an introduction and a study string quartet no. 3 by milton babbitt

An Introduction and a Study String Quartet no. 3Author(s): Stephen Arnold and Graham HairSource: Perspectives of New Music, Vol. 14/15, Vol. 14, no. 2 - Vol. 15, no. 1, Sounds andWords. A Critical Celebration of Milton Babbitt at 60 (Spring/Summer - Fall/Winter, 1976),pp. 155-186Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832633 .Accessed: 09/02/2011 15:34

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AN INTRODUCTION AND A STUDY* STRING QUARTET NO. 3

STEPHEN ARNOLD & GRAHAM HAIR

AN INTRODUCTION

Milton Babbitt's published writings refer directly to his own compositions but infrequently. Nevertheless they often point to terms of reference which are of significance in his own music, because, although they contain many references to features of the music of Schoenberg, Berg, Webern, and Stravinsky et al., they are usually couched in generalized language which directs attention to possible ramifications. Since his discussions of these features frequently centre upon apparent minutiae of their techniques, it is unfortunately only too easy to overlook the fact that the scope of their ramifications is 'comprehensive, vast and challenging',1 without a thorough and detailed knowledge of the compositional contexts (both in Schoenberg's and in Babbitt's music) to which the writings allude.

Much of the language which has evolved for the discussion of the structure of twelve-tone music, even for the discussion of Schoenberg's own music, is directly attributable to Babbitt. His articles crystallize many of the basic concepts in a manner which facilitates the under- standing of both the similarities and the differences of the music of the two composers. The nature of this relationship is perhaps best demon- strated by means of examples.

The clearest manifestations of the twelve-tone set in Schoenberg are

* Since 1967, we have been intermittently engaged on a study of Milton Babbitt's music. This has recently been completed in book form-The Music of Milton Bab- bitt (still unpublished)- and some of its material is reproduced as our contribution here.

1 Milton Babbitt: "Set Structure as a Compositional Determinant", in Perspec- tives on Contemporary Music Theory, eds. Benjamin Boretz and Edward T. Cone (New York, 1972), p. 147.

156 PERSPECTIVES OF NEW MUSIC

perhaps those which are linear statements given to a single instrument or group of instruments (see Ex. 1). Comparable instances are not all

Aaron4 . 6K I

1'' F, t4 it- . ^t4s d? ?

p, - .. w 6m.

-?

Ex. 1 Moses and Aaron, Iii: bars 124-129

that common in Schoenberg: they are usually reserved for points of

particular structural significance, e.g., the opening 'themes' in Quartet No. 4 (both first and third movements), the climactic chorus in A Survivor from Warsaw, the opening encounter of the chief protagonists in Moses and Aaron, the final denouement of the same opera, and the theme of the Orchestral Variations, op. 31.

Generally, in Schoenberg, the set is presented in some context which partitions it into more than a single layer, articulating some extracted

sequences, contiguous or non-contiguous. That is to say, there are at least classes of context in which one would wish to invoke the term

'partitioning'. Example 2 shows an instance of one such class, in which

con sord7 . ...A

toS, partitioned

Viola

P A I I I

.A-

ZViolins -A "+ T 11 u

Ex. 2 String Quartet No. 4, movement II: bar 285

the non-contiguous extracted sequences D-A-Bb-E-Ab-G and C#-F-Eb- C-Fj-B are articulated. Another such type of context is the familiar one in which two (hexachordally) combinatorially related forms of a set are presented simultaneously, articulating the two discrete hexa- chords of each. Both of the sets of such a combination may be further

partitioned in the manner shown in Ex. 2 (see Ex. 3). One may also wish to speak of the partitioning of the aggregate (of the twelve discrete pitch classes), e.g., the partitioning of the first aggregate of Ex. 3

STRING QUARTET NO. 3 157

Wood-.Wm wind nil A p ( t7S

Aaron 4H

,--Aggregate 1 - ,-Aggregate 2-J

Double- .10 - basses ti0ll

II_ P" fL -

Ex. 3 Moses and Aaron, Iii: bars 163-164

into the trichords F-A-Ab, E-B-Bb, D-C$-Eb, and G-F$-C: in this case, a 34 partitioning of the aggregate (four layers of three elements each). Analogously, Ex. 1 would be characterized as a 12' partitioning.

This combination of sets (Ex. 3) is a simple example of that poly- phonic entity in which, in Babbitt's music, the set is almost always embedded, viz., the array.2 The following is an example with a list of its structural features (Ex. 4).

Layer 1 FED G Eb C F$ A C# G$ A# B

Layer 2 Ab Bb B F? A C? G Eb C FED

Layer 3 C C$ Eb E G$ B F D Bb AG F#

Layer 4 AG F# F D Bb E GB CC C? Eb

Ex. 4 The Widow's Lament in Springtime: bars 1-6

* The array contains four rows, and each row contains twelve pitch- classes, so the dimensions would be described as 4 X 12. Many dimensions other than 4 X 12 are of course possible, and occur frequently. In the later works, especially, very large dimensions are the norm: 8 X 96 in String Quartet No. 3, 12 X 48 in Sextets and Correspon- dences, for example. In a few works, such as the Two Sonnets of Gerard Manley Hopkins and Relata I, there are irregular arrays, in

