p a i r s (including the r e p e a t ) . T h u s , the a r r a n g e m e n t of row f o r m s of the " c a n o n "

s u g g e s t s that the f i r s t A section (boxes 2 and 3) can be expected to be a r e l a t i v e l y complete or closed section, that it does not " m o d u l a t e . "

Section B, boxes 4 - 8 __

"In o r d e r to introduce new t r a n s p o s i t i o n s of the row, the c i r c u l a r scheme of the t r a n s p o s i t i o n s in Section A m u s t be b r o k e n . This is achieved by beginning the middle section with the row midway between S^ and SQ

(SQ =Si2)i i . e*, with Sg. S f o r m s continue with s y s t e m a t i c t r a n s p o s i t i o n s up m i n o r seconds through SJQ and SJJ to the o r i g i n a l form, SQ, The inverted f o r m s begin with Ij, midway between ^ a n d I JQ (boxes 2 and 3 , bottom staff), and continue with t r a n s p o s i t i o n s down m i n o r seconds through IQ and IJJ to IJQ. In box 5 the transposition of the i n v e r s i o n t u r n s out to be IQ, the same row used by the independent theme in the top staff of box 2. This row l e a d s the "canon" a t this point. In boxes 6 and 7 the b a s i c row f o r m s used (SJJ and IJJ) end with the axis.tone of the canon, F * . These r o w s a r e presented f i r s t in r e t r o g r a d e and then in their original o r d e r . The "tonic" rows in box 8 a r e presented in r e t r o g r a d e .

The s c h e m e of the row a r r a n g e m e n t in the middle section m a y be t e r m e d " s e q u e n t i a l , " It is c h a r a c t e r i z e d by (1) beginning with fresh t r a n s -positions r e l a t e d to the previous rows (in t h i s case s t a r t i n g pitches located midway between the s t a r t i n g pitches of the e a r l i e r t r a n s p o s i t i o n a l levels), (2) the consistent adherence to an o r d e r of forms (in this c a s e all S forms m i r r o r e d by I f o r m s ) , and (3) a r e p e a t e d t r a n s p o s i t i o n a l i n t e r v a l (in this c a s e m i n o r seconds upward m i r r o r e d by m i n o r seconds downward) without "common tone connections (except those n a t u r a l l y o c c u r r i n g between R I J J and SJJ and R j j a n d IJJ in boxes 6 and 7) which (4) if continued would have included a l l

-As u s e d h e r e the t e r m " s e q u e n c e " i s meant to imply the movement from one point to another through s y s t e m a t i c a l l y t r a n s p o s e d repetitions with typically t h r e e p r e s e n t a t i o n s of the m a t e r i a l .

10

possible t r a n s p o s i t i o n s before r e t u r n i n g to its s t a r t i n g point but instead (5) a r r i v e s in t h r e e steps at the " t o n i c " r o w s .

The row a r r a n g e m e n t of the middle section i s different from and in c o n t r a s t to that of the beginning section but has the appearance of leading back to a r e c a p i t u l a t i o n . It a l s o e m p h a s i z e s a c h a r a c t e r i s t i c of the "canon, " i t s c o n s t r u c t i o n about an a x i s , when the axis tone takes a prominent position in two f o r m s of the r o w .

Section A"" and coda, boxes 9-11

Because the " t o n i c " rows ending the middle section a r e in r e t r o g r a d e (box 8), the only common tone connection between the middle section and the r e c a p i t u l a t i o n is the C* of Ro and Io. The recapitulation in boxes 9 and 10

has l i t e r a l l y the s a m e row content a s boxes 2 and 3. In the final box the " t o n i c " r o w s appear in r e t r o g r a d e . Were the rows in box 11 a tritone t r a n s p o s i t i o n of those in the lower two staves of box 10, continuing the c i r c u l a r s c h e m e , the m o v e m e n t would end with SQ and I4, That it does not, but r a t h e r r e p e a t s in

r e t r o g r a d e the SQ and I JQ forms of the lower two s t a v e s of box 10, e m p h a s i z e s the " t o n i c " implications of these r o w s and c h a r a c t e r i z e s this ending a s a coda balancing the introduction.

The row a r r a n g e m e n t in relation to the m u s i c

It can be seen in the s c o r e that the formal implications of the row

organization a r e r e a l i z e d in the m u s i c . M e a s u r e s 1-5, c o r r e s p n n d i n g to box 1, a p p e a r to be a canonic introduction. In m e a s u r e s 6-15, c o r r e s p o n d i n g to b o x e s 2 - 3 , a canon continues in a l l i n s t r u m e n t s but the tenor saxophone which

c a r r i e s IQand 1^ a s a s e p a r a t e melodic l i n e . M e a s u r e s 29-37, c o r r e s p o n d i n g to boxes 9-10, constitute a recapitulation with the tenor saxophone melody now divided among all i n s t r u m e n t s but the piano which c a r r i e s the canon a l o n e .

M e a s u r e s 37b-41, corresponding to box 11, constitute a coda. To what extent sections A and A1 can actually be c o n s i d e r e d complete and " n o n - m o d u l a t o r y "

in t h e m s e l v e s will be d i s c u s s e d in Chapter two.

The middle or B section begins in m e a s u r e 16 and continues the canon. M e a s u r e s 19-21 (corresponding to box 5) a r e a t r a n s p o s e d (sequential) free r e p e a t of m e a s u r e s 16-18 ( c o r r e s p o n d i n g to box 4), In m e a s u r e s 24-27 the m a t e r i a l of the introduction a p p e a r s in r e t r o g r a d e coincident with the r e t u r n of the " t o n i c " r o w s (box 8). How this section actually leads into the r e

-capitulation will be d i s c u s s e d in Chapter two.

F i r s t Movement, Piano V a r i a t i o n s , Opus 27 The row

The row (SQ) of Opus 27 i s p r e s e n t e d in Example 3 . The first hexachord

m a y b e subdivided m two w a y s , l. e . , into t h r e e segments, each containing a second, or into two s e g m e n t s , each containing and spanned by a t h i r d , A tritone links the f i r s t hexachord to the second hexachord which m a y b e sub-divided into two s e g m e n t s , the first containing minor seconds, the second

spanned by a perfect fourth. The row contains all intervals but the perfect fourth which o c c u r s between the f i r s t and l a s t notes as the r o w ' s f r a m e . The two hexachords a r e c o n t r a s t e d in i n t e r v a l l i c content.

