RESEARCH NOTES 339

cyclical; duration and intensity are linear (and in this piece, they are ordered according to the ORMAP); and articulation is neither cyclical nor linear. The four elements correspond to a

single series, so, for example, the third entry in the series corre­ Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 A Tesseract in Boulez's Structures la sponds to order position 2, PC 9, a dotted-sixteenth duration, pianissimo, and staccato. Therefore, a twelve-tone matrix con­ PAUL LOMBARDI AND MICHAEL WESTER structed from this series can be populated with PCs, durations, dynamics, or articulations. Keywords: Boulez, Structures la, series, total serialism, Consider the twelve-tone matrices shown in Example 2. twelve-tone, matrix, cube, tesseract, hypercube, dimension, The T -matrix at the top of Example 2 has the prime form of symmetry, antisymmetry. the series in the top row, and the inverted form of the series in the left column. The I -matrix in the lower left of Example 2 has the prime form of the series in both the top row and left column, and likewise, the I -matrix in the lower right of In this article, we examine the multidimensionality that satu­ Example 2 has the inverted form of the series in both the top rates the serial design of Boulez's Structures 1 a (1952) . To do row and left column. For the duration of this paper, we will this, we will find it convenient to combine the various order­ refer to these matrices by indicating their dimensions as P ings of the series given in the piece into a single multidimen­ and!or I: the T -matrix will be referred to as P X I, the I -ma­ sional serial object that will take on the aspect of a hypercube. trix generated from P will be referred to as P X P, and the 1- A hypercube is the multidimensional analog of a cube. The matrix generated from I will be referred to as IxI. four-dimensional hypercube, commonly known as a tesseract, The P xl matrix contains all 48 P, I, Rand RI forms of is created by sweeping a cube perpendicular to its three orthog­ the series, while each of the P x P and I x I matrices contains onal axes, just as a cube is created by sweeping a square perpen­ half of those same forms. At first, the P X P and I xl matrices dicular to its two orthogonal axes or a square is created by may seem redundant because their rows and columns are du­ sweeping a line segment in a direction perpendicular to its axis. plicated in the PxI matrix. However, the orders in which the The four elements serialized in Structures la are pitch class rows are presented are different, and these orders are perti­ (PC), duration, intensity, and articulation (Ligeti 1960, 38-44; nent to the design of Structures la. For example, sometimes Brindle 1987, 56-57; DeYoung 1977-78, 27; Dallin 1974, the transpositions of the P forms of the series are presented in 213-15). Each of these four elements corresponds to a single the order of the rows from top to bottom of the P xl matrix. twelve-tone series as shown in Example 1. The order positions Other times, the transpositions of the P forms of the series are (known as the ORMAP) of the prime form of the series are la­ presented in the order of the rows from top to bottom of the P beled at the top of the table. Both the order positions and PCs x P matrix. These two orderings will be referred to as P rows in are given as the hexadecimal numbers 0-9, and A and B for 10 the order of I, and P rows in the order of P, respectively. and 11. The durations are given as the Arabic numbers 1-12, Four dimensions are required to combine the orderings of and directly correspond to the number of thirty-second notes the forms of the series as they occur in Structures la into a in the durations. The dynamics and articulations are conven­ single serial object. As shown in Example 3, a twelve-tone tionally notated. The articulations at order positions 3 and 9 series has one dimension. A twelve-tone matrix has two are crossed out because they are not used in Structures la. PC is dimensions, where each PC in the second dimension spawns 340 MUSIC THEORY SPECTRUM 30 (2008)

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

PC 3 2 9 8 7 6 4 1 0 A 5 B Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021

1 2 3 4 5 6 7 8 9 10 11 12 Duration ~ j1 )j j) ij Ji Ji. j j) j~ j,h J quasi quasi Intensity pppp ppp PI' P P mp mf f f f.f .Iff f.ff.f

