LaiTransformationalStructuresV10

8/7/2019 LaiTransformationalStructuresV10

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22 Indiana Theory Review Vol. 10opus.3 Interactions between the two sets as reflected by thestructure will also be examined. Though topics of a different naturewill come up occasionally in the course of the presentation, the focusis on the transformation networks derived from the two pc sets andtheir relationship to one another.The two pc sets mentioned above are manifested at variouslevels as melodic motives, as vertical sonorities, and as structuralelements governing the larger spans of the composition. The firstof the two sets, 3-3, is important as a vertical component in the firsthalf of the piece; in the second half of the piece it is mostlyexpressed as a linear motive, especially as an ostinato figure playedby the cello beginning at m.lS. Upon its first appearance in m.l inthe three upper parts, the set consists of the pcs B, D, and E-flat,which I designate as X; thus X = {B23}. The second set, Y, is firstheard as a linear pattern in violin I in m.S, comprising D, E-flat, andG; thus Y = {237}. Y, in contrast to X, always appears in a linearfashion either as part of a melody or as a motive.

Figure 1. Interval vectors of X and Yi n t e rva l vector of X =i n t e r v a l vector of Y = 1 0 XOO100110

The close relationship between X and Y can be seen bycomparing their interval class vectors, whose values are identical infour of the six entries. As for the nonidentical entries of intervalclasses 3 and S, the values are exchanged (Figure 1). This is whatForte calls the Rl relationship, which has maximal similarity as well

3 Although register does playa role in the structural relation between the twosets, as will be discussed later, we are mainly concerned here with the domain ofpitch-class, rather than pitch. Furthermore, the discussion of only two pc sets doesnot imply the insignificance of other pc sets which may also play a role in thestructuring of the piece. These two sets are particularly chosen because of theirmany appearances on the surface of the music (especially at the beginning) andtheir close relationship to each other. The reader might want to examine furtherother pc sets which are prominent in the piece, especially 3-8 and 4-4.


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Lai, Transformational Structures 23Figure 2. T- and I-matrices of X and Y

T B 2 3 I B 2 3B 0 3 4 B A 1 22- 9 o 1 2 1 4 S3 S B 0 3 2 S 6

T1 (X) = t 034} T1 I(X) = {AB2}T3 (X) :: : i 2S6} T2I(X) = {B03}T4 (X) = t367} T4l (X) =. ! 2S}TS(X) = t7AB} TSl(X) = {236}T9 (X) = {SBO} T6 l (X) = {347}TB(X) = {A12} TAl (X) = {7SB}Matr ices of Y

T 237 I 2372 o 1 S 2 4 S 93 B 0 4 3 S 6 A7 7 S 0 7 . 9 A 2

T1 (Y) = {34S) T2 l (Y) = {7BO}T4 (Y) = {67B} T4I(Y) = { 912}TS(Y) = {7 sol TSl(Y) = {A23}T7 (Y) = t9A2) T6 l (Y) = {B34}TS(Y) : : {AB3} T9 I (Y) = {267}TB(Y) ::: { 1261 TAI(Y) = {37S}

as exchange of interval class values.4 Besides, the closeness

4 Allen Forte, The Structure ofAtonal Music (New Haven: Yale UniversityPress, 1973), 48.


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24 Indiana Theory Review Vol. 10Example 1. X forms used in Webern's op.S, no.3. (Note: Dynamics, articulations, etc. have been removed from this andother examples to enhance readability of the segmentations.)

TB (x) To ()() Tat)() To(X) TgCX) I i eX) TsI()\) ()() T ~ ( ) : ) T"l.(X) T31(j,) Ts-CX) T'rct)(" -I Tlo


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Lai, Transformational StructuresExample 2. Y forms used in Webern's op.5, no.3

mm. 2 3 7

13 I l l)8 10 I I 12 13

It If, t7 /8

TeHyJ l 1 ( Y ) - ( - , - b ~ ~ ~ ~ :p:.-/ \\-.fi;T T1.H'O 1- n I t ' ( ) ~ T,(Y) 15I(Y)

' - . . / ----19 2.0 21 22 2 ,

Webem Funf Siitze fUr Streichquartett, op.5, no.3. 1922 by Universal Edition. Copyrightrenewed. All Rights Reserved. Used by pennission ofEuropean American MusicDistributors Corporation, sole U.S. and Canadian agent for Universal Edition.

shared between them.5

5This shared invariant subset between X and Y is in fact expressed on themusical surface, as its two pcs (2 and 3) are always presented adjacently orsimultaneously. See mm.1-5, 10-12.