Layout of row forms

12

E x a m p l e 4 . They a r e d i a g r a m m e d in a m a n n e r s i m i l a r to that of Example 2. A dotted line indicates the p r e s e n c e of a common tone between s u c c e s s i v e f o r m s of the r o w . In the m u s i c one pitch t a k e s the place of two tones; in Example 4 the s u p p r e s s e d note i s in p a r e n t h e s i s . P l a c e s w h e r e two notes a r e s u p p r e s s e d will be specially d i s c u s s e d in the text.

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It can be seen that each S and I form is always accompanied by i t s r e s p e c t i v e r e t r o g r a d e . The f i r s t row (SQ) I S followed by i t s i n v e r s i o n t r a n s -posed up a m a j o r second (1£, box 2). A common tone connection e x i s t s between SQ and R j2. The pair S Q - ^ is i m m e d i a t e l y r e p e a t e d (box 3 , bottom staff, and

box 4 , top staff). The t r a n s p o s i t i o n a l i n t e r v a l between the second row of the f i r s t pair (I2) and the f i r s t row of the second pair (SQ) is a major second

down-w a r d and a common tone connection a p p e a r s betdown-ween I2a n d R Q . BOX 5

i n t r o d u c e s both a new t r a n s p o s i t i o n a l i n t e r v a l (a perfect fourth from I2 to

Iy) and a new row form (Iy), facts which suggest that the f i r s t four boxes in t h e m s e l v e s do constitute a s e c t i o n .

The row a r r a n g e m e n t of the first four boxes fulfills t h r e e of the four c h a r a c t e r i s t i c s of a c i r c u l a r scheme as listed above: (1) a consistent

a d h e r e n c e to an o r d e r of f o r m s (in this c a s e an alternation of S and I f o r m s ) , (2) common tone connections (in this c a s e between an S form and the following RI form and between an I form and the following R form), and (3) a r e t u r n to the initial row, A problem a r i s e s , however, with r e s p e c t to the choice of

t r a n s p o s i t i o n a l i n t e r v a l s which between boxes 1 and 2 is up_ a major second while between boxes 2 and 3 i s down a major second. The question is whether this change of direction is to be r e g a r d e d as part of a s c h e m e of alternation p a r a l l e l to the a l t e r n a t i o n of S and I f o r m s and thus confirming an implied c i r c u l a r s c h e m e , or whether this change of direction is m e r e l y the r e s u l t of

the repetition of the r o w s of boxes 1 and 2 in boxes 3 and 4 and thus suggesting another type of s c h e m e . Although the l a t t e r a l t e r n a t i v e i s the m o s t simple and obvious i n t e r p r e t a t i o n , final determination of the question m u s t await examination of the r e c a p i t u l a t i o n .

Another problem allied to the question of the type of row s c h e m e h e r e employed c o n c e r n s the r e l a t i o n between the two row forms used. Is the second of each pair a " t o n i c " row, so that S Q - L = V — ^ 1 , or a "dominant" row, so that a s S Q - I2 = ( I ) — ^ V? F u r t h e r discussion of this problem m u s t

also be d e f e r r e d .

Section B, boxes 5-10

14

form initiating each p a i r , and introduces a fresh t r a n s p o s e d f o r m , Iy

(box 5). A common tone between I2 and I7 joins the f i r s t two s e c t i o n s . It is

the same note that s e r v e d a s the common tone within the first section, but this time it o c c u r s between two forward forms of the row r a t h e r than between a forward and a r e t r o g r a d e (and vice v e r s a ) f o r m . I7 is followed by S^, a transposition down a minor second. The l a s t two notes of RI7 a r e the same a s the f i r s t two n o t e s of S^. Thus the common tone bond within p a i r s i s m a i n -tained. In the m u s i c the notes a r e not r e p e a t e d ; one group is s u p p r e s s e d . The next pair, IQ and S,, (boxes 7 and 8), r e p r e s e n t s a t r a n s p o s i t i o n up a perfect fourth of the preceding p a i r , and no common tone or tones exist between the end of the f i r s t and the beginning of the second p a i r . The third p a i r of this section, I5 and S4 (boxes 9-and 10) r e s u l t s from a t r a n s p o s i t i o n up

a perfect fourth of the second p a i r . The next box (11) does not contain an I f o r m of the row, a s would be expected, but r a t h e r r e p e a t s the S form of box 10

(S4, lower staff). This change from the preceding t r a n s p o s i t i o n a l scheme and alternation of I and S forms of the row suggests that box 11 r e p r e s e n t s the beginning of a third section.

The scheme of t r a n s p o s i t i o n s in the second section (boxes 5-10) shows the same general c h a r a c t e r i s t i c s a s those found in section B of the first m o v e -ment of Opus 22 which w e r e designated a s "sequential": (1) a beginning with f r e s h transpositions related to what h a s preceded by a common tone connection,

(2) the consistent adherence to an o r d e r of forms (in this c a s e an a l t e r n a t i o n of I and S forms accompanied by t h e i r r e s p e c t i v e r e t r o g r a d e s ) , and (3) a r e p e a t e d transpositional i n t e r v a l (in this case perfect fourths upward without a common tone connection (between p a i r s ) which (4) if continued would have included all possible t r a n s p o s i t i o n s before r e t u r n i n g to its s t a r t i n g point but

instead (5) a r r i v e s in t h r e e s t e p s at the row that will be r e p e a t e d to initiate the next section.

Section A*, boxes 11-14

The third section begins with S4 (box 11) and is followed by an i n v e r s i o n t r a n s p o s e d up a major second. This is the same scheme used in boxes 1 and 2 but at a different t r a n s p o s i t i o n a l level. The next pair follows the same

s c h e m e at yet another t r a n s p o s i t i o n a l level and in r e v e r s e ; 1. e. , the inverted form ( I J J ) , beginning on E , this time p r e c e d e s the original form (So) which

s t a r t s on D . These final two p a i r s a r e joined by a common tone between the I f o r m s (boxes 12 and 13) a s was the middle section joined to the first s e c t i o n (boxes 4 and 5). The r e v e r s a l of S and I f o r m s in the l a s t pair a l s o suggests the influence of the middle section. M o r e o v e r , j u s t a s the f i r s t section is connected to the second section by a transposition of the I form up a perfect fourth (boxes 4 and 5), so the two p a i r s of the l a s t section a r e connected by t r a n s p o s i n g 1^ (box 12) up a perfect fourth to IJJ (box 13).