Articulation > ":- norm .. '1ft ~ .,- ~ X A X

EXAMPLE 1. The series .from Structures la

a transposition of the original series. Note that the result is a elements shown in Example 1 because these elements all twelve-tone matrix rather than just any serial matrix in that correspond to the same series. every row and column is a TTO (twelve-tone operation) of The lower right quadrant of Example 3 indicates a tesseract, the generating twelve-tone series. A cube, given in the lower which has four dimensions and does not exist in physical 3D left quadrant of Example 3, has three dimensions, where space; however, it can be represented as a series of transposed each PC in the third dimension spawns a transposition of cubes along the fourth dimension. Just like the twelve-tone the previous matrix (the 3D axis points out the back of the cube, the tesseract is twelve-tone as long as all four axes are page). The 2D slices of the cube along the third axis are or­ TTOs of each other. In Example 3 and subsequently, we use a dered transpositions of the previous matrix. Remember that right-handed system to consistently represent the dimensions in twelve-tone matrices, the transpositions of the generating for cubes and hypercubes as follows: the right-pointing hori­ series are in the order of the Oth column, and the Oth column wntal axis for dimension one, the down-pointing vertical axis is a form of the generating series. Any TTO of the generat­ for dimension two, and the axis pointing out the back of the ing series, including multiplication, can be combined with page (away from the reader) for dimension three.1 the generating series to produce some kind of twelve-tone As we saw in Example 2, the PxP matrix requires two P matrix. Therefore, a cube is twelve-tone as long as the or­ axes, the I xl matrix requires two I axes, and the P xl ma­ dered transpositions of the twelve-tone matrices along the trix requires one P axis and one I axis. All three of these third axis are based on a TTO of the generating series. matrices can be combined into a single tesseract. Let's start Otherwise, the cube is merely serial and not twelve-tone. twelve-tone cubes, every row and column across any In A right-handed system can be determined as follows: point the fingers pair of sub-dimensions is a form of the generating twelve­ of one's right hand in the direction of the first axis, and then curl the tone series. Even though this twelve-tone cube is based on fingers around to point in the direction of the second axis; the thumb PC structure, it can be populated with the other serialized will then extend in the direction the third axis should point. RESEARCH NOTES 341

T-Matrix n (Px1) 3 298 764 1 o A 5 B

4 3 A 9 875 2 1 B 6 0 Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 9 832 lOA 7 6 4 B 5 A 9 4 3 2 1 B 8 7 5 0 6 BAS 4 320 9 8 6 1 7 P o B 6 5 4 3 1 A 9 728 R ..... 2 1 8 7 6 5 3 0 B 9 4 A 5 4 B A 9 8 6 3 207 1 - 6 SOB A 9 743 1 8 2 8 7 2 lOB 9 6 5 3 A 4 1 076 5 4 2 B A 8 3 9 7 6 lOB A 8 5 4 2 9 3 RIi I-Matrices P! (P X P) 3 2 9 8 7 6 4 lOA 5 B (Ix 1) 3 4 9 ABO 2 5 6 8 1 7 2 1 8 765 3 0 B 9 4 A 4 5 ABO 1 3 6 792 8 9 8 3 2 lOA 7 6 4 B 5 9 A 3 4 5 6 8 B 0 2 7 1 8 7 2 .1 0 B 9 6 5 3 A 4 A B 4 5 679 0 1 3 8 2 7 6 lOB A 8 5 4 2 9 3 B 0 5 6 7 8 A 1 2 4 9 3 p650BA9743182R I 0 1 6 7 8 9 B 2 3 5 A 4 RI ..... 43A987521B60_ ..... 2389AB145706 - 1 076 5 4 2 B A 8 3 9 5 6 B 0 1 2 4 7 8 A 3 9 o B 6 5 4 3 1 A 9 728 670 1 235 8 9 B 4 A A 9 432 1 B 8 7 5 0 6 892 3 4 5 7 A B 1 6 0 5 4 B A 9 8 6 3 2 0 7 1 1 2 7 8 9 A 0 3 4 6 B 5 BAS 4 3 2 0 9 8 6 1 7 7 8 1 2 3 4 6 9 A 0 5 B Ri RIi

EXAMPLE 2. The matrices for Structures 1a 342 MUSIC THEORY SPECTRUM 30 (2008)