25


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26 Indiana Theory Review Vol. IOThe T- and I-matrices of X and Yare shown in Figure 2.According to the figure, each matrix consists of six forms whichproduce invariant pcs with the prime form. The number of invariantpes varies from one to two. Most of these operators are employed

in the piece: all T forms and three I forms from the X-matrices, andfour from each of the Y-matrices. These forms are underlined inthe figure. Examples I and 2 show the forms of X and Y which areused in the piece.6As mentioned above, X and Y have a close structuralrelationship. This closeness is reflected in the ways they areconnected in the piece. Upon examining the repetition of pcs indifferent registers spanning the first eleven measures or so, onenotices an interesting phenomenon-a pc in a low register (usually atthe beginning of a new musical segment, phrase, or idea) is repeatedin a higher registral position.7 This relationship is shown inExample 3. The first four pcs spanning mm.l-IO-D, C-sharp, C, andA-comprise the pc set {9012} which can be partitioned into two Xand Y forms, namely TA(X) and T4I(Y). However, by grouping thefirst three pcs (mm.I-8) with the F in mm.lO-ll, {OI2S} is formed,which has the same prime form as {9012} and is a transposedinversion of it. More interesting is that, upon partitioning {OI2S}into forms of X and Y, T 4I(X) and T A Y) are formed, which havethe same operators as the X and Y subsets of {9012}, except thatthey are exchanged between X and Y; so, TA(X) and T4I(Y) in{9012} vs. T 4I(X) and TA (Y) in {OI2S}. This sharing of operatorsbetween X and Y will be seen again in the following discussion.

6Examples 1 and 2 show the X and Y forms which I have found in the piece.As the discussion continues, I will demonstrate the musical and analytical importance of the X and Y forms shown in these examples.

7This pc repetition in another register is interesting in its own right. Adetailed investigation of this phenomenon using Robert Morris' concept of p-spacementioned in his book Composition with Pitch-Classes (New Haven: Yale University Press, 1987) might be fruitful.


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Lai, Transformational StructuresExample 3. Registral doubling of pes, mm.1-11

{OIlS} = T ' r l ~ ) U TA("OD /1 "*1/-,

/I ' I /1I I J I II / 7 /+ /-..., T-' i * j_ it+I Imm. I 8 10 1/

Webem Funf Siitze fur Streichquartett, op.5, no.3. @1922 by Universal Edition. Copyrightrenewed. Al l Rights Reserved. Used by permission of European American MusicDistributors Corporation, sole U.S. and Canadian agent for Universal Edition.

27

Various forms of X are presented as vertical sonorities by thethree upper parts in mm.1-6. See Example 4. They all have thesame spacing (8 + 3), with the exception of TB1(X) which has thespacing of (4 + 9). s Most of the adjacent forms share a commonpitch. By grouping the common pitches of the first three pairs of Xforms (not counting the repetitions of To(X) and Ts(X)), T21(Y) isformed (Figure 3). Similarly, by grouping the common pitches ofthe first two pairs and the T 1 X) - T4(X) pair of m.6, T21(X) isformed. (T4 X) and TseX) in m.6 do not share any common pitchnor pc.) Here again, an X form and a Y form share an operator,which is T21 in this case.Another connection between X and Y is seen in the restatements of some of their forms. Some X and Y forms stated at thebeginning of the piece (mm.1-5), T4(X), Ts(X), TB(X), T1(Y),T7(Y), T21(Y), Ts1(Y), and T61(Y), are restated (recapitulated?) ator near the end, beginning at m.19 (Examples 1, 2). These analogous placements ofX and Y forms at the same locations, in additionto the Rl relationship and the sharing of operators mentioned

8TBI(X) is unique in its spacing because it is the only inverted form of Xamong the others (TO(X), T 8(X), T1 (X), T 4(X), and T 5(X)),


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28 Indiana Theory Review Vol. 10Example 4. X forms in mm.I-6

TeCX) 1001.) 18 lX)

0

i,.......1'- '

I \4Ih i.

't- v 1'1" -v I" \ -I i -.Y . il..jfr-r 4-+ l- I'Jj.,v . I - - ' - - lA(X)- '--- --c::::::' --- T31(:1.)