C o m p a r i s o n of sections A and A*

The first and l a s t s e c t i o n s a r e r e l a t e d as A to A* by two r e s e m b l a n c e s : (1) both sections contain two p a i r s ; (2) the g e n e r a l organization within each of the four p a i r s i s the s a m e , i. e . , each pair c o n s i s t s of an S and I form of the row with the t r a n s p o s i t i o n a l level of the I form a major second above that of the S f o r m .

The connection between, the two p a i r s of each section is not the s a m e and there is no recapitulation of a t r a n s p o s i t i o n a l level of any f o r m . T h e s e

differences between the two sections suggest that the f i r s t section is in itself incomplete, that i t s row scheme is open, r e q u i r i n g later completion. The row that begins the l a s t section, S4 (box 11), is a transposition upward by a

16

major second of the row that ended the first section, L> (box 4 ) . This is a continuation of the o r i g i n a l t r a i n s positional s c h e m e which i s then completed in the final pair of the l a s t section by the changes h e r e i n noted.

S u m m a r y of the row a r r a n g e m e n t

Example 5 s u m m a r i z e s the row s c h e m e of this m o v e m e n t .

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The second row of the r e p e a t e d pair in section A begins with a pitch a major second above the beginning note of the f i r s t r o w . This scheme i s interrupted by the perfect fourth transposition into section B which follows with a

" s e q u e n t i a l " s c h e m e ending on S , , The rows within each pair in section B a r e r e l a t e d by a m i n o r second between t h e i r starting p i t c h e s , S4 is repeated at the outset of section A ' . Its beginning pitch is a major second above the beginning pitch of the row that ended section A, and it is followed by a row

whose beginning pitch is up yet another m a j o r second. This r o w scheme is again i n t e r r u p t e d by a perfect fourth t r a n s p o s i t i o n . The movement is

completed in the final row whose f i r s t note i s a step below the first note of the preceding r o w . The significance of this l a s t pair is suggested by the r e v e r s a l of S and I f o r m s and the new t r a n s p o s i t i o n a l level of the p a i r . If any row of the m o v e m e n t w e r e to be designated a s a " t o n i c " r o w , it would probably be the l a s t row of this p a i r . In any c a s e , it is c l e a r that section A" completes a t r a n s p o s i t i o n a l s c h e m e initiated in section A .

The i m p o r t a n c e of the i n t e r v a l of a perfect fourth

The importance of the perfect fourth, the interval that f r a m e s the row, in the t r a n s p o s i t i o n a l scheme of the whole movement has a l r e a d y been

suggested. It r e l a t e d the ending and beginning rows of the row p a i r s between s e c t i o n s A and B (Example 4, boxes 4 and 5) and within section A1 (boxes

12 and 13), and the beginning r o w s of the row p a i r s within section B (boxes 5, 7, and 9). In addition, it m a y be found relating rows within the p a i r s that appear in sections A and A ' , F o r example, the l a s t note of SQ (box 1) is a perfect fourth from the first note of I2 (box 2). Indeed, the only places

where the perfect fourth does not r e l a t e rows in some manner a r e within the p a i r s of section B which a r e connected by two common tones.

The row a r r a n g e m e n t in relation to the m u s i c

The m u s i c itself m i r r o r s the ABA formal implications of the row a r r a n g e m e n t . M e a s u r e s 1-18 c o m p r i s e section A, m e a s u r e s 19-36, section B, and m e a s u r e s 37-54 c o m p r i s e a recapitulation of section A, v a r i e d but not in p h r a s e s t r u c t u r e or b a s i c m o t i v e s .

In section A m e a s u r e s 1115 a r e a variation of m e a s u r e s 17 ( c o r r e s p o n d -ing to boxes 3 and 1 of Example 4), and m e a s u r e s 16-18 a r e a repetition of

m e a s u r e s 8-10 (corresponding to boxes 4 and 2). A s i m i l a r p h r a s e organization holds with r e s p e c t to section A" in spite of the changes brought about a s a

r e s u l t of row c h a n g e s . How t h e s e changes affect the tonal movement will b e d i s c u s s e d in Chapter t h r e e .

In keeping with the row manipulation section B, corresponding to boxes 5-10, contains a model and its twofold q u a s i - s e q u e n t i a l elaboration ( m e a s u r e s 19-26, 26-32, and 32-36).

4'

18

Second Movement, String Quartet, Opus 28 The row

The row (SQ) of Opus 2 83 (Example 6) c o n s i s t s of a s e r i e s of m i n o r

example it i —

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seconds s e p a r a t e d by either minor or major t h i r d s . The row m a y be divided into two hexachords or three t e t r a c h o r d s . The hexachords a r e r e l a t e d . The second hexachord of the row is the r e t r o g r a d e - i n v e r s i o n of the f i r s t hexachord t r a n s p o s e d down a minor t h i r d . A r e s u l t of this relationship is that the r e t r o -grade of the row (RQ) is identical with a transposed inversion (In). In this analysis the " I " l e t t e r i n g will be u s e d . The t e t r a c h o r d s of the row a r e also related to each other. The second t e t r a c h o r d i s either a t r a n s p o s e d r e t r o

-grade or t r a n s p o s e d inversion of the first; and the t h i r d , a simple t r a n s p o s i t i o n of the first. A r e s u l t of the relationship between the first and l a s t t e t r a c h o r d s i s that the l a s t t e t r a c h o r d of any S or I form is the s a m e as the f i r s t t e t r a c h o r d

of the S or I form found r e s p e c t i v e l y a major third below or a major third above the given f o r m . The row itself is thus r e s t r i c t e d in r e l a t i o n s h i p s with no c o n t r a s t between its s e g m e n t s . The framing i n t e r v a l of the row is a m i n o r t h i r d .

