Yale University Department of Music

Iannis Xenakis' "Nomos Alpha": The Dialectics of Structure and MaterialsAuthor(s): Thomas DeLioReviewed work(s):Source: Journal of Music Theory, Vol. 24, No. 1 (Spring, 1980), pp. 63-95Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843739 .Accessed: 06/12/2011 20:34

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IANNIS XENAKIS' NOMOS ALPHA:

THE DIALECTICS OF

STRUCTURE AND MATERIALS

Thomas DeLio

Nomos Alpha is a work for solo cello written in 1965 by the Greek composer lannis Xenakis. In it Xenakis juxtaposes two radically different types of structure, producing a fascinating discourse on the

relationship of the materials by which a structure is engendered to that structure itself. This paper will first present an analysis of the composi- tion and then consider this one significant issue which arises from that

analysis. Certain aspects of this discussion are borrowed from the

composer's own analysis of the piece in his book Formalized Music,' much of which is summarized and considerably extended here.

Nomos Alpha is in twenty-four sections which are separated into two

layers of structure (referred to in this paper as levels I and II). Level I consists of sections 1 (mm. 1-15), 2 (mm. 16-30), 3 (mm. 31-45), 5 (mm. 65-79), 6 (mm. 80-95), 7 (mm. 96-111), 9 (mm. 126-142), 10 (mm. 143-160), 11 (mm. 161-177), 13 (mm. 188-202), 14

(mm. 203-218), 15 (mm. 219-234), 17 (mm. 249-264), 18 (mm. 265- 280), 19 (mm. 281-297), 21 (mm. 317-332), 22 (mm. 333-350), and 23 (mm. 351-364). Level II consists of every fourth section-that is, sections 4 (mm. 46-64), 8 (mm. 112-125), 12 (mm. 178-187), 16

(mm. 235-248), 20 (mm. 298-316), and 24 (mm. 365-386).

63

Level I

This structural level was fashioned with the aid of that branch of mathematics known as group theory. Therefore, before moving on to the actual analysis, a few basic notions from group theory will be explained.

A mathematical group is, first of all, a collection of elements over which a single binary operation is defined. (A binary operation is one which operates on only two elements at a time.) In addition, a group must exhibit the following properties:

a) It is closed under the operation. b) It is associative. c) It contains an identity element. d) For every element, it contains an inverse. For example, the set of integers and the binary operation addition

constitute a group structure. As such, they satisfy the four group properties:

a) The set of integers is closed under addition: any integer added to any other integer always yields some integer.

b) It is associative in that for any three integers x, y, and z: (x +y)+ z =x + (y + z).

c) It has an identity element: that is, some element-in this case 0- which when added to any other element yields that other. Thus, x + 0 = x for every integer x.

d) Every element has an inverse; that is, for any element in the collection there is another which, when combined with the first under the given operation, yields the identity element. Clearly every integer (x) has an inverse (-x) under addition: (2)+ (-2)= 0.

The particular group structure which Xenakis employs is that group of twenty-four elements known as the octahedral group, so called because of its relationship to the structure of the polyhedron with eight sides. The elements through which this group structure is expressed in the composition Nomos Alpha are permutations. The binary opera- tion is composition (notated o).

The particular set of twenty-four permutations employed are listed in Table 1. The logic guiding the specific labelling system employed will become clear as the analysis proceeds. For now it should be observed that each group element is represented by a permutation of eight numbers.

It is especially useful for those unfamiliar with this branch of mathematics to think of the notation employed above for each group as a shorthand. For example,

A = 21436587

64

Table 1

I 12345678 Q1 78653421 A 21436587 Q2 76583214 B 34127856 Q3 86754231 C 43218765 Q4 67852341 D 23146758 Q5 68572413 D2 31247568 Q6 65782134 E 24316875 Q7 87564312 E2 41328576 Q8 75863142 G 32417685 Q9 58761432 G2 42138657 Q10 57681324 L 13425786 Q1, 85674123 L2 14235867 Q12 56871243

Table 2

AoD=

14235867

Table 3

BoL2=

34127856

34127856

32417685

65

is an abbreviation for

12345678

A= XXXx 21436587

Similarly,

D = 23146758

is an abbreviation for

12345678

23146758

and, of course, I = 12345678

is an abbreviation for

12345678

12345678 12345678

Thus, one should think of these twenty-four group elements as twenty- four permutation schemes represented diagramatically by sequences of arrows similar to those shown above. The twenty-four group elements are then twenty-four distinct ways of permuting.

The operation o is defined as follows. As an example let us take the binary operation AoD. AoD represents two interrelated stages in which the second permutation, D, is applied to the result of the first, A (Table 2). The result is permutation 14235867 which is equivalent to the group element L2. Thus, it may be said that AoD=L2. A similar operation BoL2 is shown in Table 3. Here the result is permutation 32417685 which, of course, is equivalent to group element G. Thus, BoL2=G.

