Morris.combinatoriality Without the Aggregate

Combinatoriality without the AggregateAuthor(s): Robert MorrisSource: Perspectives of New Music, Vol. 21, No. 1/2 (Autumn, 1982 - Summer, 1983), pp. 432-486Published by: Perspectives of New MusicStable URL: http://www.jstor.org/stable/832888Accessed: 13/01/2010 23:33

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COMBINATORIALITY WITHOUT THE AGGREGATE

Robert Morris

The attraction of specific harmonic or linear sequences of pitches and/or pitch-classes is a motivating force in composition, but the problem of finding a meaningful context for such entities is an equally familiar situation. One is often frustrated by the fact that the way to generalize or develop a pitch- pattern is often unknown or unclear. Even with the automated means to generate the members of certain interesting categories of compositional materials, a composer is all too often ironically confronted with a multitude of patterns with the desired properties but which do not satisfy his or her immediate appetite. When the return to a more ad hoc and less intuitive approach would be less fruitful, the decision either to strive for a more powerful generaliTation or to be content working within a more predictable zone becomes a necessary fact of creative life.

The development of the formalization of the TWvelve-Tone System[1] and its role in compositional structure is one example of how successive generaliza- tions can lead to greater flexibility and utility. Thus, an increasingly wider spectrum of compositional proprieties and inclinations is served. Here I am thinking of the development of the concept of chromatic completion implicit in the idea of the twelve-tone row from Arnold Schoenberg's earliest row pieces through his 'hexachordal' compositions. In turn, the generalization of Milton Babbitt and others [2], as described by the theory of hexachordal combinatoriality and secondary sets with trichordal generators, has produced the formalization of generalized combinatoriality and linear aggregate formation [3]. In this paper I will continue this line of development still further-but not in exactly the same direction-so that the contents of the rows and columns in a CM[4] or LA[5] need not be the un-, partially- or totally-ordered set of all pcs but any sub-set of the total chromatic. Specifically, the rows and columns will all be members of the same set-class (SC)[6], that is, related under Th and/or I, or of two different SCs. With the two-dimensional CM, as in tonal music, the vertical structures can be of a different SC and cardinality from the horizontal ones.

As Babbitt has pointed out, "Given a collection of available elements, the choice of a sub-collection of these as a referential norm provides a norm that is distinguished by content alone." [7] Thus, I will be concerned primarily

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with the partially ordered or unordered content of the "norms" of our LAs (henceforth called chains) and CMs. This will allow a greater flexibility with respect to the generation and transformation of these pc-structures. After some examples of CMs and chains, I will discuss certain 'simple' cases which have already been used by composers and theorists for other purposes in other contexts. The next section of the paper will deal with the ways in which CMs- simple or not-may be compared, transformed and combined. Following this, the generation of chains and CMs is described. There I show how the properties of the parent SCs and their sub-sets found in the chains' and CMs' rows and columns influence the results. After discussing further extensions and other generation algorithms, I turn to the comparison of non-aggregate and twelve-tone combinatoriality with regard to content and order.

Chains and CMs The following set of examples illustrates various kinds of chains and CMs.

Example la gives a typical chain which is a partially ordered set of pcs arranged in a linear sequence divided into positions separated by vertical slashes. [8]

0941562179B A52 (ex. la)

The order of the pcs in a position is undefined so that 04916521 B79 I A25 is an equivalent notation for the chain in example la. The union of the content of two adjacent positions is called a norm. The requirement which distinguishes a chain from just any sequence of pcs in positions is that each of a chain's norms must be a set that is a member of one SC or of one of two SCs. Thus all norms in a chain (and CM) are related to each other by Th and/or I or the norms are members of two distinct collections of sets each containing sets related by Th and/or I. The term norm should not be considered to be normative in some philosophic sense; it is merely used to denote a row or column of a CM or adjacent positions of a chain which is a member of the CM's or chain's generating CM(s). The norms of example la are included in the set-class 6-46. 6-46 cohtains the set { 0,2,4,5,6,9 } which is the union of the chain's first two positions and is designated C. The second two positions taken together give the set { 2,5,6,7,9,B } which is TBIC. The union of the last two positions is T5C.

A chain can be said to be complete or incomplete. Example la is an incomplete chain because its last position in union with its first is not a norm. From the definition implied here, a complete chain can be thought of as a cycle of positions, its first following its last such that adjacent positions are norms. In example lb such a chain is shown-one which has norms derived

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from two SCs. One of the SCs is 4-16 which contains the set D = { 0,1,5,7}, while the other is the hexachordal SC 6-22 including E = { 1,2,3,5,7,9}. Brackets show the norms and their relation to each other.

D T6D

011571123917B1611753BI(01) I ,I L E T8IE

I i i

T4IE TAE (ex. Ib)

In example lb a few pcs are written in parentheses at its end; these are the beginning pcs of the chain and are displayed to show its cyclic nature.

In example lc the contents of the positions of the chain of example lb are reordered so that even more norms can be found or joined together. These secondary norms are of interest but the norms that define a chain are always overlapped. Only the secondary norms are indicated in this example.

T2D T6ID

101571923117B1161573B1(1015719) m m I i E T2IE

I I I T4IE

T8IE (ex. lc)

I now turn to the consideration of various types of CMs. A CM is a two- dimensional set of positions arranged in I rows and J columns. A position in the CM Z is given by Z (I,J) and may be empty or filled with an unordered set of pcs. There are I times J positions in a CM. The union of any row or column of positions in a CM is a norm.

There are three types of CMs. Type I includes CMs constructed so that all its norms are from the same SC. Type II has each of its vertical (column union) norms as members of one SC while all of its horizontal norms (row unions) are member of another SC. TWo SCs also generate the norms for a type III CM, while each SC is represented at least once among its columns and rows.

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In the next group of examples, a CM is shown as a two-dimensional matrix with labels for the contents of its columns written above each column and the names of the contents of its rows written to the left of each row. The following CMs have one pc per position. Thus, their rows and columns are totally ordered.

X: T2X: T6X: T7X:

T6X 0 2 6 7

T7X 1 3 7 8

X 6 8 0 1

T2X 8 A 2 3

X = {0,1,6,8} (ex. 2a)

X: T1X: T4X: T5X:

Y 5 6 9 A

T4Y 9 A 1 2

T9Y 2 3 6 7

X = {2,5,9} Y = {5,6,9,A}

(ex. 2b)

X: T9X: T6X: T3X:

Y 0 1 7 3

T3Y 6 3 A 4

T6Y 1 9 6 7

T9Y 4 A 0 9

X = {0,1,4,6} Y = {0,1,3,7}

(ex. 2c)

Example 2a is a type I simple CM. The designation 'simple' indicates that such a CM is constructable no matter what SCs are used as norms. Examples 2b and 2c are type II CMs but only 2c is non-simple. This is inferable because the ordering of one norm in a column or row is not produced by Th alone from another lying in the same direction. This accounts for the fact that the n in the Th operators of all the rows or all the columns do not spell out a member of the SC(s) of its norms.

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In example 3a, a type I 3X3 position CM is given whose positions are either filled with trichords or empty. The whole CM is symmetric around its upper-left/lower-right diagonal under T7I.

X: T9X: T2X:

T7IX TAIX T5IX 467 123

013 4AB 456 389

X = {1,2,3,4,6,7} (ex. 3a)

The next example, 3b, shows a type III CM with either a single pc or a null set in its positions.

Y X: 0 T9IY: T6Y: 7 T9X: T5Y: T8IY: 1

T2IY T4X 4

2 8

T1IY T1Y 1

T5X 5 9

6 1 1 9

5 '7

0 2 A

6 8

X = {0,1,4,5} Y = {0,1,7}

(ex. 3b) The 4X5 CM in example 3c has norms which contain replications of pcs.

Such replications never occur within positions, however. The CM is of type II.

Y T7IY TlIY IY T8IY 127 056 5

123 678 678 12367 456 57

9 06B

X = {0,1,2,5,6,7} Y = {1,2,4,5,6,7}

AB 4 A

(ex. 3c)

Simple Chains and CMs When the generation of a chain or CM does not depend on the special

properties of its norms' SC(s) it is called simple. The format or schema for the

X: T1X: T4X: T5X:

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cases where all norms of a CM or chain are identical follows: the letters denote sets of any cardinality including the null set. Concrete examples follow each format.

The simple chain is always complete.

XIY norm is XUY

2691418 (12691418) norm is { 1,2,4,6,8,9}

(ex. 4a)

Only type I CMs are possible if all a CM's norms are to be pc-identical. Such a CM is constructed out of the pc-set Q which has N subsets which partition Q. The CMs has N2 positions and each sub-set of Q occupies one and only one position in each row and each column. The format for this kind of CM is a Roman square. Example 4b gives the format with each row or column being a cyclic permutation of any other. The example shows a 4X4 CM.

WXYZ XYZW YZW X ZWX Y

norm = WUXUYUZ

01 5 289 5 289 01

289 01 5 289 01 5

norm = {0,1,2,5,8,9} (ex. 4b)

Another (diagonal) technique for the production of type I CMs is explained later within the context of CM expansion (see example 14h).

Simple CMs of type I or II have their positions each filled with one and only one pc. A type I CM of this kind is derived from any ordering of an arbitrary pc-set, H, written (H(0) , H(1), ... H(n)). TWo kinds of CMs can be produced. The LA-type has all its norms related by Th and for every vertical norm there is a corresponding identical horizontal norm. Like the IA-type, the IB-type CM has all its vertical norms related by Th; all its horizontal norms are related one to another in the same way. However, any horizontal norm is related to any vertical norm by ThI and vice-versa. The choice of n in IhI depends on the particular norms involved.

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Schema for the IA- and LB-type simple CMs are shown below.