I",IV -

:t\::t :): i :i: ;ff::t :t #:7 -:j +"to') Tlt-lx) Tsv(JJ-:t 9- !:rt:..;l. b-t. ffi.I lb. #-t t b t = ~ T


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Lai, Transformational Structures 29Figure 3. Sharing of operator between an X form and a Y form

later discussion.

Example 5. Webern's op.5, no.3, m.7T6CO T&l{'O To It"!)\ I*::t:. / '8 CY}0 + I -Ii-

I "'It" f.t ....."\ - "'I v..... ..I -li(X)7 r -"\\. L/

I(I-' - ...1-1\. 17 I ('{) T'1-,t'(j

_I-r1'"""-.

... v ... / , ." ..I.

J=i:-+ " II \ TsIC'(J

To ICY) 1 ; ( ' t )Webem Fanf Siitze for Streichquartett, op.5, no.3. @1922 by Universal Edition. Copyrightrenewed. Al l Rights Reserved. Used by permission of European American MusicDistributors Corporation, sole U.S. and Canadian agent for Universal Edition.Two places in the music generate all the transformationaloperators of X and Y. They contain a saturation of X or Y forms.Interlocking forms of Y dominate the whole of m.7. In addition, theimitative passages between violin I and cello are inversionally related


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30 Indiana Theory Review Vol. 10Figure 4. RICH structure of Y

v ln . l

ce l loTCHT 2 ( ~ T 7 ( Y )

R1r R I C ~ R1r R I ~ T 7 I ~ O I ( ~ T 5 I ( Y ) TCH TCH

(Example 5). The structure in each of the two parts can bedescribed in terms of Lewin's RI-chain (RICH), in which the lasttwo elements of an ordered pc set are taken as the first twoelements of a transposed retrograde inversion.10 Figure 4 reflectsthe RICH structure of the two parts.According to Lewin, the (RICH)(RICH)- or TCH-interval ofa RICH structure can be calculated by the formula i = int(sl,sN) +int(s2,sN_1) where i is the TCH-interval and s1' s2' sN' and sN-1 arethe pcs of the ordered set s = sl' s2' ..., sn.1 By taking the firstordered set T6(Y) = of violin I in m.7, the TCH-interval iscalculated as int(1,8) + int(9,9) = 7. By the same token, the TCHinterval of the cello part is calculated to be 5, or the mod12complement of 7. This is so because, as already mentioned, thecello part is an inversion of the violin I part. Thus the two RICHscan be generalized and related one to the other, as shown in Figure5. As seen from the figure, the RICH- and TCH- intervals of theRICH structure of the cello part are inversions of the values of theRICH structure of violin I. Thus the entire former structure is atransposed inversion of the entire latter structure. I call the former

10David Lewin, Generalized Musical IntelVals and Transformations, 180-88.l lLewin, Generalized Musical IntelVals and Transformations, 181.


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Lai, Transformational StructuresFigure 5. RICH(Y) and RICH-\Y)

TCHRICH(Y)

TCH

RICH-I(y), and the latter RICH(Y).Figure 6. RICH and TCH operations derived from RICH(Y)and RICH-I(y).

RICH(Y) RICH-l(y)TCH = 7 TCH- l = S

RICH -1T (Y) a ... T(n+S)I(Y) T I(Y) RICH a T (Y)n n -1 '> (n-S)T I(Y) RICH b ) T(n+2) (Y) T (Y) RICH b T I(Y)n n -1 ) (n-2)T (Y) TCH T (Y) TCH)T(n+7) (Y) 'IT(n_7) (y)n nT I(Y) TCH )T(n+7)I(Y) T I(Y) TCH-

l)T(n_7)I(Y)n n

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Figure 6 shows all the RICH and TCH operations withinRICH(Y) and RICH-I(y) which transform forms of Y. Based onthese operations, a network of transformational relations can bedrawn between the various Y forms presented in the piece, as shown


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wRICH CO N

v In . 1

-IT 7 ( Y ) ~ H f u C H - ' ~ To cn~ T , ~ ' "

aq.t:1"1(D;--J

::::lr./1

SI e H

- - - ~ - -TCH ,CH

T b ~ L C . H Q ) ~ L C H a . }8('0~ / I C H b \;/RrcHh1s1C'O T Ill)