The complete range of compositions available within the group are summarized in Table 4, which contains the results of compositions of all possible pairs. Table 4, which is known as an operation table, is read in the following manner. The result of operation AoD is calculated by finding the intersection of column A and row D: yielding, as seen earlier, AoD=L2. Similarly, for BoL2 one finds the intersection of column B and row L2. Once again, as before, BoL2=G.

Next, for the purposes of the analysis which is to follow, several facts

66

Table 4

I7 A I SE I C -IL6O1 s j Q F

4A I A b c L PI J VD Q bQr 1.3 Qm qf '? .f. q"l Iv IBC TA 1LCD E~ ZFL ~ ____

B C r A L _

r 4.1 4

6 A X r'ltl~ i D 112- iz? c 912, Of sic 1,9-F rpo 413 Pt I o

PL L 24 r LP z c AL c A B 4) Q 'Pe 4w If Q e4)1j 8p i4j 4 4; 40-

c E .C1A 8 _ 2 4 l .4

q # 4z o 4 4 4 A AP

EL 4 E 14 E' o T

14 I_0 9 ip $

EE_

2/8 e24p5A oy_ Q#3QvQ4~u t L z 3 I c Z E 14 A o . T 1? ta Q. 1,4 3 ' Qr . A I4,Ox6'Or 144

x C- 'c Cr Zt 14 1? ?-l 4)p 0 ' q ! 1 4)t P96 'P3

___ _ Q 141? 14' y4)4Lc ?1 41 4. ___ 4 Pr40

LLE~y4GC8/ Az ]LL d LiZ u

t S lp Z16 Pr 9 C ,8~ ILZ~ 4 41 19-6 4*-)+ 4'0.vC L J 4f Lx- Z'P 6- 1:

L AF 14 1 Z c -trI 1 Wl4 6-16 .4 AzA lqQu ),z 4?pI

_ _ _1 _IP 10Iklg l 1L L D C _I _~i __ b

01 Pe-A46 Qr 1 O?f 02 03 A G L ,a z ~r 4

Of OV bA3 Oq F r C' L 2- j'P CLt Gt L C 101Z446'9J 4 1 I 4 _' 1 zz 4 _r

94 _, 4) ,_ 1_az Af CdLlJO ?- 1 4 1,F I A ( " 2 IS-2 o-, Q. 0,119 1 alP t? lz 1 g Z IS (5'1.r 4'1 C P'IA I -r r-

z

61a, A 0, 1P or AP6P P -Z 142 C 4

PAP I pe rgL 4141A 16 a9' q zciLl 40 2 Ar 4). A _f. g o 4- Catff

$* a , s) or I o A k- LL 14 E P

Reprinted from Formalized Music by lannis Xenakis, Indiana University Press, 1971. Used by permission.

67

should be noted. Obviously, the identity element is I = 12345678 as I is that permutation which leaves each of the eight numbers unchanged. Thus, when composed with any other element the element I leaves that other element unchanged. In addition, the following elements are inverses: D and D2, E and E2, G and G2, L and L2. As such DoD2=I, EoE2=I, GoG2=I, and LoL2=I. The reader can easily verify this fact for himself.

Moving beyond the basic structure of the group itself attention must be turned to a few simply aspects of its decomposition. First, the notion of a subgroup will be explored. A subgroup is a group which is itself contained within another group. Returning to the first example presented above we note that the subset of even integers taken from the complete collection of all integers itself forms a group under addition. The set of even integers is closed under addition and together with that operation exhibits all other group properties. Clearly, the group of even integers under addition is contained within the group of all integers under addition and, as such, constitutes a subgroup of that group.

With respect to the group employed by Xenakis we note the following two subgroups which it contains:

(I,A,B,C,D,D2,E,E2,G,G2,L,L2) = Subgroup 1 (I,A,B,C) = Subgroup 2

Each of these collections forms a group under o. Next, the question arises as to the relationship of the remaining

elements of the group to each of these subgroups. With respect to subgroup 1 the following may be noted: by re-labelling subgroup 1 as set U1 and calling the remaining twelve elements set U2 the following operation (Table 5) may be constructed where, once again, the operation is o:

U = I,A,B,C,D,D2,E,E2,G,G2,L,L2 U2 = 0102

,3, 4, 5,06

,7,08,09 ,10,011, 12 Table 5 is to be read in the same manner as discussed for Table 4. What is revealed by Table 5 is that any element from set U1 composed with any element from U2 yields some element from U2 (U1 oU2=U2) Similarly, any element from U2 composed with itself or any other from U2 yields some element from U1 (U2oU2=U1).

With respect to subgroup 2 a similar operation table may be con- structed. Here, however, the original set of twenty-four elements are partitioned into the six nonintersecting sets shown in Table 6.

