compositional spaces and other territories

Compositional Spaces and Other TerritoriesAuthor(s): Robert MorrisSource: Perspectives of New Music, Vol. 33, No. 1/2 (Winter - Summer, 1995), pp. 328-358Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/833710 .

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COMPOSITIONAL SPACES AND

OTHER TERRITORIES

ROBERT MORRIS

INTRODUCTION

HIS PAPER TAKES UP topics raised in or suggested by my book, Com-

position with Pitch-Classes: A Theory of Compositional Design.' The book is addressed to experienced composers who are interested in using pitch-class relations to make cogent, rich, and sophisticated music. It is neither a textbook nor an introduction to posttonal composition, although it can be and has been used in pedagogical settings. Since I try to meet the needs of many different kinds of composers, I do not discuss musical style, purpose, or occasion at any great length. Yet I assume my work is communal in spirit since it brings together many different compositional theories and practices into one integrated framework.2



Compositional Spaces and Other Territories

Despite its generality, Composition with Pitch-Classes does posit a specific model of the compositional process. We can compare this model with the older, "traditional" one shown in Example la. The box on the left reveals little about the cognition of composition, but nevertheless ini- tiates the chain of stages consisting of sketch, draft, score, and performance. I include performance as part of the model not only because it functions as the composer's goal or purpose, but because it has the great- est impact on future changes in the states of the other stages.

Example lb is a compositional process implied by my book. It adds two new stages: compositional design and improvisation. Compositional designs are arrays of pitch classes ready to be realized as pitches in (at least sequential) time. These designs are compositions in as much as they define classes of temporal and polyphonic sequences and alignments of pcs. Yet designs are not compositions in a different sense, since, without specifying a design's realization in at least pitch and time, one cannot actually hear its structure as music. In general, compositional designs can model anything from the simplest musical passages to entire pieces. They can also be considered as attempts to suggest or implement musical syn- tax in either the presence or absence of tonal grammar.

a. An older model:

Ideas, knowledge, -- sketch -- draft(s) -| score - performance skill

b. Implied in Composition with Pitch-Classes:

Ideas, draft(s) >-- score | Ideas, compositional / \ knowledge, > design t

performance I skill . .

"s improvisation

c. A newer model:

I score

compositional

Ideas - - design Ideas c a compositional draft(s) knowledge, spacedraft(s)

skill spae free / improvisation I performance

EXAMPLE 1: SOME MODELS OF THE COMPOSITIONAL PROCESS

329



Perspectives of New Music

But whatever their status, compositional designs are incomplete structures. Example lb indicates that a design can be completed by improvisation. "Comping" jazz chords from a lead sheet or realizing figured bass from a continuo part are examples of real-time realization. We might say that this kind of improvisation is strict.

Nevertheless, the model has the same sort of weakness as its progeni- tor. Aside from the lack of feedback from one stage to another that char- acterizes real composition, there is often a gap between "ideas, knowledge, and skill" and the "compositional design." In other words, how can a composer construct the designs he/she imagines? While there is much discussion in my book on this very point, a large part of the gap can be filled in with the concept of a compositional space. As we will see, compositional spaces are out-of-time structures from which the more specific and temporally oriented compositional design can be composed. This is shown in the model in Example Ic with another stage, labeled "free improvisation." Like a compositional design, a compositional space can be realized in real time, but the improvisation that results is not con- strained by a prearranged ordinal structure; thus it is not strict.

The contrast between the two models in Examples lb and Ic will be fleshed out in the balance of this paper. First I will discuss aspects of the realization of compositional designs. Then I will examine two basic cate- gories of compositional spaces: abstract and literal spaces. I shall illustrate and discuss two types of abstract spaces (grammars and two-partition graphs) and five examples of literal spaces (realizations of grammars, literal two-partition graphs, Room squares, Boulez multiplication matrices, and generalized Mead tiles). Along the way, when pertinent, I will take up issues that pertain to musical cognition and improvisation.

COMPOSITIONAL DESIGNS

Most compositional designs are arrays of two or more dimensions. Array cells hold pcs; ordered, partially ordered, or unordered pcsets; or even other arrays. Example 2a shows a simple twelve-tone compositional design. The design illustrates a type of four-row combinatoriality that was first introduced by Berg in his Lyric Suite. This very simple example involves P forms sequentially related by T3. The P forms each occupy a different array row while array columns hold twelve-tone aggregates. Thus the pcs within each row are totally ordered, while the pc content of each column is only partially ordered. This type of combinatoriality demands only that the four disjoint trichords of the generating row contain no ic 3s or 6s.

330




C C# F F# E B G# GA D A# D#

D# E G# A G D B A# C F C# F#

F# G B C A# F D CD# G# E A

A A#D D# C# G# F E F# BGC

column 1 column 2 column 3 column 4

The design is a "Combination matrix"3

Content of the design's rows: twelve-tone rows in row-class of < C C# F F# E B G# G A D A# D# > row 1 = P row2 = T3P row 3 = T6P row 4 = TgP

Content of each of the design's columns: a twelve-tone aggregate

EXAMPLE 2a: A SIMPLE COMPOSITIONAL DESIGN

Example 2b provides a realization of the design in Example 2a. I avoid calling it a musical realization, since it is very dull. Its literal presentation of the alignment of the array results in a blatant, note-against-note tex- ture; each aggregate is a 3/4 measure of three quarter-note chords, each chord a diminished-seventh sonority. Moreover, in addition to the lack of imagination, the uniform instrumentation and the overlap of registers between the four instrumental lines tend to obscure the rows that generate the design, rows that are surely more appealing than the chords.

y _- 1b I , I I f F - L b - ' -, ' I I I

bi #J I I r=

&h. , I 1 I r I I . I ^ I

f

EXAMPLE 2b: A VERY DULL REALIZATION OF EXAMPLE 2a

row 1

row 2

row 3

row 4

V.I

V. II

Via.