CH

.-+.....vln .2 T:2.11I ) Tq ('\) 0::::l

.......r./1.-+

vIa. tIel)RiCH a.l";cn T 7 (Y) 2na-"1(D0..(D$.(D0..ICH-' COce l l o

S-.::

2

-rcwl 7cHTL('O l ; ~

RICH-1a. / RICH-V/ K l C . H - l b ~ / I ? I C W / ~ T2 C\}

Til ('() ToH'() Ts I COT C H ~ T c H ~

:nm. 5 :--I -8 0


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v ln . l ToHY) TAHY) T2 (y ) " '"leW'

Tew'

v1n.2 TdCY) TCH T61(1)

v la . 73 IlY) TcH

'TCW'

ce l l o TAIcn T,l('O

:llm. /0 / I 12 13

-.(J'q"'1("D-......)()0r+"-"".RIc H-' b

1ToUI)

14-

r$:::

t:::""+ct:::l....

";)EC;)

ww


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J 1 n. 1

, ;In .2

" l a .

T\"'H'(y) T,(Y)T e ~ RrC-H-'o.

:::ell0

nm. 16 n

R.ICHh---T6ICY)\:R.IC.Ha.. T6 ICY}

20

I I Y)-IRIC.H II

T;(Y) /2L(Y) TsHY)

21 23

I-rj......

"'1(I)-....l--jo

" - "

w..

..

;;::s

-.

-


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Lai, Transformational Structures 35in Figure 7. Most of the forms are presented in a linear fashion,with the exception of T ley) (m.20), of which two pitches (E andD-sharp) are presented simultaneously by violin II and viola. Thisset, though it may not be recognized aurally as easily as the others,is considered important because of its location-right after the changeof ostinato patterns in violin II and viola at the same measure-andbecause it is played by the same instruments as T 6I(Y) and T BI(Y)(Example 6).

Example 6. Webern's op.5, no.3, mm.19-20, violin II and viola\ \

I 1 lliJ1 h..li f 'J' ., ".s.. l2 t. J:..:IF":; 1i ; ' i (#l

I I I I \ I I_UJ:: J.,JJJ. ..J I.J ~ - I - i . J ' i ..J ..I I vIh 11" , I " \TTC W-4 .... F

"II{,HY)Webem Funf S ~ t z e fUr Streichquartett, op.5, n . o . ~ . @1922 by Universal Edition. Copyrightrenewed. All Rights Reserved. Used by pennLSSlOn ofEuropean American MusicDistributors Corporation, sole U.S. and Canadian agent for Universal Edition.

The notes of most Y forms are played by the same instrumentexcept ToI(Y) at m.14 and T ley) at m.20. These two forms areshown inside a box in Figure 7. A few general comments precedespecific observations:1. All the RICH and TCH (including RICH- I and TCH-I ) opera-tions are employed.2. TCH operations dominate the middle of the piece (mm.lO-II),whereas RICH operations are prominent in the rest of the

mUSIC.3. Most of the forms in the graph are connected by one or morearrows, with the exception of T 2I(Y) and T 9(Y) (mm.I, 5, and21). These two forms are related to the large-scale transformations as will be discussed later.4. T6(Y) and TBI(Y) dominate mm.14-I9.


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36 Indiana Theory Review Vol. 105. The five forms which appear in mm.1-3 are restated at the end(mm.19-23).

Example 7. Webem's op.S, no.3, mm.1-3I,C'O1

11' \ t:., h { '-1 h: \ . '- I ".JT2.1(,() J" v 71 V

1\\\. 1':\ b; ' .. J...J.'1

1 _ 9 ~ ?-i-II-' I f ' \. '-1 \ ....11-1 ., III //

16H'()--1 V f l - /l,cO( ' \ , I t '..I \.""

:\j::t :; :t. 'p :t :t "#+

TsI (Y)4+ "I v

'- II I .., II

/'41- ,.-----"It' ...