With these six sets the following operation table (Table 7) may be constructed. As before, the operation is o. Table 7 is to be read in the same manner as Table 4. What it reveals is that any element from V3

68

Table 5

U, U2

U, U1 U2

U2 Ul U2

Table 6

V, =

[I,A,B,C V2 = [D,E2,G,L2 1

V3 = fD2,E,G2,LI V4 =

fQ1,Q6,Q7,Q121 V5 =Q3Q5,Q8,Q10 V6 =

fQ2,Q4,Q9,Q11J

Table 7

VI V2 V3 V4 V5 V6

VI VI V V3 V4 V4 V5

V2 V2 V3 Vi V5 V6 V4

V3 V3 VI VV V4 V5

V4 V4 V V5 V1 V3V V2

V5 V5 V4 V6 V2 VI V3

V6 V6 V5 V4 V3 V2 VI

69

composed with any element from V4 yields some element from V5 (V3oV4=V5). Similarly, any element from V5 composed with any from V2 yields some element from V6 (V50V2=V6).

The relationship between Table 7 and Table 4 may be more clearly understood if the rows and columns of Table 4 are rearranged so as to correspond visually to the groupings found in Table 7 (see Table 8).

Unfortunately, space does not permit detailed discussion of just how these sets were derived.2 For the purposes of this paper suffice it to say that they afford a somewhat simpler, more compact view of the original group structure. In a sense, they summarize certain relations found in that original structure.

Having completed this brief survey of mathematical groups the analysis may begin. First, with the help of Tables 5 and 7 two paths through the group are determined. Xenakis moves through the group tables in the following manner: first he chooses two elements with which to begin and orders them. Let us say that Q1 and Q9 will be the first and second elements respectively. He then composes Q1 with Q9 (Ql oQ9). The result is D2. He then composes Q9 with D2 (Q90oD2). The result is Q8. He next composes D2 with Q8 (D2oQ8) and so forth. This motion is reminiscent of that used in generating a Fibonacci series in that each succeeding element is fashioned from its two immediate predecessors and from this point on it will be referred to as a Fibonacci motion.

Selecting from the six sets of subgroup 2 for both the first and second sets, there are thirty-six possible Fibonacci motions. Each of these is cyclic. For example, starting with V2 and V4, the following sequence results:

V2 o V4V6 0 V3 o V5 o V6 O V2 o V4 ...

The sequence starts over with the second occurrence of V2. Of the thirty-six possibilities the composer chooses two:

V2 o V4 o V6 o V3 o V5 o V6 o (V2 o V4 ...) =path V2 o V4 and

V2 o V5 o V4 o V3 o V6 o V4 o (V2 o V5 ...)= path V2 o V5 Since each element Vi contains four elements from the original

group, further refinements in these two paths are necessary. Path V2oV4 may be expressed in sixteen possible ways:

V2 = (D,E2,G,L23 V4 =

Q12,Q7,Q1,Q6} The sixteen possibilities are illustrated in Table 9. All sixteen are, of course, also cyclic though they generate only eight different cycles,

70

Table 8

e-- V-,--- V3 W V+3 --- Vs -1-- V(-

F A 13 C o ~~L QL 'LO 1QO SI 1 8C D Ea -1 G I A L)""f 1 Q- Q4Q C9zQ O 4)

A, Af Cp ?GED LDE 7, 42 esj '1'4,qa ' q *

YC r

a Vc D 1

Di L

.

L

C2

E D ,A4,

Q,e~fc,!. Qv Q* ______ C . e A z fLaD E! I t cQQ , .Qs .

ca L 00 ,0, 9+9

, A 1 -2 E2 DA C L e f ?Ps9, Q e 4 O_ qr4 ? I

rl ? -L'C D L e D2 C' 8 z A C 9, 4, 4 4, G ,

,,

32 D D (2 L E L C'D-r c, ra _ _ ,, 9,, 9 9,

L Lc D c'Di E L c C, r, 10 co e44o

r e X : c rC 4 0 La E G ,, ,

-- Cis0 a .0 2 1 CDL Lc 8 A CE -D C

QO q 93Q

L3 C' E LD 4# C- C D L"'DFEs 9t Wa 2 , 9, It4,4, 4.