3

The basic form of the row (SQ) is taken from the beginning of the first m o v e m e n t ( m e a s u r e s 1-6), The t r a n s p o s i t i o n a l n u m b e r s of the f o r m s appearing in the second movement a r e calculated from the beginning pitch of the b a s i c f o r m .

4

To a s c e r t a i n the identical R form in the case of other t r a n s p o s i t i o n s of the I form t h r e e is added to the transposition number of the I f o r m .

Layout of row forms

The row a r r a n g e m e n t of the second m o v e m e n t of Opus'28 i s laid out in Example 7 according to\the pattern of Example 2.

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The o v e r - a l l formal schema of the movement

As a p r e l i m i n a r y r e m a r k it may be observed that the simultaneous presentation of row f o r m s o c c u r r i n g in a one to one c o r r e s p o n d e n c e in boxes 1-3 and again in boxes 6-8, and in a two to one c o r r e s p o n d e n c e in the middle boxes (4 and 5) suggest a t h r e e - p a r t design.

Section A, boxes 1-3

On the top staff of Example 7 L is followed by I .f f and I ,0, by I_. A

continuation of this scheme of transposing I f o r m s upward by major thirds r e t u r n s to I ^ a n d the whole group is r e p e a t e d . The l a s t t e t r a c h o r d of any one

20

row is the same as the first tetrachord of the following row. In the music consistently one group of pitches represents both identical tetrachords, as shown by the following diagram:

a x b^ y c z (a) : | \

I6 ^0 h

Thus, for example, tetrachord "a" begins the group of rows, begins the repetition, and ends the group. This is a circular row arrangement ful-filling all four characteristics noted with regard to the row arrangement of section A of Opus 22: (1) a consistent adherence to an order of forms (in this case all I forms), (2) a repeated transpositional interval (in this case a major third upward), (3) common tone connections (in this case common

tetra-chords), and (4) a return to the initial row. Further, tetrachord'(a'J maybe tentatively regarded as a "tonic" tetrachord since it both begins and ends the group, thus emphasizing the relative completeness of this arrangement.

Another consequence of this row arrangement is that the middle tetra-chords of the rows ("x", "y", and "z" in the above diagram) turn out to be retrograde forms of the outside tetrachords:

a c ,b. a c b a.

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The same row arrangement appears on the second staff, boxes 1-3, but the transpositional scheme descends with S forms of the row. Further,

since the I forms of the row on the top staff and the S forms of the row on the second staff are related by the fact that 1^ =Rg, I j o ^ Ri» a n d *2 = R5 » t h e

The r o w s on the third staff, boxes 1-3, a r e a t r a n s p o s i t i o n a major third higher of those on the second staff. Since the m a j o r third i s also the t r a n s -p o s i t i o n a l i n t e r v a l used on the second staff, the s a m e rows r e a -p -p e a r on the third staff in the same o r d e r but with different beginning and ending r o w s :

Second staff: S9 S5 Sj

Third staff: Sj Sg S5.

In a s i m i l a r m a n n e r a r e the r o w s on the fourth staff, boxes 1-3, the same a s those on the f i r s t staff:

F i r s t staff: 1^, IJQ I2

F o u r t h staff: IJQ I2 1^.

The f i r s t three boxes of Example 7 r e p r e s e n t a completely closed section. Since the first t e t r a c h o r d s appearing on each staff a r e identical with the final t e t r a c h o r d s of their r e s p e c t i v e s t a v e s , their v e r t i c a l combination m a y b e c o n s i d e r e d to function like a " t o n i c " e m p h a s i z i n g the closed row s c h e m e .

In s u m m a r y , the row a r r a n g e m e n t of the first section suggests the image of two concentric c i r c u l a r bands of identical t e t r a c h o r d a l content, one moving clockwise (S forms) and the other moving counterclockwise (I f o r m s ) at the s a m e r a t e of speed. In the chosen polyphonic presentation of the two b a n d s , two p a r t s partake of the clockwise band, but enter and exit at different

points while t r a v e r s i n g the band twice, and the remaining two p a r t s p a r t a k e of the counterclockwise band in a s i m i l a r fashion but in the opposite d i r e c t i o n . A d i a g r a m of this image follows this page.

Section A", boxes 6-8

The l a s t three boxes of E x a m p l e 7 a r e the same a s the first t h r e e boxes except that the polyphonic s t r a n d s enter in a different t e m p o r a l o r d e r . The

rows of the top staff, boxes 1-3, now a p p e a r on the third staff, boxes 6-8, the rows of the second staff, b o x e s 1-3, now appear on the fourth staff, boxes

6-8, e t c . Thus, the o r d e r of e n t r i e s is r o t a t e d :

4

A A» O r d e r of E n t r y : 1 2 3 4 1 2 3 4 Row G r o u p s : A B C D C D A B

Boxes 6-8 r e p r e s e n t a r a t h e r curious kind of recapitulation that in one s e n s e is l i t e r a l , and in another, v a r i e d . One effect of this change is that the two " b a n d s , " f i r s t viewed from " a b o v e , " a r e now viewed from " b e l o w . " In other words, the first section is turned "upside down, " a s will be evident in

t

the discussion of the chords of t h i s movement in Chapter four. In s u m m a r y , Sections A* and A contain s i m i l a r c i r c u l a r row a r r a n g e m e n t s . The same t e t r a c h o r d a l groupings occur at the beginning and end of both s e c t i o n s .

Section B, boxes 4-5

In the m i d d l e section of Example 7 (boxes 4-5) the six rows appearing in the f i r s t and l a s t sections a r e r e a r r a n g e d . S, is coupled with a n I form a minor second higher (I2), and S,- and Sn, moving at twice the speed of the

f o r m e r pair, a r e coupled r e s p e c t i v e l y with 1^ and IJQ. Comparing the two S and I groupings now employed, S,, Sr, Sg and L , L , IJQ, it can b e seen that the t r a n s p o s i t i o n a l scheme of the S grouping is opposite to what it was in the f i r s t section, major thirds up r a t h e r than down, w h e r e a s that of the I grouping is both the same a s it was m the f i r s t section and parallel to the p r e s e n t S group, major thirds u p . This suggests that the identity between the l a s t and f i r s t t e t r a c h o r d s within the g r o u p s i s not used a s a principle of t r a n s p o s i t i o n .