2 For Babbitt's own, more formal description of array, see his article, "Since Schoenberg", PNM, Vol. 12, pp. 3-28.


which the horizontals are of different lengths, but this is rather excep- tional. * The vertical lines separate aggregates. The horizontal lines separate 'layers' whose internal ordering is as given, but whose ordering in relation to one another requires further contextual organization (the array chart does not provide information about that). * The horizontals are usually articulated by timbre: they are described as 'layers' to avoid the connotations that the word 'line' might invoke. There may be one instrument to a layer, or where one instrument is assigned to several layers, sub-categories of timbre, arco/pizz, sul tasto/sul pont, etc., (or, as in Ex. 4, registral sub-categories: layer 1 is voice, layers 2, 3, and 4 are piano-high, middle, and low registers, respectively). In only one instance that we know of are the array horizontals articulated by dynamics (in the 'Canons for Clarinet' section of the Woodwind Quartet). In the orchestral music the timbral layers are instrumental mixtures (woodwind doubled on strings, brass doubled on percussion doubled on woodwind, etc.).

In the case of Ex. 4, the partitioning of each aggregate is into four layers, each containing three elements (34 partitioning). Schoenberg's arrays, discounting the internal partitioning of the sets themselves, are generally 2 x 12, with hexachordal (62) partitioning. The further partitioning of each of the layers is a form-defining feature.

The concept of combinatoriality is by now the subject of an exten- sive literature, both by Babbitt himself and by others. It will not be entered into here in detail, except to mention a few of the combinatorial properties of the sets which Babbitt has used. These sets are shown in Ex. 5. In a few works, it is impossible to settle for any single set as the 'basic' set, usually because, though it may be possible to identify a 'basic hexachord' or a group of 'basic trichords', these segments are concatenated in a number of different ways according to context, to form a few or many different twelve-tone successions.

All except two sets, those of Sounds and Words and Philomel, are all-combinatorial, with a predominance of first-order all-combinatorial sets. The list below gives the hexachordal combinatorial properties only. In some cases, the combinatorial properties under other partitionings play an important role in the structure of the work in question. This is particularly true of the set of Philomel, for instance.


Three Compositions for A I -"E- Piano

Composition for Four W LL6 Instruments

---- - - 7

ql* , - a A I

Instruments

The Widow's Lament A

Composition for Violave

AW 11 - and Piano rtnt

Du T1__________ __

C t ii - - . -

Woodwind Quartet ____"

___"_"______.

_

String Quartet No. 2 -

Two Sonnets of Gerard 6. Manley Hopkinstruments

Semi-Simple Variations 0 0 L am-- WI

All-Set _ _ _a-m

_ _ _L _ __-_-

__ W--

Partitions _ _ _ _ _ _ _ _ _ _ _ _ _

Sounds and Words A ..

Composition for Tenor and Six Instruments

W) 7

Ex. 5


Vision and Prayer _ _ 66

Philomel__6

Relata I and Relata II --

Post-Partitions 4 F l __ __ _

A" I

Sextets ,__

Correspondencesa _ , _ _ - _

String Quartet No. 3 -

_.=,~a?N" - .-M

Ex. 5 (cont.)

Third-Order All-Combinatorial Sets (014589)

Composition for Four Instruments (iii), (iv) Composition for Twelve Instruments Composition for Viola and Piano Vision and Prayer Correspondences (hexachord) Second-Order All-Combinatorial Sets (012678) All Set Composition for Tenor and Six Instruments String Quartet No. 3

First-Order All-Combinatorial Sets

Three Compositions for Piano (023457) Composition for Four Instruments (i), (ii) (023457) The Widow's Lament in Springtime (023457) Du (i) (012345), (ii) (024579) Woodwind Quartet (012345) String Quartet No. 2 (012345) Two Sonnets (023457) Semi-simple variations (012345) Partitions and Post-Partitions (hexachord) (012345) Relata I and Relata II (012345) Sextets (hexachord) (023457)


Non-Combinatorial Sets Sounds and Words (012568) Philomel (023578)

Arrays of many dimensions and characteristics other than 2 X 12 with 62 partitioning may be found in Babbitt. The type of partitioning shown in Ex. 4 is characteristic of a whole series of works dating from 1946-1957. The appearance in the later works of constantly varying partitioning, both symmetric and asymmetric, running the whole gamut of all the possible 77, from 121 to 112, becomes likewise a form- defining feature.

The relationship of Babbitt to Webern is not so close as to Schoen- berg. The most obvious aspect of the relationship is perhaps in the extent of use of derived sets. However, whereas Webern used a derived set in only about a half-dozen works, derived sets pervade all the early Babbitt pieces, and whereas, in Webern, a single derived set is employed as the 'basic set' throughout a piece, Babbitt commonly relates a whole complex of derived sets to a 'basic set' and to one another. Since the manner in which these derived sets are employed varies so much from work to work, the generalities about their use cannot be so readily summarised as can the role of the combinatoriality and the array.

In certain harmonic relationships in the small, Babbitt's music re- veals some concerns with which both Schoenberg and Webern were equally intimately involved. The dyads between pitches of the same order number in inversionally related sets, in particular, were of espe- cial importance to Schoenberg and Webern. The exploitation of the preservation of the dyads when the sets are transposed by complemen- tary intervals is a by-now well-known feature of such Webern works as his Opp. 21, 22, 27, and others. The changing dyads created by the identical transposition of both sets of such a combination (in order to maintain combinatoriality) is one of the contributary means of defining hexachordal 'areas' in almost all the Schoenberg twelve-tone pieces. Similar dyadic relationships play important structure-defining r61es in Babbitt's output (see Ex. 6).