-L E 0E c a C E .4 4 B D 41 E , D2 A CEN Q7 c P0 3

4 9* 9184 3 A 9*',oK , 3oscD C. I 8 E

7 L7

L D

L 4' QA , r 8 1 C4 D1 CLVE8L A cC i1

r ~I~, cs ilc

u'.Q14 Q,4Q'.Q31 D _~ t DFrOBC6 0l

C1

I I 4s 4' , q ,A .2 , Q-o C Q, OIS 0 c,0L

F 4---.,

- D' - SqlQP 403a 4, bi E(L C 4 (L2I' 8 C E L ar Q3 4.0e; Q't E

1 c - 4

I

. , , 1, 0 V9 r c 9 _A( j L 44 L V

4t. , Q r

A * jr , p tL F4a E2'-L A G

D 8

c x 4

I ,. OI

C' D 1 C, : % L (D L D C L E A C - r B

71

Table 9

DoV4. = DoQ12 DoQ7 DoQ1 DoQ6

E20V4 = E20Q12 E20Q7 E20Q1 E20Q6

GoV4 = GoQ12 GoQ7 GoQ1 GoQ6

L20V4 = L2oQ12 L20Q7 L2oQ1 L2oQ6

72

four of length eighteen (illustrated in Table 10) and four of length six (illustrated in Table 11). The sixteen original paths reduce to only eight, since each of the four longer cycles can be generated from three of the sixteen pairs listed previously. For example, the first of the longer cycles can be generated from DoQ12, E2oQ7, or L20Q7:

D Q12 Q4 E Q8 Q2E2 Q7 Q4 D2 Q3 Q4 L2 Q7 Q2 L Q8 Qll

Similarly, path V20V5 generates sixteen paths of which eight are distinct-four long and four short.

From these possibilities the composer chooses the following: for path V2V4, D Q12 Q4 ... ; and, for path V2V5, D Q3 Q7 ... These are both paths of length eighteen; that is, they proceed through eighteen group elements before they cycle around. These two paths are presented simultaneously in the course of the piece and in one-to-one corres- pondence as shown in Table 12. Note that these eighteen stages correspond to the eighteen sections of Level I listed earlier. One immediately notices that the one-to-one superimposition of these two particular paths naturally divides the series into halves. This is the case as there are only two positions at which both paths reach the very same group elements and these are the first and tenth positions.

Next we note that each of the twenty-four different elements in the octahedral group can be represented as some permutation of eight numbers. Upon considering these permutations, previously presented in "Table 1," one notices several important facts. First, all the permutations associated with the first twelve elements, I through L2, hold numbers 1, 2, 3, and 4 in the first four positions and 5, 6, 7, and 8 in the second four. Each group of four numbers are scrambled among themselves throughout the twelve permutations but no element from the set (1,2,3,4) ever mixes with the set (5,6,7,8). Conversely, in all the permutations associated with the last twelve elements, Q1 through Q12, the numbers 1, 2, 3, and 4 are held in the last four positions and 5, 6, 7, and 8 are held in the first four. In othei words,

for I through L2 (1, 2, 3, 4) (5, 6, 7, 8) Q1 through Q12 (5, 6, 7, 8) (1, 2, 3, 4).

It is revealing to translate the two paths chosen above into the notation employed in constructing "subgroup" 1:

I,A . . . L2 =U1 and Q1,Q2..

--12 = U2

v2v4 = UIU2U2UIU2U2UIU2U2UIU2U2UIU2U2UIU2U2

V2V5 = U1UU2UIU2U2UIU2U2UIU2U2U1U2U2U1U2U2

73

4?cl P

Table 10

D Q12 Q4 E Q8 Q2 E2 Q7 Q4 D2 Q3 Q4 L2 Q7 Q2 L Q8 Q11

D Q7 Q1, E Q10 Q4 G Q7 Q9 G2 Q5 Q4 E2 Q12 Qll D2 Q5 Q9

D Q1 Q9 D2 Q8 Q4 L2 Q6 Q11 G2 Q8 Q9 G Q1 Q11 L Q3 Q4

E2 Q1 Q2 E Q5 Q11 L Q6 Q4 G Q5 Q9 G2 Q7 Q4 L2 Q3 Qll

V2 V4 V6 V3 V5 V6 (V2 V4 ..

-_A

Table 11

D Q6 Q2 D2 Q10 Q2

E2 Q6 Q3 E Q9 Q3

G Q12 Q3 G2 Q2 Q3

L2 Q12 Q10 L Q9 Q10

V2 V4 V6 V3 V5 V6

Table 12

Sections 1 2 3 5 6 7 9 10 11 13 14 15 17 18 19 21 22 23

V2oV4 D Q12 Q4 E Q8 Q2 E2 Q7 Q4 D2 Q3 Q11 L2 Q7 Q2 L Q8 Q11

V2oV5 D Q3 Q7 L Q11Q6 L2 Q5 Q7 D2 Q9 Q1 G Q5 Q7 G2 Q11 Q1

Clearly, every third element belongs to subgroup 1 and all others from its coset. As a result both paths proceed as in Table 13. This procedure naturally partitions or segments the path into sixths:

UIU2U2 UIU2U2 UIU2U2 UIU2U2 UIU2U2 UIU2U2 Next, it may be noted that within subgroup 1, letters I,A,B and C are

their own inverses while the inverses of D,E,G and L are D2,E2,G2 and L2 respectively. Thus, each of the two chosen paths may be articulated through these inverses in the following way:

DQQ EQQ E2QQ D2QQ L2QQ LQQ

This fact divides the path into two halves. Each half consists of three sets of group elements with three elements in each set:

123 123 123 123 123 123 1 2 3 1 2 3

After this brief discussion of the structure of each path we move on to the specific sonic materials by which they are articulated. V2oV4 controls the transformation of sound gestures and V2oV5 controls the transformation of various internal characteristics of those gestures. In particular path V2oV4 is articulated sonically in the following way. Four different gestures are used and will be represented graphically in the manner employed by the composer in his own discussion of the piece3 (see Table 14).

