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Partition lattices in the music of Milton Babbitt Hennessy, Jeffrey James 2002

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PARTITION LATTICES I N THE MUSIC OF MILTON BABBITT by  JEFFREY JAMES HENNESSY B.Sc, Trent Univerity, 1992 B.Mus., Acadia University, 1999  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in  THE FACULTY OF GRADUATE STUDIES (School of Music; Theory Division) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 2002 © Jeffrey James Hennessy, 2002  4/9/2002 11:35 AM  UBC Special Collections - Thesis Authorisation Form  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n .  Department  o f /^\,Q^b>\C_ ^~~~Vv~<e^c^\ ^^"> v'^ ^fc><~^ <  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r , Canada  Date  >Xe^r~A  V  http://www.library.ubc.ca/spcoll/thesauth.html  V^  O  Columbia  C  ^  Page 1 of 1  ABSTRACT  This thesis explores how partitions and partition lattices can provide insight into smallscale and large-scale analytical relationships in several twelve-tone compositions by Milton Babbitt. Chapter 1 outlines the concept of partition lattices, as they have recently been described by Richard Kurth; summarizes the definitions and conventions of the lattice model; and suggests how they might be applied to Babbitt's music. Chapter 2 examines how recurring hexachords and hexachordal partitions arise from lattice interactions of the trichordal array partitions in the Woodwind Quartet (1953). Chapter 3 applies the lattice model to the first twelve measures of the Composition for Viola and Piano (1950) and explores how partitions corresponding to the surface texture are all subpartitions of a single hexachordal partition. Chapter 4 examines how partition lattices suggest various different levels of partition structure in the String Quartet No. 2 (1954). Chapter 5 summarizes the conclusions of the preceding chapters, outlines the limitations of the analytical method, and discusses the possibilities for applying partition lattices to the all-partition array works of Milton Babbitt.  ii  T A B L E O F CONTENTS Abstract  ii  Table of Contents  iii  List of Tables  iv  List of Figures  v  List of Examples  CHAPTER I  vii  Introduction 1.1 1.2 1.3 1.4  CHAPTER II  CHAPTER III  CHAPTER IV  1  Pre-history 1 Preliminaries 5 Partition Lattices 8 Partition Lattices and the music of Milton Babbitt. . 11  Trichordal arrays and recurring hexachords in the Woodwind Quartet (1953)  19  Hexachordal partitions and their surface subpartitions in the Composition for Viola and Piano (1950)  51  A partition lattice model of structural levels in the String Quartet no. 2 (1954)  CHAPTER V  Conclusions  77  . . 124  Bibliography  131  iii  LIST OF TABLES Chapter 2 2.1 2.2  Array, mm. 1-3 Lyne partitions (trichords t, and t,), mm. 1-3  20 22  2.3 2.4 2.5  Lyne partitions (hexachords h, and h ), mm. 1-3 Additional hexachord partitions, mm. 1-3 Columnar lattice interactions, mm. 1-3  23 31 32  2.6 2.7 2.8 2.9  Array, Introduction section 36 Array, Canons for Clarinet 37 Hexachordal partitions and lattice interactions, Canons for Clarinet. 41 Array, Duets for Bassoon 47  Chapter 3 3.1 3.2 3.3  Hexachordal array, mm. 1-5 Piano partitions, mm. 1-5 List of partitions resulting from trichord interactions, mm. 1-5 . . . .  58 58 60  3.4 3.5 3.6  Partitions, mm. 6-8 Partitions, mm. 8-12 Meet and join of viola partitions, mm.8-12  61 66 68  Chapter 4 4.1 4.2 4.3 4.4 4.5  Partitions, mm. 4-6 Automorphic transformations of L, L L and L (014) Trichordal array, mm. 7-18 Partitions, mm. 19-23 Partitions, mm. 24-33 and interaction with dyn  4.6 4.7 4.8  Partitions, mm. 47-64 Partitions, mm. 75-92 Partition inventories at various lattice levels  2  2  3  3  iv  4  87 89 90 99 100 106 H2 119  LIST OF FIGURES Chapter 2 2.1 2.2  Lattice L! of trichord partitions, mm. 1-3 Lattice L of hexachord partitions, mm. 1-3  24 27  2.3 2.4 2.5  Lattice L of lyne partitions, mm. 1-3 Trichord I mapping that yields s Lattice L of columnar trichord partitions, mm. 1-3  29 30 33  2.6  Lattice L of columnar hexachord partitions, mm. 1-3. .  34  2.7 2.8 2.9 2.10 2.11 2.12 2.13  Lattice L of columnar partitions, mm. 1-3 Lattice L of trichordal lyne partitions, Canons for Clarinet Lattice L of hexachordal lyne partitions, Canons for Clarinet Lattice of L of lyne partitions, Canons for Clarinet Lattice L of columnar partitions, Canons for Clarinet Lattice L„ of partitions, Duets for Bassoon Partition Memberset Connections  35 39 42 43 44 47 49  Chapter 3 3.1  Lattice L , - viola, mm. 1-5  55  3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11  Lattice L - viola, mm. 1-5 Composite lattice L - viola, mm. 1-5 Lattice L - viola, mm. 6-8 Composite lattice L - viola, mm. 6-8 Lattice L of partitions, mm. 6-8 showing viola sublattice L Hexachord lattice L - viola, mm. 8-12 Trichord lattice L - viola, mm.8-12 Composite lattice L - viola, mm. 8-12 Lattice L of partitions, mm. 8-12 showing viola sublattice, L Lattice L„ of hexachord partitions, mm. 1-12  56 57 62 64 65 69 69 70 72 74  2  3  l  3  4  5  6  7  8  9  1 0  2  3  4  5  6  4  7  8  9  1 0  8  Chapter 4 4.1 4.2  Sample lattice showing three levels of structural organization Lattice L of partitions from mm. 1-3  79 81  4.3 4.4  Structural levels, mm. 1-3 Lattice L of partitions from mm. 4-6  83 85  4.5 4.6 4.7  Lattice L of dynamic trichord partitions from O E 1 and 2 Lattice L of instrumental trichord partitions from O E 1 and 2 T cycling in d! and a  88 88 89  4.8  Lattice L of partitions, mm. 7-18 showing cyclic mapping  93  4.9  Cyclic transformational mappings of L partitions  95  4.10  Lattice L of lyne partitions, mm. 7-12  96  t  2  3  4  4  5  5  6  4.11  Lattice L of lyne partitions, mm. 13-18  4.12  Lattice L of trichordal partitions, mm. 19-25  101  96  4.13 4.14  Lattice L of trichordal partitions, mm. 24-31 Composite lattice L of lattices L and L  103 104  7  g  9  1 0  5  v  9  List of Figures, cont. 4.15  Lattice L of partitions, mm. 47-62  109  4.16 4.17 4.18  Structural levels, mm. 52-62 Lattice L of partitions, mm. 75-81 Structural levels, mm. 75-81  HI 116 117  4.19 4.20  Lattice L Lattice L  117 120  u  l 2  1 3  1 4  of partitions, mm. 75-92 of background hexachordal partitions  vi  LIST O F EXAMPLES Chapter 1 1.1 1.2  All-combinatorial hexachords All-combinatorial hexachords and their trichordal generators  2 3  1.3 1.4 1.5  Conjunct and disjunct partitions Examples of common partition formats Sample partitions p and q and their meet and join  7 8 9  1.6 1.7 1.8 1.9 1.10  Sample lattice Trichordal array Trichordal array partitions with their relative meets and joins Lattice of partitions from Ex. 1.8 Sublattices of L  10 \\ 12 13 16  Chapter 2 2.1 2.2 2.3  Woodwind Quartet, mm. 1-3 showing distribution of 03 dyads . . . . Canons for Clarinet, mm. 51-56 Duets for Bassoon, mm. 161-164  21 38 46  2  Chapter 3 3.1  Composition for Viola and Piano, mm. 1-12  53  3.2  Piano partitions, mm. 8-12  67  4.1 4.2 4.3  mm. 1-6 mm. 7-18 mm. 19-33  80 91 98  4.4  mm. 35-42  105  4.5  mm. 47-51  107  4.6 4.7 4.8  mm. 52-62 mm. 75-80 mm. 81-92  108 H3 H4  All-Partition array block, Post-Partitions Lattice of partitions from Ex. 5.2  127 128  Chapter 4  Chapter 5 5.1 5.2  vii  CHAPTER 1: INTRODUCTION  1.1. PRE-fflSTORY Since the basic pre-compositional task of a twelve-tone composer is to determine how to order the tones, it is desirable to seek rational artistic principles for doing so. Early practitioners addressed this problem by considering the aggregate to be a divisible entity, parsable into smaller pitch-class sets; and twelve-tone rows were consequently designed such that they could likewise be partitioned into smaller subsets that typically shared common intervallic and transformational features (Babbitt 1955). Initially, the most profound application of aggregate partitioning resulted from the division of the basic row into transformationally-related hexachords. Such hexachordal "combinatoriality" is a common feature in the music of Arnold Schoenberg (Babbitt 1955, 1960). It has been repeatedly demonstrated that Schoenberg was concerned as much with the large-scale preservation of combinatorial hexachords as he was with preserving pitchclass ordering in his twelve-tone rows. (Lewin 1962; Kurth 1996). Schoenberg's hexachordal design provides a window into the more general structural potential of the twelve-tone row. Early studies of the properties of hexachordally combinatorial rows led to the notion of degrees of combinatoriality, with certain hexachord types possessing a greater number of combinatorial transformations. The result was the discovery of the six hexachord set classes that exhibit the greatest degree of combinatoriality (Babbitt 1955). Example 1.1 shows these six  "all-  combinatorial"hexachords identified by the letter names given them by Babbitt, and by the corresponding set-class type.  1  Set-Class (012345)  Type A B  (023457)  C  (024579)  D  (012678)  E  (014589)  F  (02468A)  Example 1.1: All-Combinatorial Hexachords (in pitch-class integer notation).  These all-combinatorial hexachord classes extend the possibilities of aggregate partitioning in several ways. "First, as they may all be inverted onto themselves and both transposed and inverted onto their complements, a given all-combinatorial collection of six pitch classes may represent all the possible classical transformations of the interval patterns of both discrete hexachords of a row" (Mead 1994, 22). Donald Martino's landmark essay entitled "The Source Set and its Aggregate Formations" (Martino 1961) also showed how the all-combinatorial hexachord classes can be generated from a select group of trichord "sources" (Martino 1961). Example 1.2 lists the six hexachord classes with their corresponding trichordal generators, shown here as the sum of two trichord set classes. The relevance of these properties of combinatoriality to the music of Milton Babbitt cannot be overstated. The early writings of Babbitt dealt extensively with these issues, as did his compositions during this period (Babbitt 1955, 1960).  Both Babbitt  and Martino discuss the relationships among combinatorial hexachords in relation to trichordal generators. As will be seen, the early works of Milton Babbitt featured these combinatorial properties of hexachords and trichords. Furthermore, the pre-compositional 2  arrays on which the music was based featured further partitions of these hexachords into source trichords (Mead, 1994).  Set-Class  Trichord Generators (012) + (012) (013) + (013) (014) + (014) (024) + (024) (012) + (015) (013) + (014) (013) + (025)  Type A  (012345)  B  (023457)  (013) (025) (015) (024) (012) (013) (014)  + + + + + + +  (013) (025) (015) (024) (027) (037) (025)  (024579)  (025) (027) (037) (024) (027) (025) (013)  + + + + + + +  (025) (027) (037) (024) (015) (037) (025)  (012678)  (012) (015) (027) (016) (016)  + + + + +  (012) (015) (027) (016) (026)  (014589)  (014) (015) (037) (048)  + + + +  (014) (015) (037) (048)  (02468A)  (024) + (024) (048) + (048) (026) + (026)  Example 1.2: All-Combinatorial Hexachords and their trichordal generators.  Daniel Starr and Robert Morris provided the authoritative discourse on the subject of combinatoriality with their "General Theory of Combinatoriality and the Aggregate" (Starr and Morris 1977, 1978).  Morris provided a further generalized theory in  "Combinatoriality without the Aggregate" (Morris 1983). Though not used directly here, Starr and Morris's (1977) concept of a Combination Matrix, which stems largely from 1  the initial work of Babbitt and Martino, provides useful insight into Babbitt's combinatorial designs as well as those of other composers, such as Morris himself (Morris 1987). More recently, Babbitt's and Martino's insights have been developed in various ways. Steve Rouse expanded on Martino's work, providing a systematic, general treatment of the construction of hexachords from trichordal source sets (Rouse 1985). Daniel Starr also provided further insight into the possibilities of aggregate partitioning, considering the invariant properties of certain set classes as partition generators (Starr 1978). David Lewin's theory of  segmental association (Lewin 1962) deals with partition  subsets as identifiable features in Schoenberg's music and introduces the concept of the nesting of an aggregate as layers of smaller subsets. Ethan Haimo and Paul Johnson (Haimo and Johnson 1984) provided further applications of partition analysis by examining isomorphic partitioning in Schoenberg's music. Robert Morris and Brian Alegant (Morris and Alegant 1988) provided an alternate viewpoint of aggregate partitioning by disregarding the source set as an aggregate construct and focusing on a more generalized theory of the "even" partitions in twelvetone music.  2  Alegant's exhaustive dissertation on the theoretical implications of all  possible aggregate partitions provided the most generalized and thorough exposition on the topic (Alegant 1993). See Starr and Morris 1977 for the definition and application of a combination matrix (CM). Even partitions divide the aggregate into subsets of the same cardinality: two hexachords, or three tetrachords, or four trichords, or six dyads. 1  2  4  Andrew Mead, in a landmark essay (Mead 1988), further developed theoretical ideas on aggregate structuring that Morris, Alegant, and Starr had described in detail. Mead's concept of an isomorphic relation between pitch classes and order numbers lead to his own concept of a "mosaic," while Kurth (1992, 1993) adopted a methodology 3  similar to Mead's to explore aspects of Schoenberg's twelve-tone music. Kurth's notion of mosaic polyphony provided useful insight into the formal organization of Schoenberg's music on various levels. Furthermore, Kurth's application of mosaic polyphony combined with Lewin's nesting concept lead to the eventual consideration of various partitions being superimposed in the same musical space. Alegant's and Kurth's more recent work (Alegant 1996; Kurth 1996, 1999) has focused on the common feature of all of this work: on partitions, applied in analysis and related to one another in various ways and engaged in a variety of theoretical systems. Kurth's (1999) formulation of partition lattices offers one way to examine how partitions can relate to one another and how they can generate new partitions.  1.2. PRELIMINARIES In the abstract, pitch-class partitions can be studied with regards to any tuning system and any associated aggregate containing two or more pitch classes. We shall concern ourselves here with the twelve-tone context, defining the aggregate as the unordered collection containing twelve pitch classes of the equal-tempered chromatic scale. The integers 0 to 11 (modulo 12) will be used to denote the pitch classes C ascending chromatically to B. The integers 10 and 11 will be represented in hexadecimal notation by the letters A and B. The complete twelve-tone aggregate can thus be represented as: {0,1, 2, 3, 4, 5, 6, 7, 8, 9, A , B}.  There is some discrepancy as to the definition of this term. Mead's concept of a mosaic is congruous with the definition of partition used here. 3  5  Integers contained within curly brackets and separated by commas will be taken to mean an unordered collection of pitch classes: eg. {0, 3, 9, A}. A collection represented by round brackets and without commas will refer to a particular set class: eg. (014). A n ordered collection of specific pitch classes will be contained within angled brackets: eg. <23B>. The basic pitch-class transformations of transposition and inversion will be represented as follows: T  x  will be used to denote transposition of a particular pitch class  or collection by the ordered pitch-class interval x. I will be used to denote inversion x  about the 0 axis followed with transposition by T . The multiplication transformation 4  x  will not be used.  The following definitions and conventions are taken from Kurth 1999.  1.1  Definition. A pitch-class partition is a collection of subsets of the aggregate,  called the membersets of the partition, such that every pitch class in the aggregate is a member of only one partition memberset (Kurth 1999).  Partitions will be labelled with lowercase letters. The membersets of the partition will be separated by vertical lines and, by convention, the pitch classes of each memberset will be ordered numerically (even though the memberset is an unordered set). In turn, the partition membersets will be numerically ordered according to the smallest integer of each memberset: eg. p = I 0124 I 3B I 589A I 6 I 7 I Subscript integers will generally be used to distinguish partitions labelled with the same lowercase letter. The size (number of pes) of every partition memberset is listed as the format of the partition. Integers are used to specify the number of pes in each memberset and 4  This is sometimes denoted T l . x  6  exponents are used to indicate multiple occurrences of a particular memberset size. For example, the Format (FRMT) of the above partition, p, would be expressed as FRMT(p)=l 24 . 2  2  The inventory of a partition will list the set-class types of the membersets in the partition. This is of particular importance, as we will frequently be concerned with partitions comprising several membersets of the same set class. The inventory (TNV) lists the set classes in order of size with exponents again used to indicate multiple occurrences of the same set class. The lowercase letter s will be used to denote singleton sets (containing only one pc). The inventory of the above partition, p, would be expressed as INV(p) = (s) (04)(0124)(0125). 2  There are two special partitions that are assigned a specific identity. The partition with only one memberset (containing all twelve pitch classes) is referred to as the  conjunct partition and labelled conj. The partition containing twelve membersets (each containing only one pc) is referred to as the  disjunct partition and labelled disj. Example  1.3 lists these two partitions.  conj= I 0123456789AB I disj = I 0 I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 I 9 I A I B I Example 1.3: The conjunct and disjunct partitions.  In this study, certain partition formats will occur frequently and will be assigned a particular letter designation to distinguish them from other formats. Partitions containing two hexachords will feature prominently, and will be labelled h  x  with the subscript  designating specific h-type partitions. Partitions consisting of four trichords will also feature prominently and will be labelled t . Partitions with F R M T = 1 "2" will be referred x  to as s (for singleton) partitions and will be labelled s - We will return to these types in X  7  some detail later. For the present, Example 1.4 lists an instance of each of these partition formats. Other partitions or partition types may be designated by certain descriptive labels. For example, partitions that correspond to the dynamic markings of a certain passage will be labelled "dyn," and so forth.  INV=(014589) INV=(012) INV=(s) (01)  h = I014589I2367ABI t= I 012 I 345 I 678 I 9AB I s = I 01 I 2 I 34 I 5 I 67 I 8 I 9A I B I  2  4  4  4  Example 1.4: Examples of common partition formats.  1.3. PARTITION L A T T I C E S Several distinct partitions can be projected simultaneously in the same musical passage, with each partition corresponding to different features of the musical texture. The need for some method of comparing various partitions and analyzing their interaction is thus necessary. Richard Kurth's use of algebraic lattice theory (Kurth 1999) provides a useful method for comparing partitions and organizing collections of partitions. The methodology used here will closely follow Kurth's model. Partitions are combined into known respectively as partition  5  partition lattices through two binary operations  meet and partition join.  1.2 Definition. Let p and q be partitions. The  meet of p and q, written pMq, is  the partition whose membersets are the non-empty intersections of the membersets of p with the membersets of q. Equally, the pes x and y are in the same memberset of pMq if and only if x and y are in the same p-memberset and also in the same q-memberset. (Kurth 1999)  See Kurth 1999 for a detailed exposition of the definitions, theorems and derivations relating to partition lattices.  