24

L is so r e l a t e d to L Q , but LQ i s preceded by Sr (second staff) and I, is followed by Sg, its r e t r o g r a d e (third staff).

The middle section in the p r e s e n t composition is quite different from the other two examples of B s e c t i o n s . Rather than departing from the m a t e r i a l of section A, it " p e n e t r a t e s " into and r e d i s t r i b u t e s the m a t e r i a l . This m e a n s that the section does not begin with a fresh t r a n s p o s e d form, and, since it lacks the element of d e p a r t u r e , it a l s o lacks the e l e m a i of r e t u r n .

A vestige of a "sequential" a r r a n g e m e n t r e m a i n s since both the S and I groupings display (1) the consistent adherence to an o r d e r of f o r m s , (2) a repeated t r a n s p o s i t i o n a l i n t e r v a l , and (3) t h r e e s t e p s : | S j Sg Sg

U2 :6 r1 0 /

That a sequential a r r a n g e m e n t is not intended i s c l e a r from the fact that the f o u r - p a r t web of the outside sections is h e r e s e p a r a t e d into two p a i r i n g s differentiated by a row r a t i o of 1:2: ISj

b

f

5 ho

The m o s t i m p o r t a n t c h a r a c t e r i s t i c that all t h r e e middle sections s h a r e s e e m s to be the avoidance of using common tone connections a s the b a s i s of t r a n s p o s i t i o n a l i n t e r v a l s . A c o r o l l a r y to this i s that all middle sections employ a row a r r a n g e m e n t that in some sense is different from that employed in the outside s e c t i o n s . Thus, some c o n t r a s t is p r e s e n t in the middle section of the second movement of Opus 28 even though the same row forms r e c u r .

The row a r r a n g e m e n t in relation to the m u s i c

The o v e r - a l l row a r r a n g e m e n t just d i s c u s s e d c o r r e s p o n d s to the m u s i c a l form of the m o v e m e n t . Section A ( m e a s u r e s 1-18) c o r r e s p o n d s to boxes 1-3 and is unified by being w r i t t e n in the style of a f o u r - p a r t (or double) canon.

In section B ( m e a s u r e s 19-36) the tempo changes and, corresponding to the 1:2 r a t i o of rows in boxes 4 and 5, the two violins move in slower note v a l u e s than the cello and viola. After the middle of this section (from m e a s u r e s 2 7 -36) the two " r a t e s of speed" a r e no longer kept i n s t r u m e n t a l l y s e p a r a t e . The r e c a p i t u l a t i o n , section Ar ( m e a s u r e s 37-53 and boxes 6-8), i s , in a p p e a r a n c e ,

s i m i l a r to section A . The whole movement, while c l e a r l y t h r e e - p a r t , s e e m s , h o w e v e r , to differ from the t h r e e - p a r t m o v e m e n t s of Opuses 22 and 2 7 . It will be the major task of Chapter four to examine the differences between the formal conceptions h e r e involved.

Review

The main r e s u l t of the foregoing comments s e e m s to be that in the m o v e -m e n t s considered t h e r e is a definite c o r r e l a t i o n between row a r r a n g e -m e n t and the t r i p a r t i t e f o r m . In all three movements s i m i l a r t r a n s p o s i t i o n a l s c h e m e s w e r e found m their respective c o r n e r (A) sections, while a different s c h e m e w a s found in their r e s p e c t i v e middle (B) s e c t i o n s .

The c o r n e r (A) sections

The f i r s t movement of Opus 22 manifested the c l o s e s t s i m i l a r i t y between the outside sections, the rows in both sections being the s a m e . In

the second movement of Opus 28 the only difference between the two A s e c t i o n s was that in section A' the polyphonic order of e n t r y of row groupings w a s

r o t a t e d .

In both of these movements the row s c h e m e of the f i r s t A s e c t i o n s w a s c i r c u l a r , 1. e. , a selfrepeating a r r a n g e m e n t of a l i m i t e d number of t r a n s -positional f o r m s . T h e r e w e r e two important a s p e c t s to this a r r a n g e m e n t : (1) the t r a n s p o s i t i o n a l i n t e r v a l and (2) the " t o n i c " groupings. In the m o v e -m e n t fro-m Opus 22 s u c c e s s i v e S (and I) f o r -m s w e r e connected by one c o -m -m o n

26

tone through using the tritone (the framing i n t e r v a l of the row) as the t r a n s -positional i n t e r v a l with the r e s u l t that section A had two S (and I) f o r m s p r e s e n t e d horizontally. In the movement from Opus 28 s u c c e s s i v e S (and I) f o r m s w e r e connected by four common tones (a t e t r a c h o r d ) through using the major third (not the framing i n t e r v a l of the row but the t r a n s p o s i t i o n a l i n t e r v a l between the f i r s t and third t e t r a c h o r d s of the row) a s the t r a n s p o s i t i o n a l

i n t e r v a l with the r e s u l t that section A had t h r e e S (and I) f o r m s p r e s e n t e d h o r i z o n t a l l y . In the movement from Opus 22 the " t o n i c " c h a r a c t e r of c e r t a i n

row f o r m s r e s u l t e d from placing these f o r m s f i r s t in the introduction and then again at the end of section A. In the m o v e m e n t from Opus 28 the

a p p e a r a n c e of the same group of t e t r a c h o r d s both at the beginning and end of section A suggested that this group functioned a s a " t o n i c . " This closed

a r r a n g e m e n t suggested that the A sections of these movements w e r e r e l a t i v e l y complete, and this suggestion found c o r r o b o r a t i o n in their r e s p e c t i v e A"

sections w h e r e the s a m e rows r e a p p e a r e d . In the first movement of Opus 22 the " t o n i c " r o w s f i r s t r e a p p e a r e d at the end of section B, then at the end of section A", and finally in the coda. In the following d i a g r a m of Opuses 22 and 28 the " t o n i c " groupings a r e r e p r e s e n t e d by " x " :

Opus 22 Introduction A B A« Cod F i r s t m o v e m e n t x ^ |f: - - - j x : | | : x | xjH Opus 28 A B A ' '

Second m o v e m e n t | | : x — x : | | ty: x x :|| The l e a s t l i t e r a l s i m i l a r i t y between A sections o c c u r r e d in the f i r s t

movement of Opus 27. Common tone r e l a t i o n s w e r e utilized consistently in section A but the r e s u l t was the repetition of an S-I row pair and their

r e s p e c t i v e r e t r o g r a d e s . It was suggested that this is an open a r r a n g e m e n t r e q u i r i n g completion l a t e r in the m o v e m e n t . This completion s e e m e d to occur in section A* which began with its f i r s t pair continuing the t r a n s p o s i t i o n a l

scheme initiated in section A and ended with its second pair b r e a k i n g the t r a n s p o s i t i o n a l scheme and r e v e r s i n g the S-I o r d e r o c c u r r i n g in the other p a i r s in both A s e c t i o n s .