Each of these gestures may be expressed in two different ways: legato or staccato. Legato statements are generally played arco or arco tremolo while staccato statements are played pizzicato or col legno battuto. These are depicted graphically in Table 14.

These eight gestures are associated with the eight numbers of the permutations in three different ways, or mappings. These three mappings are labelled a, 0 and 7, and are listed in Table 15. Clearly, in a the four staccato textures are associated with the set (1,2,3,4) and the four legato textures with the set (5,6,7,8) and so will never mix. As a result of this mapping the partitioning of the paths into groups of three will be clearly articulated.

A Q Q (1,2,3,4) (5,6,7,8) (5,6,7,8) (1,2,3,4) (5,6,7,8) (1,2,3,4) (staccato) (legato) (legato) (staccato) (legato) (staccato) Tables 16, 17, and 18 illustrate these points in more detail using the first three sections of this piece as examples (Ex. 1).

Similarly, mappings 3 and y are so arranged as to give to each half of the series three of one type of gesture and one of the other. As a result even in 0 and 7 one type of texture, either legato or staccato,

76

Table 13

U1 (1,2,3,4) (5,6,7,8)

U2 (5,6,7,8) (1,2,3,4)

U2 (5,6,7,8) , (1,2,3,4)

U1 (1,2,3,4) (5,6,7,8)

U2 (5,6,7,8) (1,2,3,4)

U2 (5,6,7,8) (1,2,3,4)

Table 14

Gesture a. Legato b. Staccato

1. An ordered collection la lb of sounds, neither ascending nor descending

2. An ordered collection 2a 2b of sounds, either ascending or descending

3. A disordered collection 3a 3b of sounds, ascending or descending

4. Sustained sounds with 4a 4b aberrations

Table 15

1 2 3 4 5 6 7 8

a 2b lb 3b 4b 2a la 3a 4a

3 3a la 2a 3b lb 4a 4b 2b

-y 4b 3b lb la 4a 3a 2b 2a

77

Table 16. First Section

D: S2 S3 S1

S4

o-- - - - - -

s6 S7 S5 S8

Table 17. Second Section

Q12: S5 S6 S8 S7

S.

S2 S4 S3 S1S2 S4 S3

Table 18. Third Section

Q4: S6 S7 S8 S5

S2. .S. .

S3 S4 -----------S

S2 S3 S4 si

78

S2S3 Si S

$ 2

,) I$ 3

$ 17I'

I $

4 -

S6 S

S8

A 1

---1

1 "

_

56$ S8$ S7

AAgo" t_---

-- 19$1 I2 F1 HUlC1

".- - • u . -- - ; -_ : - +, ; ,,_ + ? -i - - - -- -- - : . . . .

Example 1. Nomos Alpha (beginning)

... Reprinted from Formalized Music by Iannis Xenakis, Indiana University Press, 1971. Used by permission.

00 0

S S3 S6S 11

- S.

S5 S2 . S3

I.A -....J r, ...............o.....

S4 S1

- - . . .

Example 1 (continued)

still clearly predominates within each half. In 0 (1,2,3,4) contains three legato gestures while (5,6,7,8) contains three staccato. In y (1,2,3,4) contains three staccato gestures while (5,6,7,8) contains three legato. Thus, each of these three mappings helps to articulate the division of eight numbers into two sets of four.

Finally, a, 0 and 7 are associated with path V2oV4 in the following way:

a ( a ( DQ12Q4 EQ8Q2 E2Q7Q4 D2Q3Q11 L2Q7Q2 LQ8Q11

Clearly, this particular association further articulates the natural parti- tion of the series into sixths and halves as Xenakis changes his mapping after every third section and then repeats his entire series of mappings after the ninth section, the midpoint, is reached.

Path V2oV5 is articulated sonically in the following way. The three parameters of density (D), volume (G), and duration (U) are each assigned four possible values, represented by subscripts 1, 2, 3, and 4. Thus, set D consists of values d1, d2, d3, d4, and so forth for sets G and U. From the triple product of these sets the eight points shown in Table 19 are chosen. In each of these eight, g and u always share the same subscript. From Table 19 it may be seen that the first four points, K1 through K4, are formed from all the possible combinations of the subscripts 1 and 4:

1 1 14 44 4 1

and the second four points are formed from all the possible combina- tions of 2 and 3:

22 23 33 32

Thus, by choosing these eight particular points the composer ensures that the bipartite structure of the permutations (1,2,3,4) (5,6,7,8) will also be articulated sonically in path V2oV5.