Woodwind (toS) mp i .

MP Percussion

S .Woodwind and Strings

]- d

I(tsIR) f P <

t2R (hex. 6-11)

Strings I.

_WAp tsIR (hex. 6-11)

I- 4 Ex. 6b Relata I: bar 134 (Agg. 154)

toS (hex. 0-5)

tsI (hex. 0-5)

Ex. 6a Relata I: bar 9 (Agg. 1)

StrS gIl

Percussion (tNIR)

f p Woodwind t7IR (hex. 6-11)

(t7S)

f/--- 7 pl-? p . = t4IR (hex. 6-11)

Ex. 6d Relata I: bar 310 (Agg. 303)

tio (hex. 0-5)

t7S (hex. 0-5)

Ex. 6c Relata I: bar 135 (Agg. 155)


-----

Percussion (t2S) 4Woodwind,

ff

P'-PIW -- Brass and Percussion

0 (tgR)

Woodwind f=

-- - ---- and Brass

P(t0PIR),M

I

t9R (hex. 6-11)

tpp"(hx0 -

- 5

BrassVo

(tSI) t0IR (hex. 6-11)

.'.f Ex. 6f Relata I: bar 509 (Agg. 462)

t2S (hex. 0-5)

t51 (hex. 0-5) -

Ex. 6e Relata I: bar 311 (Agg. 309)

Although the twelve-tone pitch-class system, as Schoenberg defined it, deals only with the pitch dimension, Babbitt has pointed to its 'im- manently temporal nature'. That is, it 'incorporates qualitative time properties into its very rule of formation'.3 It is Babbitt's treatment of quantitative rhythmic questions, however, that distinguishes his music from that of Schoenberg and Webern more decisively than does his treatment of qualitative ones. The development of a rhythmic syntax not dependent upon motivic organization, but upon formations and transformations which parallel operations in the pitch domain, in certain limited and specific ways, has followed a number of diverse lines, leading ultimately to the 'time-point' system. Boretz has identified these as 'sets defined by successions of different-sized bundles of evenly spaced attacks (articulated by phrasing, accentuation, etc.,: Three Compositions for Piano, 1, or defined by successions of "time intervals"

3 Milton Babbitt: "Twelve-tone Rhythmic Structure and the Electronic Medium", in Perspectives on Contemporary Music Theory, eds. Boretz and Cone (New York, 1972), pp. 151 and 150, respectively.


between attacks (Composition for Four Instruments), or defined by variable articulations of the time contents of a fixed "measure" (Com- position for Tenor and Six Instruments, All Set, Partitions, opening of Relata I and most subsequent compositions)'.' A generally applicable point in relation to the compositional deployment of all of these syn- taxes is that, to a greater or lesser extent, they function independently of the pitch structures to which they are wedded. The degree of paral- lelism in structure and deployment between the two domains is a contextual compositional decision, but, although the relationship by ana- logy between them is, on occasions, extremely close, it is not of crucial importance.

Though the rhythmic domain is independent, the pitch domain re- mains more important, and though the remaining dimensions are subject to organization within the twelve-tone system, they occupy a still lower rung on the hierarchical scale. Babbitt has supported this view by referring to the fact that

those scales of measurement which are a central issue in contemporary psychophysics, and its associated mathematics and statistics, are embodied, in their applicability to musical dimensions, in traditional musical notation itself. The instrumental or timbral equivalence class is signified by the use of the name of an instrument, thus suggesting merely a nominal scale. Customary dynamic indications suggest an ordinal scale for loudness. The durational signification of rhythmic notation suggests the intervallic scale of protensity; and the "abso- lute" notation for pitch suggests the appropriateness of a ratio scale in this dimension. Incidentally, all of this would cause one to wonder how, under any reasonable application of the word "important", it could be suggested that pitch is not the most important of the musical dimensions, since its susceptibility to musical structuring includes and exceeds that of any other dimension.5

If, in discussing Babbitt's treatment of rhythm and the other non- pitch dimensions, one is unenthusiastic about using the familiar desig- natory terms 'total' or 'integral serialism', it is not because such terms are, in themselves, wildly inaccurate or inappropriate, but simply that they have associations with a short-lived and long-since-discarded fashion of European music which flourished briefly in the early and mid-

4 Benjamin Boretz: "Milton Babbitt", in The Dictionary of Contemporary Music, ed. J. Vinton (New York, 1974), p. 47.

5 Milton Babbitt: "Contemporary Music Composition and Music Theory as Con- temporary Intellectual History", in Perspectives in Musicology, ed. B. S. Brook, E. O. D. Downes, and S. Van Solkema (New York, 1972), p. 165.


die fifties. In 1955, Babbitt recorded the disappointment which followed his early acquaintance with this fashion:

The first explicit steps in the direction of a 'totally organized' twelve- tone music were taken here some fifteen years ago (i.e., in the United States in 1940), motivated .. . by the desire for a completely autono- mous conception of the twelve-tone system, and for works in which all components, in all dimensions, would be determined by the rela- tions and operations of the system.... The specific bases.., .for achieving a total twelve-tone work were arrived at by the end of the war, and when, a short time later, there were reports of a group of young French, Italian and German composers who apparently shared like aims, their work was eagerly awaited. However, their music... eventually revealed... so very different an attitude toward the means, and even so very different means, that the apparent agree- ment with regard to ends lost its entire significance."