V'c. * x 1 6 -- r" r u ljr I

[

331

I '




Examples 3a, b, and c provide more stimulating realizations of Example 2a. Each example has a different reconfiguration of the compositional design of Example 2a written under it. If there is anything that unites these examples, it is that the partially ordered columns of the design are ordered to produce complex but local gestures and affinities.

Example 3a uses three rather than four instruments. The four array- rows are each taken by a different instrumental combination. The top row is found in notes doubled by the clarinet and the piano; the second row is articulated by pitches played by all three instruments; the last two rows are found in the pitches doubled by the 'cello and piano and those doubled by the clarinet and piano. A closer look at the realization reveals a functional distinction between notes of long and short duration. Long notes, sustained by at least one of the participating instruments, are circled in the underlying design to show that each aggregate projects its own ordered hexachord. The inside aggregates (numbers 2 and 3) project transformations of the first hexachord of the generating row; the long notes in the outside aggregates project hexachords related to one another by T1I but not to the hexachords of the P row. If this were an excerpt from an actual piece, these Qhexachords might be a reference to some other hexachord in the work or even in some other piece of music.

In Example 3b we have the members of a string quartet each playing their own row of the design; as you can see, the four rows of the design are assigned in order to violin II, viola, 'cello, and violin I. Here the quartet plays using many different means of articulation (arco, pizzicato, tremolo, ponticello, et cetera). As shown at the bottom of the example, the trichords in one aggregate determine how notes in another aggregate are articulated. For instance, the tremolo notes in all instruments in the first measure (or aggregate) are the same notes as in the second violin's music of the last measure (or aggregate). These notes are D, B l, and E . Like- wise, the ordinario notes in the first measure-F#, Db, and F-are the notes played by the viola in the last measure, to within enharmonic iden- tity. Each aggregate makes its own set of distinctions so that, for example, the fourth aggregate asks a listener to distinguish between harmonics, down-bow, and modo ordinario within naturale playing, as the previous aggregates do not. These cross-relations and the others in the example not only enrich the composition but provide a raison d'etre for the music's timbral flux which heightens the sonic entertainment. However, I should note that these associations are arbitrary, not determined by special row or array properties.

There is another feature of this realization that involves the partition- ing of each aggregate into three successive pcsets; a trichord, a hexachord, and then a trichord. The content of the hexachords in each aggregate is related to the last hexachord of the row by transposition.

332



Compositional Spaces and Other Territories 333

E ,- - .. -

,- '

v ' ?

av-v~~ o

J, ,,s ?' "'^ ^ 8 , EL '

C^i^-s', C ?C

Bp^. L, L, )

v . ^ ^

It^^^ l^ . .

A * A

g

I U 6. d

ic ri :R '



-IJ 4--

Ben Marcato. =76MM

CD '"

(n CD

CD r)

O

z CD C)

2: (D

content of content of content of content of

T4last hexofP) T41asthe of P) T(last hex of P) T4last hex of P)

c C# F F# E B G# G A D A# D#

DB E A G D B A# C F C# F# ru

F# GB C A#F D C# D#t GE A

A A# D D# C# F E F# BGC

c,g aggregate I aggregate 2

articulations in aggregate I

hfrom row I of column 4

ord. fromrow 2 ofcolumn 4

pizz. from row 3 of column 4

pont. from row 4 ofcolumn 4

articulations in aggregate 2

#fz : from row I of column 3

pizz. from row 2 of column 3

ord. from row 3 of column 3

< > from row 4 of column 3


tasto from row I of column 2

pizz. fromrow 2 ofcolumn 2

pont _ from row 3 of column 2

ord. from row 4 of column 2

EXAMPLE 3b: THIRD REALIZATION OF EXAMPLE 2a

Vn. I

Vn. II

Vn. II

Viola

V'cello

Vn. I


ord. from row I of column I

pizz. from row 2 of column I

M fromrow 3 of column

harm. from row 4 of column I

aggregate 3 aggregate 4




< i u

i ?

_ ^ \ U a

I An 0 (' ;e ( X

A n ?Cs 3 V E

AN IMN

.N-A I I ;

"- &< L I, <

U Nm

^ - O , < 4> A 3

u

/ d ? I l 1

, U :1 > _ __

___




The pattern of hexachord levels is palindromic: T4, T5, T5, T4. Pitch-class content of the trichords over the four aggregates also forms a retrograde pattern. Since neither the structure of the original array nor of its row is palindromic, the retrograde pattern of T, levels and trichords is special; indeed, it derives from a sub rosa property of the row.4

The last of my three realizations, for three violas, Example 3c, perhaps seems the simplest; yet if it were shown without its underlying array, one might not believe it was related to the design in Example 2a.