IA-type: H (1) + (0)

H(1) + H(1)

H(1) + H(n)

... H(n) + H(0)

... H (n) + (1)

... H(n) + H(n)

T4H: T5H: T3H: T1H:

T4H 8 9 7 5

T5H 9 A 8 6

norm generator

norm generator = H

T3H TlH 7 5 8 6 6 4 4 2

(4, 5, 3, 1)

IB-type: H(0)

H(0)

+

+

IH(0)

IH(1)

H(0) + IH(n)

H(1)

H(1)

+ IH(0)

+ IH(1)

H(1) + IH(n)

... H(n) + IH(0)

... H (n) + IH (1)

... H(n) + IH(n)

T4IH T8H: 0 T7H: B T9H: 1 TBH: 3

norm generator

T5IH 1 0 2 4

norm generator = H

T3IH TlIH B 9 A 8 0 A 2 0

(4, 5, 3, 1)

(ex. 4c)

H(O)

H(O)

+

+

(0)

H(1)

H (0) + H (n)

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Although the set H will not appear in these kinds of CMs unless it contains a pc 0, transforms of H under Ih and/or I will constitute the CM norms. The set H does appear as the string of ns in the labels of the CM columns and/or rows. In the IA-type the ns in the ThH column labels or ThH row labels will produce H. In the IB-type the ns in the ThIH column labels will give H where the ns in the ThH row labels will give IH.

Type II simple CMs of the one pc per position distribution are derivable from two sets H and K where H is the generator of horizontal norms and K produces the vertical norms. In the example (4d), the cardinalities of H and K are n and p, respectively.

II-type H(0) + K(O) H(1) + K(0)... H(n)+ K(0) H(0) +K(1) H(1) + K(1) .. H(n) + K(1) H(0) + K(2) H(1) + K(2)... H(n) + K(2)

H() + K(p) H(1) + K(p)...H(n)+ K(p) norm generators: H and K

T2K T1K T9K T4H: 6 5 1 T3H: 5 4 0 T1H: 3 2 A T6H: 8 7 3

norm generators: H= {2,1,9} K= {4,3,1,6}

(ex. 4d) The ns in the labels of the columns spell out the set H and the ns in the

labels of the horizontal rows of the CM produce K. Those readers acquainted with Bo Alphonse's work[9] or the article, "Fan-

fares for the Common Tone"[10], by Carlton Gamer and Paul Lansky will recognize the types IA and II CMs as 'I-invariance matrices' and type IB CMs as 'P-invariance-matrices.' If H is a twelve-tone row so that the CM has aggregates as norms, the IB-type CM is identical to the 'set-table' listing the 48 classical transforms of a row[ll]. The type IB and type II matrices can be

440

viewed as sub-matrices of the set-table. It is not surprising therefore, that aspects of the theories of set and twelve-tone row invariance are of use in the present study. I shall return to this point below.

Comparisons of CMs and Chains Since there are among any set of entities at least as many modes of com-

parison as there are available categories of differentiation, the following discussion should be considered as only an introduction to the topic of CM and chain comparison. Once again an analogy can be drawn between the topics of twelve-tone combinatoriality[12] and non-aggregate combinatoriality; it is possible to relate CMs and chains of degrees of fragmentation and pc association. On the other hand, relations based on the 'row-column' theory only apply to certain cases of non-aggregate CMs. Furthermore, pc frequency is variable among chains and CMs and is related to set-invariance and CM transformation.

1. Fragmentation FRAG(E) is a measure of the distribution of pcs over the positions of a chain or CM named E. Fragmentation varies from '0' indicating that all pcs are in one position of E to '1' where each position is filled with one and only one pc or is empty. In the formula below, chains are treated as if they were CMs with only one row. As it will be shown that chains can generate CMs, this equivalence has its utility. E(i,j) is the position in the ith row, jth column of a CM or chain E. #E(i,j) is the number of pcs in E(i,j). PAIRS(k) is a function indicating the number of pairs of pcs among a set of k pcs, {X(1),X(2),.. X(k) }, and evaluated as shown below:

k-l

PAIRS (k) = n = (k2-k)/2 n=l

Thus, PAIRS(5) is equal to 1 + 2 + 3 + 4 or (25-5)/2 = 10. T is the number of pc entries in the entire CM or chain. P equals the number of rows and Q the number of columns in E. Hence...

P Q FRAG(E) = 1 - PAIRS (#E[i,j])

i=1 j=l PAIRS (T)

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In example la the chain's fragmentation is 1 minus 1/PAIRS(12) times 4 times PAIRS(3). This becomes 1- (12/66) = 54/66 or .81. The fragmentation for example lb is .86 and is listed in Table I with the fragmentations of other examples in this paper.

Comparison of Examples

ex. FRAG SPAR SC (of U)

la .81 0 9-9 lb .86 0 9-8 2a 1 0 9-5 2b 1 0 8-20 2c 1 0 8-28 3a .88 .3 agg. 3b 1 .4 10-4 3c .94 .3 agg. 4a .6 0 6-26 4b .95 .25 6-44 5a .94 0 10-2 5b .94 .43 8-28

PFA

(222200223010>

TABLE I

2. Sparseness Fragmentation is based on the number of available pairs of pcs in a CM or chain. It is a function of the number of pairs relating pcs within different positions and is a measure of the 'coarseness' of the pc distribution. The difference between a position that is empty and one that is filled with a single pc is ignored however. Thus, a CM with only one pc in each position and a CM with positions filled with single pcs or nothing have the same fragmentation, namely, 0. Sparseness is the measure that is needed to distinguish such cases, as in examples 5a and 5b which both have a fragmentation of .94

T8IK: T1IK: T9IK:

K 34 01 58

T5K 58 9A 61

T4K 07 58 49

K = {0, 1, 3, 4, 5, 8}

(ex. 5a)

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TAIL TAIK T7IK T7IL L: 46 3A T3L: 19 67 T3K: 04 OA 34 K: 07 19

K = {0, 1, 7, 9}L = {3, 4, 6, A}

(ex. 5b)

A CM's sparseness is measured by SPAR(E) where F is the number of non-empty positions and N is the total number of positions in E.

SPAR(E) = (N-F) IN

SPAR(E) is 0 where all positions are filled. An empty CM has a sparseness of 1. See table I for the sparseness of the various examples. Sparseness and fragmentation are changed when a chain or CM is redistributed (i.e., their pcs are swapped, see below).

3. PC Fragmentation and Association The contents of a chain or CM need not exhaust the aggregate nor have an equal number of each pc. The properties of the set U, the union of a CM's or chain's pcs, are an important aspect of certain kinds of CM generation. As I will show later, a whole chain or CM may function as a norm or position in a larger CM or chain. A twelve place array called PFA (Pitch- class Frequency Array), whose positions from 0 to 11 correspond to the pcs 0 to B, is used to display the pc frequency of a CM or chain. The position k holds the amount of pcs numerically equal to k in the chain or CM. For instance, the PFA for example 5a is [220123113210] which indicates the CM has two pc Os, two pcs ls, no pc 2s, one pc 3, etc. From a PFA one can identify favored as well as omitted pcs. The set of pcs comprising the CM of 5a is a member of SC 10-3 since it has no 2s or Bs. It favors the pcs 5 and 8 which incidentally form the same interval-class as do the missing pcs. The distribution of pcs over the PFA is also of interest, ranging from flat (each pc occurs about the same number of times) to skewed (many instances of some pcs, others are slighted or missing). The PFA for various examples is given in Table I.

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The association of certain pcs in more than one position of a CM or chain is another of its distinguishing features. If a non-empty set of pcs is included in a number of positions such that every norm includes at least one of the positions, the set is called an association-set. Even a single pc can be an association-set. In a chain the association-set will occur in at least all odd or even positions, and in a CM the association-set must be found at least once in every row and column. In order for a pc, x, to be a member of an association- set, the xth position in the PFA of the chain or CM must be equal to at least half the number of positions in the chain or equal or greater than the total number of CM rows or columns, whichever is greater. As an example, the PFA of example 5a only allows pcs 5 and 8 to be members of association-sets; in fact, the association-set is { 5, 8 }. The association-sets of example 3a are {3 } and { 4 } which follows from its PFA in Table I.

Of course, the larger the CM or chain, the greater the chance of association- sets. Non-simple small non-aggregate CMs and chains tend to have few. Even if there are no association-sets, there may still be some degree of pc association as in example 3a where the sets { 1, 3 } and { 4, 6 } occur in two positions.

The PFA lists the potential of association-sets for single pcs. Larger arrays could be defined to give the number of positions in which any of 66 distinct dyads, the 220 distinct trichords, etc., could be found. Generalizing still further, certain arrays could be made to list the number of occurrences of particular SCs in positions. For example, such arrays would indicate that in example 3a interval-class 1 occurs in every filled position, ic 2 occurs once in every row and column, SCs 3-1, 3-2, and 3-5 each occur twice as the total content of a position, etc. These kinds of observations help delineate the particular sonic and linear potentials of CMS and chains.

With simple CMs of type Ia, Ib and II having fragmentations of 1 as illustrated above, the norm generator's invariance under Th and ThI determines the pc frequency. Since an IA-type CM is an I-invariance matrix, the number of occurrences of pc n within its content is determined by the number of pcs in common between the norm-generator and its transform under ThI. If the norm generator is invariant under ThI, pc n is an association-set. If the CM is of type IB, it is a P-invariance-matrix and the multiplicity of pc n is determined by the intersection of the norm generator and its transposition by n. As in the above situation, Th invariance of the norm generator forces n to be an association-set. The PFA of a type IA or IB simple CM can be defined as a listing of the self-intersections of the CM's norm generators under ThI or Ih.

In the case of a type II simple CM, the number of ns is determined by the intersection of the ThI of its vertical norm generator with its horizontal norm generator. In example 4d, the vertical norm generator, K, is { 1, 3, 4, 6} while the horizontal norm generator is { 1, 2, 9 = H. T1IK = { 0, 7, 9, A}

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which intersects in one pc with H. Since n is 1 (in TiIK), there is one pc 1 in the CM. Under T3IK (n is now 3) K becomes the set { 0, 2, 9, B } having two pcs in common with H. As predicted, the CM contains two instances of pc 3.

Transformations on CMs and Chains Any CM or chain can be changed in a number of ways. Among the possi-

bilities are transformations of its contents with or without changing the distribution and size of its pc-arrangement. Positions, rows, columns, etc. can be created and/or deleted depending on the CM or chain's structure.