5  8  1.3  Definition. Let p and q be two partitions. The join of p and q, written pjq,  is the partition whose membersets are given by the unions of any and all intersecting membersets of p and q. Equally, the pes x and y are in the same memberset of pJq if x and y are in the same p-memberset or in the same q-memberset. (Kurth 1999)  Example 1.5 shows two partitions, p and q, with their inventories and their meet and join.  p = I 012 I 345 I 678 I 9AB I q = I 013 I 245 I 679 I 8AB I pMq = s-| = I 01 I 2 I 3 I 45 I 67 I 8 I 9 I AB I pjq = hi = I 012345 I 6789AB I  INV=(012) INV=(013)  4  4  INV=(s) (01) 4  4  INV=(012345)  2  Example 1.5: Sample partitions p and q and their meet and join.  Partitions p and q above are both subpartitions of partition h i . That is, the membersets of p and q are all subsets of some memberset in h i . Likewise, partition hi is a superpartition of both p and q. Partition hi is referred to as the least upper bound for the collection {p, q}. In other words, it is the "smallest" superpartition of both p and q. Partition s is referred to as the greatest lower bound for the collection {p, q}. That is, it is the "largest" subpartition of both p and q. The four partitions above form a partition lattice, the collection of partitions being closed under join and meet:  1.4 Definition. A collection of partitions in which every pair of partitions has a least upper bound (join) and a greatest lower bound (meet) is called a partition lattice. Equally, a collection L of partitions is a partition lattice if for every p and q that are elements of L , the join pJq is an element of L and the meet pMq is an element of L.  9  1.5 Definition. A partition lattice in which every subcollection of partitions has a least upper bound (join) and greatest lower bound (meet) is called a complete lattice.  Example 1.6 graphically shows lattice L i containing partitions h i , p, q and s i . The diagram graphically represents the subpartition relations among the various relevant partitions in the lattice.  si  Example 1.6: Sample Lattice.  10  The partitions produced through the meet and join of the "original" trichordal partitions may have analytical significance for a musical passage in which p and q are analytically relevant. Certainly, the generation, through join, of the A-type allcombinatorial hexachord class (012345) - the class that is the inventory of hi - from two different trichordal generators, (012) and (013), which are the inventories respectively of p and q, is a significant manifestation of Martino's source set discourse. As we shall see, lattices of varying size and complexity can offer useful information relating to large-scale partition design and small-scale partition substructure. Partition lattices provide information of particular analytical significance to the music of Milton Babbitt.  1.4. PARTITION L A T T I C E S A N D T H E MUSIC OF M I L T O N B A B B I T T  Beginning with the Three Compositions  for Piano in 1947, Babbitt's music was  primarily concerned with the projection of aggregates derived from trichordal arrays. Example 1.7 shows a hypothetical but typical array. Both the "lynes" (horizontal rows) and the vertical columns of the array are composed of discrete trichords. Furthermore, the trichords are all representations of set class (014). Example 1.8 lists the trichordal partitions at work in this array, either in its columns or its lynes, along with their meets and joins. Example 1.9 shows the lattice that contains all of these partitions.  014  589  236  7AB  = ti  589  014  7AB  236  = t|  23B  67A  019  458  =t2  67A  23B  458  019  = t2  t3  t4  t4  t3  Example 1.7: Trichordal A r r a y 11  H = I 014 I 236 I 589 I 7AB I  INV=(014)  t2 = I 019 I 23B I 458 I 67A I t = I 014 I 23B I 589 I 67A I t4 = I 019 I 236 I 458 I 7AB I  INV=(014)  4  INV=(014)  4  tlJt3 = J1 = I 014 I 2367AB I 589 I t1 Jt4 = J2= I 014589 I 236 I 7AB I t2Jt3 = J3= I 014589 I 23B I 67A I t2Jt4 = J4= I 019 I 2367AB I 458 I  INV = (014) (014589)  3  4  INV=(014)  4  2  INV=(014) (014589) 2  INV = (014) (014589) 2  INV = (014) (014589) 2  tlJt2 = t3Jt4 =h2 = I 014589 I 2367AB I INV=(014589) J1JJ2 = J1JJ3 = J1JJ4 = J2JJ3 = J2JJ4 = J3JJ4 = h2  2  tiMt3 = mi = I 014 I 23 I 589 I 6 I 7A I B I INV=(s) (01)(03)(014) t-|Mt4 = IT12 = I 01 I 236 I 4 I 58 I 7AB I 9 IINV=(s) (01)(03)(014) t2Mt3 = m3 = I 01 I 23B I 4 I 58 I 67A I 9 I INV= (s) (01)(03)(014) t2Mt4 = 1714 = I 019 I 23 I 458 1 6 I 7A I B IINV= (s) (01)(03)(014) 2  2  2  2  2  2  2  2  t-| Mt2 = t3Mt4= S2 = I 01 I 23 I 4 I 58 I 6 I 7A I 9 I B I INV= (s)(01)(04) miMm2 = miMm3 = miMm4 = m2Mm3 = m2Mm4 = m3Mm4 = S2 4  Example 1.8: Trichordal array partitions with their meets and joins.  The lattice interactions have many analytically suggestive features. The (014) trichordal lyne partitions t i and t2 join to form hexachordal partition h2, consisting of complementary E-type all combinatorial hexachords. h2 also results from the join of the (014) trichordal column partitions t3 and t4. Indeed, as the least upper bound of the entire lattice, the h2 partition can be viewed as the overriding "background" hexachord for the array. Pairs of (014) trichords can act as trichordal generators of this hexachord as indeed they do in this array. As we will see, single background hexachord partitions will feature prominently in Babbitt's music.  12  2  2  The meets of the array partitions also produce interesting results. Individually, the lyne partitions t i and t 2 meet to form the same partition S2 as do the column partitions t3 and ty. The membersets of S2 might correspond with certain analytical properties in a musical texture built from the trichordal array. The S2 membersets may correspond to distinctions of articulation, dynamic, register etc. within the trichords. Alternatively they may have prominent roles in forming phrases, cadences or other form-defining events.  S2  Example 1.9: Lattice L 2 of partitions from Ex. 1.8  13  We shall see that s-type partitions will feature so prominently in lattices resulting from Babbitt's arrays that it is useful to consider the conditions necessary to produce an s partition from two trichordal partitions. Consider t i and t2 from Example 1.8. It is clear that the trichordal membersets of t i all contain two and only two pes in common with those of t2- The extraction of four dyads and four singletons results from the meet of t i and t2- A n s-type partition will always result from the meet of two trichordal partitions whose membersets contain two and only two pes in common with each other. This is a significant structural feature of Babbitt's aggregate constructs, as we will see that s-type partitions have a stubborn propensity to surface in the lattices produced from Babbitt's trichordal partitions. Example 1.8 indicated that when either lyne partition is combined through meet with either of the column partitions, four different partitions result that have a common inventory. These "m" partitions each retain certain trichords from the original t partitions. Of most interest, however, is how all of these meet to form the lower bound S2 partition. No extraneous partitions result. Likewise, when either lyne partition is combined through join with either of the column partitions, one of the four "j" partitions results. Each of the j partitions contains one of the hexachordal membersets of h2 along with two (014) trichordal membersets from one of the t partitions. The least upper bound of all of the j partitions is h2. Once again, no extraneous partitions are produced. We thus have a complete lattice whose partition membersets have consistent interactions and achieve closure. As we will see, not all lattices will have such neat and symmetric structures. It is a simple exercise to build an array whose partitions will not form a symmetrical and relatively small lattice such as this one. Perhaps this lends extra significance to the fact that Babbitt's arrays often generate symmetrical and complete lattices of relatively small size. 14  The lattice L , represented in Example 1.9, can also be divided into sublattices that 2  are analytically significant. The sublattice represented on the perimeter of the figure contains the trichordal partitions t i and t2 with their meet, h2, and join, s2. These two trichordal partitions correspond to the horizontal lynes of the array and could thus represent the temporal distribution of pitch classes in a musical passage derived from the array. Another sublattice comprising t3, t4, h2 and S2 represents the trichordal partitions in the columns of the array. These two sublattices are shown individually in Example 1.10 where they are labelled L3 and L4. Note that each sublattice is itself a complete lattice. In essence, each sublattice represents a specific aspect of the theoretical musical passage resulting from the array while the larger composite lattice, which includes the m and j partitions resulting from the interaction of the partitions in L3 and L4, represents the inclusion relations among all membersets of all partitions implicit in the array. Although the sublattices L3 and L4 are isographic, they are not isomorphic under T and I as there is no transposition or inversion that will map all of the membersets of ti and t2 to corresponding membersets of t3 or t4. Both sublattices are, however, individually automorphic under I3. That is, each sublattice maps on to itself under the I3 transformation because 13(t 1) = t2,13(t3) I3-automorphic as  = t4 while h2  and s2 are 13-invariant. L2 is also  I3O1) = J3,13O2) = J4, l3(mi) = m3, J-3(m2) = m4. Graphically, I3  rotates L3 about its vertical axis. This inversional property may have analytical significance in a musical passage governed by the array.  15  Example 1.10:  16  Sublattices of L2  The following chapters will use partition lattices to approach various aspects of the music of Milton Babbitt. In all cases, we will focus on passages and works in which trichordal arrays figure prominently as pre-compositional designs. Thus, we will focus on a selection of Babbitt's early works. Chapter 2 will examine the trichordal arrays used in significant passages from Milton Babbitt's Woodwind Quartet of 1953. Lattices generated from the arrays will show how the trichordal partitions interact on several levels to produce both surface and background, theoretical hexachords. Chapter 3 will look at an isolated passage of Babbitt's Composition for Viola and Piano of 1950. Certain key features of the music will be brought out by the lattice model to demonstrate how surface details of the piece are connected through a deep structure, which is graphically represented in the lattices. Chapter 4 will offer a somewhat different interpretation of lattices relevant to passges from the 1952 String Quartet no. 2. Here we will examine the concept of structural levels and prolongation, and lattices will be used to represent these notions structurally and graphically. The question of whether or not certain partitions can be said to be prolonged over longer spans of music will be discussed, and the notion of background, middleground, and foreground levels of structure will be examined in this twelve-tone context. We will conclude with a brief discussion of the potential for the lattice model in analyzing Babbitt's music beyond his early trichordal works. Specifically, we will consider the potential benefits and possible problems in using the model to access Milton Babbitt's all-partition array compositions. In general, partition lattices will be shown to be analytically productive tools for exploring the music of Milton Babbitt. In certain cases, the lattices will confirm relationships that had been postulated through other means. In other cases, new 17  relationships will emerge from partition interactions. Furthermore, the lattices will prove to be efficient graphical representations of a multitude of pitch-class relationships in the works.  18  CHAPTER 2: TRICHORDAL ARRAYS AND RECURRING HEXACHORDS IN T H E WOODWIND Q U A R T E T (1953) The Woodwind Quartet was composed in 1953, building on compositional processes established by Babbitt in his Three Compositions for Piano (1947) and the song cycle Du (1952). In all of his works to this point, Babbitt was interested in the presentation of twelve-tone rows that exhibited inversional hexachordal combinatoriality (Mead 1994, 20-25). Furthermore, the pre-compositional arrays on which the music was based featured the further partition of these hexachords into source trichords both in the lynes and columns of the array (Mead 1994, 25-30). The Woodwind Quartet follows this practice, with each of the major sections of the piece involving its own individual array. One such array will be shown shortly in Table 2.1. The Quartet is organized as one movement, but it is divided into several sub-titled sections that feature different combinations of instruments. A three-measure opening unit is followed by six substantial sections: Introduction, Canons for Clarinet, Trios for Flute, Duets for Bassoon, Cadenza and Recitative for Oboe, and Finale. Each of these six sections utilizes its own unique array. The Canons section presents two registrally distinct aggregates in the Clarinet while the Oboe and Bassoon combine to form one aggregate and the Flute projects its own aggregate (Mead 1994, 93); each of these four lynes is divided info trichords so as to form four columnar aggregates. The Trios, as the name suggests, uses a counterpoint of three lynes divided into tetrachords to form three columnar aggregates. In this chapter however, analysis will be restricted to sections that utilize trichordal or hexachordal arrays. Predictably, two "parallel" trichordal partitions in an array (i.e. two lyne partitions or two column partitions) will often "join" to produce the pair of source hexachords for a given passage. This hexachordal partition can often be readily discerned  19  on the musical surface or in the array itself. In some instances, however, the trichordal partitions join to produce a pair of different hexachords, producing a hexachordal partition that does not initially appear to be acting in the array structure, at least on the surface. We will examine this theoretical relationship, to see how it provides an interesting link between different combinatorial hexachords that have common trichordal generators. In the Woodwind Quartet, it will be shown that the array for each section utilizes subpartitions of a different hexachordal partition, but that there is also a more abstract, "background" partition linking all of the hexachords that are projected throughout the piece. In addition, certain hexachordal partitions will "meet" to form trichordal partitions that are not part of the arrays associated with the original hexachordal partitions. The result is a complex network of hexachords and partitions which are connected through their common ancestry of trichordal generators. Finally, while it will be shown that four different trichord types dominate in the passages analyzed here, the various meets of the trichordal partitions will also demonstrate that two specific dyad types emerge as primary surface interval classes. Example 2.1 shows the first three measures of the Woodwind Quartet. This brief passage is meant as a self-contained unit that foreshadows the aggregate partitioning and transformational relationships found in the ensuing Introduction passage. Each instrument exposes a complete aggregate, which is naturally parsed into trichords; together the four instruments present the trichordal array shown in Table 2.1.  Flute 97A  32B 401  865  tl  Oboe 02B 67A 598  134  t2  Clar.  586  410  23B A97  tl  Bsn.  413  589  76A B02  t2  (013)(014)(014)(013) t3  t4  t4  t3  Table 2.1 Array, mm. 1-3.  20  Bb  f^ ~~™f ===  mf^=~mp  p—==zmp  Example 2.1: Woodwind Quartet, measures 1-3 showing distribution of 03 dyads. • All instruments at sounding pitch.  21  The array can be viewed as containing two levels of organization with regard to partitions. The columnar aggregates fall into two types. The first and last involve the same partition t3, composed only of (013) trichords; the second and third columns likewise present the same partition t4, now composed only of (014) trichords. This ordering, indicated by set-class types and partition labels below the array, gives the passage a simple palindromic structure. The lyne partitions, labelled ti and t2, are distinct but both involve two (013) and two (014) trichords. These partitions, built by redistributing the same constituent trichords as the columns, are perceptible according to the same kinds of parsings of the texture. We will consider the lyne partitions first, and consider the columnar partitions a bit later. The double reed instruments both correspond with partition t2, while the Flute and Clarinet share the different trichordal partition t i . As Table 2.2 demonstrates, each partition comprises two (013) and two (014) trichords. In addition, ti and t2 are T6 and I9 related. That is, the membersets of t i will map to those of t2 under either of these transformations. The trichordal partitions are perhaps the most discernible surface entities in the musical texture. One easily sees from Example 2.1 how the trichords are distinguished both temporally and dynamically in each instrument.  Fl/Cl  ti = I 014 I 23B I 568 I 79A I  INY=(013) (014)  Ob/Bsn  t2 = I 02B I 134 I 589 I 67A I  INV=(013) (014)  2  2  2  2  t2 = T6(tl) = l9(U) Table 2.2: Lyne Partitions t i and t2, mm. 1-3. Less easily perceived - but easily seen from Table 2.1 - is the fact that the Flute and Clarinet distribute the 11 trichords so as to form the same antecedent and consequent  22  hexachords. The Oboe and Bassoon project a transformationally-related hexachordal partition from their t2 trichords. These two hexachordal partitions are shown in Table 2.3. The partition h i corresponds to the temporal distribution of pitch classes in the Flute and Clarinet while h2 represents the temporal distribution of pes in the double reed instruments. Just as ti and t2 were related by T6 and I9, h i and h2 are also related by those same operations. Both hexachordal partitions have the same inventory, with a pair of (014568) hexachords.  Fl/Cl  h i = I014568I2379ABI  INV=(014568)  Ob/Bsn  h2 = I 0267AB I 134589 I  INV=(014568)  2  2  h2 = T ( h i ) = I (hi) 6  9  Table 2.3: Lyne Partitions, h i and h2, mm. 1-3.  The (014568) hexachord is one of six semi-combinatorial hexachords that cannot be produced from a single trichordal generator but that can be formed by several different combinations of two trichords (Martino 1961, 229). In this case, it is the (013) and (014) trichords of ti or t2 that are combined to generate the (014568) hexachords. While T6 and I9 related ti and t2, and hi and h2, all four partitions are also 13-invariant individually. These three transformational operations will be shown to govern other pitch-class relationships throughout the analyzed passages. The hexachordal and trichordal lyne partitions of the array generate new partitions from their meets and joins. These are shown in Figure 2.1 with the resulting lattice L i . L i is automorphic under T6 ,19, and I3. T6 and 19 map the membersets of ti to those of t2 while h* and si are each T6 and 19-invariant. All four partitions are 13-invariant.  23  M = I 014 I 23B I 568 I 79A I  INV = (013) (014) 2  2  t2 = I 02B I 134 I 589 I 67A I  INV = (013) (014)  h* = tl Jt2 = I 01234B I 56789A I  INV = (012345)  2  2  2  S-| = MMt2 = I 0 I 14 I 2B I 3 I 58 I 6 I 7A I 9 I  INV = (s) (03) 4  4  si  Figure 2.1: Lattice L i of trichordal partitions, mm. 1-3.  