The example indicates the design's rows are articulated by dynamics- f, mf, mp, p from the top to bottom of the design. Thus the design con- trols the ebb and flow of dynamics in the excerpt. But, as in the other realizations, there is more going on. The material played by the three violas is based on three melodic figures of four, three, and five pcs in length. The chart on the bottom right shows that the melodies in each aggregate are related to the first aggregate's by T3, R, and RT3. Addi- tionally, each aggregate is totally ordered by a form of a new row X. Thus we could say either that the four forms of row X (X, T3X, T2IX, and T5IX) have been polyphonized to form a pattern consistent with the original compositional design of Example 2a, or that the compositional design has been derived (flattened out) into the four X rows.

To summarize: one simple design can generate very different kinds of music with contrasting musical properties and characteristics; the relation between a design's properties and its realizations is intimate if highly vari- able; and the relation between the features of any design (like Example 2a) and its elaborations (as in Examples 3a, b, and c) is not always obvious.

COMPOSITIONAL SPACES

As stated above, the notion of a compositional space helps bridge the gap between material and design. A definition might run: a compositional space is a set of musical objects related and/or connected in at least one specific way. But most importantly, compositional spaces are nontempo- rally interpreted-that is, they are out-of-time.

Examples 4a and 4b provide familiar instances of compositional spaces. Example 4a is the space containing the members of the set-class 4-8[0156]; these are the unordered pcsets related by Tn and/or I to {0156}. Example 4b is a row class presented in the form of a row table. The forty-eight forms are the members of a compositional space. Some of the rows in this space are those found in the compositional design of Example 2a. In addition, this space is related to the space of the previous

336




example. Each member of the row class in Example 4b starts and ends with a member of the space of Example 4a.

These sets of specific and related musical objects are compositional spaces of the simplest kind; they are undefined with respect to the ordering or connection of their objects. But even this simple a space can underlie a compositional design, for the spaces of Examples 4a and b are the background for the row combinatoriality of design 2a and its realizations.

a. The set-class 4-8[0156]:

( { C C# F F# { E F A A# { G# A C# D

I C# D F# G ) F F# A# B ) A A# D D# )

D D# G G# } { F# G B C ( A# B D# E

{ D# E G# A ) G G# C C# ) ( B C E F

b. The row-class of< C C# F F# E B G# G A D A# D# >:

C C# F F# E B G# G A D A# D#

B C E F D# A# G F# G# C# A D

G G# C C# B F# D# D E A F A#

F# G B C A# F D C# D# G# E A

G# A C# D C G E D# F A# F# B

C# D F# G F C A G# A# D# B E

E F A A# G# D# C B C# F# D G

F F# A# B A E C# C D G D# G#

D# E G# A G D B A# C F C# F#

A# B D# E D A F# F G C G# C#

D D# G G# F# C# A# A B E C F

A A# D D# C# G# F E F# B G C

EXAMPLE 4: SIMPLE COMPOSITIONAL SPACES




But we are also interested in more structured spaces, not just collec- tions of related objects. The next examples are ordered spaces.

Example 5a presents a sequence ofpcsets (chords) ordered from left to right. Since ordering is often interpreted in time, the example has a one- to-one relation with the compositional design written beneath the chord stream. Thus, Example 5a is a simple space.

C F G# C# E C# > G > A > D# > F D G# A# E F#

C C#t D F GG# A# A Gt C# D# E E F F#

EXAMPLE 5a: SIMPLE COMPOSITIONAL SPACE

In Example 5b we have a partially ordered sequence of pcs. The pcs can be realized in a number of different but related ways; a few possibili- ties are shown in the example. The first two realizations interpret the partially ordered sequence as a "transition graph"; they are two of a set of pc sequences that conform to the partially ordered sequence. The last of the three realizations uses the graph of the partially ordered set to determine a total ordering. Among other things, this last realization shows us that partially ordered sets can be treated as compositional designs. To be sure, I spent a bit of my book dealing with partially ordered sets and their relation to designs. But the first two realizations are more in the spirit of compositional spaces. Thus, partially ordered sets bridge the distinction between design and space.

The next example, 5c, presents a cyclic sequence of pcsets. Note first that the example orders its objects, the chords, but the order is not linear but cyclic; and second, that there is a beginning and end state, but no specification of how many times to loop. We realize this network by beginning on "start," following the arrows until we decide to follow to requisite arrows to "end," and stop. Since the result of our realization will be a "legal" series of chords in D major, readers may recognize Example 5c as a D-major interpretation of a simple tonal grammar.

The last examples were literal spaces involving musical objects. Now we turn to spaces involving relations among types of musical objects.

338




These are abstract spaces. Abstract spaces assert possible literal, more specific spaces.

C C F , C# start _ G# / end

\i D/,

'

o A# > B ~

C G# A# B

three J / D G# F C# realizations:

v D C G# A# F C# B

A path through the graph of the partially ordered set.

Another path through the graph.

A total ordering of the partially ordered set.