1 Dimension 2 Dimensions (series) (matrix) Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 1D "'- / 1D ) 2D

'\V

3 Dimensions 4 Dimensions (cube) (tesseract)

3D 4D \ , "'- 1D \ , , / "'- \ \ , \ , , / \ , , \ \ , , \ \ , , \ \ , , 2D r- r- r- r- etc. '\ /

EXAMPLE 3. Four dimensions in serialism

with the P xl matrix given in Example 4. Either a P or I axis points down, all axes originating from the common axis can be combined with the P x I matrix or its matrix origin of the front upper left corner of the cube. In the transpose (I X P) to produce a P X I x P cube or an I x P x I I x P X I cube, transpositions of the P X I matrix on the cube.2 In the P xlxP cube, transpositions of the P xl ma­ left face occur in slices along i, transpositions of the I X P trix on the front face occur in slices along k, transpositions matrix on the front face occur in slices along k, and of the I X P matrix on the left face occur in slices along i, transpositions of the IxI matrix on the top face occur in and transpositions of the P X P matrix on the top face occur slices along j. The P X I X P cube, where the prime series in slices along j. Note that we always describe a face ori­ is taken from Example 1, is given in ExampleS with the ented so that the first axis points to the right and the second visible surfaces populated with PCs. The three dimen­ sions are labeled in order with the coordinates ijk. In Example 6, the coordinate [ijk] = [563], which corre­ 2 A matrix transpose is different from the transposition TTO in that it swaps rows with columns and vice versa. Mathematically, this can be sponds to the PC A in the cube of Example 5, is located written ml = mji where mij designates the elements of the matrix M. as depicted. RESEARCH NOTES 343

PxIMatrix P i Standard "/ Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 twelve-tone matrix I (PxI) "-V J

PxIxPCube IxPxICube k k P (PxP) I (Ix I) Px III x Pmatrix I" . P "/1 . /1 with an added PII axis (Ix P) (PxI) (PxI) (Ix P) I P "-V "- / J J

The above cubes with an Px Ix Px ITesseract, etc. added axis

EXAMPLE 4. Four dimensions in Boulez's Structures 1a

We can add an additional axis to either of the two cubes chosen, producing axes in different orders; however, the shown near the bottom of Example 4 to produce a tesseract. same row forms and matrices would still be represented in For example, a P x I x P x I tesseract can be formed by ap­ the sub-dimensional cubes if the same numbers of P and I pending an I axis to the P x I x P cube, or prepending a P axes are selected. axis to the IxPxI cube. Transpositions of both the PxIxP Given Bouleis generating twelve-tone series S, provided in and I x P x I cubes occur along the four axes of the tesseract. Example 7(a), each of these serial objects can be represented In general, different combinations of dimensions could be mathematically. The inverted form of the series beginning 344 MUSIC THEORY SPECTRUM 30 (2008) Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 ..At P

6 7 7 6 I ~ J

EXAMPLE 5. A PxlxP cube

with the same PC is determined by Ii' Example 7(b) gives one; j corresponds to dimension two. Using similar meth­ the equation for the P xl twelve-tone matrix. A proof of the ods, we can derive formulae for the I x P, P X P and I X I ma­ formula appears in Appendix 1. The PI superscript indicates trices. These also appear in Example 7(b). Each formula has the corresponding dimensions. (Whenever possible, we sim­ been adjusted by transposition to begin with the same plifY the superscript by using PI instead of P X I; sometimes PC. Example 7( c) provides the formulae for P X I X P and the x is included for clarification.) The ij subscript indicates I X P X I cubes. The formulae are derived by setting each the coordinates within the matrix: i corresponds to dimension index (i, j, k) in turn to zero and comparing the result with RESEARCH NOTES 345

(a) Twelve-tone series .At P S = (so, 51' ... , sB)

= (3, 2,9,8, 7, 6, 4, 1,0, A, 5, B) Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021

Ii = (250 - 5;) mod 12

= (3,4,9, A, B, 0, 2, 5, 6, 8, 1, 7) (b) Twelve-tone matrices

MJI = (Si-Sj+SO) mod 12 = M!Py (-5 t + 5·+J so) mod 12 I Mt = (si + ~- so) mod 12 ! M II = (-5· - s· + 35 ) mod 12 1] t J 0 J