24  The trichordal partitions generate an s-type partition from their meet. The partition s\ contains four (03) dyads, a fact that will be especially important when we consider the significance of interval class 3 to the opening passage. The pitch-class distribution of m.l manifests the si partition in clearly patterned ways that can be seen by referring back to Example 2.1. Each instrument plays an (013) trichord that is articulated either as a single pitch followed by an (03) dyad (Flute and Oboe) or as the (03) dyad followed by the singleton (Clarinet and Bassoon). A similar pattern appears in m.2, corresponding to the second columnar aggregate of the array. There, the (014) trichords all involve consecutive (03) dyads, plus the leftover pitch before or after. In the final two columnar aggregates of this passage, ic 3 acts as the trichordal "boundary" interval with a singleton pitch class "filling in" between the "outer" pitches. As the array demonstrates, columnar aggregates 2 and 3 contain the same partition of (014) trichords while the first and last columns contain the same partition of (013) trichords. The (013) and (014) trichords share two common interval classes: i c l and ic3. The s i partition brings out the common ic3 dyads shared among the trichords of ti and t2, and our observations about mm. 1-3 have pointed the way in which those dyads are perceptible in the music. There are no i c l dyads shared by ti and t2, a fact marked by the conspicuous absence of (01) from the inventory of si. The join of t i and t2 produces an all-combinatorial hexachord partition consisting of chromatic hexachords. This result is interesting for several reasons. The chromatic hexachord is the only combinatorial hexachord that can be produced from either (013) or (014) trichords alone or from a combination of the two (Martino 1961, 229). This suggests the primacy of the chromatic hexachord in the overall design of this passage. The resulting partition is labelled h* in Table 2.4. As will be shown, this partition plays an  25  important mediating role throughout the various passages in the piece to be analyzed here, and it is therefore given a distinctive label. On the surface, this hexachordal partition offers an alternative listening strategy in terms of hexachords. Where h i and h2 corresponded to the temporal distribution of hexachords in the first three measures according to instrument pairs (double reeds vs. Flute and Clarinet), h* corresponds to the vertical aggregate partitioning in the same instrument pairs. For example, in m l , the combination of pes in the Oboe and Bassoon yields {0, 1, 2, 3, 4, B} while the Flute and Clarinet combine to form the complement {5, 6, 7, 8, 9, A}, forming the h* partition. While hi and h2 are necessarily conjoint, their meet produces an interesting result yielding the lattice L 2 as shown in Figure 2.2. The lower-bound partition of the lattice, m i , cannot be readily found on the musical surface, except that the pitch classes of the two tetrachordal membersets of the partition are exactly those found in the four dyads of si. The (03) dyads of s i are thus grouped as tetrachords in nil. In this case, however, the instrument pairing is different from the previous examples. The tetrachordal membersets of m i join the Flute dyads to the Oboe dyads and the Clarinet dyads to the Bassoon. For example, in mm. 1-2, both the Flute and Oboe exchange the dyads {7, A} and {2, B} while the Clarinet and Bassoon exchange {1, 4} and {5, 8}. Specifically, the second and third pes in m l in the Flute are (7, A} while the second and third pes in the Oboe's first measure are {2, B}. These are exchanged in m.2. A similar dyad exchange occurs with the first and second pes of the Clarinet and Bassoon in each of measures 1 and 2. The second half of the passage similarly exchanges the dyads but retains the same instrument pairing. Here, it is the first and third pes of each instrumental trichord that are exchanged. Specifically, the Flute's third trichord contains <41> as its first and third pes while the fourth trichord contains <85>; these are exchanged in the third and fourth trichords in the  26  Oboe, and a similar exchange occurs with the Clarinet and Bassoon. The m i partition is therefore important since it offers an alternative listening strategy, this time changing the way pairs of instruments are related by finding the common properties among the ways that hi and h2 organize the pes and instruments.  INV hi = I 014568 I 2379AB I INV h2 = l0267ABI 1345891 mi = hiMh2 = I 06 I 1458 I 27AB I 39INV I hlJh2 = conj  = (014568)  2  = (014568)  2  = (06) (0347) 2  2  conj  L2  mi Figure 2.2: Lattice L 2 of Hexachord partitions, mm. 1-3.  27  Before examining the columnar partitions of the array, it is interesting to observe the results obtained when L i and L2 are combined to make the larger lattice L3. Figure 2.3 illustrates how h* interacts within hi and I12, how m i interacts with ti and t2, and how no new partitions are generated thereby. The meets of h i and I12 with h* simply reproduce the two trichordal partitions t i and t2 found in the opening array. In addition, si resurfaces as the meet result between m i and each of those two trichordal partitions. L3 confirms the importance of h* as an abstract hexachordal partition that, in a sense, serves as a mediator among the various trichordal and hexachordal partitions of the lattice. In effect, h* reduces (via meet) the hexachords of hi and h2 to their corresponding trichords of t i and t 2 that ultimately reduce to s i . It is important to note that while the two kinds of parallel hexachordal combinatoriality are at work (the Flute and Clarinet presenting h i combinatoriality, the Oboe and Bassoon presenting h2 combinatoriality), the trichordal combinations involving tl and t2 (and, as will be examined shortly, the two columnar partitions) is what generates (via join) the h* partition and its chromatic hexachords. It is useful at this point to consider the possible automorphisms of L3. As Figure 2.3 shows, L3 is automorphic under T6, I9, and I3, as were L i and L2. These transformations provide some insight into the possible theoretical origin of the s i partition. Each trichord in t i maps onto a trichord in t2, (and vice versa) under both T6 and I9. The si partition is also invariant under both T6 and I9. In addition, t i and t2 are each invariant under I3 since  I3 = T6I9. As Figure 2.4 shows, I3 also maps the ic3 dyad 6  subsets of the t i membersets onto corresponding ic3 dyad subsets in the t2 membersets.  6  T6 is commutable with l . n  28  The preserved dyads (shown in bold in Figure 2.4) correspond to the dyad membersets of si, so the example demonstrates that si is also 13-invariant.  h* = tlJt2 = I 01234B I 56789A I hi =tiJmi=l014568l2379ABI r>2 = t2Jmi= I 0267AB I 134589 I ti = h*Mhi = I 014 I 23B I 568 I 79A I 1 2 = h*Mh2 = I 02B I 134 I 589 I 67A I mi = hiMh2= I 06 I 1458 I 27AB I 39 I S 1 = t-|Mt2= tiMmi = t2Mmi  INV = (012345)  2  INV = (014568)  2  INV = (014568)  2  INV = (013) (014) 2  2  INV = (013) (014) 2  2  INV = (06) (0347) 2  2  I 0 I 14 I 2B I 3 I 58 I 6 I 7A I 9 I INV = (s) (03) 4  4  conj  hi  L3  t1  L3  Figure 2.3: Lattice L 3 o f Lyne Partitions, mm. 1-3.  29  J6_ l9. l3  L3  H  014  23B  568  79A  t2  32B  104  A97  865  Figure 2.4: Trichord I3 mapping that yields s i .  Let us now consider the partitions inherent in the columns of the array. As previously shown in Table 2.1, these partitions are homogeneous, with t3 containing (013) membersets and t4 consisting of (014) trichords. Columnar hexachordal partitions are also evident when considering the assignment of dynamics in the first three measures. Each trichord is assigned a dynamic gesture consisting of a crescendo or decrescendo between two dynamic values. These dynamic gestures link the trichords into hexachordal units. For example, the first trichord articulated by the Clarinet crescendos from a dynamic level of tap to mf. This crescendo is then taken up by the Flute, which continues to a level of  forte. The six pitch classes involved in this crescendo {5,6,7,8,9,A}  correspond to one of the membersets of h*. The Oboe and Bassoon involve a similarly combined decrescendo gesture, from f to mf and m f to mp, and they produce the complementary h* memberset. Thus h* does, in fact, appear temporally on the musical surface by linking dynamic gestures. The pairings just described parse the first columnar aggregate into the hexachords of h*. The other columnar aggregates can also be parsed into h* hexachords, again on the basis of dynamic envelope, although each aggregate involves a slightly different kind of pairing. In the second columnar aggregate, for instance, the Clarinet decrescendos from f to mf. This is reversed by the Flute, which crescendos from mftofand the combination of pes produces the {0,1,2,3,4,B} hexachordal memberset of h*. The complementary h* memberset is produced in the Oboe and Bassoon by the same decrescendo/crescendo 30  gesture from mf to mp and mp to mf. The Flute/Clarinet and Oboe/Bassoon pairings produce the membersets of h* in each columnar aggregate and are related in each column by "connectable" dynamic gestures. Other instrument pairings may also be made, on the basis of different dynamic linkages. In the second columnar aggregate, the Clarinet and Bassoon yield an overall decrescendo. Specifically, the Clarinet decrescendos from fto mf while the Bassoon completes the decrescendo from mf to mp. Concurrently, the Oboe and Flute execute an overall crescendo, with the Oboe beginning at mp and ending at mf where the gesture is taken up by the Flute, completing the crescendo to f. The trichords of each pair combine to form a new hexachordal partition, h3, which contains (014589) membersets. In the fourth column, the Clarinet and Bassoon contain the same dynamic assignments, a decrescendo from fTto mf, while both the Flute and Clarinet decrescendo from mp to pp. These pairings produce the hexachordal partition h4, which contains (023457) membersets. These partitions are shown in Table 2.4. It is interesting to note that the same instrument pairing in the first and third columns also produces the I13 and h4 partitions even though there is no dynamic relationship between the members of each pair in these columns.  h3 = 1014589 I 2367AB I  INV=(014589)  h4 = I 0279AB I 134568 I  INV=(023457)  2  2  Table 2.4 Additional Hexachord Partitions, mm. 1-3  Both h3 and h4 contain all-combinatorial hexachord types. The (014589) hexachord can be generated by (014) trichords, but not by (013) trichords while the (023457) hexachord is easily generated by (013) trichords, but not by (014) trichords (Martino 1961, 229).  As will be shown shortly, the h* partition retains its role in the 31  lattices as mediator between these two additional hexachord partitions, because its (012345) hexachords can be generated by (013) and (014) trichords or by a pair of each type. For this reason, the h* hexachords can synthesize the mixed (013) and (014) trichords of t i and t2, as in lattice L i , or the strictly (013) or (014) trichords of t3 and t4 respectively, as in lattice L 4 to be shown shortly. Table 2.5 lists all of the lattice interactions of h*, h3,114, t3 and t4. Here we see the resurgence of si as the common denominator of the various partitions and the creation of a different, but related, meet partition labelled m2. The partition m2 has the same tetrachords as m i , but has two (03) dyads instead of two (06) dyads.  All hexachordal partitions are conjoint. INV=(013) t3= I02BI 134 I 568 I 79A I INV=(014) t4= I 014 I 23B I 589 I 67A I INV=(014589) h3 = l014589l2367ABI INV=(023457) h4 = I 0279AB I 134568 I INV=(012345) h* = I 01234B I 56789AI h*Mh3 = t4 h*Mh4 = t3 h3Mh4 = rr»2 = I 09 I 1458 I 27AB I 36 I INV =(03) (0347) t3Jt4 = h* t3Jm2 = h3 t4Jm2 = h4 t3Mt4 = S1 = I 0 I 14 I 2B I 3 I 58 I 6 I 7A I 9 I INV=(s) (03) t3Mm2 = si t4Mm2 = si 4  4  2  2  2  2  2  4  Table 2.5: Columnar Lattice Interactions, m m 1-3.  32  4  Figures 2.5, 2.6, and 2.7 show the lattices produced from the interactions listed in Table 2.5. L4 and L5 are sublattices of Le in the same way that L i and L2 interact through join and meet to produce L3.  h* = t3Jt4 = I 01234B I 56789A I  INV=(012345)  t3 = I 02B I 134 I 568 I 79A I  INV=(013)  t4 = I 014 I 23B I 589 I 67A I  INV=(014)  2  S1 = t3Mt4 = I 0 I 14 I 2B I 3 I 58 I 6 I 7A I 9 I  4  4  INV=(s) (03) 4  4  h*  L  4  S1  Figure 2.5: Lattice L 4 of columnar trichord partitions, mm. 1-3.  33  h.3 = I 014589 I 2367AB I  INV = (014589)  h4 = I 0279AB I 134568 I  INV = (023457)  2  2  m2 = h3Mh4 = I 09 I 1458 I 27AB I 36 I INV = (03) (0347) 2  2  conj  h  4  m2 Figure 2.6: Lattice L 5 of columnar hexachord partitions, mm. 1-3.  34  h* = t3Jt4= I 01234B I 56789A I h3 = t3Jm2= I 014589 I 2367AB I h4 = t4Jm2= I 0279AB I 134568 I t3= h*Mh4= I 02B I 134 I 568 I 79A I t4 = h*Mh3= I 014 I 23B I 589 I 67A I m2 = h3Mh4 = I 09 I 1458 I 27AB I 36 I si = t3Mt4 = t3Mm2 = t4Mm2 = I 0 I 14 I 2B I 3 I 58 I 6 I 7AI 9 I  INV = (012345)  2  INV = (014589)  2  INV = (023457)  2  INV = (013)  4  INV = (014)  4  INV = (03) (0347) 2  INV = (s) (03) 4  4  conj  h3  t4  h4  m2 t3  T6  S1  Figure 2.7: Lattice Ltf of columnar partitions, mm. 1-3.  35  •Le  2  It is interesting to compare the positions of h* and si in L 3 and L 6 . The h* partition acts as the common, mediating partition in both cases, and behaves and interacts in analogous ways, while si is the lower bound partition of both lattices. L 6 does not, however, retain all of the automorphisms that apply to L 3 . O f those three automorphic transformations, only T 6 is an automorphism on L 6 owing to the invariance of each partition in the lattice under Tg. Every partition in L 6 has a distinct inventory, so no operation maps one partition onto any other. The only automorphisms on L 6 will be those operations under which every partition in L 6 is invariant. Only T 6 meets this criterion.  Let us now examine the  Introduction  section. Its array is shown in Table 2.6.  When compared with Table 2.1, it is immediately obvious that this array is, in fact, nothing more than a 90° rotation of the array used for the opening three measures, so that ti and t2 now appear as the column partitions, and t3 and t4 as the lyne partitions. Thus, L 1 - L 6 retain their significance as applicable lattices for the  Introduction section as well. In  addition, the relevance of the earlier columnar hexachord partitions I13 and h4 is further confirmed by this array, since their component hexachords are now the consecutive hexachords appearing in the lynes of the array.  104  598  23B  32B A76 014 586  314  79A B02 tl  t2  A67  t4  589  t4  A79 02B  t3  865  t3  tl  431 t2  Table 2.6: Array, Introduction Section. 36  However, where the theoretically constructed s i , m i , and m2 retain a certain analytical significance in mm. 1-3, these partitions do not seem to reassert themselves in the Introduction  section by associating with repeating patterns in one or more  parameter(s). Instead, they remain as implicit partitions or entities, and ultimately act as abstract echoes of the previous passage.  The Canons for Clarinet section begins in measure 52, directly following the Introduction section. This passage also conforms to a trichordal array. In this case however, the Clarinet articulates two registrally distinct aggregates (see Example 2.2) while the Oboe and Bassoon combine to form one aggregate and the Flute projects its own aggregate (Mead 1994, 93). The array is shown in Table 2.7. Once again, the lynes of the array offer two trichordal partitions. The registral trichords are exact pitch inversions of each other. In mm. 51-53, the trichords are symmetrically opposed about the G4/Ab4 axis, while in mm. 54-56, the trichords invert about the A4/Bb4 axis. This is shown in Example 2.2. The trichordal partitions are shown in Figure 2.8 along with their meet and join and the resulting lattice Lj. (The column partitions are labelled on the array, but these will be discussed later.)  Flute  341  856  0B2 7A9  t3  Ob/Bsn 02B  A79  134  t3  Clar.  576  OBI A89 342  Clar.  A89 342 t7  t8  576 t8  568 (lower register)  15  OBI (upper register)  t5  t7  Table 2.7 Array, Canons for Clarinet.  37  Example 2.2: Canons for Clarinet, measures 51-56. All instruments at sounding pitch.  Registral Trichords of Clarinet shown in boxes.  38  t3 = I 02B I 134 I 568 I 79A I  INV = (013) Others  t5 = I 01B I 234 I 567 I 89A I  INV - (012) Clarinets  4  4  h* = t4Jt5= I 01234B I 56789A I  INV = (012345)  2  S2 = t4Mt5= I OB I 1 I 2 I 34 I 56 I 7 I 8 I 9A I  INV = (s) (01) 4  4  Figure 2.8: Lattice L 7 of trichordal lyne partitions, Canons for Clarinet.  39  Both lynear Clarinet aggregates are partitioned into the (012) membersets of t5 while the other instruments revisit the (013) trichords of t3. Once again, the join of these two partitions produces the h* partition. The chromatic hexachord membersets of h* can be generated from either a pair of (012) or a pair of (013) sets, but not by one of each (Martino 1961, 229).  This means that t3Jt5 = h* because t3 and t5 respectively  subdivide the hexachords of h* into a pair of (013) or (012) trichords. The meet of the two trichordal partitions produces a new s-partition. The S2 partition contains four (01) dyads, as opposed to the four (03) dyads of si. As with the previous arrays, hexachordal partitions other than h* arise from the temporal succession of pitch classes in each lyne. In fact, two more hexachord types emerge as surface entities in this passage, as shown in Table 2.8. Both of the new hexachord types are all-combinatorial. The (013) trichords of t3, as they are presented in pairs by Flute and by Oboe/ Bassoon, produce a pair of type B (023457) hexachords constituting the new partition I15, while the (012) trichords presented in pairs by the Clarinet generate a pair of D-type (012678) hexachords in the new partition h6. Table 2.8 lists the lattice interactions of h*, I15, h6, t3 and t5 with resulting lattices shown in Figures 2.9 and 2.10. L 8 differs from previous lattices involving surface hexachords (L2 and L 5 ) in that the join of the hexachordal partitions is not an "m-type" partition but another homogeneous trichordal partition, t6- The t£ partition is comprised of four (015) trichords that can act as single generators of both (012678) and (023457) hexachords (Martino 1961, 229). The lattice generation of t6 is particularly interesting, because its (015) membersets are at odds with the (012) and (013) trichords of the array, even though it is derived from composite hexachords of the array.  40  h5= I 0279ABI 1345681 h6= I 01567BI23489AI  INV=(023457)  2  Others  2  INV=(012678)  Clarinet  All hexachordal partitions are conjoint. h Mh6 = t6 = I 07B I 156I29AI 348 I 79AI h*Mh5 = t3= I 02B I 134 I 568 I h*Mh6 = t5= I 01B I 234 I 567 89AI I  INV=(015)  t Jt5 = h* = I 01234B I 56789A I t Mt5 = S2 = I OB I 1 I 2 I 34 II 756I I t3Jt6 = h5 = I 0279AB I 134568 t3Mt6 = S2 t5Jt6 = h6 = I 01567B I 23489A t5Mt6 = S2  INV=(012345)  5  Table 2.8: Hexachordal partitions and Canons for CI.  4  INV=(012)  4  2  3  3  INV=f013)  4  8 I 9A I  INV=(s) (01) 4  4  INV=(023457)  2  INV=(012678)  2  Lattice Interactions,  L9 (Fig. 2.10) combines L 7 and L8 into a single, complete lattice that is isographic to L3 and L6. Like the previous composite lattice (L6), L9 is automorphic under T6 only, owing to the invariance of each partition under that operation. Every partition in the lattice has a different inventory, so there are no operations that map the partitions onto one another. Notice the recurring position of h* in this lattice, just as in L3 and L6. The chromatic h* partition seems yet again to be mediating the various other trichordal and hexachordal partitions of the Quartet.  41  h =: I 0279AB I 134568 I  INV=(023457)  h =: I 01567B I 23489A I I 07B I 156 I 29A I 348 I t = h5Mh6= l  INV=(012678)  5  6  6  2  2  INV=(015)  4  conj  t6 Figure 2.9: Lattice L g of hexachordal lyne partitions, Canons for Clarinet.  42  INV=(023457) h5 = t3Jt6 = I 0279AB I 134568 I INV=(012678) h6 = t5Jt6 =l 01567B I 23489A I t6 = h5Mh6 = I 07B I 156 I 29A I 348 I INV =(015) INV=(013) t3 = h*Mh5 = I 02B I 134 I 568 I 79A I 15 = h*Mh6 = I 01B I 234 I 567 I 89A I INV=(012) h* = t3Jt5 = I 01234B I 56789A I INV=(012345) S2 =t3Mt5 = t3Mt6 tsMt6 = S2 = I OB I 1 I 2 I 34 I 56 I 7 I 8 INV=(s) I 9A I (01)  2  2  4  4  4  2  4  conj  Figure 2.10: Lattice L9 of lyne partitions, Canons for Clarinet.  43  4  An examination of the columnar partitions from the array will confirm the structural importance of h* in this passage. Figure 2.11 lists the columnar trichordal partitions along with their meet and join and the resulting lattice Lifj.  h* = t7Jt8 =1 01234B I 56789A I  INV = (012345)  2  t7 = I 02B I 134 I 567 I 89A I  INV = (013) (012)  ts = I 01B I 234 I 568 I 79A I  INV = (013) (012)  S2 = I OB I 1 I 2 I 34 I 56 I 7 I 8 I 9A I  INV = (s) (01)  2  2  2  4  2  4  h*  MO  S2 Figure 2.11: Lattice Lio  of columnar partitions,  Canons for Clarinet.  