EXAMPLE 5b: PARTIALLY ORDERED SET OF PCS

C# D A -> B F# G

/ F# E A -- * > D -> C# ->* F# --end

- \ B A D

EXAMPLE 5c: CYCLIC NETWORK OF PCSETS-

AN INTERPRETATION OF A SIMPLE TONAL GRAMMAR

start -

339




One class of abstract spaces are musicalgrammars. In Example 6a we have the musical grammar that underlies Example 5c. Of course, the example is far too simple and merely didactic, for it is only a cyclic transition system. For real adequacy it needs other kinds of rules-rules such as: substitution (e.g., VII for V); embedding (secondary dominants); transformation (relative and parallel minor substitutes); and realization (voice leading and chord registration and doubling). Nevertheless, the grammar as it now stands is the basis for a more complete grammar.

III - IV

start I * > VI V * I - end

EXAMPLE 6a: THE MUSICAL GRAMMAR THAT UNDERLIES EXAMPLE 5c

Another class of abstract spaces are the two-partition graphs of the kind found in my book and in my earlier article on non-aggregate combinatoriality.5 Some instances of two-partition graphs are found in Examples 6b to 6d.

Example 6b illustrates what I call the "Babbitt network," trichords that partition the six all-combinatorial hexachords. Many of the long-range processes in Milton Babbitt's compositions from 1948 to about 1960 based on hexachordal and trichordal combinatoriality are modeled by this space.6 The graph's nodes (circles) are trichordal set classes (or set types), the arcs (lines) are associated with hexachordal set classes. We read the space by seeing which trichord types are connected by arcs. Trichords connected by arcs can partition an all-combinatorial hexachord; the arc's label indicates the resulting hexachord type. If a node is connected to itself, the associated trichord can partition a hexachord with two

340




members of its class. Note the arcs do not order but only associate the trichords and hexachords. That is, there is no order of succession implied by the space as in the partially ordered sets and grammars. However, when we trace a path on the graph, we create a sequence of trichords modulating one to another via all-combinatorial hexachords, or vice versa.

E

D

D

F hexachords: A = 6-1[012345] B = 6-8[023457] C = 6-32[024579] D = 6-7[012678] E = 6-20[014589] F = 6-35[02468A]

A

B

nodes = trichords (all) arcs = hexachords (all-combinatorial) arcs connecting two nodes = the (two) trichords can partition the hexachord

EXAMPLE 6b: AN ABSTRACT COMPOSITIONAL SPACE:

THE "BABBITT NETWORK." TRICHORDAL SET CLASSES

THAT PARTITION ALL-COMBINATORIAL HEXACHORDS

341




A number of interesting, if familiar facts are illustrated by the structure of this space. For instance, note the following trichordal associations.

There are two independent groups of trichord-types related by the A, B, and Chexachords. These are: 3-1[012], 3-4[015], and 3-9[027]; and 3-2[013], 3-3[014], 3-7[025], 3-11[037]. Trichords 3-5[016], 3-8 [026] are connected solely by the D and F hexachords. Trichord 3-6 [024] associates only with itself, but with four hexachords. The same obtains for 3-12[048] but with only two hexachords. 3-10[036] is inert; it is not in the graph. There is also a unique case involving the A and C hexachords; both are partitioned by the 3-2 and 3-7 trichords.7

The next example, 6c, illustrates an abstract space involving trichordal set classes and the all-trichord hexachord, 6-17[012478], found often in music by Elliott Carter. The interpretation of the graph of this space is the same as in the Babbitt network with one notational addition; the bars on the hexachordal arcs indicate invariance of the nearest trichord.8

3-9 3-1 3-10 [027] [012] [036]

TnI TnI T, I

[037] [014] [026] [024O

I35 - ( 3-2 \ _ 34 \ _ 3-7

[016]j [013] l[015] l[025]

TnI, T~n+4, Tn+8I T4, T8

3-12

nodes = trichords arcs = all-trichord hexachord: 6-17[012478] nodes connected by arcs = 6-17 can be partitioned by the trichords associated with the nodes

arcs cut by bar ( - ) = nearest trichord is invariant under the operations indicated.

EXAMPLE 6c: AN ABSTRACT COMPOSITIONAL SPACE:

TRICHORDS THAT PARTITION ALL-TRICHORD HEXACHORDS

342




For instance, the node for 3-9 has a nearby bar labeled TjI, which indicates that 3-9[027] is invariant under TnI. Thus 3-9 has two operations of invariance: TnI and To. Note that the 3-12 circle (the augmented chord) has six operations of invariance-the five labeled on the graph plus To.

Example 6d is an abstract space involving ics 3, 6, and the all-interval tetrachord-type. The all-interval tetrachords (4-15[0146] and 4-29 [0137]) are a hallmark of Carter's music from circa 1950-65. This space reflects the fact that any all-interval tetrachord can be partitioned by a tri- tone and a minor third.

T6r

TnI@ TnI TnI

Tn+6I

nodes = unordered interval 3 or 6 arc indicates an all-interval tetrachord: (4-15[0146] or 4-29[0137]) can be partitioned into unordered intervals 3 and 6. Solid bar indicates invariance as in Example 6c.

EXAMPLE 6d: AN ABSTRACT COMPOSITIONAL SPACE:

UNORDERED INTERVALS 3 AND 6 PARTITION

ALL-INTERVAL TETRACHORDS

Examples 7a and b are compositional spaces that "realize" the last two abstract ones. These spaces are literal and deal with actual sets of notes, not classes of sets, as in Examples 6a-d. Example 7a, derived from Example 6c, is the space of a set of all-trichord hexachords and trichords. This graph has been studied by James Boros.9 The complexity and symmetry of this graph vis-a-vis its abstract precursor in Example 6c is based on the invariance of the trichords. For instance, where in Example 6c the 3-12 class was connected to the 3-5 class in one way, in this graph the 3-12 member ID#, G, B is connected to six distinct members of 3-5.