The differences which distinguish Babbitt's treatment of the concept of the twelve-tone system as related to all the musical dimensions from that of the European practitioners of 'integral serialism' are fundamen- tal and far-reaching and can hardly be overstated. In particular: * his treatment does not require the perception of intervallic scaling

or ratio scaling in dimensions which cannot sustain them, and, for this reason (not because of an ideological stance), the pitch dimension is accorded first importance as a structural determinant;

* his pitch and rhythmic structures are both articulated explicitly in the surface of the music, and are not numerologically manipu- lated at a 'pre-compositional' stage so that they give rise to a 'polyphony' whose relationships are unforeseeable and impercepti- ble; pre-composition is a means of characterizing or discovering general systematic relationships, not an early stage of the compositional process.

Babbitt's own critical comments on European 'integral serialism' un- derline the divergences of compositional practice:

Mathematics-or, more correctly, arithmetic-is used.., .as a compositional device, resulting in the most literal sort of 'programme music', whose course is determined by a numerical, rather than a narrative or descriptive 'programme' ... rhythm is independent of and thus separable from the pitch structure; this is described and justified as a 'polyphony' of components, though polyphony is

6 Milton Babbitt: "Some Aspects of Twelve-Tone Composition", in The Score and IMA Magazine (June 1955), p. 55.


customarily understood to involve, among many other things, a principle of organized simultaneity, while here the mere fact of simultaneity is termed 'polyphony'. The most crucial problems of twelve-tone music are resolved by being defined out of existence; harmonic structure in all dimensions is proclaimed to be irrelevant, unnecessary, and perhaps, undesirable in any event; so, a principle, or non-principle, of harmony by fortuity reigns. Finally, the music of the past-and virtually all of that of the present as well-is repudiated for what it is not, rather than examined-if not celebrated-for what it is; admittedly, this is a convenient method for evading confrontation by a multiple of challenging pos- sibles, including-perhaps-even a few necessaries.7

A STUDY: A descriptive overview of STRING QUARTET NO. 3

The sectional divisions of the String Quartet No. 3 break the work down thus: * into four main parts (for each of which the array dimensions are

8 X 96), with further subdivisions into * eight sections (two per part), defined by the completion in all

eight horizontals of four set statements; * thirty-two subsections-

within each section it seems reasonable to identify four subsections of unequal length, in which each of the eight horizontals completes one set statement, even though the delineation is, in fact, only approximate (repetition of terminal and anticipation of initial set elements extends across the subsection divisions to effect the desired aggregate partitionings) ;

* two-hundred and eighty aggregates (either eight or nine per subsection, thirty-five per section, and seventy in each of the four main parts of the quartet).

The twelve-tone set of the quartet is shown in Ex. 7. This set is one of the possible orderings of the unordered hexachord pair (E-F-F?- An-B-C) (G-G?-A-C#-D-Eb), which, under the usual transformations of the twelve-tone system, can generate only three different hexachordal contents. The distinction among these three contents is an important basis of the structure of the quartet. The three contents referred to are:

'A': (E-F-Fj$-Bb-B-C) (G-Gi$-A-Ct-D-Eb) 'B': (F-F$-G-B-C-C#) (G#-A-Bb-D-Eb-E) 'C': (F$-G-G$-C-C$-D) (A-Bb-B-Eb-E-F)

7 Ibid., p. 55.


-It

Ex. 7

Within each of the four parts of the quartet, each pair of adjacent layers of the array maintains the same hexachordal content. The rela- tions between these pairs, within and across the parts of the work, form the pattern shown in Ex. 8.

Part One Two Three Four

Layers 1/2 A B C A 3/4 A B C A 5/6 B C B C 7/8 C A A B

Ex. 8

Each of the four 8 X 96 arrays of the quartet partitions into seventy aggregates, moving through all seventy partitionings of the aggregate into 1..... 8 strata. The arrays for the first two parts of the work, showing these partitions, are displayed in Fig. 1.

The four parts are parallel in structure, since the arrays are identical to within transposition, inversion, retrogression, and combinations thereof. The pattern of their succession (assuming, arbitrarily, that the first part is toS of the array as a whole) is as follows (Ex. 9) :

Bar Array Part Aggregates references transformation

1 1- 70 1-139 toS 2 71-140 140-270 t2R

3 141-210 271-390 t,,o 4 211-280 391-end tIR

Ex. 9

As in all the later works, from Ensembles on, the array of the first part is subject to complete reinterpretation, from the point of view of in- strumentation, registral assignment, harmony, dynamics, and rhythm


each time it is restated, so that a quite radically different textural surface results each time.