44  The various partitions and lattices of the Canons for Clarinet section demonstrate a consistency in array design from previously analyzed passages. A l l of the trichordal arrays up to this point have featured two trichordal partitions in each of the columns and lynes of the array. Moreover, in each case, one pair of partitions (either the columns or the lynes) featured two trichord memberset classes while the other pair contained four of the same trichord memberset type. The partition t3, which contains (013) trichords only, has also been present in each of the arrays to this point, as has h*, which continues to function as the primary hexachord partition in each of the arrays. The structural importance of h* will be further confirmed with a closer look at another passage of the Quartet.  The first four measures of the Duets for Bassoon section of the Quarter (mm. 161164) are shown in Example 2.3. As the name suggests, this passage comprises two contrapuntal lines of overlapping hexachords. The Bassoon always forms a complete aggregate while the other instruments contribute hexachords in turn to form the counteraggregate. The hexachords are all (012345) sets and furthermore, all conform to the h* partition. Here the h* partition appears on the horizontal musical surface for the first time. Table 2.9 shows the two-lyne array that dictates the pitch distribution for the Bassoon's first aggregate and its accompaniment. While the musical texture itself does not strongly induce a trichordal organization, it is still useful to construct the array in terms of trichords in order to see how the source hexachord is generated. As with the first sections of the quartet, the hexachords of the h* partition are here formed by combining (013) and (014) trichords in both lynes of the array. Figure 2.12 shows the two trichordal partitions as well as their meet and join and the resulting lattice L i 1. The join produces the h* 45  partition. The meet, however produces a new inventory of s-partition than seen previously. The S3 partition contains both dyad set classes of the previous two spartitions, (01) and (03), each type now occurring only twice. L\1, like L3 is automorphic under T6,19 and I3. Here, the automorphism results because of the invariance of both trichordal partitions under T6 and from the fact that r.9 and tioare I3 and I9 images of each other.  DUETS FOR BASSOON J~»6  Flute Oboe Clarinet in Bb Bassoon  3J Flute Oboe Clarinet in Bb Bassoon  Example 2.3: Duets for Bassoon, mm. 161-164. 46  (4J)  034 B12 6A9 857 = h*,t9 956 7A8 30B421  =  h*,  tio  Table 2.9 Array, Duets for Bassoon.  tg = I 034 I 12B I 578 I 69A I  INV=(013) (014) 2  2  t10= I 03B I 124 I 569 I78AI  INV=(013) (014)  h* = tgJtlO= I 01234B I 56789AI  INV=(012345)  2  2  2  S 3 = tgMt-lO = I 03 I 12 I 4 I 5 I 69 I 78 I A I B I  INV=(s) (01) (03) 4  Figure 2.12: Lattice L11 of partitions, Duets for Bassoon.  47  2  2  To summarize our observations about trichordal arrays and the various hexachords that have been extracted from them, Figure 2.13 lists all of the characteristic memberset types of the partitions seen so far. This figure, while not a lattice in itself, demonstrates the high degree of interaction among the characteristic membersets of the partitions. Whenever two partitions have appeared together in any of the preceding lattices, a line segment on Fig. 2.13 links their characteristic memberset types. Equally, two memberset types of different cardinalities are linked when they sustain an (abstract) subset/superset inclusion relation. Consequently, the trichords are always mosaic constituents (in Martino's sense of "mosaic") of the hexachords to which they are linked. Clearly, the central, structural role of h* is paramount, with all other hexachords relating to it by lattice association. That is, h* is the only hexachordal partition that appears in all of the larger lattices (L3, L6 and L9). It is also interesting to see how the other hexachordal partitions are each derived from their respective trichordal generators which in turn also act to generate h*. The lattices L 3 , L6 and L 9 show that h* is further partitioned into (012), (013), (014) and (015) trichords.  7  These trichords are then "regrouped" to form  other hexachordal partitions. O f the four trichords, the (013) and (014) trichords can also be said to occupy central positions linking the various trichords and hexachords in a similar way as h* does. The m and s partitions provide details pertaining to the musical surface itself. It is clear that interval class 3 is predominant during the opening and Introduction passages while interval class 1 abounds in the Canons for Clarinet section. The Duets for Bassoon combines the previous surface tendencies by containing an equal balance of both (01) and (03) dyads.  Only the (024) and (025) trichordal generators are missing from the complete list of trichords that can generate the A-type, chromatic hexachord (Martino 1961, 229). 7  48  49  The partition lattice model is thus analytically useful when considering the compositional design of the Woodwind  Quartet. While the surface trichords and  hexachords are easy to discern, the lattice model effectively brings out other trichords and hexachords in the music's sub-structure. A complex web of relationships among hexachords and their trichordal generators emerges, with certain trichords and hexachords especially (013), (014) and (012345) - playing a more fundamental, structural role than others, demonstrating their particular combinative properties, as outlined by Martino. The lattice analysis of these passages from the piece also serves to demonstrate the compositional system of Babbitt's earlier works. Throughout the period when the Woodwind  Quartet was composed, Babbitt was primarily interested in inversional  combinatoriality of source hexachords constructed from various trichordal generators. The trichords were clearly the source cells of the musical process, while the hexachords governed the distribution of trichords throughout the musical texture. The lattice model demonstrates, however, that there is more to this music than the mere scattering of like trichords into hexachords in an array structure. The joins of the various trichordal partitions demonstrate that a unifying hexachordal partition is acting as a central mediator among the various trichords and hexachords. In turn, the meets of certain hexachordal partitions reveal additional trichordal generators acting in the background while the trichordal partition meets yield m and s partitions which favour particular interval classes (namely ics 1 and 3). The result is a more detailed picture of the theoretical relationships existing among the surface entities in the music. It is also clear that the music does not just present hexachords and trichordal generators one at a time. Multiple combinatorial properties are often all operating at once, with various trichords generating several kinds of hexachords in different dimensions of the music. The partition-lattice model helps bring out these multiplicities and aids in their systematic exploration. 50  CHAPTER 3: H E X A C H O R D A L PARTITIONS AND THEIR SURFACE SUBPARTITIONS IN T H E COMPOSITION FOR VIOLA AND PIANO (1950). Milton Babbitt's years before the  Composition for Viola and Piano (1950) was composed three  Woodwind Quartet. While Babbitt's interest in inversional hexachordal  combinatoriality is also at work in this piece, the compositional method of pitch-class distribution is quite different from that of the  Quartet. In the Quartet, we observed the  derivation of multiple hexachord types from shared trichordal generators. As Mead has shown, the  Composition for Viola and Piano relies on one focal, all-combinatorial  hexachord type that serves as the source for all of the resulting trichordal partitions (Mead 1994, 76-89). But beyond the use of one primary hexachord type, a partition-lattice analysis of the first twelve measures of the work will reveal a single "background" hexachordal partition that governs all of the pitch material. This is tantamount to saying that the source hexachord and its complement exist in this piece at only one transpositional level. Thus, Babbitt was primarily concerned with extracting maximal subset diversity from a single hexachordal source. While the lattice model served in the previous chapter to divulge a multiplicity of hexachord types that participated, in a more or less theoretical sense, at a background level in the  Woodwind Quartet, the lattices here will serve an increasingly practical  purpose. That is, the partitions in many of the lattices generated here will be shown to exhibit analytical significance in the music itself. The various hexachordal, trichordal, and s-type partitions will elucidate surface and background details active in the musical texture. Example 3.1 shows the first twelve measures of the  Composition for Viola and  Piano. Aggregate partitioning of both the viola and piano parts suggests multiple 51  interpretations based on the temporal and registral ordering of the pitch classes. In all cases, a partition comprising two E-type (014589) hexachords dominates, and is articulated through various subpartitions. Figure 3.1 demonstrates how the first aggregate in the viola (measures 1-5) can be parsed as two separate trichordal partitions (Mead 1994, 79), and also shows the join and meet of these two partitions with the resulting lattice L i . Partition ti corresponds to the distribution of pitch classes according to the temporal order in which they appear. The result is an (014) trichordal partition with 4  trichord pairs combining temporally to form the (014589) hexachords of partition h i . Partition t2 corresponds to the registral distribution of concurrent lines in the viola's twopart polyphonic texture. This is easily discerned in the music by the contrasting stem directions, which indicate two distinct registral strands both parsed as a new partition of four (014) trichords, h i , the join of t i and t2, corresponds to the larger temporal division of the aggregate into an antecedent and consequent pair of E-type (014589) hexachords. This result is far from trivial. The E-type hexachord can be produced from four different trichord types, and likewise, the (014) trichords can combine to produce other types of combinatorial hexachords. This unique ordering and division of pitch classes serves not only to present an interlocking string of (014) sets but, perhaps more importantly, articulates the unique E-type hexachord partition h i . As will be seen, the E-type hexachords (and indeed the partition hi) will control the majority of the work's material.  52  J =  r3  64  con sordino  Viola  X-X  L  •  J  Piano  ^,  ^T-J-T  Example 3.1: Composition for Viola and Piano, mm. 1-12. (accidentals are valid only for notes they immediately precede) 53  The meet of t i and t2 is an s-type partition. Partition s i provides additional information pertaining to the musical surface. In essence, it is a further refinement of both trichordal partitions. Notice that the pitch classes are ordered in time and register so that the si dyads are the registrally paired pes within the t l trichords. In measures 1-2, the lower register contains pes 0 and 1 while the upper register completes the trichord with pc 9. The next t i trichord is articulated as the upper-register dyad <58> over the sustained pc 4 in the lower register. The result corresponds exactly with the si partition. The resulting lattice L i is shown in Figure 3.1. Here is an example of how a lattice can neatly sum up the various ways in which aggregate partitions can be superimposed. The four partition members of L i all have musical significance in the given passage. Figure 3.1 also indicates the inherent automorphisms of the lattice and, consequently, the transformational automorphisms of the passage. L i is automorphic under 11,17 and T6Specifically, Il and T6 map ti to t2 and vice versa; while h i and si map onto themselves under Ii or T6; all four partitions map to themselves under I7. The E-type (014589) hexachord has been shown to play an important role in the organization of pitch-classes to this point.  Other hexachords are also brought out  registrally in this passage, however. Figure 3.2 shows that the registral distribution of hexachords in the viola corresponds to the partition h2, which contains hexachords of two different types, both being symmetrical and invariant under I7. The join of the two hexachordal partitions is (necessarily) the conjunct partition, and the meet produces the expected partition t2 . Figure 3.2 groups these partitions as a simple lattice, L 2 , which is automorphic under the 17 transformation only (since each partition has a different inventory, but is 17-invariant). I7 maps the hi hexachords and ti trichords onto one  54  another; the h2 hexachords, being of different types, map onto themselves under I7. The I7 automorphism is significant as it is a common feature of both L i and L 2 .  INV = (014) (by temporal order) ti = I 019 I 238 I 458 I 67A I INV = (014) t2 = I 014 I 2AB I 367 I 589 I (by register) INV = (014589) h-|= f|Jt2 = I 014589 I 2367AB I si = MMt2 = I 01 I 2B I 3 I 4 I 58 I 67 I INV 9 I =A(s) I (01) (03) 4  4  2  4  t2  t1  L1  T6 11, 17  Figure 3.1: Lattice L i - viola, mm.1-5.  55  2  2  hi = I 014589 I 2367AB I  INV=(014589)'  h2 = I 013467 I 2589AB I  INV=(013467)(012369)  12 = h-iMh.2 = I 014 I 2AB I 367 I 589 I  INV=(014)<  hi Jh2 = conj  conj  L2  L2 ^  •  L2  t2  Figure 3.2: Lattice L 2 - viola, mm.1-5  Figure 3.3 combines L i with L 2 into a composite lattice, L3, which sums up the various partitions of the viola in measures 1-5. Not surprisingly, L3 is 17-automorphic since all of its partitions are 17-invariant. This lattice offers a productive visual model of the partition hierarchy in this passage. The central nexus positions are occupied by h i and t2, which mediate among all other partitions. By contrast, the partitions h2 and t i reside on peripheral vertices of the lattice. Thus, the lattice representation gives an accurate picture of the influence of the various hexachord types in the viola's first 56  aggregate. Clearly, the (014589) hexachords control all aspects of pitch class distribution in the passage while the peripheral hexachord types have a sphere of influence limited to register alone. Also the distinct registral lines modelled by t2 (and by h2) seem to play a more prominent role than the simple temporal unfolding of pes and trichords, modelled by ti. Certainly the contrapuntal unfolding of the viola aggregate as registral lines is more interesting and musically compelling than is the simple one-after-another unfolding of the pes.  h-|= f|Jt2 = I 014589 I 2367AB I h2 = l013467l2589ABI H = I 019 I 23B I 458 I 67A I  S1 Figure 3.3: Composite Lattice L3 - viola, mm.1-5. 57  While the viola part in this passage can be clearly partitioned as (014) trichords in two different ways, the piano part presents a slightly more complex picture. As with the viola, the aggregate divides temporally into (014589) hexachords and thus the piano and viola together involve hexachordal combinatoriality. Needless to say, the division corresponds to the hi partition, with its complementary hexachords juxtaposed against those of the viola. Table 3.1 shows this as a simple, two part-array block.  Viola Piano  091485 23BA67 2B37A6 810495  Table 3.1 Hexachordal A r r a y , mm. 1-5.  Also like the viola, the hexachords in the piano can be further divided into constituent trichords. In this case, however, the first two trichords are (014) sets while the second hexachord contains two (015) subsets. Table 3.2 labels the resulting partition t3(by temporal order) (by register)  13 = I 018 I 23B I 459 I 67A I  S2 = I 0 118 I 2B I 3 I 49 I 5 I 6 I 7A I  INV=(014) (015) 2  2  INV= (s) (03) (05) 4  2  Table 3.2: Piano Partitions, mm. 1-5.  The trichords of t3 correspond to the temporal distribution of pitch classes. Register also plays a part here. In this case, however, the trichord is not the basic unit of registral division. Instead, each t3 trichord is presented as a dyad and a singleton in different registers and thus the registral partition of the piano part corresponds to the stype partition labelled S2 on Table 3.2, which contains two (03) dyads and two (05) dyads. 58  2  The si and S2 partitions are interesting because their dyad subsets, (01) , (03) and 2  3  (05) respectively, are readily discernible on the musical surface. These dyad types are 2  precisely the three dyad types that can evenly divide the E-type hexachord. That is, any (014589) hexachord can be evenly partitioned into three (01) dyads, three (03) dyads, or three (05) dyads. The s-partitions indicate how Babbitt featured each of these three dyad types in the passage, giving a slight preference to the (03) sets that are found in both the viola and piano parts. The partitions of the piano part are simply subpartitions of each other; the dyads of S2 are subsets of the trichords of t3, which are further subsets of the h i hexachords, so their meet and join interaction produces no new partitions ( h i , t3 and S2 thus already form a simple lattice). O f note is the fact that two of the t3 trichords, {2,3,B} and {6,7,A}, are also membersets of the ti partition, while their other trichords differ somewhat. This leads to a much more complex picture when the interactions among the three trichordal partitions in the viola and piano are considered together. Their joins will all produce the expected hi hexachordal partition, but heir meets do not always generate s-type partitions. Table 3.3 begins by listing the series of partitions derived from the meets of t i , t2, and t.3. As Table 3.3 shows, the three trichordal partitions interact to produce a complicated set of partitions. The lattice that would incorporate all of these partitions becomes somewhat unwieldy as a single structure. Furthermore, the list of partitions in Table 3.3 is not complete. The meets of many of the various n-partitions (not in the list) will fail to produce the corresponding s-type partitions that originally "joined" to create them. The result is a rather complex lattice that contains numerous non-symmetric partitions. This indicates that the interaction of the viola and piano parts is lacking in symmetry. 59  Some partitions in the lattice - especially those on the lower two-thirds of the table - fail to model any significant aspects of the musical passage.  O f special note  however, is that all of the partitions listed in Table 3.3 preserve the dyad {2,B} as a memberset, or memberset subset, (that is, pes 2 and B are found in the same memberset in every partition). These pes are significant to the passage as they are the first two sounding pitches in the music and are played as a simultaneity in the piano in measure 1. Partition x2 = s]Ms2 brings these two pes especially to the fore, since they form its only non-singleton memberset.  tlMt2 = S1 = I 01 I 2B I 3 I 4 I 58 I 67 I 9 I A I  INV= (s) (01) (03)  S2 = I 0 I 18 I 2B I 3 I 49 I 5 I 6 I 7A I  INV= (s) (03) (05)  4  2  4  2  2  2  t2Mt3 = S3 = I 01 I 2B I 3 I 4 I 59 I 67 I 8 I A I INV=(s) (01) (03)(04) tiMt3 = mi = I 01 I 23B I 45 I 67A I 8 I 9 I INV= (s) (01) (014) m-iMsi = xi = I 01 I 2B I 3 I 4 I 5 I 67 I 8 I 9 I A I INV= (s)<(01) (03) miMs2 = miMs3 = xi s-|Ms2 =X2 = I 0 I 1 I 2B I 3 I 4 I 5 I 6 I 7 I 8 I 91 Al INV= (s) (03) si Ms3 = X1 4  2  2  2  2  2  ,0  S 2 M S 3 = X2  s-|Js2 = ni = I 0158 I 2B I 3 I 49 I 67A I INV=(s)(03)(05)(014)(0347) I (s) (01) (03)(014) s-|Js3 = n2 = I 01 I 2B I 3 I 4 I 589 I 67 I AINV= S2JS3 = n3 = I 018 1 2B I 3 I 459 1 67A 1 INV=(s)(03)(014) (015) miJsi = 114 = 1 01 I 23B I 458 I 67A I 9 I INV= (s) (01)(014) INV=(014) (015) m i J s 2 = t3 = I 018 I 23B I 459 I 67A I m-|Js3 =n5 = I 01 I 23B I 459 I 67A I 8 I I N V - (s)(01)(014) (015) (s)(03)(014)(014589) n i J n 2 = Q1 = I 014589 I 2B I 3 I 67A I =INV n2Jr>3 = n i J n 3 = qi INV= (014) (014589) niJri4 = Q2 = I014589I23BI67AI n-|Jt3 = niJri5 = n3Jri4 = n3Jn5 = n4Jt3 =P2 n3Jt3 = t3 t3Jri5 = t3 INV= (01)(014) (0145) n4Jri5 = q3 = = I 01 I 23B I 4589 I 67A I 3  2  2  3  2  2  2  2  2  Table 3.3: List of Partitions resulting from trichord interactions, mm. 1-5. 60  2  The second aggregate in both instruments spans measures 6-8. The musical texture retains the same type of pitch-class distribution as in the previous passage. The viola consists of two trichordal partitions that parse the aggregate by register or temporal order, while the piano again divides its surface trichords in correspondence with an s-type partition. Table 3.4 lists these partitions.  INV=(014) Viola - register t2 = I 014 I 2AB I 367 I 589 I INV=(037) Viola - temporal t4 = I 049 I 158 I 27A I 36B I INV=(014)*(015) Piano - temporal t5 = I 089 I 145 I 267 I 3AB I Piano - register S5 = I 0 I 15 I 27 I 3 I 4 I 6 I 89 I AB I INV=(s)(01)(04)(05) 4  4  2  4  2  Table 3.4: Partitions, mm. 6-8.  