(c) Twelve-tone cubes EXAMPLE 6. Coordinate [563J in the P X I X P cube CrIP = (si - Sj + sk) mod 12

CJfI = (-Si + sr Sk + 250) mod 12 the matrix equations, as we can see when we consult Example 7(b) and Example 8. Example 7(d) defines the (d) Twelve-tone tesseract P X I X P X I tesseract. Given the series and any [ijkl] coordinate in the tesseract, HtIr =(Si-Sj+Sk-S/+SO) mod 12 Example 7(d) determines the PC at that coordinate. In Example 9(a), for instance, the PC at coordinate [91.1\4] is 9. EXAMPLE 7. The formulae for the series, matrices, cubes and Setting a single coordinate in the tesseract to zero gives a 3D tesseract in Structures la cube. For instance, as shown in Example 9(b), setting the I index to zero gives the P X I X P cube, which is a 3D slice of the tesseract. Setting two of the coordinates in the tesseract to zero gives a 2D matrix. In Example 9(c), we see that set­ to indicate order positions B-O. So, inspecting Example ting either the kl or the jk indices to zero, for instance, 9(e), we see that the coordinates [~OOO] or [OO~O] indicate gives the P X I matrix. Incrementing one of the zero coordi­ the prime form of the series. Given the P series [~OOO], in­ nates transposes the sub-dimensional cubes or matrices. In crementing either the j or I indices transposes P in the order Example 9(d), incrementing the I index transposes the sub­ of I (Example 9[ f]), and incrementing the k index transposes dimensional P xl matrix in the order of P. P in the order of P (Example 9[gJ). A double arrow (=» in­ We will use a right arrow ( ~) to indicate an entire row of dicates a row of order positions O-B for each order position order positions O-B, and a left arrow (f-) for its retrograde of a single arrow. So, for example, the coordinates [=>~OO] MUSIC THEORY SPECTRUM 30 (2008)

(a) ct5f = MJf=(so-Sj+Sk) mod 12 s = (so' SI'···' sB) = (3, 2, 9, 8, 7, 6, 4, 1,0, A, 5, B) C/~f = M~P= (Si -so + Sk) mod 12 Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 CPIP = Ml'I=(s-s.+s)mod12 HtffI = (Si-Sj+Sk-S/+SO) mod 12 yO y ! J 0 HP!P! = (s9 - sl + SA - s4 + SO) mod 12 91A4 = (A - 2 + 5 - 7 + 3) mod 12 EXAMPLE 8. Boundary conditions for the P X I X P cube in Example 7(c) =9 H PIPI _ C l'IP (b) iikO - yk indicate ([~OOO], [~100], [~200], ... , [~BOO]), which is (e) H l'01P01 = M l'I 1j lJ equivalent to all of the P forms of the series in the order of lor otherwise the entire span of the P xl matrix (Example 9[h]). Hf'orcf = MtI HPIPI = T (Ml'I) For the remainder of this paper, we will use the leftmost in­ (d) ijlO (sl-s0)mod 12 y dices whenever possible. That is, while the P xl sub-dimen­ P!PI - (H PIPI H PIPI H PIPI H PIPI\ sional matrix occurs on both the ij and the il axes, we will use (e) H -->000 - 0000' 1000' 2000" .., BOOQl the ij axes because they are leftmost (Example 9[c]). = (3, 2, 9, 8, 7, 6, 4, 1,0, A, 5, B) As previously stated, the P xl matrix contains the Prows =s in the order of I and the I columns in the order of P, while the P X P matrix contains the P rows in the order of P and Htl~o = (Htl6l, Htlil, Htlfl,·· .,Htl~~) the I X I matrix contains the I rows in the order of 1. The tesseract contains all these scenarios, that is, the P and I rows =s in both the orders of P and I; and this accounts for the or­ (f) derings of the forms of the series as they progress in Structures la. Example 10 provides all the forms of the series in the piece for the PCs and durations (Brindle 1987, (g) 30-31). The piece has two parts, and the table shows the H PIPI - (HPIPI HPIP! HPIPI HPIPJ ) forms of the series for both parts and both pianos. (h) =>->00 - -->000' -->100' -->200' ... , -->BOO An I -matrix is symmetric with respect to its main diag­ = HPJP! onal (top-left, bottom-right) and so is equal to its matrix ijOO transpose. (Symmetry is depicted on the left half of = Ml'J y Example 11 and defined in Example 12[a].) That is to say, every PC in the upper-right triangle is mirrored in the lower-left. Antisymmetrical objects, shown in Example = Hfltf 12(b), are equal to the negative of their matrix transpose. T -matrices are almost antisymmetrical, but the antisym­ metry is transposed by the constant offset 2so. In trans­ posed antisymmetry, shown in the right half of Example 11 EXAMPLE 9. Examples from the tesseract RESEARCH NOTES 347