In the set D (density) three different sets of values are actually employed. These will also be called a, 0, and 7 (Table 20). The four elements in y are formed by steady increments of .5 and those of p by steady increments of 1. The four elements of a are formed exponen- tially. Therefore, we actually have three sets of eight Ki's for path V2 o V5 (Table 21).

The values for G and U are shown in Table 22. 81

Table 19

K1 = dlglU1 = d1 (gu)l K5 = d2g2u2 = d2 (gu) 2

K2 =d1g4u4 = d (gu)4 K6 = d2g3u3

= d2 (gu)3

K3 = d4g4u4 = d4 (gu)4 K7 = d3g3u3

= d3 (gu)3

K4 = d4glu1 = d4 (gu)1 K8 = d3g2u2 = d3 (gu)2

Table 20

Set D

a f y

d1 .5 1 1

d2 1.08 2 1.5

d3 2.32 3 2

d4 5 4 2.5

(Numbers represent attacks per second)

Table 21

Kla = d glU K2a = d g4u4

K1 = d1 glUl K2 = d g4u4

Klr = dl glUl K2 = dl g4u4

Table 22

G U

gl - mf U1

- 2"

g2 - f U2 - 3" g3 - ff u3 - 4"

g4 - fff u4 - 5" 82

The eight points Ki are associated with the eight numbers which are, in turn, associated with each element of the group: K1 = 1, K2 = 2. ..

The three mappings a, 1 and y are associated with path V2oV5 in the following way:

a (3 a ( y DQ3Q7

LQllQ6 L2Q5Q7 D2Q9Q1 GQ5Q7 G2Q11Q1

Once again, the particular association employed articulates the natural partition of the series into sixths and halves.

Table 23 summarizes the analysis presented thus far. Here all sets are unordered. Also, numerical values represent total density per event; that is, density per second multiplied by the number of seconds.

One final observation with respect to level I should be considered. The durations of the eighteen sections reveal the scheme presented in Table 24. This scheme also suggests a division of the entire piece into halves, each of which is characterized by a temporal expansion.

The remaining six sections of the piece which constitute level II are striking in their apparent dissimilarity with those of level I. These sections-4, 8, 12, 16, 20, and 24-are not based on a group structure. Rather, they are based on one continuous motion of registral evolution.

First of all it should be noted that these six sections are given both a different dynamic range and a different registral space than the sections of level I. In particular, level I uses registers in the middle range of the cello and closely related dynamic levels (mf, f, ff, and fff) while level II focuses on the extremities of the instrument's range and striking dynamic contrasts (pppp, pp, ff, and fff).

The scheme of durations for the six sections is shown in Table 25. This scheme suggests a downward-upward motion similar to that heard on level I, sections 1-11. But, most significantly, the piece, once again, seems divided into two halves.

As stated earlier, however, the primary structural evolution deter- mined over the course of these six sections is registral. The gestures used are very simple-long sustained tones and glissandi. The former is used to establish registral plateaus and the latter to shift from one of these plateaus to another. Within each section all of the gestures employed are completely fused to form one or more broad sweeping registral motions. For example, in section 8 there are two such continuous motions, each moving in contrary motion to the other. This section, in particular prepares the two expansive gestures, also in contrary motion, found at the very end of the piece.

The overall registral motion of level II is described in Example 2.4 Sections 4, 8, and 12 create one large registral descent from d+5 to CC

83

Table 23

Sections 1 path V2oV4 (bl b2 b3 b4) ~(a a2 a3 a4) 2 (al a2 a3 a4)

(b1 b2 b3 b4)

3 (a1 a2 a3 a4) (bl b3 b4)

1 path V2oV5

mf fff ] [f fff

2.25 22.5 1 10 3.72 7.98 2.83 6.08

2 f ff mf fff 3.72 7.98 2.83 6.08 2.25 22.5 1 10

3 f ffmf fff 3.72 7.98 2.83 6.08 2.25 22.5 1 10

5 path V2oV4 (a, a2 a3 b3) (bl b2 b4 a4) 6 (bl b2 b4 a4) (a1 a2 a3 b3) 7

(b1 b2 b4 a4) , (a1 a2 a3 b3)

5 pathV2oV5 mf fff /f

ff 2 14 8 4.5 .5.24 10.32 7.86 6.88

6 f ff 1mf ff 5.24 10.32 7.86 6.88 ([2 14 8 4.5 7

f ff mf fff

L 5.24 10.32 7.86 6.88 ] 2m 14 8 4.5

9 path V2oV4 (bl b3 b4 a,) (a2 a3 a4 b2) 10 (a2 a3 a4 b2) (b1 b3 b4 a,) 11 (a2 a3 aa4 b2) (bl b3 b4 al)