The  viola's registral trichords of the previous passage (the t2 partition) are  retained in the viola in mm. 6-8. Temporally, however, this passage dispenses with the earlier temporal (014) trichords of ti, preferring instead a new collection of (037) sets that make up t4. The (037) trichord is one of the four trichordal generators of the E-type hexachord (Martino 1961, 229). The use of a different temporal partition is interesting when one observes that the hexachordal partitions h i and h2 from the previous passage are here retained in their previous roles, hi is a superpartition of t4 and governs the temporal distribution of pitch classes, while h2 is a superpartition of t2 and once again controls the registral division. (The h2 hexachord that was stemmed up in the first viola aggregate is stemmed down in the second viola aggregate, and vice versa.) Even so, t2Jt4 = hi.  Since t2 is a subpartition of both h i and h2, L 2 retains its significance throughout  these measures. Figure 3.4 shows the meet and join results of t2 with the new trichordal partition, t4 and the resulting lattice L 4 . 61  t2 = I 014 I 2AB I 367 I 589 I  INV = (014)  t4 = I 049 I 158 I 27A I 36B I  INV = (037)  4  4  h-|= t2Jt4 = I 014589 I 2367AB I  INV = (014589)  2  S4 = t2Mt4 = I 04 I 1 I 2A I 36 I 58 I 7 I 9 I B I  INV = (s) (03) (04)  Figure 3.4: Lattice L 4 - viola, mm.6-8.  62  4  2  2  L4 is analogous to L i in some ways. The musical texture features held single notes in one register accompanied by dyads in the opposite register and the distribution corresponds to S4, the lower bound of L4. Unlike L i , however, L4 has no automorphic transformations. The symmetries inherent in the previous passage are compromised in these measures by the appearance of (04) dyads. The (04) sets are important subsets of the E-type hexachord but cannot divide it equally because they overlap cyclically. The (04) dyads of both the viola and piano parts, brought out by the S4 and S5 partitions respectively, suggest a change in intervallic character from the previous passage. Since L2 remains relevant to the passage, Figure 3.5 combines L2 and L 4 into a composite lattice, L5. Once again, this lattice neatly sums up the series of partitions that govern the passage. The hexachords retain their previous hierarchical status as does t2. The only change from L3 is the substitution of t4 and S4 (for ti and si respectively). Another prominent difference from the previous passage can be observed by comparing the representative trichords of the viola and piano parts collectively. In this case, no trichords from the viola part are preserved in the piano as they were in the earlier passage. The representative lattice will therefore be much simpler than that which would have resulted from the long list of partitions shown earlier in Table 3.3. Figure 3.6 lists the series of interactions and displays the resulting lattice, L6.  63  h 1 = t2Jt4 = I 014589 I 2367AB I  INV = (014589)  2  h2 = I 013467 I 2589AB I  INV = (013467)(012369)  t2 = h i Mh2 = I 014 I 2AB I 367 I 589 I  INV = (014)  t4 = I 049 I 158 I 27A I 36B I  INV = (037)  S4 = t2Mt4 = I 04 I 1 I 2A I 36 I 58 I 7 I 9 I B I  INV = (s) (03) (04)  4  4  4  2  conj  S4  Figure  3.5:  Composite Lattice  64  L5  - viola,  mm.6-8.  2  12 = I 014 I 2AB I 367 I 589 I  INV=(014)  4  14  = 1049 I 158 I 27A I 36B I  INV=(037)  4  15  = 1089 I 145 I 267 I 3AB I  INV=(014) (015) 2  2  S4 = t2Mt4 = I 04 I 1 I 2AI 36 I 58 I 7 I 9 I B I  INV=(s) (03) (04)  56 = t2Mt5 = I 0 I 14 I 2 I 3 I 5 I 67 I 89 I AB I  INV=(s) (01) (03)  5 7 = 14M15  = I 09 115 I 27 I 3B I 4 I 6 I 8 I A I  4  4  4  L6  S6  L4  hi  t2  Figure 3.6: Lattice L6 of partitions, mm. 6-8 showing viola sublattice L4  s 65  2  3  INV=(s) (03)(04) (05)  t5  disj  2  4  2  It is difficult to make a case for the analytical relevance of L 6 beyond its upper bound h i partition and its inclusion of the viola sublattice, L 4 . L 6 is not automorphic under any canonical transformation as S4, S6, and S7 have different inventories, and therefore can't map onto one another. None of these s partitions have any invariance properties either. The s partitions individually do not represent any complete aspect of the musical surface. Collectively they yield some of the surface dyads of the passage, namely <15>, <89> and <AB>. Surprisingly, the S5 partition that represents the registral distribution of pes in the piano is not part of any of the above lattices as it is not generated from the meets of any trichordal partitions. Rather, S5 is a subpartition of t5 and represents a registral refinement of the trichordal partition.  The third aggregate begins at the end of measure eight and continues until the beginning of measure twelve. Table 3.5 shows the relevant partitions.  Viola and piano - temporal = h-| = I 014589 I 2367AB I INV=(014589) Viola - registral polyphony = h3 = I 023457 I 1689AB I INV=(023457) Viola - register t6 = I 045 I 189 I 237 I 6AB I INV=(015) Viola - temporal t7 = I 015 I 23A I 489 I 67B I INV=(015) Piano - temporal t8 = I 018 I 236 I 459 I 7AB I INV=(014) (015) Piano - surface si 1 = I 08 I 1 I 23 I 456 I 7B I 9 I A I INV=(s) (01) (04) 2  2  4  4  2  J  4  2  Table 3.5: Partitions, mm. 8-12. The temporal distribution of pes in the viola corresponds once again to the partition, h i . However, the viola's two-voice polyphony is here governed by a different hexachordal partition than the two previous viola aggregates. The new partition, I13, projects a pair of (023457) hexachords. Trichordal partitions in the viola are again defined by register and time but now consistently present only (015) trichords. 66  2  The third and fourth t6 and t7 trichords of the viola part overlap temporally. That is, two pes (7 and A) have attacks that occur at the same time point. There are therefore two possibilities for temporal trichordal partitions, only one of which has been shown in Table 3.5. The trichords of t7 were chosen so as to distinguish them from the registral trichords of t6  The other possible trichordal partition would share two trichordal  membersets of t6- There is, admittedly, a certain amount of forced artificiality to this decision. However, previous passages have demonstrated two clearly different trichordal partitions in each viola aggregate. This offers some justification for the decision to choose a temporal trichordal partition that doesn't share any common trichordal membersets with the registral trichordal partition. The piano offers the familiar combination of (014) and (015) trichords and the overall temporal distribution of pes is controlled by h i . The piano trichords are again articulated as dyads and singleton pes corresponding to an s-type partition (su). Rather than controlling registral distribution as in previous passages, the dyads of si l correspond to dyad simultaneities on the musical surface. Example 3.2 shows the three superimposed partitions of the piano part.  Dyads and Singletons (in columns)= su Trichords (circled) = t8 Hexachords (separated by line) = hi  Example 3.2: Piano partitions, mm. 8-12.  67  Table 3.6 lists the partitions that result from the various meets and joins of the viola partitions in Table 3.5. The two hexachordal partitions meet to yield the partition t6 that controls the registral distribution of trichords in the viola. Consequently, the resultant lattice L7, shown in Figure 3.7, is isographic and analogous to L2. L7 also retains the same three automorphisms of L2, as each of the partitions are invariant under 11,17, and T6.  hi =I014589I2367ABI  INV=(014589)  h3= I 023457 I 1689AB I  INV=(023457)  2  2  hiJh.3 = conj h-|Mh3 = t6 = I 045 I 189 I 237 I 6AB I  INV=(015)  t6Jt7 = h-| = I 014589 I 2367AB I  INV=(014589)  4  2  t6Mt7 = S8 = I 05 I 1 I 23 I 4 I 6B I 7 I 89 I A I  INV=(s) (01) (05) 4  2  2  Table 3.6 Meet and Join of viola partitions, mm. 8-12.  Figure 3.8 displays the lattice L s resulting from the meet and join of the viola trichordal partitions, t6 and t7. It is isographic and analogous to both L1 and L4 and is automorphic under T6,15, and I n . The composite viola lattice L 9 shown in Figure 3.9 is T6-automorphic, since all of the partitions in L7 and L8 are T6-invariant. L7, L 8 , and L 9 , which represent the various viola partitions in measures 8-12, retain the same analytical significance as their isographic counterparts: L i , L2, L3, L4, and L5. The background partition, h i , still occupies the central position as the source for the trichords. The partition h3 (in L 7 and L9) remains on the periphery, influencing the registral hexachord distribution. The S8 partition (in L s and L 9 ) aptly describes the surface  pitch detail of the passage consisting  68  of dyads  against singletons.  h i = I 014589 I 2367AB I  INV=(014589)  2  conj  t6 Figure 3.7: Hexachord Lattice L 7 - viola, mm.8-12.  t6 = I 045 I 189 I 237 I 6AB I  INV=(015)<  17 = I 015 I 23A I 489 I 67B I  INV=(015)  t J t 7 = h 1 = I 014589 I 2367AB I  INV=(0145 89)  4  2  6  I6M17 = S8 = I 05 I 1 I 23 I 4 I 6B I 7 I 89 I A I  INV=(s) (01) (05) 4  hi  Figure 3.8: Trichord Lattice Ls - viola, mm.8-12. 69  2  2  hi =I014589I2367ABI  INV=(014589)  h 3 = l 0 2 3 4 5 7 l 1689AB I  INV=(023457)  2  2  hi Jh3 = conj hiMh3 = t6 = I 045 I 189 I 237 I 6AB I 17=  INV=(015)  I 015 I 23A I 489 I 67B I  4  INV=(015)  4  t6Jt7 = h-| = I 014589 I 2367AB I  INV=(014589)  T6M17 = S8 = I 05 I 1 I 23 I 4 I 6 B I 7 I 89 I A I  INV=(s) (01) (05)  2  4  2  2  conj  T6 Lg  S8  Figure 3.9: Composite Lattice L9 - viola, mm.8-12.  70  •  Figure 3.10 shows the lattice interactions of the three trichordal partitions in the viola and piano in measures 8-12 and the resulting lattice Lio  The three trichordal  partitions join to produce the hi partition and yield a series of s-partitions from their meets. O f note is the fact that si i, which projects the surface dyad simultaneities in the piano, is not reproduced by any of the trichordal partition meets. As with previous passages, this surface s-type partition acts in isolation from the viola part and is a surface refinement and subpartition of t8. The resulting s-partitions (s -s ) all contain varying numbers of (01) and (05) 8  10  dyads (slO also contains one (04) dyad). This is significant since the musical surface abounds with i c l and ic5 simultaneities. The s partitions are not disjoint; their pairwise meets all preserve the {2,3} dyad found as the sole non-singleton memberset in the partition X3. Lio summarizes these partition relationships in Figure 3.10. Once again, the significance of this lattice is less clear than lattices that consider the viola in isolation. Note that Figure 3.10 shows the inclusion of the viola sublattice, Ls as part of the larger lattice, Lio.  71  t6 = I 045 I 189 I 237 I 6AB I INV=(015) t7 = I 015 I 23A I 489 I 67B I INV=(015) t8 = I 018 I 236 I 459 I 7AB I INV=(014)(015) h-| =t6Jt7 = t6Jt8 = t7Jt8 = I 014589 I 2367AB I INV=(014589) S8 = I 6 M I 7 = I 05 I 1 I 23 I 4 I 6B I 7 I 89 I A I INV=(s) (01) (05) sg = i6Mt8 = I 0 I 18 I 23 I 45 I 6 I 7 I 9 I AB I INV=(s) (01) (05) S10 = t7Mt8 = I 01 I 23 I 49 I 5 I 6 I 7B I 8 I A I INV=(s)(01)(04)(05) 4  4  2  2  2  4  4  3  4  X3 = S8M9=S9Msio=S8Msio  = 10111 23 I 4 I 5 I 6 I 7 I 8 I 9 I A I B I  rNV=(s) (01) ,0  hi  72  2  2  2  The three passages analyzed indicate a consistency in the application of pitchclass sets throughout the musical texture. Indeed, the first twenty measures of the piece, which form the first section of the composition, maintain a common method of pitchclass distribution. The viola part always contains two overlapping trichordal partitions delineated by register and temporal distribution and two overlapping hexachordal partitions with the hi partition retaining sole influence over the distribution of hexachords in time. The piano part always contains trichords that are temporally linked into h i hexachords. In all cases, these trichords are articulated as s-type partitions. Three types of trichord predominate throughout this section: (014), (015) and (037). These represent three of the four possible trichordal generators of the E-type hexachord (Martino 1961, 229) that forms the background partition for the entire piece (Mead 1994, 78-79). The s-type partitions produce (01), (03), (04) and (05) dyads which are common entities on the musical surface of the piece. All of these dyads except for (04) can "evenly divide" the E-type hexachord. The (04) dyads are cyclical and will combine within this hexachord to form the (048) (augmented) trichord, the fourth trichordal generator of the (014589) hexachord. While h i is the common link among all of the various partitions present in the observed passages, two other hexachordal partitions assert a certain influence over the viola in particular. Partitions h2 and I13 collectively contain three other hexachord types: (023457), (013467), and (012369). Figure 3.11 lists the partitions that result from the interaction of these three hexachord partitions and the resulting lattice L n . L n is automorphic under I7 as all of its partitions are 17-invariant.  73  hi = I 014589 I 2367AB I  INV=(014589)  h.2 = I 013467 I 2589AB I  INV=(013467)(012369)  h.3 = I 023457 I 1689AB I  INV=(023457)  hiMh2 = t2 = I 014 I 2AB I 367 I 589 I  INV=(014)  4  hiMh.3 = t6 = I 045 I 189 I 237 I 6AB I  INV=(015)  4  h2Mh3 = m2 = I 0347 I 16 I 25 I 89AB I  I N V = (03)(05)(0123)(0347)  2  2  t-j Mt6 = t i Mm2 = t6Mm2 = S 1 2 = I 04 I 1 I 2 I 37 I 5 I 6 I 89 I AB I I N V = ( s ) ( 0 1 ) ( 0 4 ) 4  2  2  conj  Figure 3.11: Lattice L u  of hexachord partitions, mm.  74  1-12.  The partitions in L i l serve to summarize the various partition membersets found throughout the viola part. The two trichordal partitions, t2 and t6, represent the (014) and (015) sets respectively. Partition m2 contains an (0347) tetrachord which further contains (014) and (037) trichords as subsets. Thus, the "middle" partitions contain all of the observed trichordal memberset types, with the (014) trichord type being represented twice (just as there are two (014) partitions, ti and t2). The s u partition contains (01) and (04) dyads while m2 contains (05) and (03) dyads. The (0347) m2 tetrachord can also be divided into pairs of (03) or (04) dyads or as a combination of an (01) and (05) dyad, while the (0123) m2 tetrachord can be divided into pairs of (01) or (02) dyads or as a combination of an (01) and (03) dyad. Therefore, the partition membersets of L11 contain all of the dyadic, trichordal and hexachordal partition memberset types found throughout the viola part. In addition, it retains the I7 automorphism inherent in many of the viola lattices.  The various partitions and lattices analyzed in this passage serve to connect the various dyadic, trichordal, and hexachordal entities that abound in the musical texture. The lattice model neatly summarizes much of the surface detail pertaining to the viola part. The viola contains overlapping hexachords which meet to yield trichordal partitions that govern the registral distribution of trichords. These registral partitions, combined with their overlapping temporal trichordal partitions, meet to yield s-type partitions which accurately describe the musical surface. I7 automorphisms are retained throughout each passage in many of the viola lattices while Ii and T6 also frequently assert their automorphic influence.  75  The situation with the piano part is much more complex. While a consistent method is, more or less, applied to the distribution of pitch classes and one partition class is used extensively, the lattices fail to yield much productive information besides their sole dependence on the h i superpartition. The s-type partitions are not relevant to the piano passages except in the case of some isolated dyads. If nothing else, this suggests an independence of the two parts that act, essentially, in isolation. Their common ancestor is the E-type hexachord but each part essentially follows a different path. Nevertheless, the presence of the h i partition as a type of repeated background structural entity is powerful. The smaller partitions divide the hi partition in different ways throughout the passage. Thus, maximal surface diversity is achieved from a single background source partition. The Composition for Viola and Piano therefore differs from the Woodwind Quartet in that the latter piece was composed of a variety of hexachord types that acted as background partitions. The chromatic A-type hexachord occupied a central, but more abstract position in the Quartet while other hexachords controlled the surface of much of the music. In the case of the Composition for Viola and Piano, the E-type hexachord is the only true background hexachordal partition. The sphere of influence of the other two hexachordal partitions, h2 and I13, are limited to governing registral distribution in the viola alone. The concept of the hexachordal partition as a background entity is suggestive for its relation to the Schenkerian idea of background structure and structural levels in general. It would be difficult to make a case for a consistent structural hierarchy in this piece. A closer look at another work by Milton Babbitt composed during this period, however, will reveal how the concept of structural levels in partition lattices may be constructive.  76  CHAPTER 4: A PARTITION L A T T I C E M O D E L OF STRUCTURAL L E V E L S IN T H E STRING QUARTET NO. 2 (1954). Being confronted with the String Quartet No. 2 (1954) is much like being handed a complex, multi-faceted translucent crystal. An exterior view of any one face provides an angular picture of the adjacent faces and an oblique image of each of the non-adjacent faces in the same visual plane. Looking through the crystal from the same vantage point provides a unique set of images of the faces on the other side, coloured by the angular relationships and the translucent quality inherent in the construction of the crystal. Changing the perspective by rotating another face into view changes the overall impression of the structure. As we turn the crystal, we accumulate these impressions in an orderly fashion, and our appreciation of its structure develops and takes on a more distinct shape. In this process we discover that each face provides a unique window on the centre of the crystal, and we develop an understanding of how the centre affects the exterior design (Zuckerman 1976, 85).  Mark Zuckerman's analogy between Milton Babbitt's second String Quartet and a complex crystal introduces his detailed analysis of the first ninety-two measures of the work. Zuckerman's analysis demonstrates the relationships between the characteristic set-classes for each sub-section and their derivation from an abstract background hexachord. While this hexachord is not presented explicitly in the piece until m. 276, it is analogous to something essential about the crystal, hidden amongst its multitude of exterior faces and vertices. The crystal analogy can be carried further. Every section and sub-section of these first ninety-two measures projects a small number of pitch-class partitions that are derived from the various arrays Babbitt used to design the piece. As will be shown, the meets and joins of these various partitions produce lattices that demonstrate a complex web of relationships and illuminate the quartet's structure. As we will see, the graphic depiction of these lattices will also suggest that the quartet can be conceived as multiple 77  "crystals" each having partitions at its vertices. Three levels of structural organization in the quartet are elucidated through the lattice model. As will be seen, the uppermost stratum of the lattices will generally consist of combinatorial hexachord partitions, while the middle layers consist of partitions built from trichords or dyads. The lower layers of the lattices are generally comprised of stype partitions or partitions with varying numbers of dyad and singleton membersets. Figure 4.1 illustrates the generic lattice structure. Several passages that project these three types of partitions concurrently in the musical texture will be examined. The model in Fig. 4.1 suggests a quasi-Schenkerian analogy to background, middleground and foreground levels of structural organization, and the idea that certain partitions might be "prolonged" over a period of time. It will be shown, for example, that the first thirty-one measures of the quartet are derived from a single hexachord partition and that this partition is therefore "prolonged" as a background structure throughout that passage. The hexachordal, trichordal and dyadic partitions are derived from the precompositional arrays that Babbitt used to design the quartet. In many cases, however, it is the lower bound partitions of the lattices which are most readily audible on the musical surface. While these "foreground events" are sometimes not easily related to any array design, the lattice model will show that many of these "foreground events" result from meets of partitions deployed in middleground and background structural levels. The "complex crystal" analogy gains added significance with this analytical approach since it will model the quartet as a network of lattices containing partitions that delineate multiple levels of structure.  