Perhaps the most obvious transformation is simply transposing and/or inverting the whole CM or chain. Since a CM or chain may possibly not include all 12 pcs, the choice of a pc operator can influence the preservation or change of its content. For instance, the set { 0, 2, 4, 5, 6, 7, 9, A, B }, which is the content of the chain in la, can map into itself under T4I so that the resulting chain shuffles its contents to produce 4701BA21975 6B2. In the same way, one is able to preserve a norm or position by transforming it into itself or into another norm or position (which must be a member of the same SC as the untransformed position or norm). Example 6a shows example 3a transformed by TAI which preserves its lower-left corer position.

346 789 79A 06B 456 127

(ex. 6a)

Example 3b can be transformed under T9I so that its top row's content (norm) is interchanged with its third column's content (norm) and its first column's norm is interchanged with its second row's norm. The rest of its columns and rows are altered however. The result of this transformation is given in example 6b.

9 5 8 4 7 1 0

2 3 8 8 0 7 B

4 9 3 8 2 1

(ex. 6b)

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Despite the fact that the U of a CM or chain is invariant under ThM or ThMI, its sub-section's pcs or SCs may or may not be preserved. Below, in example 6c, the result of the transformation of example 5a under T4M is written. [13] 5a's norms are all members of SC 6-14 which is ThM/TnMI invariant. Thus, the norms in 6c are of the same SC. In addition, the dyads in each position of 5a are found exchanged or unchanged in 6c. In fact, 6c is the same as its parent CM with its first and third columns as well as its second and third rows interchanged.

07 58 34 49 16 58 58 9A 01

(ex. 6c) A different state of affairs occurs when example 5b is changed into exam-

ple 6d under T1M which keeps U invariant. 6d's norms are of the same SCs as 5b's, but the contents of many of the former's positions are not shared by positions in the latter CM-only { 1, 9 , { 3, 4 }, and { 0, 7 are found in both CMs.

79 34 6A 70

19 13 49 01 6A

(ex. 6d)

In addition to pc transformations, CMs can be structurally reordered without making any changes in the content of their parts. Complete chains can be cyclically permuted and/or retrograded, but this can be considered only a change of notation rather than a change in any of the chain's position's adjacencies. A CM column may be interchanged with any other column without a change in any of its norms. The same is true of its rows. The group of CMs in 7a are related by these interchanges. Furthermore, a CM can be rotated by 90 degrees. This is shown in example 7b. (Other rotations are either retrogrades or retrogrades and rotations by 90 degrees together.) These operations may seem trivial in the sense that they are obvious and apply to any CMs. Quite a few transforms are possible, however. For a CM of X columns and Y rows there are X! ? Y! ? 2 distinct results. This amounts to 1,036,800 ways to transform a 6X6 CM. Nevertheless, the compositional assignment

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of CM rows and columns is highly influenced by the order of a CM's positions so that flexibility in this regard is quite desirable.

089 13B 16A 79B 12A 049 089 13B

16A 79B 12A 049

089 13B 049 12A 12A 049 13B 089 79B 16A 16A 79B

(ex. 7a)

027 345 9AB 027 9AB 168 168 345

(ex. 7b)

In some cases a CM may exhibit an invariance so that operating on it with a pc operator is equivalent to reordering its columns and rows. Example 5a was shown to have this kind of invariance under T4M. Another example is given by example 3a in which interchanging its rows for its columns (retrograding the order of its rows and rotating 90 degrees counter-clockwise) is the same as operating on it with T7I.

CMs may have their pc distributions radically changed via swapping-a subject that has been previously treated in the context of twelve-tone aggregate structure. Because the order of positions of a norm can reordered in any way whatsoever as explained above, we can regard the norms of a CM to be at most partially ordered, if ordered at all. Swapping pcs allows one further to disorder and transform the entire CM structure.

If two distinct positions of a CM E contain the same set, the sets can be swapped in the following manner. Let the two positions be E (i, j) and E (k, m). The set in E (i, j) can be moved to E (i, m) if the set in E (k, m) is moved to E (k, j). This swap is illustrated below. (The movements of the set from E (i, j) -~E (k, j) with E (k, m) -E (i, m) and E (i, j) - E (i, m) with E (k, m)- E (k, j) are the same because the sets moved are identical.) If

447

m = j or i = k, the swap has no effect; thus swapping does not change the structure of chains.

col. j row i

col. m

set x- -

row k -set x

- or -

col. j row i set x

'I row k

col. m

t set x

(ex. 8a)

Unlike swapping in twelve-tone CMs, the swapping need not be horizontal and between adjacent columns because the contents of the CM positions and rows are not ordered.

Example 8b reveals how the CM in example 7a can be swapped to change its fragmentation and sparseness.

089 12A

16A 79B

89 0 16A

89 OA 16

013B 129A 4 7B 9

1B 29 7A

13B 049

03 14 9B

(ex. 8b)

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The PFA can be used as a significant diagnostic of a CM's swapping potential. The higher the PFA entry, the more possibilities there are for swapping the pc corresponding to the entry. A moment's reflection will indicate why an association-set can occupy any position in a CM via swapping alone. It is important also to observe that the position and content of norms in a CM are not affected by swapping.

If a CM is a type-I, it may be possible to swap out one of its rows and columns if the CM contains a row and column whose contents (norms) are equivalent. Then either the content of the row or column can be swapped into the intersecting position. This position is then deleted with its row and column from the CM. This process occurs in example 8c-the first CM is the same as in example 4b.

01 5 289 5 289 01

289 01 5 289 01 5

015 289 012589

01289 5 289 5 01

015 289 01289 5

289 5 01

(ex. 8c)

It is possible to add pcs to CM positions. If there are two positions E (i,j) and E (k,m) which share the set of K, K or its sub-sets can be placed in E (k,j) or E (i,m). This does not change the constitution of any of CM's norms, it only duplicates pcs. Example 5b's position containing the pcs 04 (third row, first column) was added to a sparser CM in exactly this way. Since the PFA of example 5b is < 320230220220 > indicating that every pc in it is duplicated at least once, it is possible to fill in every empty position in 5b. This is done in example 8d.

449

46 3A A 4 6 19 67 9 04 3 OA 34 07 9 0 19

(ex. 8d)

If a CM possesses an association set, the set can be placed in every position in the CM. This maneuver has implications for building hierarchical structures as an outcome of non-aggregate combinatoriality. At any rate, it is clear that the pc associations of a CM are of consequence for its structural elaboration.

By taking advantage of the addition of pcs to positions in a CM one is able to swap out a horizontal or vertical norm. [14] Just removing a row or column of a CM would not affect the norms lying in the same direction as the removed norm; only the other intersecting norms would have some of their pcs removed. If these intersecting norms had pc duplication where one of the duplicated pcs were situated in the norm to be removed, the removal of the norm would have no effect on any other norm's content. Therefore, if it is possible to duplicate every pc in a norm Q, so that every intersecting norm has identical pcs to the ones in the shared positions with Q, we may remove Q. Such a series of operations is accomplished in example 8e where the first CM is derived from 3b with its lowest row's norm duplicated by pcs (underlined) in intersecting norms. Then the lowest row is removed and the process is repeated twice more in order to delete a vertical then another horizontal norm. Finally the resulting CM's columns are re-arranged to place all duplicating pcs in adjacent positions.

0 4 1 5 2 8 8 9

7 7 6 1 1 9 2 A

1 5 0 6 1 7 8

0 4 1 5 2 8 8 9

7 7 6 1 1 9 2 A

1 5 0 6

450

0

7 1 0

0

7 1 0

2 7

4 8

9 1 5

2 7

4 8

9 1 5

0

7 1 0

2 71

45 8

9 1 5

0

7 1

5 6 9 A

2 71

1 1 5 8 9

6 1 6 2 A

0 6

1 8 1 2

1 8 1 2

45 1 8 8

9 9 2

54 1 1

8 8 9 7

5 9 6 A 6

5 9 6 A 6

5 9 6 A

0 17 7 2

1

(ex. 8e)

Combining CMs If two or more different CMs have compatible norms, they may be com-

bined to form larger CMs. Type I and II can only result when the smaller CMs to be combined are of the same type. When type II CMs are put together to form a type II CM, they must have their vertical norms related by Th and/or I which entails that their horizontal norms will be likewise related. Type III CMs result when a type I and II are combined with each other or a type III. Thus the arbitrary combination of CMs tends to degenerate into a type III CM.

The combination of CMs is accomplished by merging the smaller CMs then swapping. To merge CMs, each small CM must manufacture empty posi-

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tions in which to place the other small CM. For this purpose one CM is placed below and to the right of the other producing the empty positions. The CMs then can be considered to be the residents of one large CM. In example 9a two CMs are written side by side, then shown placed in the proper arrangement producing the larger CM (dashes show the large CM's empty positions).

1 367 42 7 46 125 03 4 1AB 58 79A 4 456 89 3 234 5 OB

1 367 42 - - 03 4 1AB - - 456 89 3 - - - --

- - 7 46 125 58 79A 4

- - - 234 5 OB

(ex. 9a)

6 3 7 12 4 3 1B 4 0 A 4 68 3 5 9 1 7 4 2 5 6 5 9 A 8 4 7 0 4 2 3 B 5

(ex. 9b)

1 - 367 - 42 - 03 - 4 - 1AB - 456 - 89 - 3 -

- 7 - 46 - 125 - 58 - 79A - 4 - 234 - 5 - OB

(ex. 9c)

452

1 37 6 42 0 3 4 A B1 6 45 8 9 3

7 46 12 5 5 8 9 7A 4 34 2 5 B 0

(ex. 9d)

By the means of swapping, the 6X6 CM of 9a produces example 9b. Exam- ple 9c shows 9a's large CM re-aligned to interlace the generating CMs. 9c is swapped to produce example 9d.

The swapping of 9a into 9b produces a situation where the bottom three and top three rows of the CM are no longer independent CMs and cannot be detached from each other. On the other hand, by making sure that one swaps into adjacent empty columns in the same way that 9c produced 9d, a CM in which the bottom three and top three rows are still CMs in their own right is formed. (In 9d take the top three rows' first two positions together to get the first column of the old CM. Continued in groups of two positions to produce the rest of the CM-likewise for the bottom three rows.) Example 9d is an over-layed CM and can be used to produce a variety of compositional stratifications. Overlayed CMs effect an partial ordering of the positions of their component CMs.