The B sections

The middle (B) sections of the m o v e m e n t s w e r e found to u s e different r o w a r r a n g e m e n t s than their r e s p e c t i v e A s e c t i o n s . In the m o v e m e n t s from

Opuses 22 and 27 new forms w e r e introduced and followed by f o r m s governed by a new t r a n s p o s i t i o n a l i n t e r v a l . In both c a s e s had the t r a n s p o s i t i o n s c h e m e b e e n continued the starting rows would have r e a p p e a r e d after the row s u c c e s s i o n

had passed through all possible t r a n s p o s i t i o n a l l e v e l s . Only two t r a n s p o s i t i o n a l i n t e r v a l s would give this r e s u l t , the minor second and the p e r f e c t fourth. In Opus 22 the minor second was used; in Opus 27, the perfect fourth (the f r a m i n g interval of the row). In neither c a s e was a common tone connection u s e d .

In the movement from Opus 22 this a r r a n g e m e n t led in t h r e e s t e p s to the

" t o n i c " rows that r e a p p e a r e d before section A ' . In the movement from Opus 27 the l a s t row of the third pair was found to be the s a m e a s the f i r s t row a p p e a r -ing m the r e c a p i t u l a t i o n .

The B section of Opus 28 did not follow this pattern but r a t h e r redeployed the same rows found in the A s e c t i o n s . They n a t u r a l l y tie in with the r e

-capitulation.

F i n a l s u m m a r y

It does s e e m , t h e r e f o r e , that t r a d i t i o n a l tonal movement has a kind of analogue in the row a r r a n g e m e n t s of these c o m p o s i t i o n s . The row a r r a n g e

-m e n t s in their own right w e r e found to point u n -m i s t a k a b l y toward t r i p a r t i t e f o r -m s . In two c a s e s the A sections were deemed to be r e l a t i v e l y complete, w h e r e a s in

28

one c a s e the f i r s t A section was deemed to be incomplete. All B s e c t i o n s showed some kind of a c o n t r a s t with the initial A sections and in t u r n connected with the r e c a p i t u l a t i o n s . In two of these movements the t r a n s -positional s c h e m e s of the B s e c t i o n s suggested sequential t r e a t m e n t . The f o r m a l i m p l i c a t i o n s of the row a r r a n g e m e n t s found c o r r o b o r a t i o n in a

g e n e r a l s u r v e y of the m u s i c of these m o v e m e n t s . In the following c h a p t e r s the o r g a n i z a t i o n of the individual m o v e m e n t s will be examined in d e t a i l .

TENOR SAXOPHONE, AND PIANO, OPUS 22

In this chapter each p a r t of the f i r s t movement of Opus 22 will be examined in t u r n . The a n a l y s i s of the individual p a r t s will include, f i r s t , a d e s c r i p t i o n of their r e s p e c t i v e phrase organizations in such t e r m s a s rhythm, m e l o d i c contour, i n s t r u m e n t a t i o n , and f o r m a l function, and, second, a d i s -c u s s i o n of their r e s p e -c t i v e pit-ch s h a p e s . Although the row f o r m s will be mentioned when a p p r o p r i a t e1 to clarify a p h r a s e or pitch organization, the

chief e m p h a s i s in the d i s c u s s i o n of the pitch shapes will lie with d i s c e r n i n g s u c h r e l a t i o n s h i p s a s exist, for e x a m p l e s , (1) among the pitches beginning a n d ending p h r a s e s , s e c t i o n s , e t c . , (2) among the high and low notes of

s u c c e s s i v e p h r a s e s , and (3)among pitches r e p e a t e d in identical octave locations within and between p h r a s e s . The o v e r - a l l r e s u l t s of the a n a l y s i s will be

d i s c u s s e d in a s u m m a r y at the end of the chapter which will also include a c o m p a r i s o n of this movement with a t r a d i t i o n a l example of the t r i p a r t i t e form

that s e e m s b a s e d on a f o r m a l analogue.

It m a y be r e c a l l e d that in Chapter one mention was made of two

p a r t i c u l a r a s p e c t s of this movement, i t s p o s s e s s i n g an introduction and being b a s e d on canonic writing, and that furthur clarification of these two a s p e c t s

w a s p r o m i s e d . P r i o r to the detailed examination of the p a r t s of the composition, and by way of a s u r v e y of the movement's g e n e r a l outline, these two a s p e c t s

w i l l now r e c e i v e s p e c i a l attention.

The r e a d e r is r e m i n d e d that the row a r r a n g e m e n t of this m o v e m e n t is d i s c u s s e d in Chapter one, pp. 5-11.