78  4  ure 4.1: Sample Lattice showing three levels of structural organization.  79  Example 4.1 shows the first six measures of the quartet. This is one of four passages in the quartet's first ninety-two measures (herein referred to as the first movement) in which pairs of instruments play the same pc dyad, in isorhythm, but using 8  different pitches. Zuckerman refers to these passages as "Octave Episodes" (OEs) and this terminology will be preserved here. The first three measures complete an aggregate with the dyad partition labelled d i on Figure 4.2. This first aggregate can also be partitioned by dynamic level into the trichord partition labelled dyni. Figure 4.2 shows the dyadic and dynamic partitions along with their meets and joins and their inventories as well as the lattice L i that results from these partitions.  Violin 1  Violin  2  Viola  Cellc  4J p  tf  SEP  2  ^  P  1'  JL PEP  m  ^  1  pm-C^1 -  i  Iff  I.J i' -  t.  "  '  1  p =-mp '•'  i  1  r  ''li'i  i  p  jjgp p  JHBP  r j  j9f Example 4.1: mm. 1-6. The piece is in one movement, but there are four large sections which are set off by double bars and pauses. The four sections are based on differing pitch materials and structural divisions. 8  80  dl = I 09 I 14 I 2B I 36 I 58 I 7A I dym = I 089 I 145 I 2AB I 367 I mp ff pp mf h = di Jdyni = I 014589 I 2367AB I si = diMdym = 109 114 12B 136 I 5 I 7 18 IA I n  6  INV = (03)' ,4  INV = (014)  INV = (014589)  2  INV = (s) (03)  Figure 4.2: Lattice L i of partitions from mm. 1-3.  81  4  4  It is difficult to make a clear case for distinct structural levels in this lattice. Arguably, the partition most discernible on the musical surface is the dyadic partition, while the most abstract partition is certainly the dynamic partition, d i and dyni are shown on the same level in L i not as a means of equating them in terms of structural perception but to show how they combine to produce the two other significant partitions. The join of d i and dyni produces a partition consisting of two (014589) hexachords. In fact, these hexachords are projected in the musical texture as the lynes governing the pitch material of each individual instrument. In mm. 1-3, the first violin and viola both play the hexachord {0,1,4,5,8,9} while the second violin and cello both play the second hexachord {2,3,6,7,A,B}. The meet of di and dyni produces a partition (si) consisting of four singletons and four (03) dyads. As in the previous chapter, this type of partition will figure significantly throughout the first movement. The s i partition can be viewed as a "foreground" refinement of the d i partition. The dyads that are preserved in si - {0,9}, {2,B}, {1,4}, {3,6} - are the first four and last four pitches respectively of the opening three measures; each of these dyads has a different uniform dynamic mapping. By contrast, the middle four pitches {5, 8, 7, A}, even though they are articulated as two instrumental dyads, are differentiated by opposing dynamic levels, thus "disconnecting" the dyads into singleton pitch classes that constitute the four singleton membersets of s l . A case can be made that the h i , d i , and s i exist on three different structural levels with the dyni partition acting as "differentiator" among the three. This idea is represented graphically by Figure 4.3.  82  background  hi  middleground  foreground  Figure 4.3: Structural Levels, mm. 1-3.  Measures 4-6 can be viewed as a consequent response to the three-measure antecedent "phrase" of the opening aggregate. Octave simultaneities are still very prominent, as are surface dyads, but the musical texture is much more complex. Simultaneous or successive dyads of interval class 4 predominate in measures 4-6, in contrast with the ic3 dyads that defined the first three measures. But unlike measures 1-3, measures 4-6 do not project a dyadic partition; indeed it is impossible to design a dyadic partition containing only 04 dyads. The 04 dyads of measures 4-6 are best derived from 83  an augmented trichord partition, a = I 048 I 159 I 26A I 37B I. Each instrument plays two (04) dyads and two single pitches in alternation. This corresponds to the format for an s-type partition (1 2 ). Each (04) dyad and the adjacent singleton form an (048) 4  4  trichord. The texture in mm. 4-6 actually uses two distinct s-type partitions. Figure 4.4 shows the two partitions consisting each of four (04) instrumental dyads and four singletons as well as their meet and join. It is no surprise that the join of S2 and S3 produces the augmented trichord partition, here labelled a. The disjunct partition arises from their meet since no dyad membersets are shared between s2 and S 3 . The hi partition is also clearly present in these measures and again arises as the lynes projected by each instrument. In fact, the instrumental assignment of the hi hexachords in measures 1-3 is reversed in measures 4-6. That is, where the first violin and viola originally played {0,1,4,5,8,9} in the first three measures, they now play the complementary {2,3,6,7,A,B} hexachord in measures 4-6. The corresponding reversal is applied to the second violin and cello. A new hexachordal partition is also evident when considering the temporal distribution of pitch classes in all four instruments in mm. 4-5. The first six pitch classes (represented by twelve pitches) comprise the hexachord {0,2,4,6,8,A) while the next six pitch classes (again represented by twelve pitches) correspond to the complementary {1,3,5,7,9,B} hexachord. The resulting partition is labelled ha in Figure 4.4. This partition comprises F-type (02468A) hexachords, and this hexachord class is the only other all-combinatorial hexachord class, besides the E-type hexachord, that can be generated from pairs of (048) trichords (Martino 1961, 229). Indeed, the partition h interacts with h i , via join, to yield partition a. Figure 4.4 shows a  the lattice L2 that results from these partition interactions.  84  Violins Via &Vc  S2 = 108 11 12 I 3 14 I 59 16A I 7B I INV S3 = I 04 115 I 2A I 37 I 6 I 8 I 9 I B IINV  S2JS3= S2MS3 =  a = I 048 1159 I 26A I 37B I  disj  lynes temporal h-|Jha= h-|Mh=  hi = I 014589 I 2367AB I h = I 02468A 113579B I conj a=l048M59l26AI37BI a  a  = (s) (04) 4  4  = (s) (04) 4  4  INV = (048)  4  INV = (014589)  2  INV = (02468A) INV = (048)  4  conj  hi  ha  L2  S3  S2  disj Figure 4 . 4 : Lattice L 2 of partitions from mm. 4 - 6 . 85  2  It is clear that h i is present as a "background" structure for the first six measures, uniting the seemingly disparate halves of this first octave episode. The h partition plays a  a somewhat subservient role to hi as it is only presented in the music in mm. 4-5. The two halves also share another element in common. Measures 4-6 can also be parsed by dynamic levels into (014) trichords yielding the partition dyn2 as shown in Table 4.1. Thus, both halves of the Octave Episode contain two overlapping partitions of instrumentation and dynamics. It is useful to compare the partitions of both halves. Table 4.1 lists all of the partitions from the first six measures along with their meets and joins. Table 4.1 confirms the prominent role of h i in the passage as it is the least upper bound of all the "middleground" partitions; d i , dyni, dyn2, and a. The lattice that includes all of the listed partitions is somewhat unwieldy to represent graphically as a single structure. The four "middleground" partitions (a, dyni, dyn2 and d i ) meet to produce seven s-type partitions that are not all disjoint. Certain dyads are preserved from the meets of some of the s-partitions. This larger, theoretical lattice can, however be separated into components that describe either instrumentation or dynamic assignments. Figures 4.5 and 4.6 show two of these sublattices, L 3 and L 4 respectively. L 3 compares the two dynamic trichord partitions, dyni and dyn2, while L 4 projects the instrumental partitions, di and a. Both lattices show the interaction of a partition for mm. 1-3 (dyni or d i ) with the corresponding type of partition from mm. 46 (dyn2 or a). In both cases, hi is the join or least upper bound. The greatest lower bound of each lattice is distinct, however. The meet of dyni and dyn2 yields S5 which introduces a new inventory class to the s-type partitions. S5 contains two (01) dyads and two (03) dyads, and it is interesting  86  to note that the (04) dyad, so prevalent in the instrumental assignments of measures 4-6, does not surface as a common feature in the dynamic partitions even though it is a subset of the (014) trichords that define both dyni and dyn2.  di = dym = dyn2= a=  I 09 I 14 I 2B I 36 I 58 I 7A I I 089 1145 I 2AB I 367 I I 019 I 236 I 458 I 7AB I I 048 I 159 I 26A I 37B I  INV = (03)  6  INV = (014)  4  INV = (014)  4  INV = (048)  4  INV di Jdyni = hi = I 014589 I 2367AB I di Jdyn2 = hi dyniJdyn2 = hi diJa = hi dyni Ja = hi dyn2Ja = hi INV h = 02468AI 13579B hiMh = a INV 52 = I 08 11 I 2 I 3 14 I 59 I 6 A I 7B I INV 53 = I 04 I 15 12A I 37 16 I 8 I 9 I B I diMdyni = si = I 09 I 14 I 2B I 36 I 5 I 7 I 8 I A IINV diMdyn2 = s = I 09 11 I 2 I 36 14 I 58 I 7A I BINV I dyniMdyn2 = S5 = I 09 11 I 2 I 36 1 45 I 7 I 8 I ABINV I diMa = disj a  = (014589)  2  = (02468A)  2  a  4  = (s) (04) 4  4  = (s) (04) 4  4  = (s) (03) 4  4  = (s) (03) 4  4 /  = (s) (01) (03) 4  2  dym Ma = S6 = I 08 115 12A I 37 I 4 I 6 I 9 I B I  INV = (s) (04)  dyn2Ma = S7 = I 0 119 I 26 I 3 148 I 5 I 7B IAI  INV = (s) (04)  s<|Ms4 = xi =10911 I2I36I4I5I7I8IAIBI  INV = (s) (03)  4  4  8  4  4  2  S1MS5 = xi S4MS5 = xi  S2MS6= yi= I08I1I2I3I4I5I6I7I9IBI S2MS7 = y2= I0I1I2I3I4I5I6I7BI8I9IAI S3Ms6 = z= I0I15I2AI37I4I6I8I9IBI All other s partitions are disjunct. Table 4.1: Partitions, mm. 4-6. 87  INV = (s) (04) 10  INV = (s) (04) ,0  INV = (s) (04) 6  3  2  hi  w  '  dym =1089 1145 1 2AB I 367 I dyn2 = l 0 1 9 l 2 3 6 l 4 5 8 l 7 A B I S5 = dyniMdyn2 = I 09 11 12 I 36 145 I 7 I 8 I AB I  Figure 4.5: Lattice L3 of dynamic trichord partitions from mm.  Figure 4.6: Lattice L4 of instrumental 88  trichord partitions from mm.  1-6.  1-6.  L 4 shows that the meet of the instrumental partitions, d l and a, is the disjunct partition. This is as expected since di consists of (03) dyads while a consists of the (04)derived augmented trichords. It is interesting to observe, however, that L 4 is automorphic under T e n and I dd as h i , d i , a, and disj are all invariant under these transformations. e v  0  The transpositions map each of the "middleground" partitions onto itself in an interesting way. Half of the dyads of d i map onto each other through a T e n cycle while the other e v  half do the same. This is shown in Figure 4.7, applying the T 4 transformation. The dyad cycling under T 4 provides an interesting link with the augmented trichord partition, since each of its trichord membersets can also be viewed as the result of a T 4 (or Ts) cycle of its constituent dyads. This is also shown in Fig. 4.7. As will be seen, the T 4 cycle will also feature prominently in the next passage. (Applying T 4 : )  09 - • 4 1 - ^ 8 5 — ^ 0 9 2B—^63 — ^ A 7 - • 2 B  d i dyads  04-^48~^08 15—•59—^91  048 159  atrichords  26-^6A-^A2  26A  37-^7B-^B3  37B  Figure 4.7: T4 cycling in d i and a. It is useful at this point to consider the automorphisms of each of the four lattices produced so far. Table 4.2 lists the automorphisms of each lattice. Lattice  Automorphisms (aside from To)  Ll  13  L2  15  L3  T ,13,19  L4  Teven, lodd  6  Table 4.2: Automorphic Transformations of L i , L2, L3, and L4. 89  The automorphisms demonstrate the symmetrical nature of the pitch material used in measures 1-6. Partition h i , the least upper bound of all four lattices, is comprised of the highly symmetric (014589) hexachords. Like L 4 , hi is itself invariant under T e n , ev  lodd- For example, I5 preserves each hexachord of hi while I3 exchanges them. It is therefore interesting to note how the lattices highlight certain degrees of symmetry inherent in the "background" hexachordal partition. The twelve degrees of symmetry in L 4 and h i will figure prominently as lattice automorphisms in subsequent sections of the quartet.  Example 4.2 shows measures 7-18, the next section of the quartet. This section parses into trichords, with each of the four instruments playing independent (014) sets. This is one of four passages in the first movement in which the four instruments play pitch material derived from trichordal arrays. Zuckerman refers to these passages as "Four-Part Episodes" (FPEs). Table 4.3 shows the array (in pitch-class integer notation) used for measures 7-18.  tl  tl  t2  t2  t3  t3  tl  tl  Vln 1  908  415  3B2  6A7  091  584  263  B7A  Vln2  B7A  263  140  859  A2B  736  415  908  Via  415  908  6A7  3B2  584  091  B7A  263  Vc  263  B7A  859  140  736  A2B  908  415  Table 4.3: (014) Trichordal A r r a y , mm. 7-18.  90  The columnar aggregates of the array figure prominently in the musical texture, as there is no temporal overlap between consecutive columns. (Measure 7 contains the first column which is completed before the second column begins; the latter spans the next two measures etc.). For this reason, the columnar aggregates will take precedence over the lynear aggregates of the array in determining the relevant partitions in each passage. We shall therefore examine the columnar partitions first, and consider the lyne partitions later. We see these three columnar partitions engage some powerful transformational symmetries. Figure 4.8 lists the columnar partitions for measures 7-18 as well as their meets and joins and portrays the resulting lattice L5. The method of interaction among the three partitions differs significantly in this passage compared to the preceding Octave Episodes. In the previous passage, two partitions were presented congruently at a given time, each controlling a single aspect of the musical texture (instrumentation vs. dynamics). Here, the three trichordal partitions each have their own autonomy over a particular temporal domain as no two partitions are presented simultaneously. It is interesting to note how the trichords are articulated by each instrument in this passage. The trichords are divided into articulated dyads contrasting with single pitches. For example,  in measure 7, the violins play slurred (04) dyads and repeated staccato  single pitches while the viola and cello play staccato (04) dyads and legato or tenuto singletons. Once again, the trichords are further refined in the foreground as s-type 9  partitions. During this passage the s-type partitions associated with differences in articulation often involve (04) dyads, but not exclusively. On the other hand, the partitions s5, s8 and s9, generated in Figure 4.8, involve (01) and (03) dyads, but not (04) dyads. see Zuckerman 94 for an explanation of how these articulations derive from hexachords which mirror the overall hexachord progression which governs the first movement.  9  92  mm. 7-9, 15-18 t i = I 089 1145 I 236 I 7AB I  INV = (014)  4  mm. 10-12 t2 = I 014 I 23B I 589 I 67A I  INV = (014)  mm. 13-14 t3= I 019 I 2AB I 367 I 458 I  INV = (014)  4  4  M = t l Jt2 = t i Jt3 t2Jt3 = I 014589 I 2367AB I INV = (014589) S8 = tiMt2 = I 0 114 I 23 I 5 16 I 7A I 89 I B I INV= (s) (01) (03) S5 = tlMt3 = I 09 11 I 2 I 36 I 4 5 I 7 I 8 IAB I INV= (s) (01) (03) sg = t2Mt3 = I 01 I 2B I 3 I 4 I 58 I 67 I 9 IA I INV= (s) (01) (03) 2  S5MS8 =s5Msg = S8MS9  4  2  2  4  2  2  4  2  2  = disj hi  disj  Figure 4.8: Lattice L 5 of partitions, mm. 7-18 showing cyclic mapping.  93  Of the s-type partitions found in L5, two have literal foreground significance. In measure 12, arco vs. pizzicato divides each trichord into a dyad and singleton. The resulting partition is S9, a partition that results from the meet of the partition t2, which defines mm. 10-12 and of t3, which defines mm. 13-14. Measure 12 thus acts as a bridge between the two trichord partitions. In like manner, the articulation assignments from the end of measure 16 through to the end of measure 18 correspond to the S5 partition. This is particularly interesting considering that same S5 was also found earlier as the meet between dyn 1 and dyn2, the two dynamic trichord partitions in measures 1-3 and 4-6 respectively. S5 was previously a theoretical abstraction, but it now has a more literal foreground significance. Moreover, it appears at an appropriate place with regard to the array; the music has just left a t3 area (mm. 13-14) and is now in a ti area (mm. 15-18); S5 precisely captures the interaction between the two, since S5 = tiMt3. The association of ss with the articulation during this t i area thus forges a relationship back to the preceding t3 area. L5 has another interesting property in that it is a self-contained transformational network. The three trichordal partitions and the three s partitions map onto each other cyclically under T 2 , T 4 , T s , and T i O - (The h i partition maps onto itself under these transpositions as does the disjunct partition). Figure 4.9 shows these mappings. The net result is that the partitions of each stratum of L5 are transformed by a counter-clockwise, rotational network through the cyclic application of the T 4 and T8 operation and by a clockwise, rotational network through the cyclic application of T 2 and T i O - This is shown in Figure 4.8.  94  T4, T  8  Figure 4.9: Cyclic transformational mappings of L5 partitions. The cyclic transpositional structure provides a connection between this trichordal passage and the previous Octave Episodes. In the OEs, the T 4 cycles mapped the individual instrumental partitions (a and d 1) onto themselves through relating the internal dyad membersets and subsets. In this case, the transposition cycles connect the three trichordal partitions ( t i - t 3 ) , mapping them onto each other as well as connecting the partitions produced by their meets and joins. In the case of the trichord partitions, the T 4 (and T 8 ) cycle is literally present in the music itself as the distribution of trichord partitions throughout the array corresponds to the T 4 mapping. The first two columns of the array correspond to the partition, t l while the next two contain t 2 followed by t3 and then a return to t i . Thus, the counter-clockwise rotation of L 5 accurately describes the distribution of partitions throughout the passage. The transpositions T 2 , T 4 , T s , and T 1 0 are not the only automorphisms on L 5 . L 5 is automorphic under all T e n , lodd e v  The inversion operations do not rotate the lattice, however; instead, they exchange two of the t-partitions, and leave the other invariant, and they do likewise for the s-partitions. All of the partitions are invariant under To and T 6 . It is also useful to consider how the (014) trichords are dispersed in the lynes of the array for mm. 7-18. Figures 4.10 and 4.11 list the two distinct lyne partitions for each six-measure section of the passage as well as their meets and joins and the resulting lattices L 6 and L 7 . 95  14 = I 089 I 145 I 23B 167A I t5 = I 014 I 236 I 589 I 7AB I  INV = (014) INV = (014) 4  4  hi = UJt5 = I 014589 I 2367AB I INV = (014589) S8 = t4Mt5 = I 0 I 14 I 23 I 5 I 6 I 7A I 89 IA I B I INV = s (01) (03) 2  4  2  2  Figure 4.10: Lattice L g of lyne partitions, mm. 7-12. 16 = I 019 I 236 I 458 I 7AB I  INV - (014)  t7 = I 089 I 145 I 2AB I 367 I  INV = (014)  M  4  4  = I 014589 I 2367AB I INV = (014589) S5 = t6Mt7 = I 09 I 1 I 2 I 36 I 45 I 7 I 8 I AB I INV = s (01) (03) =t6Jt7  2  4  Figure 4.11: Lattice L 7 of lyne partitions, mm. 13-18. 