Folded CMs are those that are formed by merging and swapping two CMs which are related by a pc operation. Examples 9b and 9d are indeed folded since the original CMs in 9a are related. The left CM in 9a is a swapped version of 3a with its second and third columns interchanged and the right CM is the T8I transform of the left CM. The T8I operator was chosen after con- sulting 3a's PFA. The PFA indicated which pcs occurred only once in 3a form- ing a set { 0, 8, 9, A, B } which is invariant under T8I. By combining the CM with its T8I transform, pcs which could not be swapped in each separate CM were available for swapping in the merged CM. Pc 4, an association-set in the original CM, became the only association-set for the folded result since 4 is invarant under T8I.

Another way to combine CMs involves finding two CMs with compatible norms that have at least one position with identical contents. The CMs are merged so they share the identical position as shown below (example 9e) which merges examples 3a and 6a-both have their lower-left corer position filled with the { 4, 5, 6 }. In the example, the CM of 6a is positioned with its own first and third columns exchanged and its own rows also similarly exchanged; it is located in the lowest three and left-most three positions of the merged

453

CM. As can be seen, this kind of merging places a chain in the rows and columns of the CM. The chain will of course have the same norms as the whole CM if the CM was generated from two pc operator related CMs.

~- - 467 123 - 013 4AB 127 456 389

06B 79A - - 789 346

(ex. 9e) The Generation of CMs from Chains

Non-simple CMs are generated from complete chains with an even number of positions. These CMs are basic because they cannot be broken down into smaller sub-CMs since they are not made by merging and swapping. If one wishes to use a chain with an odd number of positions, the chain must be lengthened by inserting one new position between two adjacent positions. The added position will contain the union of the sets in the positions it is inserted between. If the adjacent positions are PIQ, PIPUQIQ is the result. Another way to make an odd chain even is to repeat it twice: XIYIZ - XIYIZIXIYIZ.

In any case, given a chain of z positions, C, whose positions are C(O) IC(1) ... IC(z) and z is even, a CM can be constructed subject to the following constraints: 1) the assignment of chain positions to CM positions is one to one; 2) no more than two positions of C may occupy a row or column of the CM; 3) ...

If we assign C(n) > E(i, j) and n is odd

C(n-1 (mod z))- E(i, m) and C(n + 1 (mod z))- E(k, j)

else if n is even, then C(n- 1 (mod z)) - E(k, j) and C(n+ 1 (mod z)) - E(i, m)

One must accomplish these assignments for all n from 0 to z while obey- ing constraints 1) and 2). Thus the choice of the last position, C(z), will be in the same row or column as the first, C(0). By interchanging the resulting CM's rows and columns, it can be transformed into another CM derived by different (but legal) assignments. The assignment requirements can also be

454

modified if the words 'odd' and 'even' are interchanged but this is not necessary since the resulting CMs can be derived by row and column interchange from those generated by the assignments as given.

Tvo out of many possible assignments for the chain a|blc|dleif are: 1) a - E (0,0)

b > E (0,1) c >E (1,1) d

--E (1,2) e --E (2,2) f > E (2,0)

2) a ->E (0,0) b > E (1,0) c ->E (1,2) d

--E (2,2) e ->E (2,1) f -*E (0,1)

The resulting CMs are shown in example 10a.

1) 2) a b a f

c d b c f e e d

(ex. 10a) It should be noticed that the second CM can be derived from the first

by interchanging its rows and rotating it by 90 degrees. In fact, due to row/column transformations, one need only use one assignment which threads the chain through the CM positions in a zig-zag diagonal path.

The basic schema for generating CMs from chains is given in ex. 10b. CMs must have at least four positions so chains of less than four positions must generate CMs as illustrated in example lOc. The only deviation from the assignment rules given above will be the assignment of 'branched chains' or the inclusion of chains within CMs.

455

C(O)jC(l)jC(2)j. .IC(n)

C(O) C(l) C(2)

C(n) -... C(n- 1)

(ex. 10b)

Chains: alb cldle-* cldlcle

CMs: a b c d b a e c

(ex. 10c)

Generating Chains Now that the relation of chains to CMs has been explained, I am able

to show how chains are composed out of the SCs whose sets make up the norms of the chain. I will initially confine my discussion to the generation of chains that can produce type-I CMs. Therefore, only one SC provides the repertoire of sets for norms. The generation of type II and III CMs will be easily understood if the principles of type I chains and CM generation is grasped.

The fundamental principle behind the generation of chains is simple. Con- sider three successive positions in a chain, X, Y and Z. X in union with Y forms a set T and Y with Z forms V; V and T are norms and related by Th or ThI. The middle position Y is the intersection of two members of the same set-class. Thus, a chain is dependent on the shared subsets of sets within a SC-or in the case of type II and III chains and CMs two SCs.

To begin constructing a chain, it is useful to select one set out of the SC chosen for generating the norms and list every way it can be divided into two parts. If our set were of four elements { a, b, c, d }, its 2-partitions (simply 'partitions' herein) would be albcd, blacd, clabd, dlabc, ablcd, aclbd, and adlbc. One should notice that the order of the two parts is not a factor in the partition's identity. However, if I have a partition W defined as alb, I can treat it as a partially-ordered set so that bla is RW.

Combinatorial analysis tells us ...

Z = K!/(P!(K-P)!)

456

...where Z is the number of distinct (unordered) sets of size P from a set of size K. Now for each of the Z pc sets of size P, there is another set of size (K-P) which is its complement with respect to the whole set, except where K is even and P = K/2. In the latter case, the list of Z sets is paired with itself to provide complement-pairs. Thus, there are 2-partitions of a set of size K into complementary subsets of size P and K-P; where P is equal to K/2, Z becomes Z/2. The total partitions of a set can be listed as shown in example 1la. There the partitioned set is a member of SC 5-15 and is { 0, 1, 2, 6, 8 }. There are five 115 partitions and 10 213 partitions each labeled by a letter. The SC represented by each half of each partition is written immediately after its entry. For partition F, 0, 1 } is a member of SC 2-1 and { 2, 6, 8 } is a member of SC 3-8, for instance. In many cases, a number or an asterisk (' * ') follows a partition's entry. This label indicates the partition's special properties as described below.

Partitions for SC 5-15 set = { 0, 1, 2, 6, 8}

A 011268 1-1 4-16 1 F 011268 2-1 3-8 1 B 110268 1-1 4-25 2 G 021168 2-2 3-9 C 210168 1-1 4-16 * H 061128 2-6 3-5 1 D 610128 1-1 4-5 1 I 081126 2-4 3-4 1 E 810126 1-1 4-5 * J 121068 2-1 3-8 *

K 161028 2-5 3-8 1 L 181026 2-5 3-8 * M 261018 2-4 3-4 * N 281016 2-6 3-5 * 0 681012 2-2 3-1

(ex. 11a)

Example 1 la contains all the necessary information to generate an incomplete or complete chain whose norms are members of SC 5-15. Any partition may be chosen to begin. I will start with F. { 0, 1 } will be placed in the chain's first position and { 2, 6, 8 } will be placed in its second position. To continue the chain I will need to find a partition which when transposed and/or inverted will have one of its parts equal to { 2, 6, 8 }. Since the second half of F is a member of SC 3-8, I will choose another partition that includes a part that is a member of the same SC. Out of four possible partitions (F, J, K and L) I choose K. Under T6 K becomes 071268. Thus, { 0, 7} can be the third position in the chain. Now I need to look for a partition that begins or

457

ends with a member of the same SC of which { 0, 7 is a member. I can stop this process any time I like, or continue until I find a transformed partition that ends or begins with { 0, 1 } at which point I have a complete chain. Exam- ple 1 lb shows the whole series of partitions begun above and their transforms that produce a complete chain, followed by the chain and two CMs derived from it.

F RT6K T1IK RT7IF T6F RK T7IK RT1IF

011268 268107 07115B 15B167 671028 028116 16157B 57B101

011268107 15B1671028 16157B

01 268 07

57B

1 B 0 57

02 7 8 6

15B 67

6 15 7 B

028 16

8 0 26 1

(ex. lib)

Although the generation of chains from a listing of the partitions of a set of a SC is straightforward if a little tedious, several questions remain: How many distinct chains can be generated and of what lengths? What role does invariance play in this process? What mode of notation will best display and summarize all of the generable chains?

Partition Invariance The previous discussion has focused on the partitions of an SC member.

Every set in a SC has the same properties of invariance under pc-operators (with changes in the n in Th and IhI). These properties influence the genera-

458

tion of chains. Let the partition be defined XIY, XUY is K and F, F, and G are pc-operators. (F' is the inverse operator of F; F'F is equivalent to the identity operator T(0).)

1. Total invariance: From F(K) = it follows that for every partition XlY there is another

CID where F(X)-->C and F(Y) - D so that CID is not a unique partition but only a transform under F. For example, if K equals { 0, 1, 2, 3, 4, A, B } and XIY is 04A1 123B, under F = T2I becomes CID = 24A1 103B which is another partition of K. This means that XIY which can be defined as G(CID) is the same as GF(XIY): consequen- tially, operating on CID or XIY gives equivalent answers. One of the two partitions is termed a redundant partition.

It is often possible that where F (K) = K, F(X) -X and F(Y) Y. F is equal to F' since F(F(X)) = X. This is of no interest since the partition is unchanged in content under F. For example, 012134AB has this property under T2I. If #X = #Y, it is also possible that F (X) = Y and F (Y) = X which implies that F = F' since F' (F(Y)) = Y. But once again, this invariance does not change the partition's content.

It is concluded on the basis of these observations that invariances associated with K have no effect on or actually limit the number of useable partitions for a chain.