Example 10 shows the way in which layers are instrumentally assigned throughout the work, and also the way in which two independent kinds of timbral and registral re-interpretation are superim- posed upon them; during the course of the second and third parts of the quartet, the sixteen constituent subsections are articulated by:

(i) all sixteen (24) possible dispositions of the alternative timbres 'con sordino' and 'senza sordino' amongst the four instruments

(ii) all sixteen possible dispositions amongst the four instruments of the two alternative pairs of indications 'pizzicato/arco' and 'upper register/lower register'.8

The rhythmic structure of the quartet derives from an array which is isomorphic with the first two parts (aggregates 1-140) of the pitch structure.9 At the opening, the rhythmic aggregates unfold at the ratio of exactly 1:3 in relation to the pitch aggregates, as far as pitch aggregate 27/rhythmic aggregate 9 (ending in bar 53) ; thereafter, the ratio increases immediately to under 1: 2, so that, by the end of the piece, it has averaged out at exactly 1:2.

Unlike Sextets, for example, it is the rhythmic subsections, not the pitch subsections, which are sharply defined by the work's frequent changes in 'velocity'. At the opening rhythmic subsection one, (rhythmic aggregates 1-9), the rhythmic interval 'one' is assigned the durational value of a demisemiquaver,* but this is re-interpreted with each

8 In the published score of String Quartet No. 3, the first violin does not remove its mute at m. 377, nor at any subsequent point. Example 10, however, shows the mute removed between m. 377 and the end of the work. It seems that things should have worked out as in the example, since the removal of the mute at that point provides the sixteenth and last disposition of the con/senza sordini timbres amongst the four instruments. It seems to represent a more problematic inconsistency than the occasional rhythmic/dynamic discrepancy, such as occurs on the first page of the printed score. It would be helpful to have the composer's view of these questions.

9 The given 140 aggregates of the pitch array chart (Fig. 1) may be read as a rhythmic array chart by substituting time-point numbers for the given pitch classes, F? being equivalent to time-point zero, F$ to one, etcetera. (For a full explanation of Babbitt's time-point system, see the composer's article "Twelve-Tone Rhythmic Structure and the Electronic Medium", PNM, Vol. 1, No. 1.) Each of the eight layers of the rhythmic arrays, as is usual when Babbitt employs this system of rhythmic organization, is defined by being articulated by its own dynamic level. There are eight such levels, ranging from ppp to fff.

* (demisemiquaver = a 32nd-note; crotchet = a quarter note)

Part 1 (bar 1) Part 2 (bar 140)

Pitch Sect.:"I 1-2 1 3 4 Subsection: 1-8 9 10 11 12 13 14 15 16 Aggregates: 1-70 71 79 88 97 106 115 124 132 Bar refs: 1-139 140 151 175 190 204 218 233 249

Layer Vln.I - Cello arco arco arco upper arco arco upper arco arco C C C C S S S S S

Layer Vln.I --- Cello

2 pizz. pizz. pizz. lower pizz. pizz. lower pizz. pizz. C C C C S S S S S

Layer Vln.II -~ Viola

3 arco arco arco arco arco lower arco lower arco C C C S C C C S S

Layer Vln.II -

Viola,_ 4 pizz. pizz. pizz. pizz. pizz. upper pizz. upper pizz. C C C S C C C S S

Layer Viola Vln. I

5 arco arco upper arco arco arco arco arco upper C C C C S C C C S

Layer Viola - 0 Vln. I

" 6 pizz. pizz. lower pizz. pizz. ' pizz. pizz. pizz. lower C C C C S C C C S

Layer Cello - Vln. II

7 arco arco arco arco lower ' arco arco lower lower C C S S S S C C C

Layer Cello -~ Vln. II

8 pizz. pizz. pizz. pizz. upper pizz. pizz. upper upper C C S S S S C C C

Key: 'upper' = upper register; 'lower' = lower register; 'C' = con sord.; 'S' = senza sord.

Ex. 10

Part 4 Part 3 (bar 271) (bar 391)

Pitch Sect.: 5 6 7-8

[Subsection: 11 17 18 19 20 21 22 23 24 25-32

Aggregates: 141 150 158 167 176 185 194 203 211-280

Bar refs: 271 287 297 317 ' 330 341 358 377 391-end.

Layer Vln. II ,

Viola

pizz. pizz. upper upper pizz. upper upper upper upper C C S S C C S S S

Layer Vln. II 0 Viola arco arco lower lower arco lower lower lower lower 2 C C S S ' C C S S S

Vln. Ii ' Cello Layer arco upper arco upper upper upper arco upper upper

3 S S S S S C C S S S S Si

Layer Vln. I Cello

Ly pizz. lower pizz. lower lower lower pizz. lower lower S S S S S C C S S

Cello Vln. II--- Layer lower pizz. lower lower lower pizz. lower lower upper S C C C C C S S S

Cello a Vln. II--- Layer 6 upper arco upper upper upper arco upper upper lower

S C C C C C S S S

Layer Viola I Vln. I-- upper upper pizz. pizz. upper upper upper upper upper

7C C C S '

S S S S S Viola ' Vln. I Layer lower lower arco arco lower lower lower lower lower

8 C C C S S S S S S

Key: 'upper' = upper register; 'lower' = lower register; 'C' = con sord.; 'S' = senza sord.

Ex. 10 (cont.)

STRING QUARTET NO. 3 171 new rhythmic subsection. There are eight different interpretations of the rhythmic interval 'one', arising from the division of the crotchet into two, three, four, five, six, seven, eight, and twelve parts. Each kind of 'metre' occurs once per main rhythmic part following the pattern detailed in Ex. 11.