9 path V2oV5 mf fff 1 ff 2 5 4.5 11.25 1.3.93 5.24 5.15 6.88 J

10 [ f ff [mf fff 3.93 5.24 5.15 6.88 1] [2 5 4.5 11.25

11 f ff mf

fff 3.93 5.24 5.15 6.88 2m 5 4.5 11.25

Similarly for sections 13, 14, 15, 17, 18, 19, 21,22, and 23 84

Table 24

Sections 1 2 3 5 6 7 9 10 11 13 14 15 17 18 19 21 22 23

Durations 48 48 48 42 36 44 65 66 62 48 49 51 46 46 49 62 70 54 in seconds

Average 48 40.6 64 49.3 47 62 for every group of three sections

64 62

49.3 48

47 40.6

00oo tA

Table 25

Sections 4 8 12 16 20 24

Durations 60.8 44.8 29.6 44.8 60.8 70.4 in seconds

70.4 60.8 0.8

44.8 44.8

29.6

-4 8-- -- 1 2 -

4 8 12 16 20 24

Example 2

86

with one contrary ascending motion in section 8 beginning on CC, the goal of the descent. This secondary motion serves at once as a prepara- tion for and foil to the final descent. In the second half, sections 16 and 20 focus, respectively, on the cello's high and low extremes. However, in the first half these extremes are smoothly connected to one another as the end points of one sound continuum. In the second half the con- nection between them is severed as each is presented as a separate plateau, drawing attention to the two registers in their isolation from one another. Significantly, section 16 employs no glissandi and section 20 only one. This is the case since glissandi were the means through which the different plateaus were connected to one another and such connection is not desired here. The final section of the second half, and of the entire piece, section 24, once again unites the registral extremes and so recaptures the motion that integrated sections 4, 8 and 12. Section 24 begins on a unison gl in the middle of the cello's range and proceeds to travel three octaves and a fifth in both directions simul- taneously, recapturing both the highest and the lowest points of the piece, the d+5 reached at the beginning of section 4 and the CC reached at the end of section 8. Section 24 summarizes the entire registral motion of the piece on level II and once again unites the two registral plateaus isolated in sections 16 and 20. This is summarized in Figure 1.

Before concluding, some comments concerning the pitch structure of the composition might be appropriate. However, any detailed dis- cussion of the pitch structure of Nomos Alpha involves an explication of a branch of theory introduced by Xenakis and called by him Sieve Theory. Unfortunately, space does not permit a detailed explanation of this theory which is necessary background for any study of this aspect of the composition.

For the purposes of this paper suffice it to say that a series of twelve interrelated pitch collections are used in the piece. These are presented in Table 26. From just this much information it may be observed that the assignment of pitch collections to sections serves both to further articulate the partitioning of the sequence of eighteen sections on level I into groups of three, and, to intensify the separation between the two levels. The former is accomplished through the use of a single pitch collection for every three sections of level I. The latter is accomplished simply through the introduction of a new collection only at those moments when the composer shifts from one level to the other and at every such moment without exception.

In comparing his work to that of his younger colleague, Milton Babbitt, Edgard Var6se once observed: "It seems to me that he wants to exercise maximum control over certain materials, as if he were above them. But I want to be in the material, part of the acoustical vibration,

87

Sections: 4 8 12 16 20 24

I-

Figure 1

88

00

Table 26

Level I - 1 2 3 5 6 7 9 10 11 13 14 15 17 18 19 21 22 23 Level II- 4 8 12 16 20 24

Pitch 1 2 3 4 5 6 7 8 9 10 11 12 collections

so to speak."5 Through the juxtaposition of its two structural levels Xenakis's Nomos Alpha, within its single framework, explores precisely the same differences of structural morphology as those rather poetically formulated by Varese.

Nomos Alpha defines and explores a profound dialectic concerning the relationship of a structure to the materials by which that structure is engendered. The two levels employed in the composition are fundamentally different in their morphology. One presents a structure borrowed from mathematics and determined apart from any interaction with the sonic materials employed. The other, in contrast, is a direct and organic outgrowth of the composer's manipulation of his sound materials.

With respect to level I, a structure is created prior to and indepen- dent of the choice of specific sonic materials employed. More precisely, as may be clear from the analysis presented above, the materials are chosen and shaped only that they may clearly articulate various aspects of the mathematical group structure from which the sonic structure is derived.

Specifically, one permutation group is applied to a variety of vastly different types of materials. The two paths employed are not only structurally identical but, in fact, share many elements in common. They control, independently, on the one hand, the parameters of volume, duration, and density, and on the other hand the nature and timbral quality of each gesture.