A simple way to use this space is to play one trichord after another following the connecting arcs; the sequence of concatenated trichords will articulate a series of concatenated and/or overlapped all-trichord hexachords. So, as in the case of the output of the tonal grammar of Example 5c, we can improvise "harmonious" music directly from the space itself

343




3-11 3-11

3-9

3-11 3-11

3-8

3-11 ] L -'~ ;I[ 3-11

CEG GDGA#

3-8

c#F#G#

3-2 3-4 3-7

EF#G- CC#A# --|E|

EXAMPLE 7a: LITERAL COMPOSITIONAL SPACE OF ALL-TRICHORD

HEXACHORD EXPANDED FROM EXAMPLE 6c (AFTER JAMES BOROS)

344




by following the arcs of the graph. This practice resembles the music generated by the optionary scores of the avant-garde of the 1950s. Pieces such as Karlheinz Stockhausen's Klavierstiick XI, or Earle Brown's Avail- able Forms may come to mind. But, in addition, we have the maxim of "unity in diversity" in microcosm; the trichords can vary but the hexachords are of only one type.

Incidentally, there are three other literal spaces that are transpositions of 7a (under T1, T2, and T3). All other transformations yield one of these or the original. These observations tell us that the set of transformations among the trichords and hexachords in Example 7a form a mathematical group; the distinct transpositions of the graph's transformations are cosets.

Example 7b is derived from the abstract space in Example 6d-the space of all-interval tetrachords and members of ic 3 and ic 6. The lower- case letters show how arcs are to be connected in a wrap-around pattern; one can jump from the graph's bottom to top or left to right by arcs connected by the same letter. Thus, unlike the previous graph, this one would be best written on a torus (or a doughnut). Its complexity derives from the invariances of the intervals 3 and 6, as indicated in its abstract generator, Example 6d.

h a b c d e

C#G DG#l D#A

F#A G A# BD

a I b c d e /f

Each pair of connected pcsets forms an all-interval tetrachord. Letters designate connected lines.

EXAMPLE 7b: IITERAL COMPOSITIONAL SPACE

(EXPANDED FROM EXAMPLE 6d) OF ALL-INTERVAL

TETRACHORDS (ON SURFACE OF A TORUS)

345




These last two spaces can be mapped onto their abstract generators via group-theoretic epimorphisms.'0 Such mappings are often necessary when a literal graph is too complex to be of much use in clearly display- ing its network of relations. Such is the case with the Babbitt network.

For the balance of the paper I shall introduce three other kinds of literal spaces that contrast with two-partition graphs even though all of them imply a torus topology. The three spaces are Room squares, Boulez multiplication matrices, and generalized Mead tiles.

In Example 8a and b, we have Room squares, which take their name from the mathematician T. G. Room, who first studied them." Room squares are n-by-n matrices (where n is odd) whose cells all contain integers or are empty. Here n is 11 and the pairs of integers are interpreted as pc intervals. The Room squares in the examples have two properties: (1) Each of the sixty-six distinct pitch-class dyads is found only once; (2) Each row and each column is an aggregate.12 This type of Room square is therefore a special two-dimensional generalization of the linear all-interval row.13 One can move through the square by playing complete columns, then taking a ninety-degree turn to start a row and continuing likewise, wrapping around the square as necessary for as long as one wants. This march on a doughnut can be used in either composition or improvisation.

CC# FG# DB EA F#G D#A#

EB CD F#A C#D# FA# GG#

C#F CD# GA# ED F#B G#A

AA# DF# CE G#B D#F C#G

DG# A#B D#G CF C#A EF#

FG D#A C#B EG# CF# DA#

F#G# EA# C#D FA CG D#B

GA FB DD# F#A# CG# C#E

G#A# C#F# D#E GB CA DF

D#F# AB DG EF C#G# CA#

EG C#A# D#G# FF# DA CB

EXAMPLE 8a: A ROOM SQUARE

346




CC# D#A# F#G EA DB FG#

CD EB GG# FA# C#D# F#A

GA# CD# C#F G#A F#B DE

G#B CE DF# AA# C#G D#F

C#A CF D#G A#B G#D F#E

DA# CF# EG# C#B D#A FG

F#G# D#B CG FA C#D EA#

FB GA C#E CG# F#A# DD#

D#E C#F# G#A# DF CA GB

EF DG AB D#F# CA# C#G#

DA FF# D#G# C#A# EG CB

EXAMPLE 8b: ANOTHER ROOM SQUARE

Pierre Boulez has employed the principle of "multiplication" in his serial music since 1950. Multiplication is explained and illustrated by Boulez in his book Thoughts on Music, by Lev Koblyakov in his analytic work on Le Marteau sans maitre and other Boulez pieces, and more recently, by Stephen Heinemann.14 Independently, and to different ends, multiplication has been studied by Howard Hanson (under the name "projection") and by Richard Cohn, who has more directly termed the principle transpositional combination and put it to analytic work in a wide spectrum of twentieth-century music. In the following I will use Cohn's term or its abbreviation, TC.15

Essentially, transpositional combination involves "building" one pcset on the notes of another. Example 9a gives the formal definition of TC. As shown in the Example, for pcsets X and Y, one takes the union of transpositions of X starting on each of the notes of set Y. The result Z is called the transpositional combination of X and Y (In 9a { C C# ) is X and ( E F G } is Y. We transpose ( C C# } to the notes of Y(E, F, and G) to get the result Z, the pcset ( E F F# G G# }. Since we are transposing, Boulez's "multiplication" is actually modular addition. But in group theory, the term multiplication is used to denote the binary operation of a group so that modular addition is a particular instance of multiplication in the group-theoretic sense.