2. Partial partition invariances Now where F (K) does not yield K, two other conditions can arise.

1. F(X) = X and/or G(Y) = Y. G(K) + K and F + G otherwise F(K) would equal K. F (XIY) --XIC G (XIY) -DIY Examples: T6I (0361145) = 0361125 TBI (146129) = 57A129 T7I (3410189) = 34167AB T9I (3410189) = 5610189 2. If #A = #B, and if F (X) = Y and F' (Y) = X, then F + F' otherwise F (K) = K. (F cannot be ThI because ThI (ThI(a)) = a.) X and Y are of the same cardinality and members of the same SC. F(XlY)-- YIC F'(XIY)- DIX

459

Examples: T4 (0131457) = 457189B T8 (0131457) = 89BI013

Partial partition invariances produce richer possibilities for chain generation. Even in the case where only one partition ends or begins with a particular SC member, if the partition possesses partial invariance, that partition can generate a chain alone. Such partitions will therefore be termed multi-generators. Using the previous example, the partition 0131457 can link to its T4 version, 457189B, which, in turn, can link to the T8 transform of the former, 89B1013, which completes the chain.

In a listing of partitions, as in example 1 la, multi-generators will be assigned labels on the basis of their properties (see Table II). Redundant partitions are given ' * ' labels. Two other partition listings are given in examples lc and 1ld based on the complementary SCs 6-13 and 6-42.

Property Label

F(X) =X 1 F(X) = X; G(Y) = Y 12 G(Y) =Y 2 F(X) = Y; F(Y) # X 3

Table II

Partitions for SC 6-13 set = {0, 1, 3, 4, 6, 7} AA 0113467 1-1 5-10 1 AG 0113467 2-1 4-3 1 AB 1103467 1-1 5-16 1 AH 0311467 2-3 4-13 1 AC 3101467 1-1 5-19 1 AI 0411367 2-4 4-15 1 AD 4101367 1-1 5-19 * AJ 0611347 2-6 4-12 1 AE 6101347 1-1 5-16 * AK 0711346 2-5 4-10 AF 7101346 1-1 5-10 * AL 1310467 2-2 4-29 1

AM 1410367 2-3 4-18 1 AN 1610347 2-5 4-17 AO 1710346 2-6 4-12 * AP 3411067 2-1 4-9 2 AQ 3610147 2-3 4-18 * AR 3710146 2-4 4-15 *

460

AS 4610137 2-2 4-29 * AT 4710136 2-3 4-13* AU 6710134 2-1 4-3 *

3-2 3-2 3-3 3-3 3-5 3-3 3-5 3-2 3-3 3-5 * 3-10 3-10 3 3-11 3-7 3-8 3-8 3-11 3-7 * 3-5 3-2 *

(ex. lic)

Partitions for SC 6-42 set = { 0, 1, 2, 3, 6, 9 } 1-1 5-38 1 1-1 5-31 1 1-1 5-31 * 1-1 5-38 * 1-1 5-4 1 1-1 5-4 *

3-1 3-2 3-5 3-3 3-2 3-8 3-7 3-10 3-10 3-10

BG 0112369 2-1 4-18 1 BH 0211369 2-2 4-27 1 BI 0311269 2-3 4-20 BJ 0611239 2-6 4-5 1 BK 0911236 2-3 4-4 1 BL 1210369 2-1 4-28 12 BM 1310269 2-2 4-27 BN 1610239 2-5 4-13 1 BO 1910236 2-4 4-12 1 BP 2310169 2-1 4-18* BQ 2610139 2-4 4-12* BR 2910136 2-5 4-13 BS 3610129 2-3 4-4 BT 3910126 2-6 4-5 * BU 6910123 2-3 4-1

3-10 12 3-11 3-5 3-3 3-11 * 3-8 3-7 3-4 1 3-4 * 3-1 *

(ex. lid)

Al A2 A3 A4 A5 A6 A7 A8 A9 A10

0131467 0141367 0161347 0171346 0341167 0361147 0371146 1461137 0471136 0671134

BA BB BC BD BE BF

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

0112369 1102369 2101369 3101269 6101239 9101236

0121369 0131269 0161239 0191236 0231169 0261139 0291136 0361129 0391126 0691123

461

Partition Graphs A listing of the partitions of a member of a SC can be aptly summarized

as a graph. Points representing SCs are connected by lines which correspond to and represent the various partitions of the listing. Points are written as circles enclosing a SC name. A line is furnished with an arrow indicating which direction the partition is written. In example 12a, four (arbitrary) partitions are graphed. Redundant partitions (marked on the listing with' * ') are not graphed; multi-generators are written with a marker showing which side(s) of the partition have partial invariance. The entire graph of a listing is broken into several sub-graphs which reflect the cardinality and multiplicity of the SCs represented by the sets in the listing's partition halves. A graph for the listing in 1 la is constructed in example 12b.

S 05112A 2-5 3-3 1

Q 3410189 2-1 4-7 12 N 0151237 3-5 3-5 3 (

P 0151489 3-5 3-5 () P

(ex. 12a) 4?-16

4(-5 - 1-1 4-25 D B

(3'> e I

F

O< (O (G ) O2-(- K H

(ex. 12b)

462

Following a path on the graph by moving from point to point via the lines provides the relevant information needed to generate a chain. Thus, the lines transversed by a 'walk' on the graph will indicate which partitions occur in the chain in what order. Moving against the direction of a line's arrow is permitted; this indicates that the R of the partition is used. The partition usually needs to be transformed under a pc-operator so its first half is the same as the last half of the previous partition.

The classification of the kinds of chains that can be formed from a partition listing is greatly simplified by its graph. First of all, there are 'trivial' paths which produce complete but simple chains of the alb variety. These are produced by oscillating between two points. If the line is marked at one end to show a multi-generator, then the chain of the alblclb type can be generated. If both ends are marked, a '12' (see table II) multi-generator is present and a complete chain of 2 to 12 unique positions can be made. Sometimes a line begins and ends at the same point denoting that the partition it represents has the same SC in both of its halves. Such a line is called a loop. Marked loops represent type 3 multi-generators and can be traced over and over to produce non-simple chains. (The line that forms a loop has no arrow-a choice of whether to take the partition in a forward or reverse direction is not needed unless the loop is a multi-generator.) In addition, paths on the graph that form cycles (closed paths) can be used to create complete chains. These cycles involve three or more points and lines and can include crossings, recursions and sub-cycles as part of their make-up. One can walk up to a point and then retrace one's steps to the point of departure thereby constructing a chain with a series of positions followed by its retrograde. If one turns back at a multigenerator or loop the retrograde can be transformed under the partial- invariance operator of the turning point.

I have already indicated above that the listing of the partitions derived from the graph walk does not immediately generate the chain; one has to transform each partition so its first half has the same pcs as the preceding partition's second half. When one returns to the point of origin in a graph, the chain may not be complete. This would mean that the untransformed initial partition's first half is not the same as the chain's last partition's second half. The original partition on its second appearance must be adjusted by Th and/or I to link with the previous partition. Thus, the path will have to be transversed at least one more time. Letting H equal the pc-operation that this partition has to take for the linking to be successful, the amount of perfor- mances of H so that it equals the identity operator gives the number of times the path has to be taken to produce a complete chain. These recursive walks produce chains that have whole segments related to others by H or H performed so many times. Hence, CMs can be made to exhibit invariances like

463

the one illustrated in example 3a. Sometimes the first position of the chain is encountered before the chain is finished, but his only occurs when the path crosses its first point in its windings through the graph. It should be clear that the greater the number of points in the graph walk, the greater the variety of SCs in the chain's and CM's positions.

Examples of each of the situations described in the last two paragraphs will be illustrated below. For this purpose in addition to the listing in 1 la and its graph in 12b, the listings in lc and lid are graphed in 12c and 12d.

Graph of 6-13 .~AC

AA AB

2-3

AJ

2-5) ANAK

S) A6QC0)6 A7 A8

??5

(ex. 12c)

464

Graph of 6-42 H

BO

2-3

BN

4-4 BK

5-4( ) (5-38

BE BA

1-136

BB

5-31

465

are connected to other points by other lines-and BI and BU which are partitions in a larger sub-graph.

To produce larger chains continuing with the graph of 6-42, it is necessary to use one of the sub-graphs in which two points are connected by one marked line. The partition BO (1910236), shown by a line connecting the points 2-4 and 4-12, will be used. Starting with R(BO), 0236119 is obtained as the chain's first two positions. The invariance associated with 2-4, illustrated by the marked line, allows BO to have its first part equal to the chain's second position under TAI. TAI(BO) = 191478A. Now the chain is 02361 191478A. Since the set 478A has no invariance, the chain must continue with RTAI(BO), which can only be followed by BO, thus completing the chain: 02361 191478A1 191(0236).

Larger and more complex chains which still include some identical positions can be made out of the sub-graphs of 6-42 with 3 or more lines and points. The sub-graph that connects 4-1, 2-3, 4-20, and 4-4 with partition lines BI, BK and BU will be used to produce a chain starting with partition RBU. One can trace a path over the graph as follows: RBU, BK, RBK, BU, through the SCs, 4-1, 2-3, 4-4, and 2-3. These partitions are then adjusted by Th and/or ThI to link the chain.

R (BU) = 0123169 T9(BK) = 69103AB

RT9 (BK) = 03AB169 (BU) = 6910123

Chain: 0123169103AB169

Because of (BK)'s invariance, the second and third positions of the chain can be substituted by T6I (BK) = 6910345 and RT6I (BK) = 0345169. The chain could be also changed into 0123169103AB16910345169 or the branched chain, shown in example 12e, followed by a CM realization. Branched chains produce CMs with chains in their rows and columns.

103451 01231691 169

103ABI

0123 69 0345 69 03AB 69

(ex. 12e)

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Another interesting partition is shown as the sub-graph BL which is marked on both sides indicating it is a '12' multi-generator. See the following sequence of positions.

(BL) = 1210369 RT3 (BL) = 0369145 T6I (BL) = 4510369 = T3 (BL) RT6 (BL) = 0369178 T9I (BL) = 7810369 = T6 (BL) RT3I (BL) = 0369112 = R (BL)

From the above, the chain 12103691451036917810369 is formed. A branched chain given in example 12f generates a CM maximally filled with chains. Such a CM can be the source of chains by making any complete cyclic walk through it moving between adjacent but not diagonal positions. From these chains smaller CMs can be made.