96  2  2  As expected, the joins of the lyne partitions in each array block yield the h i partition. A surprising result occurs, however, from their meets. The meets of the lyne partitions reproduce two of the s-partitions ( s s and s$) that were originally produced as meets of the columnar partitions. These s partitions, in effect, serve as a link between the pitch-class organization of the lynear trichords and that of the columnar ones. They are acting on several levels as "foreground" refinements of the array trichordal partitions. Of even more interest is the fact that S5 again resurfaces as a lower bound partition of "middleground" trichordal partitions. This s-type partition has persisted throughout the first 18 measures of the piece both as an implicit, theoretical partition and as a literal "foreground" one. Several other s-type partitions are found in this passage that are not accounted for by L5, L6, or L7. As explained previously, the articulation designations in measure 7 correspond to an s-type partition. This partition is not present in L5, L6, or L7, however. These lattices therefore do not summarize all of the events of the passage, but nevertheless they do suggest the three levels of structure inherent in the quartet to this point. Certainly the (014589) hexachordal partition continues to be prolonged as a 2  "background" structure throughout the passage and the s partitions continue to suggest a foreground refinement of the trichords which govern the passage, even if the lattice fails to include all of the s partitions found on the music's surface.  Measures 19-33 of the quartet comprise the second octave episode of the movement and are shown in Example 4.3. Like the first octave episode, this passage presents a dyadic partition, now involving only (05) membersets, and a dynamic trichordal partition, now involving only (015) membersets (Zuckerman 1976, 91). Table 4.4 lists these partitions. 97  98  d2 = I 05 118 I 27 I 3A I 49 I 6B I dyn3 = I 04B I 129 I 378 I 56A I mf ff  pp p mp fff  INV = (05)  6  INV = (015)  4  ppp f  d2Jdyn3 = conj d2Mdyn3 = disj Table 4.4: Partitions, mm. 19-23.  Unlike in the first octave episode, there is no subpartition relationship between the dyadic partition (d2) and the dynamic partition (dyn3) in this passage, these two partitions being both conjoint and disjoint. Moreover, d2 is a subpartition of h i , but dyn3 is not. In fact, dyn3 is the first trichordal partition we have posited which is not a subpartition of h i . The dynamic partition does, however, come into play when considering the instrumental trichordal partitions for measures 24-33 (see ex. 4.3). Table 4.5 lists these partitions, which all have the same (015)" inventory as dyn3, but are distinct partitions. The trichords of measures 24-33 are again articulated as dyads and single pitches in this passage. Once again, this s-type distribution refers to the articulation assignments (staccato vs. legato) of pitch classes within the trichords. Most of these s-type articulation partitions are not related to the s-partitions in Table 4.5. However, si3 does define the articulation designations of measures 24-25. This partition results from the meet of the relevant trichord partition (t7) and the dynamic partition (dyn3) of the immediately preceding passage (mm. 19-23). As seen previously, this meet partition acts essentially as a bridge between the two passages, relating the dynamic and trichordal partitions simultaneously. Figure 4.12 shows the simple lattice L8 that results from the  99  meet and join of dyn3 and tj. L% is automorphic under T6, II, and I7 as all partitions are invariant under those transformations. What stands out about S 1 3 is that it - alone in this context - has no (01) or (05) dyads. The trichord partitions, t8 and dyn3 both contain (015) membersets, which feature interval classes 1, 4, and 5. These two trichord partitions interact, via meet, to bring out the (04) dyads of S 1 3 . While ic4 is featured in S 1 3 , the other s partitions, sio, s u , and si2 all feature interval classes 1 and 5.  dyn3 = I 04B I 129 I 378 I 56A I  INV = (015)  4  24-25 t8=l045M89l237l6ABI  INV = (015)  4  26-27 t9 = I 015 I 23A I 489 I 67B I  INV = (015)  4  28-31 t10= I 018 I 267 I 3AB I 459 I  INV = (015)  4  t8Jt9 = hi = I 014589 I 2367AB I t8Mtg = S 1 0 = I 05 I 1 I 23 I 4 I 6B I 7 I 89 I A I  INV = (014589)  2  INV= (s) (01) (05) 4  2  2  tsJtiO = hi t8Mtio = s i 1 = I 0 I 18 127 I 3 I 45 I 6 I 9 I AB I  INV= (s) (01) (05) 4  2  2  tgJt-io = hi tgMtio = S 1 2 = I 01 I 2 I 3A I 49 I 5 I 67 I 8 I B I dyn Mt8 = S13 = I 04 119 12 I 37 I 5 I 6A 18 I B I 3  INV= (s) (01) (05) INV= (s) (04) 4  2  4  4  dyn3Jt8 = conj  dyn Mtg = disj 3  dyn Jtg = conj d y n M t i o = disj dyn3Jtio = conj All s partitions are disjoint. 3  3  Table 4.5: Partitions, mm. 24-33 and interaction with dyn3. 100  2  dyn3 = I 04B I 129 I 378 I 56A I t8= I 045 I 189 I 237 I 6AB I S13 = dyn3Mt4 = I 04 119 12 I 37 I 5 16A I 8 I B I  INV = (015)  4  INV = (015)  4  INV= (s) (04) 4  conj  L8  S13  Figure 4.12: Lattice L g of trichordal partitions, mm. 19-25.  101  4  Figure 4.13 shows the lattice L a that results from the partitions of measures 2433. L a is isographic with L 5 , and also shares the same lattice automorphisms with its isographic relative, namely T e n , lodd- The automorphic mapping is similar as well. e v  Specifically, all partitions are invariant under To and T 6 while each of the inversions exchange two of the t and two of the s partitions while leaving the remaining one of each type invariant. For example, I i maps to to t i o and leaves t8 invariant. L 9 is also a selfcontained transformational network as was L 5 . In this case, however, T 2 and Ts map the t and s partitions onto each other via a clockwise rotation of the lattice while T 4 and T i 0 provide the counter-clockwise mapping. This rotational mapping is shown in Figure 4.13. It is clear that the partition h 1 continues to act as a background structure for all of the material investigated thus far despite our observation that dyn3 is not one of its subpartitions. This is shown in Figure 4.14 which links the lattices L 5 and L 9 into one (incomplete) composite lattice. This lattice ( L 1 0 ) clearly shows the h i background, and also delineates the two broad subsections of the first 33 measures which are defined by their "middleground" trichords and their contrasting inventory families (014) and (015) . 4  4  The dyadic partitions are omitted from the diagram as well as the dynamic partitions. Many of the s-type partitions are also present in the musical texture as foreground refinements of the trichords. A case can therefore be made that three structural levels exist throughout these 33 measures.  There are two passages in this movement in which only two instruments are involved. Zuckerman refers to these as two-part episodes (TPEs) and they are found between mm. 34-42 and measures 63-74. Zuckerman (95-96) clearly demonstrates how these passages are also derived from trichordal arrays. Unlike previous passages, 102  however, the TPEs do not project their trichords as discrete units in the musical texture. Example 4.4 shows measures 35-42. While the lynes in the first violin and cello can be temporally chunked into trichords, the musical texture does not offer these trichords as isolated entities. The resulting trichordal partitions are therefore not analytically (or aurally) relevant with regard to the lattice model. It would therefore be prudent to restrict the focus of this analysis to the Octave Episodes and Four-Part Episodes. t = | 045 I 189 I 237 I 6AB I  INV = (015)  tg = I 015 I 23A I 489 I 67B I  INV = (015)  tl0= I 018 I 267 I 3AB I 459 I  INV = (015)  h i = t8Jt9 = 8JtlO = t9Jt10 = I 014589 I 2367AB I  INV = (014589)  4  8  4  4  S1 o = t8Mtg = I 05 I 1 I 23 I 4 I 6B I 7 I 89 I A I  2  INV= (s) (01) (05) 4  2  2  511 = t8Mt10 = I 0 I 18 I 27 I 3 I 45 I 6 I 9 I A B I  INV= (s) (01) (05)  512 =tgMtl0 = I 01 I 2 I 3A I 49 I 5 167 I 8 I B I  INV= (s) (01) (05)  4  4  2  2  hi  disj  Figure 4.13: Lattice L9 of trichordal partitions, mm. 24-31.  103  2  2  Figure 4.14:  Composite lattice Lio of lattices L5 and L9.  104  Examples 4.5 and 4.6 show measures 47-51 and 52-62, the third O E and F P E respectively. Table 4.6 lists all of the partitions in this passage, along with their relevant joins and meets. The lattice of the non-conjoint and non-disjoint partitions is shown in Fig. 4.15.  47-51 d3 = I 03 I 1A I 2B 147 I 58 I 69 I dyn4 = I 029 I 14B I 368 I 57A I mf ff  p fff  INV = (03)  6  INV = (025)  4  ppp pp f mp  52-62 INV = (025)  ti 1 = I 035 I 18A I 247 I 69B I  4  d3Jdyn4 = conj d3Mdyn = disj ti 1 Jdyn4 = conj ti 1 Mdyn = disj ti 1 Jd3 = h2 = I 01358A I 24679B I INV = (024579) t i l M d =S14 = I 03 I 1A I 2 I 47 I 5 I 69 I 8 I B I INV = (s) (03) 2  4  3  Table 4.6: Partitions, mm. 47-64.  This passage involves a dyadic partition (d3) with an (03) inventory that echoes 6  the same inventory of d i , establishing a strong connection back to mm. 1-3. However, while these dyadic partitions share the same inventory they are not transformationally related. That is, no single canonical transformation will map the dyad membersets of di to those of d3.  106  4  Example 4.5: mm. 47-51 107  108  d3 = I 03 I 1A I 2B I 47 I 58 I 69 I  INV = .(03)  6  tl 1 = I 035 I 18A I 247 I 69B I  INV = (025)  h2 = tl 1 Jd3 = I 01358A I 24679B I  INV = (024579)  4  2  S14 = t l l M d 3 = I 03 I 1A I 2 I 47 I 5 I 69 I 8 I B I INV = (s) (03) 4  h2  S14 (mm. 52-53)  Figure 4.15:  Lattice L n of partitions, mm. 47-62.  109  4  The passage also differs from previous passages in many respects. As L n demonstrates, the "background" hexachord partition (h2) for this section consists of complementary diatonic (024579) hexachords rather than (014589) hexachords. It is of some interest that this shift in hexachord design comes exactly at the mid-way through the "movement." The dynamic partition ( d y n 4 ) also offers a new trichordal inventory class (025) . As well, in contrast to previous sections, the four-part episode (mm. 52-62) 4  is based on only one trichordal partition (ti l). This results because of the more limited ability of the partition h2 (in comparison with h i)to generate trichordal partitions having the same inventory but different membersets. However, like previous sections, the inventory of the FPE trichordal partition(s) is the same as that of the dynamic partition which precedes it (in this case (025) ). The dynamic partitions of the Octave Episodes 4  foreshadow the trichordal partition inventories of the Four-Part Episodes that follow. In the previous passage we saw that the dynamic partition ( d y n 3 ) was projected into the FPE that followed by interacting with the trichordal partition t8. In this case, it is the dyadic partition (d3) that is projected into the Four-Part Episode as opposed to the dynamic partition. The join of tl 1 and d3 produces the background h2 partition while the meet produces another s-type partition, s i 4 , which also has an inventory s (03) that 4  4  echoes back to the first passage. Specifically, si and S4 also had this inventory; even so, none of these partitions are transformationally related. Partition s i 4 aptly describes the foreground events taking place between measures 52 and 62. For example, in measures 52-53, the violin plays pes 6 and 9 simultaneously while pc B is isolated 3 and 1/2 beats later. Similarly, the second violin states pes 1 and A with the same pp dynamic, while pc 8 is played ff>f<ff; the viola and cello in these measures play <03> and <47> as simultaneities while isolating pes 2 and 5. These 110  distinctions correspond exactly with the membersets of si4. This partition has practical "foreground" significance throughout this 10-measure passage, often delineated by articulation, dynamic or register. Here is a clear case of a foreground partition which serves to "embellish" its middleground trichords of tl 1. Ln  has an isographic relationship to L i . Like the previous lattice, L n also  contains one partition (d3) which though not actually present in measures 52-62, is projected into this passage from previous measures (mm. 47-51), and acts as a mediator among the three concurrent partitions of the passage in question. Figure 4.16 illustrates this idea, and its three structural levels. As with Figure 4.3, this graphic is not a lattice, but it does show how three partitions, h2, ti 1 and s 14, are acting in the same musical space in three structural levels, while d3 is projected into the passage as a differentiator among the three.  background  h2  middleground foreground Figure 4.16: Structural Levels, mm. 52-62.  Ill  L n also has three (non-trivial) automorphisms; two inversions I i and I7, as well as the T6 operation. Once again, these automorphic transformations are derived from the degrees of symmetry inherent in the background hexachord for this section which is invariant under To, T6, II, and I7. Each partition maps onto itself under these operations. (They each have a different inventory, so none can map onto any other.)  The final section to be analyzed is shown in Examples 4.7 and 4.8. Measures 7580 comprise the final dyadic Octave Episode of the movement while mm. 81-92 contain the final trichordal Four-Part Episode. Table 4.7 summarizes all of the partitions found in this passage along with their relevant joins and meets.  75-80 d4 = I 01 I 23 145 I 67 I 89 I AB I dyn5 = I 01A I 235 I 467 I 89B I ppp f  p fff  mf ff  INV  (oi)  INV  (013)  6  4  pp mp  81-92 INV tl2 = I 023 I 1AB I 457 I 689 I INV d4Jdyn5 = h.3 = I 0189AB I 234567 I d4Mdyn5 = S15 = I 01 123 14 I 5 I 67 I 89 IA I B I I N V t-|2Jdyri5 = h3 12Mdyn5 = s-|6 = I 0 11A 123 I 47 I 5 16 I 89 I B I I N V ti2Jd4= h3 t«|2Md= S17 = I 0 11 I 23 145 16 I 7 I 89 I AB I I N V INV S15MSI6 = X 2 = I0I1I23I4I5I6I7I89IAIBI si5Ms-|7=x2  (013)  4  (012345)  2  (s) (01) 4  4  (s) (01) 4  4  (s) (01) 4  4  4  SI6MS17 = X 2  Table 4.7: Partitions, mm. 75-92.  112  (s) (01) 8  2  Example 4.7: mm.75-80 113  114  Once again, the Octave Episode (mm. 75-80) comprises a dyadic partition that outlines the temporal succession of pitch classes and a trichordal partition that corresponds to the distribution of dynamic levels. The dyadic partition d4 features a new inventory of (01) . As with previous OEs, the (013) inventory of the dynamic trichordal 6  4  partition foreshadows the inventory of the trichordal partition in the ensuing FPE. As with the previous passage, this FPE utilizes only one trichordal partition, t\2The Octave Episode of mm. 75-80 is similar to the first Octave Episode in that the meet of the dynamic partition and dyadic partition results in an s-type partition which is analytically significant. The four (01) dyads of this s partition outline the first and last pitches of the passages, <67> and <89>, as well as the two dyads which sound together as simultaneities, {0,1} and {2,3}. For this reason, these four dyads are brought out more as discrete units than the other two dyads of d4. Figure 4.17 shows the resulting lattice, L 1 2 , for this octave episode. Once again, a case can be made for three structural levels in this passage with the dynamic partition acting as the "differentiator" among the three. Fig. 4.18 demonstrates this relationship as the hexachordal, dyadic and s-type partitions are shown as the structurally (and aurally) relevant partitions while the dynamic partition mediates among the three. In other words, the dynamic partition  (dyns) interacts  with the dyadic partition (d4) to produce the other partitions  (I13  and  si5), which act on different structural levels. This section also introduces a new "background" hexachordal partition, h3, consisting of chromatic (012345) hexachords. As in the previous Four-Part Episode, measures 81-92 utilize only one trichordal partition, 112 The lattice that models all the partitions in mm. 75-92 is L13 (Fig. 4.19), which is isographic to L 3 . Two of the s-type partitions of the lattice ( s i 6 and si7) are not particularly significant to the musical texture. However, it is again interesting to 115  discover that the meets of all of the s partitions are not disjoint. In fact, another x-type partition (x2) results in which two dyads are preserved, {2,3} and {8,9}. These dyads feature significantly in the passage as simultaneities (as in mm. 77 and 79 in the violin 1 and viola) or as consecutive pitches (as in mm. 80 in the violin 2 and cello). L 1 3 is also automorphic under T6 as well as I5 and 111.15 preserves the dyads of x2 while T6 and Il 1 exchanges them.  d4 = I 01 123 I 45 I 67 I 89 I AB I dyn5 = I 01A I 235 I 467 I 89B I h3 = d4Jdyn5 = I 0189AB I 234567 I S15 = d4Mdyn5 = I 01 I 23 14 I 5 I 67 I 89 I A I B I  r N V = (oi)  6  INV = (013)  4  INV = (012345)  2  INV = (s) (01) 4  h3  L12  dyns  d4  S15  Figure 4.17:  Lattice L12 of partitions, mm. 75-81.  116  4  background middleground foreground Figure 4.18: Structural Levels, mm.  75-81  h3  tl2 = I 023 I 1AB I 457 I 689 I *2 d4 = I 01 I 23 I 45 I 67 I 89 I AB I dyn5 = l01AI235l467l89BI h3 = d4Jdyn5 = ti2Jdyn5 =tl2Jd4 = I 0189AB I 234567 I 515 = d4Mdyn5 = I 01 I 23 I 4 I 5 I 67 I 89 IA I B I 516 = ti2Mdyn5 =l 0 MA I 23 I 47 I 5 I 6 I 89 I B I 517 = ti2Md4= I 0 I 1 I 23 I 45 I 6 I 7 I 89 I AB I X2 =S15MS16 =  S15MS17 = S16MS17 = I 0 I 1 I 2 3 I 4 I 5 I 6 I 7 I 8 9 I A I B I Figure 4.19: Lattice L13 of partitions, mm. 75-92. 117  The series of lattices presented in the preceding discussion model the various aggregate partition interactions which govern most of this first movement of the quartet. Each lattice strongly suggests the presence of three concurrent structural levels in each section. Table 4.8 (next page) summarizes the inventories of the various partitions which delineate these structural levels. Compare this table with Fig 4.1, the general lattice initially proposed as a summary of the structural levels of the quartet's first movement. Both figures strongly suggest that the progression of hexachordal partitions outlines the background structure for the movement. The table shows that these hexachordal partitions are prolonged over larger sections of the musical sub-structure. The progression of hexachordal partitions represents the overall background structure for the "movement" as modelled by the lattices.  As there are only three hexachord types and three hexachordal partitions that govern the pitch material for the entire "movement," a comparison of these hexachordal partitions, via meet and join, may prove useful. Figure 4.20 lists these partitions with their meets and joins and displays the resulting lattice, L14.  118  119  All h partitions are conjoint. hi = I 014589 I 2367AB I h2= I01358AI24679B I h3 = 10189AB I 234567 I mi =hiMh2 = I 0158 I 267B I 3A I 49 I m2 = hiMh3 = I 0189 I 2367 I 45 I AB I m = h2Mh = I 018A I 2467 I 9B I 35 I n =miMm2 = miMm3 = m2Mm3 = I 018 I 267 I3I4I5I9IAIBI 3  3  INV = (014589)  2  INV = (024579)  2  INV = (012345)  2  INV = (05) (0158) 2  2  INV = (01) (0145) 2  2  INV = (02) (0135) 2  2  INV = (s) (015) 6  2  conj  L14  n  Figure 4.20: Lattice Li4 of background hexachordal partitions.  