In many works the processes of transformation which guide the structure's evolution are in some way engendered by the specific materials employed. As such, their unfolding is tied in very specific ways to the qualities and characteristics of those materials. On level I of Nomos Alpha, in contrast, these processes (the permutations schemes) are broad and inclusive and are capable, therefore, of operating, as they do, on a wide range of dissimilar materials. 6 As such, the structure of level I is suggestive of certain qualities of abstraction which are quite in keeping with the abstract nature of the mathematical structures themselves.

On level II, however, one finds a striking contrast to this composi- tional approach. The structure of this part of the work is as rooted in its sonic materials as that of level I is independent of them. Here form emerges from the direct manipulation by the composer of his materials. Indeed, on level II form is identified with those very processes by which the materials are shaped and transformed. This is strikingly dissimilar to the abstract, precompositional constructs employed in the other sections of the piece. The fashioning of level II, therefore, involves not so much the shaping of materials to fit some predetermined mold as, rather, the unfolding of a continuous evolutionary process in which the nature of the materials themselves plays a crucial formative role. 90

It is interesting to note that the structure of level I is an example of a discrete structure while that of level II is an essentially continuous one. On level I eight distinct elements are defined for each parameter and then used to articulate the permutation schemes. In contrast, on level II the sonic elements seem tied to one another within an unbroken temporal/spatial continuum. In fact, with respect to the structure of level II it would seem more correct to speak of a single continuously evolving shape than a collection of discrete elements united in some atomistic manner. Indeed, the two rather disparate notions of discrete and continuous structure seem apt metaphors with which to consider the essential dialectical thrust of the piece.

The form of Nomos Alpha, then, would appear to have as its source the juxtaposition of two radically different methodologies. From their interaction a magnificent dialogue evolves and over the course of this dialogue some of the deepest issues concerning the evolution of struc- ture are addressed and illuminated.

R/

91

Appendix

For those familiar with group theory it may be interesting to recall that the twenty-four permutations employed as group elements on level I of Nomos Alpha can be derived from the twenty-four isometries of the cube as, indeed, Xenakis's writings suggest he has done. The following chart (Fig. 2) is borrowed from Formalized Music and depicts all twenty-four possible rotations of a cube. Labellings are the same as those used in this analysis. Thus, by virtue of the position in which the eight end points of the cube are found after it has moved about in space, a rotation with a particular label is associated with that permutation bearing the same label.

It is also fascinating, in light of this analysis, to recall certain relationships between the cube and the tetrahedron. The natural partitioning of each eight element permutation into two groups of four elements (see above, pages 73 and 76) has its roots in the fact that the tetrahedron may be inscribed within the cube in only two different ways (see Fig. 3, a & b). This, of course, suggests that as the cube is rotated in space certain adjacencies are preserved. Given the labelling employed above, if the cube is rotated in any of the following twelve ways, I, A, B, C, D, D2, E, E2, G, G2, L, L2, all the points labelled 1, 2, 3, or 4 will only exchange positions among themselves. Similarly, those labelled 5, 6, 7, or 8 will also only exchange positions among themselves.

In contrast, if the cube is rotated in any of the remaining twelve ways, Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q9, Q10, Q11, Q12 all the points labelled 1, 2, 3, or 4 will exchange positions with those labelled 5, 6, 7, or 8 and, of course, vice versa. Thus, some of the most important properties of the twenty-four permutations explored by the composer in Level I of Nomos Alpha have as their source certain geometric properties of the cube.

92

toct

A N

...

.~-5

i.. el iCIO

..

C• t -

INN k~ (3

'I)

.Y -. r, I•

•

....." . ,

I"

r Pn

I-,

Reprinted from Formalized Music by lannis Xenakis, Indiana University Press, 1971. Used by permission.

93

Figure 3

94

NOTES

1. lannis Xenakis, Formalized Music (Bloomington, Indiana: Indiana University Press, 1971). pp. 219-236. Whenever possible, the composers own charts have been employed in this study. I am grateful to the Indiana University Press for their permission to reprint material from Formalized Music.

2. Those readers with some background in mathematics will immediately recognize that subgroups 1 and 2 are both normal; that sets Ui (i=l1, 2) and

Vj (j=1, 2... 6) are the cosets of these subgroups; and that Tables 5 and 7 are, respectively, the quotient groups for subgroups 1 and 2.

3. Ibid., pp. 226-227. 4. All tones below C shown in Example 2 are produced by scordatura, with the

C string of the cello tuned down one octave to CC. 5. Edgard Varese, "Conversation with Varese" (Varese in conversation with

Gunther Schuller), Perspectives on American Composers, ed. by Benjamin Boretz and Edward Cone (New York: W. W. Norton, 1971), p. 38.

6. Indeed, this very same structure of permutations has been used by the

composer in a very different piece entitled Nomos Gamma for large orchestra

(see Formalized Music, pp. 236-241). In this work the same group is applied to an even broader range of materials and textures.

3

95