/ '> ''-1' - - - - - - '

=t "* +-Webem Funf Siitze for Streichquartett, op.5, no.3. @1922 by Universal Edition. Copyrightrenewed. Al l Rights Reserved. Used by permission of European American MusicDistributors Corporation, sole U.S. and Canadian agent for Universal Edition.Figure 8. RICHa and RICH-1b transformations, mm.1-3

Now here are the details. Before Y is clearly stated in m.S, itis implied by its various forms which are embedded within thevertical X forms (Examples 4, 7, mm.1-3). Four of these five formsappear later as part of the RICH(Y) and RICH-\Y) structures atm.7. Two of these four forms are related by RICHa, whereas theother two are related by RICH-1b (Figure 8). A comparison ofthese two pairs with the analogous ones in m.7 shows that theirtransformations are identical. Thus RICH(Y) and RICH-\Y) are


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Lai, Transformational Structures 37anticipated at the beginning of the piece before their full presentations. This structural relationship between the two pair forms is alsoreflected in mm.19-23, a l t h o u ~ h the transformation from T I (Y) toT 6I(Y) is reversed (Figure 7). 2

Another pair of Y forms from RICH(Y) which dominates amajor portion of the composition is T 6(Y) and T BI(Y). These twoforms are linked by RICHa and RICH-Ia (mm.14-19). (Thisalternation between RICHa and RICH-Ia may also be related to thestasis created by the ostinato figure in this passage.) It is interestingto note that these two forms, which are used at the beginning of theRICH structure at m. 7, are also employed here at the beginning ofthe last section of the piece, at m.14 (violin II, T 6(Y) andm.16(viola, TBI(Y)). By the same token, the pair T7(Y) and TsI(Y)mentioned above as the last two forms of the RICH structure, arepresented at the end of this section (or the whole composition) atmm.20-21, 23. By extending the idea of RICHa/RICH-la pairingbetween T6(Y) and TBI(Y) to TCH and TCH-I operations, therelationship occurs at other locations as well (mm.1-6, 10-12).

Figure 9. TCH-Ias a link between RICH-I(y) and TCH transformations.

mm.7-8 mm.lO-11

A link between RICH-I(y) and the TCH transformations atmm.10-11 is shown in Figure 9. This link is formed by T sI(Y) ofRICH-\Y) with TAI(Y) of the cello part in mm.10-11 and is crucial

12There is a process going on in the presentations of these two pair forms. Atmm.1-3, they are "hidden" within the vertical X forms, whereas at m.7 they areheard as part of the RICH structure, although each form is still interlocked withits par tner within the RI-chain. The forms, upon their last presentation at mm.19-23, are separated from one another, with T(Y) (mm.20-21) and T SI(Y) (m.23)heard most clearly in the high register played by violin 1. Thus there is a gradualaural realization of these two pair forms in the course of the composition.


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38 Indiana Theory Review Vol. 10Figure 10. Transformation network formed by RICH-\Y) andTCH transformations.

because it extends from the end of RICH-\Y) to include a TCHtransformation network in mm.10-12. Further examination of thisTCH network with RICH-1(y) reveals other relationships-the Yforms of the TCH network "reverse" their directions back toRICH-1(y) (Figure 10) with T3I as the pivot, and three pairs of Yforms are identical in the whole network. This structure is relatedto the musical surface as the middle of m.12 signifies the entry of anew musical idea, and the measure before it has imitative figureswhich "recur" one after the other.

Figure 11. Transformation networks of Y; 18a) mm.1-5, violinI; 18b) mm.19-20, violin II and viola.

(b ) 'rCH

( : ~ b ) : ~ { : : l a ' ~ ' r l ( Y ) l RICH a?

Violin I in mm.1-5 reflects a transformation network as shownin Figure 11a. A similar network occurs at mm.19-20 between violinII and viola (Figure 11b). If the RICH-1a arrow is reversed inFigure 11b to give RICHa, an inversional structure of Figure 11awould result. The arrow cannot be reversed for the only purpose ofobtaining a close relationship between Figure 11a and Figure 11b,


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Lai, Transformational Structures 39as the meaning of Figure lIb would be totally different.13 Thisambiguity between RICHa and RICH-1a seems to imply a structuralphenomenon which is reflected elsewhere in this piece, for example,the continuous transformations ofT6(Y) and TBI(Y) via RICHa andRICH-1a in mm.14-19.

Figure 12. Transformation network formed by T 2I(Y), T 9(Y)'and T7I(Y), mm.1-6.