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Principle of TC (transpositional combination)

For pcsets X, Y, and Z,

Z = TC(X,Y)

such that Txy = Z(mod 12) for z Z, for all x E Xand y E Y

Example of TC

x= (c# C ; = {EFG ;Z= (EFF#GG#

TC( { C C# }, I E F G } ) = E F F# G G#

I-matrix representation of TC

numeric representation:

4 5 7

0 4 5 7 1 5 6 8

pitch-class name representation

E F G

C E F G C# F F# G#

Set classes and TC

Let TC(X,T) = Z

(1) if X = TJIXand/or Y = TmIY(for any m and n), then Z SC(W) for all X E SC(X) and Y E SC(Y).

(2) if neither X = TIXnor Y = TmIT, then Z E SC( W1) or Z E SC( W2) for all X E SC(X) and Y E SC(Y).

EXAMPLE 9a

We can write TC most effectively in a matrix form, what Bo Alphonce calls an I-matrix.16 The middle of Example 9a shows the numeric and

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pitch-class name versions in the matrix representation of TC. The numeric representation shows the transpositional derivation of Z from X and Y most clearly. Here we will call the matrix representation a Boulez multiplication matrix.

The bottom of Example 9a, under the heading "Set classes and TC," reveals what happens to the set-class of Z under transformations of X and T The two cases are presented less formally below:

1. If either X or Tare TJI invariant, then TC from any member of the set class of X and any member of the set class of r always yields a pcset Z of the set class W

2. If neither X nor T have TnI invariance, then the Z result of TC of any member of the set class of X and any member of the set class of Tis a member of one of only two set classes, W1 or W2.

Example 9b presents a six-by-six Boulez matrix especially designed to maximize TC redundancy. The two sets X and Y are now ordered. X is the vertical pcset and is a whole-tone scale. Tis the horizontal pcset and is a member of 6-20[014589], the E hexachord. Note that all of the adjacent trichords of X are of the same type-set-class 3-6[024]. The same is true of the trichords of T; these are 3-3[014]. Note also that the trichords of X are each inversionally invariant. Thus, by case 1 of TC, any three-by-three block in the matrix is of the same set class, 8-2 [01234568]. If we switch our attention to three-by-two blocks, we find these are of two set-classes, which follows from case 1 also. Here we have 6-1[012345] and 6-8[023457] respectively generated by: minor seconds from Tand 3-6 trichords from X; and minor thirds from Tand 3-6 trichords from X. Similarly, two-by-three blocks are of 6-2[012346] generated by TC from major seconds from X and 3-3 trichords from T

In Example 9c we have a twelve-by-twelve matrix. This is more in keeping with Boulez's invocation of TC, involving complete twelve-tone rows as the X and r sets. But here the two rows are not related by twelve-tone operators. Nevertheless, the two rows are similar because they have the same nonoverlapping hexachords (6-7[012678], the D- type) and are both trichordally constructed. The nonoverlapping trichords of X (the vertical row) are of 3-9[027] ( TI invariance), while the nonoverlapping trichords of the horizontal r are of 3-5[016]. As a result, every three-by-three box shown in Example 9c is of the same set class 7-7[0123678]. In addition, the four six-by-six blocks emanating from the matrix's corners are also of the same set class.

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c c#

c C#

E

E

F G# A

F G# A

B D D# F#

A C C# E

G A# B D

F G# A C

D D# F#

G

F

D#

C#

The horizontal pcset is made out of imbricated trichords from set-class 3-3[014].

The vertical pcset is made out of imbricated trichords from set-class 3-6[024], whose members are invariant under

TnI.

G A# B

All three-by-three "boxes" of pcs in the matrix are of the same set class. A few of these boxes are shown below.

All three-by-two "boxes" of

pcs in the matrix are of two set classes. Two-by-three "boxes" are of one SC.

C C# E F G# A C C# E F G# A

A# B D D# F# G A# B D D F G

G# A C C# E F I G# A CI C# E F L

F# G A# B D D# F# G_ A# B D D#

E F G# A C C# E I F G# A C C#

D D# F# G A# B D ' D# F# GI A# B L I

EXAMPLE 9b: A SIX-BY-SIX BOULEZ MULTIPLICATION MATRIX

Unlike the other spaces, in Boulez multiplication matrices we consider

overlapped boxes of positions rather than distinct nodes connected by arcs. Nonetheless, it is easy to use them to generate uniform pc materials for composition or free improvisation. If we imagine a three-by-three mask passing over the matrix of 9b, moving adjacently from one three-

by-three box to another (wrapping around as necessary), we will generate a series of overlapped 8-2 set classes that intersect in four or six pcs.