SECTION DURATIONAL VALUE RHYTHMIC BAR

(RHYTHM) OF INTERVAL 'ONE' AGGREGATES REFERENCE

IA (1) 1-9 1

IA (2) 10-17 54

IA (3) 18-26 78

IA (4) 27-35 132

IB (1) 36-44 162

IB (2) L--3

45-53 200

IB (3)

__

54-61 226

IB (4) 62-70 266

IIA (1) 71-79 289 L--3 IIA (2) 80-87 308

IIA (3) 88-96 338

IIA (4) 97-105 367 L-3J

IIB (1) 106-114 401

IIB (2) 115-123 425

IIB (3) 3 124-131 461

IIB (4) -3

132-140 483

Ex. 11


Composed for and Dedicated to The Fine Arts Quartet:

Leonard Sorkin - Abram Loft Bernard Zaslav - George Sopkin

String Quartet No.3 duration: ca. 181/2 minutes MILTON BABBITT

1969/1970 Szca. 72/1 $ con sord. / j - ?

Violin 1 o d

SP p P ppp N :- -= r f con sord. pizz. 6r Violin 2 gig con AV"-pp

sord.

pizz- ViolaME f con sord. Pz.

Violoncello ,1 ,- f .

pizz F41

pizz

pp ppp

Sarco f Imp f p ff ff anf

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Note: Accidentals, without exception, apply only to those notes which they immediately precede.

Babbitt: String Quartet No. 3, mm. 1-11

PART ONE [bar 1]

Aggregate 1 Aggregate 2 Aggregate 3 Aggregate 4 Layer 1 G D: D G: C: A Layer 2 F Ft B A# C E Layer 3 F E B C A AS F? Layer 4 D? G G? D A C? Layer 5 B D# E AS E A Layer 6 F? D C? G C Gj Layer 7 C C# F# F G B Layer 8 A G D$ E D

Aggregate 5 Aggregate 6 Aggregate 7 Aggregate 8 Layer 1 C C E Ft F A# Layer 2 E G D? G? D C A Layer 3 F D? G D G? A C Layer 4 C? A? Ft E F C B Layer 5 A FI D C C? G G Layer 6 G? B E F F E A Layer 7 D A DI Layer 8 AS G B B

Aggregate 9 Aggregate 10 Aggregate 11 Aggregate 12 Layer 1 B D C ( Gf A G DI Layer 2 F# At B Layer 3 A? F? F B E C D G Layer 4 D D Layer 5 G At A E F D# Layer 6 AS G G( C? C D Layer 7 D? A G? E F C? C F# B Layer 8 F# C C$ F E G A

Fig. 1

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Aggregate 13 Aggregate 14 Aggregate 15 Aggregate 16 Layer 1 C E B F F? At Layer 2 F C Layer 3 A G? C# D Layer 4 DI G( G A' Cj E C Layer 5 B G C G C I D Fj Layer 6 F A F A? E DI B Layer 7 G A I D E DI G( A Layer 8 D) At D B G F F CSf

Aggregate 17 Aggregate 18 Layer 1 A C? D G? D? G E C A# Layer 2 E C# A G G# D# D B Layer 3 Layer 4 F B A# F# Layer 5 Layer 6 Layer 7 F# F Layer 8 C

Aggregate 19 Aggregate 20 Layer 1 B Layer 2 Layer 3 Layer 4 At F F# E C A C ( G# D Layer 5 A F E AS D? B D F# G# G Layer 6 G Layer 7 C C#t Layer 8 D#

Fig. 1 (cont.)

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CI)

CI

Aggregate 21 Aggregate 22 Aggregate 23 Aggregate 24 Layer 1 F# F G# Layer 2 D D# G Layer 3 B C F E F# A# A# C# A D G$ G D# Layer 4 C# A Gj D G D# F# At C B E F F Layer 5 C C C# Layer 6 B D F# Layer 7 C# B G E G# D# A A# Layer 8 E A

Aggregate 25 Aggregate 26 Layer 1 Layer 2 E Layer 3 DI) Layer 4 Layer 5 Layer 6 F? G C# G# C A F D# E B A# Layer 7 D Layer 8 A G# A# D F C# F# C B G

Aggregate 27 Aggregate 28 Aggregate 29 Aggregate 30 Layer 1 G? A D C#

D#F Layer 2 E C B F Af F) Layer 3 C E F B F#f A# Layer 4 G# G D D# C# A Layer 5 A# B E D# F A Layer 6 G F# C# D C Layer 7 C? F F# C G B G# Layer 8 D A# A D# G# E G

Fig. 1 (cont.)

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Aggregate 31 Aggregate 32 Aggregate 33 Aggregate 34 Layer 1 D# G A# F? F# B F Layer 2 A C? D? D G G? Layer 3 G G D C D Layer 4 F? A? F B C E Layer 5 C C G? G# Layer 6 G# F A E E At Layer 7 E D D D$ A? A A Layer 8 B B C C C F Ft

Aggregate 35 Aggregate 36 Aggregate 37

Layer 1 F E C Layer 2 C E F Layer 3 A G#G B Layer 4 G A D C D G Ag F Layer 5 C? G F D Layer 6 AL B D#I Layer 7 Layer 8 B G F C F C E Gg A A D D$

Aggregate 38 Aggregate 39 Aggregate 40 Layer 1 C? A

G(: D G Layer 2 F B F# A#( G D#t C# Layer 3 A# F F? E C C A Layer 4 B F Layer 5 E Layer 6 G? C C? G D F# D B Layer 7 Layer 8 A Aft DIf D E G#

Fig. 1 (cont.)