30

The Function of the M a t e r i a l in M e a s u r e s 1-5

As suggested in the p r e v i o u s c h a p t e r , the m a t e r i a l in the first five m e a s u r e s of this movement a p p e a r s to function as an introduction. T h e s e m e a s u r e s a r e s e p a r a t e d from the f i r s t main division of the movement (section A) by the repeat m a r k s which include only m e a s u r e s 6-15, and a r e balanced by a coda which i s s e p a r a t e d by r e p e a t m a r k s from the second main division of the movement (sections B and A", m e a s u r e s 16-37, a second ending provid-ing an overlappprovid-ing of section A* with the coda). In this s e n s e , then, the m a t e r i a l in question functions as an introduction. However, the m a t e r i a l in the f i r s t five m e a s u r e s lacks c e r t a i n c h a r a c t e r i s t i c s usually a s s o c i a t e d with traditional introductions, such a s , for example, a slower tempo than the main m o v e m e n t and, m o r e i m p o r t a n t l y , a " l e a d i n g - t o w a r d " h a r m o n i c plan (i.e.",

sequences or s i m i l a r p r o g r e s s i o n s concluding on a chord p r e p a r i n g for the main tonality). R a t h e r than acting a s a prologue to the main m a t e r i a l of the

movement, these m e a s u r e s s e e m to s t a t e , a s a unit, the main m a t e r i a l . This point will be elucidated during the detailed a n a l y s i s that follows. M o r e o v e r , this m a t e r i a l r e c u r s , m o r e o r l e s s v a r i e d , throughout the entire m o v e m e n t . In section A the m a i n melodic line is c a r r i e d by the tenor saxophone. Toward t h e end of the section the m a t e r i a l of the first five m e a s u r e s r e a p p e a r s a s a background to the tenor saxophone melody ( m e a s u r e s 12-15), Section B i s b a s i c a l l y devoted to this m a t e r i a l . (See e s p e c i a l l y the t r a n s p o s e d s t a t e m e n t s in m e a s u r e s 16-18 and 19-21; in m e a s u r e s 24-27 the original m a t e r i a l a p p e a r s in r e t r o g r a d e . ) The m a t e r i a l r e t u r n s again n e a r the end of section A1 a s a

background to the melodic line that was in the saxophone in section A but i s now divided among all i n s t r u m e n t s but the piano, this a p p e a r a n c e of the m a t e r i a l c o r r e s p o n d i n g to i t s appearance in section A (see the piano part, m e a s u r e s

34-37). Finally, the m a t e r i a l a p p e a r s in the coda in r e t r o g r a d e ( m e a s u r e s 37b41). However, the m a t e r i a l undergoes continual change; in i t s final s t a t e -ment, in the coda, only a minimum of i t s initial design r e m a i n s . In the following diagram of the movement "x" r e p r e s e n t s this m a t e r i a l , and " y " r e p r e s e n t s the above mentioned melodic line o c c u r r i n g in s e c t i o n s A and A":

Introduction Section A .Section B Section A" Coda M e a s u r e s : 1-5 6-15 16-27 28-37a 37b-41

II: y-"— :|1:

a b - *

:\\

•

Thus, the m a t e r i a l of the first five b a r s of this movement i s not only not typical of an introduction but contains the m a m material of the piece and r e -a p p e -a r s repe-atedly -at import-ant points surrounding the m -a i n f o r m -a l p -a r t s -a s

a frame and playing an important role within section B . Hence, it can be n o t e d that this m a t e r i a l behaves a s a r i t o r n e l l o .

It would probably be m o s t a c c u r a t e to say that the plan of t h i s m o v e m e n t d e m o n s t r a t e s a combination of a t r i p a r t i t e form with an introduction and coda, and r i t o r n e l l o p r o c e d u r e . For t e r m i n o l o g i c a l e a s e , the introduction and the r e c u r r e n c e s of its m a t e r i a l will be called the " r i t o r n e l l o , " and the melodic line appearing in the tenor saxophone in m e a s u r e s 6-14 and again among all i n s t r u m e n t s but the piano in m e a s u r e s 28-36, the " t h e m e , "

The canon

A second p r o c e d u r e s e e m s to be u s e d throughout the m o v e m e n t , that of a t w o - p a r t canon by i n v e r s i o n . All the m a t e r i a l of the movement except the theme p a r t a k e s of this apparent canon. Thus the r i t o r n e l l o is in canon; the

2

This procedure is called canonic by, among o t h e r s , Rene Leibowitz. See his Schoenberg and His School; The C o n t e m p o r a r y Stage of the Language of Music^ '..translated by Dika Newlin (New York: Philosophical L i b r a r y , I n c . , 1949), p . 218.

32

canon continues in section A a s an a c c o m p a n i m e n t to the theme and t u r n s into the r i t o r n e l l o that ends the section, continues throughout section B, throughout section A* a s in section A, and concludes at the end of the coda.

In some w a y s , however, the p r o c e d u r e is not s t r i c t l y canonic. In describing the actual p r o c e d u r e four a s p e c t s will be examined: the pitch a x i s , s e g m e n t a l superimposition, voicing, and time i n t e r v a l s . While most i l l u s t r a t i o n s of t h e n a t u r e of this procedure will be drawn from the opening r i t o r n e l l o , the

points made apply to the whole "canon. "

The pitch a x i s

The r i t o r n e l l o begins with a t h r e e - n o t e group in the saxophone which i s followed b y an> overlapping inverted imitation in the violin ( m e a s u r e s 1-2),

After a r e s t another t h r e e - n o t e group o c c u r s in the left hand of the piano which i s followed by an overlapping inverted imitation in the right hand

( m e a s u r e 3). These two units will be called the f i r s t and second m e m b e r s of the r i t o r n e l l o .

An examination of the pitch r e l a t i o n s between c o r r e s p o n d i n g notes of the initial and imitating n o t e - g r o u p s of these m e m b e r s r e v e a l s that the

notes a r e r e l a t e d through being equidistant from an axis on r " . Thus, while the first note of the initial group of the first m e m b e r of the r i t o r n e l l o in m e a s u r e 1 i s d °u, a perfect fifth above the a x i s , the first note of the

imitating-3

When, in the text, it i s d e s i r a b l e to identify a pitch not only by its l e t t e r name but a l s o by its octave location the designations given by Hindemith have been u s e d . (Paul Hindemith, E l e m e n t a r y Training for Musicians [New