C

C

A#

G#

F#

E

D

A#

G#

F#

E

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C C# F# D G G#

C C# F#

A# B E

F F# B

D G

C F

G C

D# E A F A# B

E

D

A

A

G

D

F A#

D# G#

A# D#

B

A

E

F# G C G# C# D A A# D# B E F

E F A# F# B C G G# C# A D D#

B C F C# F# G D D# G# E A A#

A A# D# B E F C C# F# D G G#

G G# C# A D D# A# B E C F F#

D D# G# E A A# F F# B G C C#

D# E A F A# B F# G C G# C# D

C# D G D# G# A E F A# F# B C

G# A D A# D# E B C F C# F# G

The horizontal generator's nonoverlapping trichords are members of 3-5[016].

The vertical generator's nonoverlapping trichords are members of 3-9[027] and are TnI invariant.

EXAMPLE 9c: A BOULEZ MULTIPLICATION MATRIX DIVIDED INTO

THREE-BY-THREE BOXES EACH CONTAINING SET-CLASS 7-7[0123678]

Generalized Mead tiles are derived from the trichordal tiling patterns in Andrew Mead's article on the pitch-class/order number isomorphism in twelve-tone theory.17 The properties of generalized Mead tiles are given in Example 10. They are four-by-four arrays with the wrap-around topology-the torus surface again.18

C

C

A#

F

F#

E

B

A

G

D

D#

C#

G#

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In a generalized Mead tile:

1. each diagonal holds the universal set which may be the twelve-tone aggregate or a subset thereof;

2. each square block of four adjacent positions holds a universal set;

3. the content of adjacent positions (a king's move away) do not intersect.

Thus, a generalized Mead tile can be generated by a twelve-tone row or any other series by placing the series (suitably partitioned) in a diagonal string of positions. The different diagonals in a Mead tile can therefore associate different series unrelated by T7, I, and/or R.

Examples lOa and b show tiles with a small catalogue of different set types. Although the examples show positions filled with dyads or tetrachords, there is no restriction on cardinality with generalized Mead tiles. Thus, some tiles may have empty positions.

When we perform music from a generalized Mead tile by moving dia- gonally or circularly we get the universal set. When we move horizontally or vertically we get pc duplication, but only after two moves. Zigzag paths offer different degrees and rates of pc saturation.

Examples lOa and b are related. In fact, generalized Mead tiles come in fours. Example 1 la shows the underlying group structure. The permuta- tion operators called flip and exch and the pc involutions X and Y form a four-group. The members of the group act on a tile to form three others. Associating pc operations cycles with X and/or Ycan produce invariant tiles or tiles related by twelve-tone operations. Example l1b shows the four group-related tiles with the octatonic scale as the universe.

CONCLUSION

Compositional spaces can be the substance of free improvisation. In many ways, spaces that are cyclic, partially ordered graphs resemble the melodic concepts of maqam and raga in Arabic and Indian music. In composition, spaces help construct compositional designs with special properties. They might also be used in the pedagogy of twentieth- century ear-training and keyboard harmony.

Spaces have an important role in musical analysis. David Lewin, in his recent book, Musical Form and Transformations: Four Analytical Essays, writes "rather than trying to make ... transformations denote phenome- nological presences in a blow-by-blow fashion, we can more comfortably

352




a.

C E G# G C c# F# A D F G# A#

B C# F# A D# A# D F B E D# G

E G# C G C# F# G# A C D F A#

# D# F# AD F A# B D# E G B

content of adjacent horizontal positions: content of adjacent vertical positions:

4-19[0148] 8-19[01245689] 6-5[012367] 6-34[013579]

6-32[024579] 6-25[013568] 6-47[012479]

b.

C G E G# C c# D F F# G# A A#

C# D F B D# F# A A# G B D# E

G G# C E C# D F G# C F# A A#

C# D D# F F# AA# B D# G E B

content of adjacent horizontal positions: content of adjacent vertical positions:

4-19[0148] 8-19[01245689] 6-2[012346] 6-15[012458]

6-9[012357] 6-12[012467] 6-41[012368] 6-30[013679]

Properties of generalized Mead tiles: (1) (wrap-around) diagonals hold an aggregate. (2) each box of four adjacent positions holds an

aggregate (wrapping up/down and/or left/right). (Note: there are no restrictions on cardinality of pc content of cell.)

EXAMPLE 10: GENERALIZED MEAD TILES

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flipY exchX

aef Icj ie a cgj A fka Ich kai gch

ikd bhgb dkf lbh invZ i ed gbj fed bjl

fcj ael icj gea invZ fch bIka ich akg

bhi gkd fbh dkl bji ged fbj led mu m m mm m q m

t m m

exchX cgh

flipY dij bgh def bkl

cef ak cij agh

bij dgh bef dkl

position permutations: flip = flip tile pattern around nw-se diagonal exch = flip tile pattern around ne-sw diagonal inv = do both flip and exch

element permutations: X = (al)(bi)(cg)(df)(ek)(hj) Y = (af)(bg)(ci)(dl)(ek)(jh) Z = XY = Tr = (ad)(bc)(e)(k)(gi)(lf)(h)(j)