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Aggregate 41 Aggregate 42 Aggregate 43 Layer 1 G D$ F? Layer 2 D Layer 3 A Ct G( D Layer 4 E C C Layer 5 E D$ A? B A F D F C? G G? Layer 6 B A At F E Layer 7 D# Layer 8 G? B G C F? F C A?

Aggregate 44 Aggregate 45 Aggregate 46 Layer 1 F? A? C B E F D D$ G? G A C$ E C Layer 2 D A G G? F Layer 3 DI G F? F? At B F Layer 4 G Layer 5 Layer 6 C? C? Layer 7 C B Layer 8 Ag D D$ A E

Aggregate 47 Aggregate 48 Aggregate 49 Layer 1 F B Layer E B C A A? F# Layer 3 C E C? A G G( D# D Layer 4 D? D G? C# A Layer 5 C D? B A? E A F Layer 6 C( D G G F? Layer 7 F? Layer 8 G F

Fig. 1 (cont.)

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Aggregate 50 Aggregate 51 Aggregate 52 Aggregate 53 Layer 1 B At F? D# Layer 2 D# G D G A C? Layer 3 D F Layer 4 A C E F? F AS B Layer 5 G C D C? F? Layer 6 G C D# B E E A? A F Layer 7 G F C# A# D A D? E G? G Layer 8 C# B C G F?

Aggregate 54 Aggregate 55 Aggregate 56 Aggregate 57 Layer 1 D# Dj G Layer 2 A? F? F B B Layer 3 F F# B A? C C Layer 4 D C? G# A A G D? Layer 5 G E E F Layer 6 C G? G C C Layer 7 B C F C? F D A? G? A E Layer 8 D# D D A A? G? E

Aggregate 58 Aggregate 59 Aggregate 60 Layer 1 G G# G# Layer 2 E C D# G A G? C? D Layer 3 E Layer 4 C E B F F A? Layer 5 A? A B D F: D G C C Layer 6 F? D F A B A# D# Layer 7 Layer 8 C F

Fig. 1 (cont.)

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Aggregate 61 Aggregate 62 Aggregate 63 Layer 1 D A C? A? F? E F C B G# G D Layer 2 Layer 3 G D# G# D C? A Layer 4 At Layer 5 G( F F A A# E B D#t Layer 6 E C C Layer 7

DI: Ff Layer 8 C F? G B

Aggregate 64 Aggregate 65 Aggregate 66 Aggregate 67 Layer 1 D# C A

F: A? F Layer 2 B C F Layer 3 E C B F A F? A Layer 4 A C? D G? D? G E Layer 5 C G? F? G D Layer 6 G G? F# D B D# At E Layer 7 G C B C F Layer 8 GD E D# A D A# Cj

Aggregate 68 Aggregate 69 Aggregate 70 Layer 1 B C E Layer 2 E F? A? C? A D G? G D# Layer 3 C D? D G Gj Layer 4 C A? B Ft F Layer 5 C? Layer 6 F A Layer 7 G# E A D# D A? Layer 8 F G F? B C

Fig. 1 (cont.)

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PART TWO [bar 140]


Layer 1 F# D C# Layer 2 F A A# E B D? C G? F# Layer 3 A? A E F D# Layer 4 G G C C D Layer 5 D#: Layer 6 B G Layer 7 C E F B F A# Layer 8 D C G# A G

Aggregate 74 Aggregate 75 Aggregate 76 Aggregate 77

Layer 1 G C G# B D# F Layer 2 G D C# Layer 3 B G C G C# D F# Layer 4 F? A F AS E D# B Layer 5 E A G? A? D Layer 6 F? C F C? E G# A# A Layer 7 G D# C? D A Layer 8 D? C E B F F# A#


Layer 1 E A A C? D G F# G# C D# B E Layer 2 Layer 3 B D# E A? F A Layer 4 C Layer 5 F C? F? C B G G At Layer 6 D D? F? Layer 7 G# F Layer 8 C# A G? D

Fig. 1 (cont.)

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Aggregate 81 Aggregate 82 Aggregate 83 Aggregate 84 Layer 1 A# A# A A Layer 2 E D# A# B A F D F$ C# Layer 3 F#t D Layer 4 C G# G C# F# D F Layer 5 D D# A E G# F C? B C Layer 6 F C C# B G E G# G# Layer 7 F#t B A# C E G D#: Layer 8 G D#:

Aggregate 85 Aggregate 86 Aggregate 87 Aggregate 88 Layer 1 F F Layer 2 C# G G# C Layer 3 D C C# G# G Layer 4 A B B A# D# E C# Layer 5 G F# F# A G# D# E D Layer 6 D D# A A# D G B C Layer 7 G# D C# A At F# F Layer 8 F# A# C B E E F

Aggregate 89 Aggregate 90 Aggregate 91 Aggregate 92 Layer 1 F G# C Ct G Layer 2 D# B A# E A F G# C Layer 3 G E E F A# A B D# F# D Layer 4 C G G#t F# D B Layer 5 A# Layer 6 F# F# C? F D A# Layer 7 B E C D# G A Layer 8 G# A D C# D#

Fig. 1 (cont.)