York: A s s o c i a t e d Music P u b l i s h e r s , I n c . , 19491, p. 3o et p a s s i m . ) Thus, for e x a m p l e s , pitches in the octave from middle C to the B above a r e indicated by s m a l l l e t t e r s followed by one " a p o s t r o p h e " (c1 - b1), pitches in the next

higher octave a r e indicated by s m a l l l e t t e r s followed by two " a p o s t r o p h e s " ( c, r - b, r) , and pitches in the octave d i r e c t l y below middle C a r e indicated

group in m e a s u r e 2 i s b, a perfect fifth below the a x i s . The second n o t e s of the initial and imitating groups of the f i r s t m e m b e r a r e , r e s p e c t i v e l y , m a j o r t h i r d s above and below the a x i s . These relationships continue throughout the whole "canon" and r e s u l t in a set of seven dyads such that when one note of a dyad o c c u r s its c o r r e l a t e a p p e a r s on the opposite side and at an equal d i s t a n c e from the a x i s . T h e s e dyads a r e shown in Example 8, The n u m b e r s above and below the dyads r e p r e s e n t the distance of their r e s p e c t i v e pitches from

a i o w

D , the f i r s t note of the original row form (SQ). The fact that the sum of the numbers of each dyad equals ten (or ten plus twelve) shows why the sum of the number r e p r e s e n t i n g the initial note of a row of the "canon" plus the number r e p r e s e n t i n g the initial note of the accompanying t r a n s p o s e d inversion l i k e

-4 # wise equals ten (or ten plus twelve). In the "canon" pitch F always a p p e a r s

on the axis position except in m e a s u r e 24. The c* - c1 1 dyad forms a s e c o n d a r y

a x i s ; c o r r e s p o n d i n g l y pitch C occurs only in these two octave r e g i s t e r s except in m e a s u r e 22 and again in m e a s u r e s 3 2 3 3 . The octave positions of the

-remaining dyads shift throughout the "canon, " Segmental superimposition

An examination of the c o r r e l a t i o n of the row and the accompanying

t r a n s p o s e d inversion of the row with the initial pitch groups and their r e s p e c t i v e 4

This r e l a t i o n s h i p between S and I f o r m s of the row was mentioned in Chapter one,

34

a n s w e r s in the f i r s t two m e m b e r s of the r i t o r n e l l o r e v e a l s that in the f i r s t m e m b e r the initial pitch group i s a segment of the row while the answering pitch group is the corresponding segment of the transposed inversion of the row, w h e r e a s in the second m e m b e r the initial pitch group is a segment of the transposed inversion of the row while the answering pitch group is the corresponding segment of the row. This t e m p o r a l shifting of segments of the S and I f o r m s of the row, which continues throughout the "canon, " suggests that the basic procedure h e r e involved is a quasi-simultaneous presentation of original and inverted f o r m s of the row whereby either segments of the original form precede corresponding s e g m e n t s of the inverted form or vice v e r s a .

Overlapping two- or t h r e e - n o t e initial and answering row segments is the general "canonic" p r o c e d u r e throughout the movement. There a r e non-overlapping segments (for example, at the beginning of m e a s u r e 22), four-not segments (for example, in m e a s u r e s 10-11), and two places where the row segmentation differs from the imitative pattern (measure 28:

^ \ Qf 2 *»% , and likewise in m e a s u r e 31).

% •

S

6 '•! ©

Voicing

In addition to the t e m p o r a l shifting of S and I segments the voicing is also shifted. F o r example, in the first two m e m b e r s of the ritornello the higher pitch group - the "top voice" - is the answering pitch group, w h e r e a s in the third m e m b e r of the r i t o r n e l l o (end of m e a s u r e 3 through m e a s u r e 4) the higher pitch group is the initial pitch group. Thus, there a r e four possible a r r a n g e m e n t s of two pitch groups r e l a t e d by inversion ("x" and " y " ) : (i

-Time i n t e r v a l s

That the type of segmental manipulation d e s c r i b e d above is not,

s t r i c t l y speaking, canonic is disclosed when the time interval between initial and answering row segments is examined. W h e r e a s , for example, the time i n t e r v a l between the initial and answering groups in the first m e m b e r of the r i t o r n e l l o i s two eight-notes (two beats), that in the second m e m b e r i s one eight-note and that in the third m e m b e r i s one sixteenth-note. The variation in time interval that occurs throughout the movement suggests that the canon i s only "apparent, " not " r e a l , "

In s u m m a r y , it may be said that what has been called a canon is

actually an out-of-phase superimposition of kindred segments of S and I forms of the row identical in rhythm and r e l a t e d by m i r r o r inversion about a pitch a x i s .

The detailed analysis that follows will take, in o r d e r , the opening r i t o r -nello, section A, section B, section A', and the coda,

The Ritornello, M e a s u r e s 1-5 P h r a s e design .

-The r i t o r n e l l o m a y be divided into four m e m b e r s ( m e a s u r e s 1-2, 3, 4, and 5), each m e m b e r containing an initial t h r e e - n o t e pitch group (the dux) followed by an overlapping inverted answer (the comes) and s e p a r a t e d from the

5

following m e m b e r by a r e s t . The f i r s t two m e m b e r s form an antecedent

^The r e a d e r may wonder whether the dux and comes a r e to be r e g a r d e d , as they seem to be, of equal melodic status or whether they a r e related in some m a n n e r as " t h e m e " and " a c c o m p a n i m e n t . " Actually, they a r e subordinate to a m e m b e r a s a whole which becomes an integrated unit chiefly, a s will be shown in the text, through m e t r i c placement and the use of d y n a m i c s . Thus, the dux and c o m e s do not function a s melodic equals; r a t h e r , one initiates while the other ends a m e m b e r . The melodic line of the first m e m b e r , for example, follows the first two notes of the saxophone and the three notes of the violin. The claims of the unified m e m b e r s a s against the imitative p a t t e r n s r e s u l t s in a certain amount of a u r a l ambiguity, a type of ambiguity which, a s will be shown, r e c u r s within each movement analyzed in this p a p e r .

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p h r a s e ( m e a s u r e s 1-3) while the third and fourth m e m b e r s form a consequent p h r a s e ( m e a s u r e s 4 - 5 ) .

The rhythmic s t r u c t u r e of the dux of the first m e m b e r of the r i t o r n e l l o a p p e a r s to be made up of two durational motives, motive (a), a

sixteenth-note followed by an eighth-sixteenth-note ( * • ), and motive (b) which is s i m i l a r to motive (a) except for the substitution of a s i x t e e n t h - r e s t for the initial sixteenth-note ( J ! b e c o m e s J* ). The second member of the r i t o r n e l l o a p p e a r s after a dotted e i g h t h - r e s t . The durational motive of the dux of the second m e m b e r , (c), a p p e a r s , like motive (b), to be a varient of motive (a) through subdividing

the eighth-note of motive (a) into two sixteenth