1121 1451

103691 1781 |ABI

0369 12 45 0369 0369 78 AB 0369

(ex. 12f)

A comparison of the graphs of the listings of 6-13 and 6-42 shows that the former has fewer isolated points with loops, larger sub-graphs and a more fertile set of multi-generators. For instance, the marked loop corresponding to the multi-generator A6 (0361147) will generate a twelve position chain by successively transposing A6 by T1: 036 14712581369147A1 58BI0691 17A128B10391 14A125B

Another chain involving trichordal divisions of 6-13 is derived from the sub- graph with lines Al, A2, A3 and A4. The order and the transformations of the partitions derived from a walk on this sub-graph are:

467

A1 = 0131467 RTAI (A4) = 467139A T9 (A3) = 39A1014 A2 = 0141367 RTAI (A3) = 367149A T9 (A4) = 49A1013 013 467139A10141367149A

If it is desired that a chain should quickly be chromatically saturated, the following conditions must hold.

1. The norm's cardinality is 6 or greater. 2. The SC of which the norm is a member is not SC 7-12 or not a "Z-

related" Hexachordal SC. 3. The norm is a partition of its parent SC such that one of partition's

halves is a member of the SC's complementary SC.

If the partition is XIY, where X is a member of the norm's complementary SC, then the chain may continue with Z which is the literal complement of the union of X and Y. The union of Y and Z will be a partition of the generating SC since it is the literal complement of X which was defined as a member of the complementary SC. Hence, F(XUY) = (YUZ) and X, Y, and Z taken together produce the aggregate. This situation is termed an complement embedding. If F is not reflexive, the chain may continue without returning to Y for its fourth (and last) position.

As an illustration, let the generating SC be 9-9 whose member is partitioned 138102567A. Since { 1, 3, 8 } is a member of SC 3-9, the complement of the partition, { 4, 9, B }, which must also be a member of 3-9, is the chain's third position. The chain reads so far: 138102567A149B and the union of its second and third positions is the TOI or T8 of its first two positions. Defining F to be T8 allows the chain to continue, 138102567A149B| 12368A1057. The content of any three successive positions is an aggregate.

Where the norm-generating SC is hexachordal, the partition X|Y has Y as a null-set. When the resulting chain is made into a CM and where F is reflexive, the potential for a two row twelve-tone CM is produced (a potential twelve-tone CM, since it will need to be swapped and have its norms ordered).

Up to now, only type I chains and CMs have been developed. In order to produce types II and III it is useful to combine the graphs of the two SCs

468

involved. In 12g the listings of 6-13 and 6-42 are combined to make one large graph. Solid and dotted lines are used to identify from which SC their corresponding partition is dervied. To produce a type II chain, one must follow paths which alternate dotted and solid lines. A type III CM or chain has no special restraints associated with the graph paths used to generate them, except that both dotted and solid lines must be used.

The number of chains that can be developed from two SCs can be used as an assessment of similarity between the SCs vis a vis other SC pairs. The measures of similarity provided by John Rahn in his article, "Relating Sets" [15], are based on the number of SCs shared (included) in two SCs. This is the factor which underlies chain production. Graphs of pairs of SCs can show differences as well. In the present case, the Z-related SCs 6-13 and 6-42 are differentiated in detail by the comparison of their graphs.

TWo chains, their partitions and type II CMs that can be derived from 12g are found in examples 12h and 12i. In the first of the two examples, every position holds pcs that are members of SC 3-3, while the latter is composed of two representatives of SCs 3-2, 3-7 and 3-11. In 12h the vertical norms are members of 6-13 and the horizontal norms house members of 6-42. 12i reverses this assignment of SCs to norms.

(A2) 0141367 T6 (B4) 3671089 T8 (A2) 089123B T2 (B4) 23B1458 T4 (A2) 45817AB TA (B4) 7ABI014

01413671089123B145817AB

014 367 089 23B

7AB 458

(ex. 12h)

469

Graph of 6-13 and 6-42

J AB - - '

5-38 (5-19) BB

5-31

BG BI BL 41 -

(4-28) I AH >4 4-27 4-13

AL )

AJ (--12 - (2-6 4j

AI

B6O -

B9 3-13) (3-10 -- 3-(3-4

?, > A6

B7 Al B3 A2

XA73 (3B 2

, A4 5 A3 B'

(ex. 12g)

470

(B2) 0131269 T9I (A7) 2691358 T5I (B7) 358124B

RTA (A7) 24B 15A RT7I (B2) 15A1467 T7I (A1) 457(013

01312691358124B| 15A1467

013 269 358 24B

467 15A

(ex. 12i)

Since the generation of type III chains and CMs presents no further com- plications, no more examples of chain generation of this species will be presented.

Generating CMs by Operator-Cycles I have delayed the presentation of the following method for generating

CMs up to now since it is based on different premises from those underlying what has been discussed so far. These CMs are not generated from chains. The properties of a CM's norms no longer are factors for its construction. Rather, the properties of the union of all pcs in the CM, designated U above, provide the relevant conditions for CM creation.

In order for such a CM to be made, U must be invariant under some pc-operator F This is equivalent to saying that U is a union of complete cycles of F. [16] The size of the CM is n2 where n is the size of the longest F-cyles. For each position of the CM E, designated E(i, j), containing a number of pcs or a null-set, there is a diagonally adjacent position E(i + 1, j + 1)-i andj taken mod n-that holds the contents of E(i, j) transformed under F. This means that the norms lying in the same direction are all members of the same SC. After the CM has been completed it can be swapped and row/column interchanged. Since all its norms are related under F (or F taken m times), this category of CM can be easily identified. In addition, the norms involved can be considered ordered and related to others in the same direction by rotation.

471

A further property which distinguishes this kind of CM generation from chain derived CMs is that if no single-element cycles are used and all cycles are used only once, the CM has no associated pcs or association-sets.

Examples 13a-d show various CMs, their U sets, pc-operators with cycles, and their norm generators; in the last two examples, a swapped and row/column interchanged version of the CM is also displayed.

In example 13a, F is T1I and is reflexive making a 2X2 CM. E(0,0) contains both elements of the F-cycle (0-1) and one element of the F-cycle (5-8). This obligates E(1,1) to hold the other element of (5-8), namely 8, and the entire (0-1) cycle [because F(E(0,0)) = E(1,1)]. E(0,1) contains { 2, 5, 9 } which is half of the (4-9) cycle, half of the (5-8) cycle and half of the (2-B) cycle. Since F(E(0,1)) = E(1,0), E(1,0) = 4, 8, B}. 4, 8, B } is the union of the other halves of the cycles used in E(0, 1).

U = {0, 1,2,4,5,8,9,B} SC 8-17 F = T1I F-cycles (0-1) (2-B) (4-9) (5-8)

H norm generator = { 0, 1, 2, 4, 9 } SC 5-37 V norm generator = { 0, 1, 4, 5, 8, B } SC 6-44

015 259 48B 018

(ex. 13a)

U = aggregate F = TOI F-cycles (0) (6) (1-B) (2-A) (3-9) (4-8) (5-7)

H norm generator = {0, 1, 2, 3, 4, 5, 6} SC 7-1 V norm generator = { 0, 2, 4, 5, 6, 9, B } SC 7-29

0245 136 69B 078A

(ex. 13b)

472

U = {0, 1,2,4,5,6,8,9, A} SC 9-12 F = T4 F-cycles (0, 4, 8) (1, 5, 9) (2, 6, A)

H norm generator V norm generator

0 A 19

9A 01

= {0, 1, 5, 6} SC 4-8 = {0, 1,9,A} SC 4-3

15 4 2

6 59 8

45 56 89 12

(ex. 13c)

U = aggregate F = T2 F-cycles (0-2-4-6-8-A) (1-3-5-7-9-B)

H and V norm generators = 4 B 0

6 1 8

1 8 9

48 1

3 A

3B 6

{0, 4, 7 2 3 A

7, B} SC 4-20

9 4 5 0

5

04 1 2 9

5 08

9 A

A 37

5 2

(ex. 13d)

B 6 7 2

7B 6

473

Further CM Extensions Further techniques I have described may be used to develop CM struc-

tures in a number of different directions. Out of many such possibilities, I will discuss those extensions which involve norm and partition embedding.

If a position or norm of a CM, E, has a invariance, it may be made into a CM by considering it to be a U-set. The norms in these smaller embedded CMs can become norms in the larger CM. The CM in example 14a is derived from the twelve position chain based on the multi-generator A6 in the listing of 6-13. Each norm is of course a member of 6-13 and has ThI invariance. For instance, Tl(a6) = 1471258 which maps into itself under T9I. Consequen- tially, a small 2X2 CM may be derived from T1(A6) and forms example 14b. Note that its horizontal norms are from SC 3-2 and its vertical norms are members of 3-3. By embedding transformations of this small CM in each row of the CM in example 14a, the CM of example 14c is produced. It has twelve horizontal norms, all members of 3-3 which, when taken in pairs, are members of 6-13. For contrast, in example 14d, the first two columns of 14a have been likewise embedded.

26 0 3B 5 4 5 7 8 12 67 3 9A 4 0 9 58B 6

8 A 17 2 B 13 4 09 A

(ex. 14a)

2 OB 56 3

(ex. 14b)

474

26 0 3B

4 7 8 5

5

12 9A

3 8B

9 5 A

8

4

6 1 7

2

4 0 9

B

A

(ex. 14c) 26

4 5 67

0 9 8

3 1

(ex. 14d)

3

CM positions can also produce CMs if their contents have some invariance. Example 14e is related to 1 lb by swapping, and the former's positions either hold single pcs or members of SC 4-25. Since any member of 4-25 has T6 invariance, these positions are converted via the cycles of T6 into mini CMs whose verticals and horizontals are ic 2 and ic 4, respectively. This CM whose columns and/or rows taken in groups are members of 5-5, and are members of 2-2 or 2-4 (or 1-1) when taken alone, is presented in example 14f. The CM is then swapped into example 14g which disturbs its small CMs but preserves its larger structural features.