EXAMPLE lla: GROUP STRUCTURE OF MEAD TILE

PATTERN TRANSFORMATIONS

fka cel | aibh jcg

dih gbj fkd clb

ckf a el cihagj

hib \djg bkf led

aeflckl laij

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flipY

01 13 116713A

46719A 104 19

03 1 3 6 7 1A

I

I

exchX

6 7 9 4 A 09 4

0 1 3A 17 3A A 01 36 1 7 3 6 invZ

47 169 104 169 4iZ 7 9A 04 9A

0 3 A 1 3 7 A invZ 0 3 611 3 6 711

6 7 9 4 0 6 9 4 \ 79A 4 0 9 A 4 * m m . ~ y ~ i i u

exchX

01 3 1 7 A3 6

k 47A6 9 0 4 9 I

0 3 1 3 7 A 1 6

7 9 A 4 60 9 4

flipY

Substitutions for variables in Example 1 la:

a = 1 b = 9 c = 3 d = 4 e = null f= 0 g = null h = 6 i = 7 j= A k = null I = null

EXAMPLE lib: OCTATONIC MEAD TILES DERIVED FROM EXAMPLE lla

1I

I

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regard them as ways of structuring an abstract space ... through which [a] piece moves."19 For Lewin, musical form arises out of the way a composer dances, as it were, through a space. And while Lewin tends to consider only the part of an entire space that is verifiably used in a specific composition, there is no essential structural difference between transfor- mational networks uncovered by analysis and those constructed for composition.

Without doubt, the cognitive status of compositional spaces could be the subject of a series of papers in music psychology. It is, nevertheless, important to say here that if spaces are not musical grammars, they sub- stitute for such grammars by using association rather than chunking as their underlying psycho-cognitive mechanism. This reflects the fact that today's music need not be primarily hierarchic in order to have the rich- ness of affiliation and scope of reference so often associated with the tonal music of earlier musical periods. Spaces and the designs they under- gird not only help provide a resource for composers who wish to con- tinue and expand the tradition of Western art music but provide a heuristic for those of us who are looking for new, even radical ways of reconstruing and reconstructing the musical matrix.

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NOTES

1. Robert D. Morris, Composition with Pitch-Classes: A Theory of Com- positional Design (New Haven and London: Yale University Press, 1987).

2. In addition, a major concern of my book is to place pitch-class relations in their appropriate musical contexts. To that end, it discusses the relations between and among contour, pitch, and pitch class and their temporal analogues, sequential, measured, and cyclic time.

3. See Daniel Starr and Robert Morris, "A General Theory of Com- binatoriality and the Aggregate," parts 1 and 2, Perspectives of New Music 16, no. 1 (Fall-Winter 1977): 3-35; no. 2 (Spring-Summer 1978): 50-84.

4. RT3MIP = P

5. See Morris, "Combinatoriality without the Aggregate," Perspectives of New Music 21 (1982-83): 432-86.

6. Babbitt has his own way of discussing this space using "planar graphs" in Milton Babbitt, "Since Schoenberg," Perspectives of New Music 12 (1973-74): 3-28.

7. The graph could be written so that its symmetry under the M (M5) operation would become obvious.

8. The trivial invariance under To is not shown.

9. See James Boros, "Some Properties of the All-Trichord Hexachord," In Theory Only 11, no. 6 (1990): 19-41.

10. Epimorphisms map the members of a group onto one of its sub- groups; therefore they are many-to-one mappings.

11. These are also known as Howell's Movements. See T. G. Room, "A New Type of Magic Square," Mathematical Gazette 39 (1955): 307.

12. In general, Room squares of order n have their nonempty cells filled with each member of aXb such that 0 < a, b < n and a ? b and each row and column contains each integer from 0 to n.

13. There is a published algorithm for generating Room squares by Erie Glen Whitehead, Jr. which I used to write a computer program to generate the squares. See Erie Glen Whitehead, Jr., Constructive Combinatorics (New York: Courant Institute of Mathematical Stud- ies, New York University, 1973).

357




14. See Pierre Boulez, Thoughts on Music (Cambridge: Harvard Univer- sity Press, 1970), Lev Koblyakov, Pierre Boulez: A World of Harmony (Chur: Harwood Academic Publishers, 1990), and Stephen Heine- mann, "Pitch-class Set Multiplication in Boulez's Le Marteau sans maitre," DMA dissertation, University of Washington, 1993. Heine- mann's paper "Boulez's Elegant Operation: Pitch-Class Set Multipli- cation in Le Marteau sans mattre," was delivered at the 1992 National Convention of the Society for Music Theory in Kansas City, Missouri.

15. See Howard Hanson, The Harmonic Materials of Modern Music (New York: Appleton-Century-Crofts, 1960) and Richard Cohn, "Inversional Symmetry and Transpositional Combination in Bart6k," Music Theory Spectrum 10 (1988): 1942, and "Transpositional Combination in Twentieth-Century Music," PhD dissertation, East- man School of Music, University of Rochester, 1986.

16. Koblyakov and Hanson did not use a matrix model for multiplication (although it is implied by Boulez).

17. See Andrew Mead, "Some Implications of the Pitch Class/Order Number Isomorphism Inherent in the Twelve-Tone System: Part I," Perspectives of New Music 26, no. 2 (1988): 96-163.

18. General Mead tiles are arrays of the type called general array class in my book.

19. David Lewin, Musical Form and Transformation: Four Analytical Essays (New Haven and London: Yale University Press, 1993), 34.

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compositional spaces and other territories

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