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Aggregate 93 Aggregate 94 Aggregate 95 Layer 1 D F? D# B A A# F E Layer 2 C D C# F# Layer 3 G C# C G( Layer 4 B DI A# E F Layer 5 G B F$ C C# F D Layer 6 G? A A E D? Layer 7 G( Layer 8 G A#

Aggregate 96 Aggregate 97 Aggregate 98 Aggregate 99 Layer 1 G F# C# D C G# G# F A Layer 2 G A# A# B E Layer 3 G( A F E A# D$ Layer 4 A D D F? Layer 5 A# A D G# E G B C# C Layer 6 D? D# F# G C B C? Layer 7 C( D F Layer 8 F? B F E C

Aggregate 100 Aggregate 101 Aggregate 102 Layer 1 A E A# B D# Layer 2 E D# F A C G? C? G Layer 3 B B D D F# G? Layer 4 G Layer 5 F F# Layer 6 C: Layer 7 Layer 8 C D? G G# D A C# A A# F# E F C

Fig. 1 (cont.)

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Aggregate 103 Aggregate 104 Layer 1 Layer 2 G F# D Layer 3 G C C? Layer 4 C# G( C A F DI E B Aj Layer 5 F? Layer 6 F Gj E A D# D A# Layer 7 Layer 8 B

Aggregate 105 Aggregate 106 Aggregate 107 Layer 1 D F# G G Cj Layer 2 Aj A Layer 3 A? B E D$ F Layer 4 F# D Layer 5 E G( A DI Layer 6 F Cj C C Layer 7 F E B C A# F? D? G D G? A C$ B Layer 8 G#

Aggregate 108 Aggregate 109 Aggregate 110 Aggregate 111 Layer 1 G? G# C A F Layer 2 E F DI B G# C G Cj Layer 3 A A Layer 4 C# G C Gjt B D# F E Layer 5 A Aj D D Layer 6 F# F? B G A D Layer 7 B C F E E Fj A# Layer 8 G D D# D# C C# A Fj

Fig. 1 (cont.)

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Layer 1 F D# E B A# Layer 2 D FI Layer 3 C G# C? G F D Layer 4 A A# Layer 5 B G F F? C? C Layer 6 E D# G? A Layer 7 C# A D Gj G D# Layer 8 Aj? F B C E

Aggregate 115 Aggregate 116

Layer 1 Layer 2 FI Layer 3 F Layer 4 Layer 5 Layer 6 C C? F? F G B D A# D# A Gj Layer 7 E Layer 8 A C# D G? D# G E C As B


Layer 1 Aj G G( Layer 2 Layer 3 F A A# E B D C Layer 4 G G F# C? D C Gj F A E Layer 5 D# D D Layer 6 G( E C Layer 7 C B F A# F? A Cj D# Layer 8 B F?

Fig. 1 (cont.)

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Aggregate 120 Aggregate 121 Layer 1 C# Layer 2 A F E E A? D# B D F? G? G C Layer 3 C G( F ( G D C# Layer 4 AjL B D# Layer 5 A Layer 6 Layer 7 Layer 8 F

Aggregate 122 Aggregate 123 Layer 1 C D F? A F A? E DI B Layer 2 C# Layer 3 Layer 4 Layer 5 A A? G? E C? F C F# G B Layer 6 Layer 7 DIF D G G( Layer 8

Aggregate 124 Aggregate 125 Aggregate 126 Aggregate 127 Layer 1 C

G#: G

Layer 2 E F A# A B D F$ D Layer 3 E Layer 4 G# C C# G D F D? B A A# F Layer 5 G# E DI A D At C Layer 6 C# F F# C G B Layer 7 B A# F F? Layer 8 D D# G# G A C# E C

Fig. 1 (cont.)

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Aggregate 128 Aggregate 129 Aggregate 130 Aggregate 131 Layer 1 Cj F# D F A B A? D? E Layer 2 G Cj C G(j Layer 3 Dj A# B A F D F# C# G G? C Layer 4 E Layer 5 F G F? B C Layer 6 Gj E D D# A A Layer 7 E C A C# G? D D? G Layer 8 F B A? F#

Aggregate 132 Aggregate 133 Aggregate 134 Aggregate 135 Layer 1 C# C G G# F# D Layer 2 B D# E A? F Layer 3 DI B A? E A F Layer 4 E C# D G F# G? C Layer 5 A A A I D# D E G( Layer 6 C B F G Layer 7 F# A? B F F C Layer 8 G D? D G C C# A

Aggregate 136 Aggregate 137 Aggregate 138 Aggregate 139 Aggregate 140 Layer 1 D B D# A? E F A Layer 2 A Fj F D C Cj( G# G Layer 3 Gj G# C C D C# F G Layer 4 D# D# B E A# A F Layer 5 B B F# C F# F C# Layer 6 G F# C# A# A# D A D# E G# Layer 7 E C# A G G# D# D Layer 8 C E F# F A B

Fig. 1 (cone.)

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an introduction and a study string quartet no. 3 by milton babbitt

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