6 7 0

3 1

475

0268 157B

7 6

1

0268

(ex. 14e)

0 8 2 6

1 5 B 7

7

6

(ex. 14f)

0 8

1 1 5 B

2 6 0

7

7

6 2

0

6

(ex. 14g)

0

157B

1

0 6 2 8 0

5 1 7 B

7

5 B

1

476

Simple type I CMs can be derived by placing norms in an empty CM's diagonal positions and then swapping. For example, in 14h a CM is developed according to this method; its norms are members of SC 6-42. Since this diagonal method allows an entire norm to occupy one position, the position can hold a CM if the norm has an associated invariance. Then the whole CM can be swapped. This is done to the first CM of example 14h to produce 14i.

012369 14567A

23458B 134569

09 16 23

16 57A 4

23

8B 45

4 5 0369

(ex. 14h)

06 2

17 5

28 4

1 39

6 4A

3 5B

39 5

6 1

17 6 A

2 3

0

8 5 4

4 60

0 239

5 4

B 9 4 3

5 6

(ex. 14i)

477

Merging and folding CMs with or without embedded chains and combined with embedded CM development makes the possibilities of non-aggregate combinatoriality quite extensive. As only a token representative of this vast potential, I show a CM where a position of one of two CMs is expanded by the other. TWo CMs are given in example 14j and are reproduced from example 12h and the T7I of example 12i. The first column of the left CM contains the same pcs as the top row of the right CM. By properly partitioning the former, it can be used as the substitute for the latter as shown in the next example (14k).

014 367 lAB 047 089 23B 136 029

7AB 458 245 38B

(ex. 14j)

1 04 367 089 23B

AB 7 458 136 029

245 38B

(ex. 14k)

Twelve-tone and Non-Aggregate Combinatoriality Although modified forms of many of the concepts, techniques and strategies

of general twelve-tone combinatoriality have been used in the foregoing, the ordering of norms and their content has kept the two types of combinatoriality distinct. The fact that a CM's or chain's U-set or norms need not exhaust the total chromatic is as important as it is obvious, and especially in a 12-pc (possibly atonal) context. Moreover, all the norms in a CM or chain are not necessarily pc-identical but may partake of the interval-content of one or two SC's sets. While the invariance of the norm's parts and of U is of consequence to the generation of non-aggregate pc-structures, the norm's own self-mapping is of no interest. The Babbitt quotation cited earlier warrants the definition of a norm as unordered precisely because it is a sub-set of the aggregate. As a result, the unordered norm allows the formulation of partitions of a SC representative to generate a large amount of CMs and chains. The row and column

478

interchanges, rotations, and pc-swapping among any column or row in a CM provide a very large array of compositional designs.

These considerations are reinforced by imagining that the rows in a twelve- tone CM are unordered sets. Then any distribution of pcs could manufacture linear and vertical aggregates in the manner of example 4b. The ordering of the sub-sets of the rows of a twelve-tone CM is precisely what gives the CM its meaning above and beyond its aggregate formation. In the formulations of Babbitt[17] and Donald Martino [18], much importance is given to the breakdown of CM aggregates into particular hexachordal and trichordal units. This adds a harmonic or at least vertical dimension to what is otherwise a basically contrapuntal arrangement of ordered sets within a series of aggregates. In fact, the method of generating a CM from the invariances associated wth U with or without pc saturation via complement embedding can be seen as an extension of these authors' procedures. In contrast, this paper has emphasized the 'harmonic' unordered norm at the expense of precompositional linear relations and structures. Indeed, what virtues would ordering the norm generators of a chain or CM entail?

First of all, the ordering of a group of pcs induces a set of ordered intervals measured from one pc to the next. This helps identify the set and distinguish it from other sets with the same elements in another order. Thus the n! orderings of a set of pcs are no longer members of the same equivalence class, making the number of norms far exceed the 4,096 unordered sets of up to 12 pcs. Ordering a set also imposes a unique nesting of its subsets. The role of content invariance under pc-operators is now one of permutation rather than identity. However, along with distinctions produced by permutation come greater restrictions. For instance, swapping pcs in CMs may only occur between adjacent columns (or ones separated by null-sets in the rows where the swap- pable pcs reside) and only between the ends of a ordered set. The interchange of columns is no longer permitted and 90 degree rotation must likewise be avoided. Aside from other obvious limitations, the generation of chains from listings becomes problematical since a SC member of cardinality k that is ordered and produces the listing can only be broken up in k ways without disturbing its order; only k 2-partitions are available for use. Thus, chain development is limited to those chains that alternate two partitions, as in 12h, but are ordered. Such a chain need only have every other position reordered by a pc-operation, which means it can be generated by two partitions only one of which will be the horizontal ordered norm. When generating the CM from the chain at least one R form needs to be used. (See example 15a for two realiza- tions of 12a's chain.)

479

S: 041 637 T8S: 809 2B3 RT4S: B7A 584

J: 367 809 T8J: B23 485 RT4J: 140 BA7

(ex. 15a)

Naturally, multi-generators can produce CMs with ordered norms, as illustrated in example 15b, where the twelve-position chain that generated 14a is utilized. This example also interpolates more than one R-form. Such CMs as in examples 14a and 14b can be termed canonic CMs due to their make-up and in analogy to the structure of twelve-tone canonic CMs.[19]

K: 063 471 RT2K: 396 582 T4K: 4A7 8B5 T6K: 609 A17 RT8K: 930 B28 RTAK: B52 14A

(ex. 15b)

The role of redundant partitions where F(X14) = RYIRX does not change with the introduction of order into the norm. Distinctions between two partitions related by F can be produced by permutation alone. This is exactly the same situation that occurs when a row can map into its retrograde under F; either there are no distinct R forms, or there are no forms involving F. Of course, the ordering of pc-operator-related complementary hexachords in a twelve-tone row is a familiar use of two orderings of a norm that are not identical under F.

Where U is invariant under F and F is reflexive, a 2X2 CM can be generated where its bottom row is the RF of its top row. Example 15c is the ordering of the CM of example 13a so its rows are related by RT1I.

480

Y: 510 2B5 RT1IY: 82B 108

(ex. 15c)

The order operator R has been presumed throughout this paper. One can use other (perhaps arbitrary) order operators as well. Rotation seems a good candidate, since it preserves most of the interval succesion of a segment of pcs and it is of importance in the works of Stravinsky and other composers. By including it, a greater number of partitions of a listing could be invoked by the rotation of a ordered norm. In addition, U-sets, where F produces invariance but is not reflexive, would be permitted as CM generators (see above); norms would be related under CnF.[20]

Another way to produce CMs with ordered norms is to use the (trivial) method as shown in 14i. One would want to order the norm generator in such a way that it had invariant segments under F, or a segment would be related to another under F. These segments would be placed at the norm generator's beginning and end. This would increase the opportunity for the necessary swaps into the rest of the CM's positions. Naturally the "end" and "begin" sets of norms designed for this kind of CM generation would best be multi-generators.

In any case, the ordering of norms helps limit and focus the contrapuntal aspects of chain and CM usage in compositional strategy.

Final Remarks If I have only begun to realize the potential implied by this paper's intro-

duction, I hope I have shown how the (pitch-class) abstraction of a harmonic/ melodic complex, defined by content alone, can be generalized and extended within or without a twelve-tone context. It has been my purpose to present the matter in an uninterpreted, somewhat formal manner, avoiding examples on a staff or references to extant compositions. In this way, those to whom this paper may be of use are free to take from it whatever they can without having to fathom the subjective interpretations that satisfy my own personal compositional needs. [21]

481

Notes [1] It is assumed that the reader is familiar with the basic terms and concepts which pervade the set-theoretic and twelve-tone literature. Those terms which are not universally found in this literature are defined in the text or in foot- notes. For an introduction to the subject of twelve-tone combinatoriality, see John Rahn, Basic Atonal Theory (New York: Longman, 1980) or Charles Wuorinen, Simple Composition (New York: Longman, 1979).

[2] See Milton Babbitt, "Some Aspects of 'Ielve-tone Composition," The Score andI.M.A. Magazine, 20 (1955), "Set Structure as a Compositional Determinant," Journal of Music Theory 5/1 (1961), "Since Schoenberg," Perspectives of New Music (Fall-Winter 1973/Spring-Summer 1974) and Donald Martino, "The Source Set and its Aggregate Formations," Journal of Music Theory 5/2 (1961).

[3] See Daniel Starr and Robert Morris, "A General Theory of Combinatoriality and the Aggregate, Part I," Perspectives of New Music (Fall-Winter 1977), and "A General theory of Combinatoriality, Part II," Perspectives ofNew Music (Spring- Summer 1978).

[4] A CM (Combination Matrix) is a two-dimensional array which aligns special- ly selected related twelve-tone rows to form a series of vertical aggregates. The pitch-classes in CMs are written in integer notation (see [8]). A CM for the row 014392A857B6 is:

Y: 01 4 392A8 57 B6 RT4IY: A59 B8 627 1034 T3Y: 347 6 0 5 1B 8A29 RT7IY: 1082 B95A43 67 T9IY: 985 607B1 42A3 T1Y: 1254 A3B 9680 7 RT5IY: B6 A0973 8 21 4 5

[5] The kind of LA (linear aggregate) referred to here is formed by two or more concatenated twelve-tone rows selected so that the pcs at the end of one row in union with the pcs at the beginning of the next produce an aggregate. The sequence of these pitches is a LA. Babbitt's "secondary set" is this type of LA.

[6] Set-classes (SCs) are collections of unordered sets related by Th and/or I. I use Allen Forte's system of SC identification and nomenclature; see his The

482

Structure of Atonal Music, (New Haven: Yale University Press, 1973). Appen- dix A contains a listing of all SCs mentioned in this paper.

[7] See Milton Babbitt, "'lelve-tone Invariants as Compositional Determin- ants," inProblems ofModemrMusic, ed. P. H. Lang (New York: WW. Norton, 1960).

[8] Pitch-classes (pcs) are written in 'fixed-do' notation where all C naturals, B sharps and D double-flats are denoted by the pc 0; pc 1 denotes all C sharps, B double-sharps and D flats; pc 2 denotes all D naturals, C double-sharps and E double-flats; etc. Pc 10 and pc 11 are written as A' and 'B' respectively, to avoid confusion with the pc sequences, 10 and 11.