120  The hexachordal partition meets yield three partitions of format 2 4 . These m2  2  partitions contain tetrachords, and although these have not been modelled by any previous lattice, they do offer some analytically productive information. The three tetrachord types contain, as subsets, all of the trichord-types of the t and dyn partitions, with the exception of the augmented trichord partition a. In addition, all of the dyad memberset types of the d and s-type partitions are also subsets of the three m partitions of L14. The dyadic membersets of the m partitions also yield two of the dyad types, (01) and (05), found in the previous d and s partitions. The d and s partitions contain, as membersets, all of the possible dyad types with the exception of (06) and (02). It is curious to note therefore that m3 contains (02) dyad membersets. These (02) dyads are subsets of the (025) and (013) trichords that persist throughout the second half of the movement. Surprisingly, (02) dyads were not modelled by any dyadic or s-type partition, so it is significant that they arise here. It is also curious that the m partitions all share a common meet, labelled n in I44. This partition contains two (015) trichords as membersets which were featured previously as membersets of t i o but no other trichord or dyad type is preserved. The relevance of L14 is thus less clear in comparison to previous lattices as it fails to produce many of the relevant partition classes found in the lattices and in the music itself. Table 4.8 also shows how the t, dyn, and d partitions differentiate each smaller section of the movement by partition inventory. The analogy to a Schenkerian "middleground" is compelling. The various dyadic and trichordal partitions serve to prolong the background, hexachordal partitions from which they are extracted and they control the pitch relationships in smaller formal divisions of the "movement." The situation with s partitions is more complicated. We initially posited these stype partitions as foreground entities that served to embellish their respective 121  "middleground" and "background" partitions. There is a certain amount of evidence to support this in the lattices, since most lattices yielded s-type partitions from the meets of "middleground" partitions. In addition, many of these s partitions were associated with specific aspects of the musical texture: register, instrumentation, articulations and dynamics. However, other s partitions produced in the lattices were not found to have any musical relevance beyond their preservation of certain dyad classes as membersets. As well, the S5 partition was produced in several lattices and was actually "prolonged" itself through the first eighteen measures of the piece. In a sense, it could be seen as existing on a higher structural level than its parent trichordal partitions, as it was "prolonged" over a larger formal section. It is clear that s-type partitions are prevalent throughout the movement in the music itself, as several examples have demonstrated. It is not clear, however, if these can be considered as true foreground entities. Their presence as a theoretical foreground in the lattices is certainly clear, but the fact that some of these s partitions are purely theoretical and not musically evident questions their true foreground relevance. It is clear from the lattices used to model the pitch relationships in this piece (and indeed in the previous works analyzed) that Babbitt's early trichordal array works contain various levels of structural organization. Certainly, the hexachordal partition is the overriding background source for pitch relationships in large sections of the piece. The "middleground" trichordal and dyadic partitions control smaller structural divisions and embellish the background hexachordal partitions. The s-partitions may not present a clear foreground level of structure but are certainly important smaller partition types in this work and in the other works analyzed. The analogy used to describe the quartet as a complex crystal gains new significance with this analytical approach. The crystal can be imagined as a network of  122  lattices which connect the various aggregate partitions. Three possible levels of structure are evident in each lattice and in the overall crystal itself. A true strength of the lattice approach is that it relies less on pre-compositional evidence (i.e. derivation from an overall 12-tone series) and, for the most part, models the most aurally significant pitch relationships. The trichords and s partitions shape the detailed character of the musical surface while the hexachords control the overall pitch structure of each formal section. Thus, the crystal analogy becomes less of an abstract metaphor and more of an apt description of the structure of the movement: a series of facets each with a unique perspective on the crystal's sub-structure, that outline a complex web of vertices which influence the overall shape and character of the complete structure.  123  CHAPTER 5: CONCLUSIONS  The discussion in the preceding chapters has illustrated how partition lattices can be an effective tool for accessing the trichordal array works of Milton Babbitt. Chapter 2 demonstrated how the lattices bring out multiple hexachordal, trichordal, and s-type partitions, which are superimposed in the textures of the Woodwind Quartet. A complex web of combinatorial hexachords with their trichordal generators interact with each other and control various aspects of the musical texture. The lattices show the relationships among all of these hexachordal and trichordal partitions and provide a graphical representation of the organizational complexity of the work. The partition lattices posited in Chapter 3 revealed how the various surface trichordal and s-type partitions in the Composition for Viola and Piano are all extracted from a common hexachordal superpartition. Furthermore, the lattices modelled much of the surface pitch detail by projecting partitions that described various aspects of the musical texture. Partitions that played a more pivotal role in the musical texture were generally found in central positions in the lattices while partitions that had a more limited influence were generally found on the periphery. In Chapter 4 , lattices were used to suggest a structural hierarchy of partitions in the String Quartet no. 2. Three hexachordal superpartitions are prolonged over the first ninety-two measures, and their progression constitutes a type of "background" structure. Trichordal and dyadic partitions exist at a "middleground" level while partitions consisting of varying numbers of dyads and singletons comprise a possible "foreground" level. The partition lattice model, therefore, demonstrated the relationships among these three works, each being composed from arrays with aggregates that project multiple partitions. In each case, however, the lattices were used to bring out different aspects of partition organization and to explore the different ways that Babbitt  124  realized compositions from his trichordal arrays. While the lattice model certainly helps access the complex musical textures of Milton Babbitt, the method has certain limitations when applied to this music. Certain musical passages are not easily represented by partition lattices. In general, the "neatest" results are obtained when the set of partition generators is limited to three or four partitions and when none of these source partitions have membersets in common. In other words, simple, closed lattices result when the meets and joins of the source partitions do not preserve any of the original membersets of the partition generators. By contrast, Tables 3.3 and 4.1 show results obtained from source partitions that have common membersets. Such large lattices are not easily represented in graphic form. In both of these cases, however, sublattices of the theoretical superlattices were analytically productive. In general, though, these sublattices projected a limited set of partition generators and ignored interactions between source partitions that had common membersets. Thus, the partition lattices are efficient models of limited aspects of the musical texture but may fail in graphically representing the complete musical picture. As Chapter 4 demonstrated, the lower bound partitions of many of the lattices were not representative of any discernible aspect of the musical surface. Several of the s and x partitions that were modelled by the lattices were not found in the music itself. The conclusions that can be drawn from these partitions are, in many cases, purely theoretical. Concrete conclusions can be drawn from the production of trichordal and hexachordal partitions via meet and join because, in most cases, the partition interactions correspond to the pre-compositional design of the music. In other words, Babbitt's arrays are constructed so that they can project these hexachordal and trichordal partitions. This is not necessarily the case for the lower-bound s and x partitions. However, this observation does add a certain significance to those cases where s and x partitions do, in fact, model clear aspects of the music itself. 125  The lattice model is clearly useful when applied to the simple, even-partition array constructs of the three works analyzed. The scope of this discussion was, in general, limited to trichordal and hexachordal arrays, though equally productive results can be obtained by analyzing tetrachordal partitions in relevant sections of Babbitt's compositions. We saw in Chapter 4 how the lattices were analytically useful in dealing with dyadic partitions. Thus, the method is very appropriate to Babbitt's early works, which feature even-partition arrays, especially trichordal arrays. Beyond this era, however, the partition lattice approach presents some problems. Example 5.1 shows one block of the all-partition array used in Milton Babbitt's  Post-Partitions  (1966). It is immediately obvious that the array block is much more  complex than the trichordal and hexachordal arrays studied in the preceding chapters. The columns are designed so that they project seven partitions, each with a different format. (Mead 1994, 274). A comparison of these partitions, via meet and join, will not be very productive as most of the columnar partitions are conjoint and their meets will preserve only a small number of dyad membersets. However, the lynes of the array block project three distinct hexachordal partitions as shown in Example 5.1. Example 5.2 lists the partitions produced from the interactions of these hexachordal partitions and displays the resulting lattice.  126  67  79  4  0  8B  5  2  A  31  1  8  AB  3  6  9  954  2  2  7  7  4  89B  B6  6  3  AO  5  5  2  = h  2  0  5  32  A  Al  8  B  76  4  4  9  = h  2  B  6  89  1  2  7  54  0  = h  3  = h  3  01  732  4  6AB  9  2'  3  2  2  3  642  = hl  0  = hl  1A 3  3  5  0 8  18  543  42  4  31 3  3  Example 5.1: All-Partition array block,  127  3 2 1 2  2  2  Post-Partitions.  hi = I 0138AB I 245679 I I N V = (023457) h2 = I 01235A I 46789B I I N V = (023457) h3 = I 023457 I 1689AB I I N V - (023457) hiMh2 = mi = I 013A I 25 I 4679 I 8B I I N V = (0235) (03) hi Mh3 = m2 = I 03 I 2457 I 18AB I 69 I I N V = (0235) (03) h2Mh3 = IT13 = I 0235 I 1A I 47 I 689B I I N V = (0235) (03) m-|Mm2 = m-|Mm3 = m2Mm3 = S1 = I 03 I 1A I 25 I 47 I 69 I 8B I I N V = (03) 2  2  2  2  2  2  2  2  6  conj  h3  mi  rri2 irti3  S1  e 5.2: Lattice of hexachord lyne partitions from E x . 5.1.  128  2  It is clear that a combinatorial hexachordal design is at work in this array block. All three hexachordal partitions comprise complementary (023457) hexachords. Their meets produce m partitions of common inventory and a dyadic (03) partition is the greatest 6  lower bound of the entire lattice. These results, however, do not seem to relate to other partitions in the array block. There is a dyadic partition in the first column of the block and thus the first columnar aggregate of the piece will feature surface dyads. However, this dyadic partition does not correspond to s i . In fact, si is never presented once in the entire array. The all-partition arrays are designed so that they project one representative of all of the possible partition formats. The 2  6  format is, therefore, only presented once  at the beginning of the work. Some of the dyads of si are found as membersets in other partitions or as subsets of partition membersets, but s i itself is not particularly analytically relevant to the array. In addition, none of the membersets of any of the m partitions are found as partition membersets in the array block. Many of the all-partition array works are, indeed, based on rows that parse the aggregate as combinatorial hexachords (Mead 1994, 124-136). However, the arrays are constructed such that maximal partition diversity is extracted from these hexachordal partitions. Therefore, even if a subpartition that result from the meets of its hexachordal superpartitions can be found elsewhere in the array, it will only be found once in the entire array and no other partition will share its format. Consequently, it will not map to any other partition in the array under any canonical transformation. Therefore, while simple lattices can be constructed from hexachordal source partitions in the array, the lattices themselves will not usually offer further analytical detail. Any lattice constructed from the set of partitions in any all-partition array block will be very complex and lacking in symmetry. The membersets of these partitions are designed to share little in common with each other. Certain dyads may be found as 129  membersets or subsets of more than one partition, but the preservation of trichords or larger membersets will be very rare. The lattice model appears not to offer much in the way of analytical potential for such works. The partition lattice model clearly does provide a useful analytical method for approaching the early works of Milton Babbitt. The three works analyzed here project a multitude of complex pitch relationships involving partitions which are effectively summarized by the lattice approach. Other works that would benefit from a partition lattice approach include:  Three Compositions for Piano (1947), Composition for Four  Instruments (1948), Composition for Twelve Instruments (1948, 1954), The Widow's Lament in Springtime (1950), Du (1951), Two Sonnets (1955), Duet for Piano (1956), Semi-Simple Variations' (1956), All Set (1957), Partitions (1957), Sound and Words 0  (1960) ,  Composition for Tenor and Six Instruments (1960), and Vision and Prayer  (1961) .  See Kurth 1999 for a partition lattice approach to the opening measures of Variations.  10  130  Semi-Simple  BIBLIOGRAPHY Alegant, Brian. 1993. "The Seventy-Seven Partitions of the Aggregate: Analytical and Theoretical Implications." Ph.D. dissertation, University of Rochester, Eastman School of Music. . 1996. "Unveiling Schoenberg's op. 33b." Music Theory Spectrum 18/2: 143-166. Babbitt, Milton. 1950. Composition for Viola and Piano. New York: C.F. Peters. . 1953. Woodwind Quartet. New York: Associated Music Publishers. . 1954. String Quartet no. 2. New York: Associated Music Publishers. . 1955. "Some Aspects of Twelve-Tone Composition." The Score and LMA Magazine 12: 53-61. . 1960. "Twelve-Tone Invariants as Compositional Determinants." Musical Quarterly 46(2): 246-59. . 1961. "Set Structure as a Compositional Determinant." Journal of Music Theory 5(1): 72-94. . 1971. "Contemporary Music Composition and Music Theory as Contemporary Intellectual History." Perspectives in Musicology, ed. Barry S. Brook, Edward O.D. Downes, and Sherman J. van Solkema. New York: Norton. 72-94. . 1976. "Responses: A First Approximation." Perspectives of New Music 14(2)/15(1): 3-23. Barkin, Elaine. 1967. " A simple Approach to Milton Babbitt's Semi-Simple Variations." The Music Review 28: 316-322. Bazelow, Alexander R., and Frank Brickie. 1976. "A Partition Problem Posed by Milton Babbitt." Perspectives ofNew Music 14(2)/15(1): 280-293. Berger, Arthur. 1976. "Some Notes on Babbitt and His Influence " Perspectives of New Music 14(2)/15(1): 32-36. Borders, Barbara Ann. 1979. "Formal Aspects in Selected Instrumental Works of Milton Babbitt." Ph.D. dissertation, University of Kansas.  131  Dubiel, Joseph. 1990. "Three Essays on Milton Babbitt, Part One: Introduction, 'Thick Array / Of Depth Immeasurable'." Perspectives of'New Music 28(2): 216-261. . 1991. "Three Essays on Milton Babbitt, Part Two: 'For Making This Occasion Necessary'." Perspectives of New Music 29(1): 90-123. . 1992. "Three Essays on Milton Babbitt, Part Three: 'The Animation of Lists'." Perspectives ofNew Music 30(1): 82-131. Gibbs, Jason. 1989. "Prolongation in Order Determinate Music." Ph.D. dissertation, University of Pittsburgh. Haimo, Ethan, and Paul Johnson. 1984. "Isomorphic Partitioning and Schoenberg's  Fourth String Quartet." Journal of Music Theory28: 47-72. Howe, Hubert S. 1965. "Some Combinational Properties of Pitch Structures."  Perspectives ofNew Music 4(1): 45-61. Hush, David. 1982-83 and 1983-84. "Asynordinate Twelve-Tone Structures: Milton Babbitt's Composition for Twelve Instruments." Perspectives of'NewMusic 21(12): 152-208; and 22(1-2): 102-116. Kassler, Michael. 1967. "Toward a Theory That is the Twelve-Note-Class System."  Perspectives ofNew Music 5(2): 1-80. Kowalski, David. 1987. "The Construction and Use of Self-Deriving Arrays."  Perspectives of'New Music 25(1-2): 286-361. Kuderna, Jerome George. 1982. "Analysis and Performance of Selected Piano Works of Milton Babbitt." Ph.D. dissertation, New York University. Kurth, Richard. 1992. "Mosaic Polyphony. Formal Balance, Imbalance, and Phrase Formation in Schoenberg's Suite, Op. 25." Music Theory Spectrum 14(2): 188208. . 1993. "Mosaic Isomorphism and Mosaic Polyphony: Balance and Imbalance in Schoenberg's Twelve-Tone Rhetoric." Ph.D. dissertation, Harvard University. . 1996. "Dis-Regarding Schoenberg's Twelve-Tone Rows: An Alternative Approach to Listening and Analysis for Twelve-Tone Music." Theory and Practice 21: 79-122.  132  . 1999. "Partition Lattices in Twelve-Tone Music: An Introduction." Journal of Music Theory 43(1): 21-82. Lake, William E. 1986. "The Architecture of a Superarray Composition: Milton Babbitt's String Quartet no. 5." Perspectives of New Music 24(2): 88-111. Lewin, David. 1962. "A Theory of Segmental Association." Perspectives of New Music 1: 89-116. . 1976. "On Partial Ordering." Perspectives ofNew Music 14(2)-15(1): 252-257. . 1977. " A Label-Free Development for 12-Pitch Class Systems." Journal of Music Theory2\{\).  29-48.  . 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. . 1995. "Generalized Interval Systems for Babbitt's Lists, and for Schoenberg's String Trio." Music Theory Spectrum 5: 89-109. Lewin, Harold. 1965. "Aspects of the Twelve-Tone System: Its Formation and Structural Implications." Ph.D. dissertation, Indiana University. Lieberson, Peter G. 1985. "Milton Babbitt's Post-Partitions:' Ph.D. dissertation, Brandeis University. Martino, Donald. 1961. "The Source Set and Its Aggregate Formations." Journal of Music Theory 5: 224-273. Mead, Andrew. 1983. "Detail and Array in Milton Babbitt's My Complements to Roger" Music Theory Spectrum 5: 89-109. . 1984. "Recent Developments in the Music of Milton Babbitt." Musical Quarterly 70(3): 310-331. . 1988. "Some Implications of the Pitch Class/Order Number Isomorphism Inherent in the Twelve-Tone System: Part One." Perspectives of New Music 26(2): 96-163. . 1989. "Twelve-Tone Organizational Strategies: An Analytical Sampler." Integral 3: 93-170.  133  . 1994 An Introduction to the Music of Milton Babbit. Princeton: Princeton University Press. Morris, Robert D. 1977. "On the Generation of Multiple-Order Function Twelve-Tone Rows." Journal of Music Theory 21: 238-262. . 1987. Composition with Pitch Classes. New HavensYale University Press. , and Brian Alegant. 1988. "The Even Partitions in Twelve-Tone Music." Music Theory Spectrum 10: 74-101. , and Daniel Starr. 1974. "The Structure of All-Interval Series." Journal of Music Theory 18(2): 364-389. Rahn, John. 1975. "How Do You Du (by Milton Babbitt)?" Perspectives of New Music 14(2)/15(1): 61-80. Rothstein, William. 1980. "Linear Structure in the Twelve-Tone System: An Analysis of Donald Martino's Pianissimo" Journal of Music Theory 24: 129-165. Rouse, Steve. 1985. "Hexachords and Their Trichordal Generators." In Theory Only 8(8): 19-43. Samet, Bruce. 1987. "Hearing Aggregates." Ph.D. dissertation, Princeton University. Starr, Daniel. 1978. "Sets, Invariance and Partitions." Journal of Music Theory 22(1). 142. . 1984 "Derivation and Polyphony." Perspectives of New Music 23(1): 180-257. , and Robert Morris. 1977 and 1978. "A General Theory of Combinatoriality and the Aggregate." Perspectives of New Music 16(1): 3-35 and 16(2): 50-84. Straus, Joseph N . 1986. "Listening to Babbitt." Perspectives of New Music 24(2). 10-24. . 1987. "The Problem of Prolongation in Post-Tonal Music." Journal of Music Theory 22(1): 1-42. Swift, Richard. 1976. "Some Aspects of Aggregate Composition." Perspectives of New Music 14(2)/15(1): 236-248.  134  Taub, Robert. 1986. "An Appreciation of Milton's Piano Music.  Perspectives of New  Music 24(2): 26-29. Westergaard, Peter. 1966. "Toward a Twelve-Tone Polyphony."  Music 4(2):  Perspectives ofNew  90-112.  Winham, Godfrey. 1970. "Composition with Arrays." Perspectives  ofNew Music 9(1):  43-67. Wintle, Christopher. 1976. "Milton Babbitt's Semi-Simple Variations."  Perspectives of  New Music 14-15: 111-154. Zuckerman, Mark. 1976. "Derivation as an Articulation of Set Structure: A Study of the First Ninety-Two Measures of Milton Babbitt's String Quartet No. 2." Ph.D. dissertation, Princeton University. . 1976. "On Milton Babbitt's String Quartet No. 2." Perspectives  New Music 14(2)/15(1): 85-110.  135  of  


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