Figure 13. Transformation network of Y, mm.14-20, violin IITCHT 6 ( Y ) ~ T l (Y )

\OICH a /.ICH b~ T B I ( Y )

T2I(Y) and T9(Y) at mm.1-6, which are not connected withany other Y forms via RICH or TCH in Figure 7, can be linked withT7I(Y) of RICH-1(y) to form a transformation network which isvery similar to Figure 11a (Figure 12). A comparison of Figure 12with Figure 11a shows that the two networks are basically the same,exceptine he reversal of order of the intermediate transformations(RICH- a and RICH-1b). In contrast to Figure 11a, which does nothave its own inverse structure, the inverse structure of Figure 12 ispresented in the music. Figure 13 shows the transformation networkformed by the Y forms in violin II of mm.14-20. The operators in13T6I(y) generating T1(Y) via RICH-1a is totally different from T1(Y)generating T 6I(Y) via RICHa.


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40 Indiana Theory Review Vol. 10this network are exactly the inverse of the ones in Figure 12, and theorder of the respective transformations is the same. More precisely,the network of Figure 12 is the inverse of the one in Figure 13.This relationship could in fact be treated as an extension of theRICH structure.14

Figure 14. "X-Matrix"T1(X) T4 (X) TS(X)

T4 (X)T1(X)TS(X)

T6 I(X)Having exhausted the investigation of Y and its transformations, it is time to examine X and its transformations. As with Y,there is a location in the music at which a saturation of X formsoccurs, namely m.6. The three upper parts from the second to thefourth beat comprise three forms of X: T I (X), T4(X), and Ts(X).These three forms are presented vertically as well as linearly

(Example 4). A matrix is formed as shown in Figure 14 by takingthese nine pitches and labelling them with pc numbers. One moreform of X, T 6I(Y), is projected by the numbers forming the diagonalfrom the lower left corner to the upper right corner. lS Furtherinvestigation of this "X-Matrix" reveals another interesting aspect:the operands of the three T forms (1, 4, and 5) comprise a form ofset X (T2(X)). Because of its structural significance, this matrix isassumed to be a kind of generator of all the X forms in the piece.

14This inversional relationship, of course, is different from the one mentionedin Lewin's article, which deals with inversion about an axis of symmetry. SeeLewin, "Transformational Techniques in Atonal and Other Music Theories."15Each of the members of T6I(X), 3,4, and 7, is presented twice in the matrix.

The remaining pcs belong to set 3-8 which, as mentioned before, may be anotherimportant set in this piece.


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Lai, Transformational Structures 41Figure 15. Operations derived from the X-Matrix

T1(X) T3 )T 4 (X)T_ 3 )T1(X)

Tl T_lT4 (X) )TS(X) >T4 (X)T4 T_4T 1 (X) )-TS(X) )oT1 (x)T1(X) T7

I >T6 I(X)T7 l '>T1(X)

TAl TAlT4 (X) )T 6 l (X) )T 4 (X)TEl TElTS(X) )T 6 l (X) )rTS(X)

A series of operators which connect the four forms of X withone another is derived from the matrix and shown in Figure 15.This figure shows that the operators transforming Tm(X) to T6l(X)and vice versa (where m = 1, 4, or 5) are identical, since they areall T nl operators (T71, TAl, and TBI), and they are involutions.16By connecting all the X forms in the piece with these operators,another transformational structure is formed (Figure 16). The graphgroups the presentations of X forms into two categories, linear andvertical. Although the X forms in parentheses are not as easilyaudible as the others, they are not unimportant. According to thegraph, X is always presented as a vertical component in the first halfof the piece, whereas in the second half it is always linearly presented. T B(X), though not connected with other X forms in two of itsthree presentations, plays a significant role in the "framing" of thepiece, as it is presented at the beginning, the end, as well as at thestatement of the melodic idea in m.9. 17

At first glance, rnA seems to be an interruption of the verticalpresentations of X, due to the imitative figures played by the violinand viola (Example 4). Closer study reveals that this measure

16Morris, 127.17T6I(X) at m.13, which is also not connected with other X forms, has aminor role to play, since one of its pitches (E-flat) only spans a duration of asixteenth-note. This sonority is thus hard to be perceived in the context, especiallywith J= 84.


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42 Indiana Theory Review Vol. 10Figure 16. Transformational structure derived from X

x ...... r

--..


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Lai, Transformational Structures 43Figure 16 (cont.)

25 0-\-'='---..x

- ( ~ t.!. 25 00

" - ( ~ .. 6 r "-0 6 25 t - l::: I p>

LaiTransformationalStructuresV10

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