[9] See Bo Alphonse, The Invariance Matrix, (Ph. D. dissertation, Yale Univer- sity, 1973).

[10] Paul Lansky and Carlton Gamer, "Fanfares for the Common Tone," Per- spectives of New Music (Spring-Summer 1976/Fall-Winter 1977).

[11] See Babbitt, "Since Schoenberg."

[12] See Starr and Morris, 'A General Theory.. .Part II."

[13] M is the pc-operator known as M5 or the "cycles of [ascending] fourths" operator. M multiplies pcs by 5 mod 12. It is commutative with I to form the M7 operator. See Hubert H. Howe, Jr., "Some Combinational Properties of Pitch Structures," Perspectives of New Music (Fall-Winter 1965).

[14] This transformation (in a twelve-tone context) was first brought to my attention by the composer, William Book.

[15] John Rahn's paper, "Relating Sets," was delivered as a response to other papers delivered during the "set theory" panel at the Society for Music Theory's National Convention in New York City on November 1, 1979. See Perspectives of New Music (Fall-Winter 1979/Spring-Summer 1980).

[16] A pc-operator F can be described as one or more cycles of pcs. For example if under F, a - b, b -> c, and c -> a, then the cycle (a-b-c) is defined. The cycle is an ordered end-connected set such that for a successive pair of pcs in the cycle, the right pc is the result of the operation F on the left. The operator T4 has four cycles of the form (a-b-c) - they are (0-4-8), (1-5-9), (2-6-A),

483

(3-7-B). T1 has one twelve element cycle: (0,1,2,3,4,5,6,7,8,9,A,B). A ThI operator has six two-element cycles if n odd, five two-element cycles and two one- element cycles if n is even. See Starr and Morris, 'A General Theory.... part I," and Daniel Starr, "Sets, Invariance and Partitions," Journal of Music Theory 22/2 (1978).

[17] See Babbitt, "Since Schoenberg."

[18] See Martino, "The Source Set..."

[19] See Starr and Morris, "A General Theory.. .Part II."

[20] CnFn is an operator where the pc-operator, F, is performed n times on an ordered set which is subsequentially cyclically permuted n places to the left.

[21] I would like to acknowledge Ellen Koskoff, John Rahn, Nola Read and Wayne Slawson for reading the various drafts of this paper and offering many helpful suggestions for its improvement.

Appendix A Set-classes Found in this Paper. To read the table: 1. 'Name' consists of two numbers separated by a hyphen. The second of

these numbers indicates the position of a particular set-class on the list and the first number gives the set-class's member's cardinality. For example, 4-5 is the name of the set-class whose cardinality is 4, and is fifth on the list of set-classes of that cardinality.

2. 'M/MI' indicates the set-class whose members are related under ThM or ThMI to those of present set-class. This number is the second of the two numbers in 'name' of the former set-class. If this number is identical to the second number of the present set-class's name, the set-class has sets that are invariant under ThM and/or TnMI.

3. 'Z-ZZ' tells whether or not the SC has a unique interval-class-vector. If there is an entry in the column it indicates the SC with the same current vector. (A vector is only shared by two SCs at most.)

4. SC member is simply one of the sets included in the current set-class. 5. 'Interval-Class-Vector' gives the interval-class content for any set within

the current set-class. Six successive numbers occur within brackets. The

484

left-most number gives number of interval-classes of size 1, the second number from the left gives the number of interval-classes of size 2, and so forth until we get to the last (sixth, right-most) number which indicates the amount of ic 6's in any set within the set-class. For example, [011010] indicates that any set within the set-class with which it is associated has no ic l's, one ic 2, one ic 3, no ic 4 or 6 and one ic 5.

6. 'Invariance-Vector' indicates the properties of the sets within a set-class. The first four of the vector's eight positions show whether or not the SC's sets have Th, ThI, ThM, and ThMI invariance respectively. The last four positions tell if the member sets map into their complements under Th, ThI, ThM, and ThMI respectively. If a position is zero, the sets with the set-classes do not have the associated property. When the position is greater than zero, the number within the position gives the number of different 'n's in Th in the operator associated with the position.

For instance, the SC 6-38 has the invariance-vector 11000011 which tells us that the sets within this SC have (total) invariance under one value of n in Th, one value of n in ThI, - that they map into their complements under ThM and ThMI with one value of n in each case. The sets with 6-38 have no other special properties of this kind.

Note: All SCs have a '1' in the first position of the vector which indicates (trivial) invariance under TO.

Name M/MI Z-ZZ SC-member Interval- Invariance- Class- Vector Vector

1- 1 1 {0 } [000000] 1111BBBB

2- 1 5 01 [100000] 11009988 2- 2 2 {02 } [010000] 11119999 2- 3 3 {03 } [001000] 11119999 2- 4 4 {04 } [000100] 11119999 2- 5 1 {05 } [000010] 11009988 2- 6 6 {06 } [000001] 2222AAA

3- 1 9 {012 } [210000] 11007744 3- 2 7 {013 } [111000] 10005655 3- 3 11 {014 } [101100] 10005655 3- 4 4 {015 } [100110] 10105656 3- 5 5 {016 } [100011] 10016776

485

3- 7 2 {025 [011010] 10005655 3- 8 8 {026 [010101] 10016776 3- 9 1 {027 [010020] 11007744 3-10 10 {036 [002001] 11118888 3-11 3 {037 [001110] 10005655

4- 1 3 {0123 [321000] 11005511 4- 3 26 {0134 [212100] 11003322 4- 4 14 {0125 [211110] 10001323 4- 5 16 {0126 [210111] 10002432 4- 8 8 {0156 [200121] 11114444 4- 9 9 {0167 [200022] 22226666 4-10 10 (0235 [122010] 11113333 4-12 27 {0236 [112101] 10002432 4-13 13 {0136 [112011] 10012442 4-15 29 29 {0146 [111111] 10000331 4-16 5 {0157 [110121] 10002432 4-17 17 {0347 [102210] 11113333 4-18 18 {0147 [102111] 10012442 4-20 7 {0158 [101220] 11003333 4-25 25 {0268 [020202] 22226666 4-27 12 {0258 [012111] 10002432 4-28 28 {0369 [004002] 44448888 4-29 15 15 {0137 [111111] 10000331

5- 4 29 {01236 [322111] 10000200 5- 5 14 {01237 [321121] 10000111 5-10 25 {01346 [223111] 10000110 5-15 15 {01268 [220222] 11112222 5-16 32 {01347 [213211] 10000110 5-19 19 {01367 [212122] 10010220 5-31 31 {01369 [114112] 10010330 5-37 17 17 {03458 [212320] 11001122 5-38 18 18 {01258 [212221] 10000110

6-13 50 42 {013467 [324222] 11000000 6-14 14 {013458 [323430] 10101010 6-22 22 {012468 [241422] 10100101 6-26 4 48 {013578 [232341] 11000000 6-42 29 13 {012369 [324222] 11000000

486

{ 0 12569 { 012469 {10123456 { 0123479 { 0 124679 { 01345689 {101245789 { 0134679A { 012346789 { 01234678A { 01235678A { 01245689A {1012345678A { 012345679A {1012346789A

} [313431] } [233331] } [654321] } [444342] } [344352] } [546652] } [545662]

[448444]

} [766674] } [676764] } [676683] } [666963] } [898884] } [889884] } [888984]

12-1 1 { ~0123456789AB } [CCCCC6] cc0 0

19 24

36

19 10

35 12 4

17 7

28

5 8 1

12

6-44 6-46

7- 1 7-12 7-29

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Article Contentsp.[432]p.433p.434p.435p.436p.437p.438p.439p.440p.441p.442p.443p.444p.445p.446p.447p.448p.449p.450p.451p.452p.453p.454p.455p.456p.457p.458p.459p.460p.461p.462p.463p.464p.465p.466p.467p.468p.469p.470p.471p.472p.473p.474p.475p.476p.477p.478p.479p.480p.481p.482p.483p.484p.485p.486

Issue Table of ContentsPerspectives of New Music, Vol. 21, No. 1/2 (Autumn, 1982 - Summer, 1983), pp. 1-616Front Matter [pp.590-590]The Exhaustion of Western Art Music [pp.1-14]Notes from the Timbre Space [pp.15-22]Listening to The Eskimos of Hudson Bay and Alaska [pp.24-25]Forum: Improvisation [pp.26-111]Morton Feldman: One Whose Reality Is Acoustic [pp.112-113]The Reception of Arnold Schoenberg in the German Democratic Republic for H. R. [pp.114-137]Alexander Tcherepnin's Thoughts on Music [pp.138-144]Ussachevsky on Varse: An Interview April 24, 1979 at Goucher College [pp.145-151]Asynordinate Twelve-Tone Structures: Milton Babbitt's Composition for Twelve Instruments: Part One [pp.152-208]Relationships of Symmetrical Pitch-Class Sets and Stravinsky's Metaphor of Polarity [pp.209-240]Ren Leibowitz [pp.241-256]A Tonal Analog: The Tone-Centered Music of George Perle [pp.257-284]Luciano Berio, Sequenza VI for Solo Viola: Performance Practices [pp.286-293]"Any Bunch of Notes": A Lecture [pp.295-310]Transformational Techniques in Atonal and Other Music Theories [pp.312-371]Report from Venice (1982) [pp.372-377]The Eighteenth Annual Festival/Conference of the American Society of University Composers: The Composer in the University Reexamined [pp.378-392]Report from Buffalo - The North American New Music Festival 1983 [pp.393-401]"-ISMS": New York: "Horizons '83" [pp.402-406]George Rochberg: Progressive or Master Forger? [pp.407-409]Luigi Dallapiccola: Review of New Recordings [pp.410-416]untitled [pp.417-424]untitled [pp.425-430]Combinatoriality without the Aggregate [pp.432-486]Roland-Manuel and the 'Poetics of Music' [pp.487-505]Pythagoras and Pierrot: An Approach to Schoenberg's Use of Numerology in the Construction of 'Pierrot lunaire' [pp.

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