Part A PartB P series in the order of I Rl series in the order of Rl PC H PIPI H,PIPI Piano 1 ~--->oo 0<= Of-- Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 Rl series in the order of Rl I series in the order of R Duration H,PIPI PIPI 0<= Of-- H f--=>oo I series in the order of P R series in the order of R PC H PIPI H PIPI Piano 2 --->=>00 <=Of--O R series in the order of R P series in the order of Rl Duration PIPI PIPI H<=Of--O H =>f--OO

EXAMPLE ro. Forms ofthe series in Structures 1a

and defined in Example 12(c), every pair of PCs mirrored at T6I, which is the axis that passes through PC 3, the first around the main diagonal makes a symmetrical pair of PCs PC in the series. So, around the main diagonal, 3 mirrors with respect to the index 2so. For the remainder of this 3, 2 mirrors 4, 1 mirrors 5, etc. The symmetrical pairs are paper, we will simply refer to this transposed antisymmetry connected by curved lines in the PC wheels shown in as just anti symmetry. In Structures la, P and I are symmetrical Example 11.

(a) Symmetry

Symmetry Transposed antisymmetry M;i=Mij (Px Pand Ix I) (PxI) (b) Antisymmetry l'vfT =-M A@BOI2 -9 3- 8 4 M;;=-Mij 7 6 5 (c) Transposed antisymmetry

Mij = (si - S; + so) mod 12, so

O M;i = (Sj - Si + so) mod 12 A®BI 2 -9 3- 8 4 = (- (si - Sj + so) + 250) mod 12 7 6 5 =T2 So (-My .. )

EXAMPLE II. Symmetry vs. transposed antisymmetry EXAMPLE 12. Types ofsymmetry MUSIC THEORY SPECTRUM 30 (zo08)

(a) Generalized symmetry (PxIvs. IxP) its axes, and produces various reflections and rotations.4 By applying retrograde to a matrix axis, the type of symmetry MtI ='(Si-Sj+SO) mod 12, so does not change, but the origin and/or direction of the sym­ M;[ = (-.y + Si + so) mod 12 metry does. In other words, retrograding one or both axes Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 = (si - .y + so) mod 12 produces equivalent matrices. The resulting symmetries, = MtI however, are rotated or reflected in 3D space. For example, (b) Generalized anti symmetry (PxPvs. Ix!) P xl, when rotated in 3D space around a vertical axis that passes between columns 5 and 6, is equivalent to R x!. So, MtF = (si + Sj - so) mod 12, so when comparing symmetry within and among matrices, we M'j; = (-Sj - si + 350) mod 12 can ignore the retrograde operation because the types of symmetry will still exist; however, the symmetry will not = (- (si + Sj - so) + 250) mod 12 necessarily emanate from the origins of the matrices. = T2so (-Mr) The generalized symmetry and generalized antisymmetry that is inherent within the tesseract plays out in Structures 1 a. Each pair of matrices shown in Example 10 is identical, gen­ EXAMPLE 13. Generalized symmetry/antisymmetry between pairs of matrices eralized symmetrical, generalized antisymmetrical; or asym­ metrical. Example 16 provides all of the generalized sym­ These examples show symmetry and antisymmetry metrical and generalized antisymmetrical pairs of matrices within matrices. We can extend this discussion of types of contained within Structures la (the generalized symmetrical symmetry to pairs of matrices. Example 13 carries this out: P pairs are connected by solid lines and the generalized anti­ x I is generalized symmetrical to I x P, and P x P is general­ symmetrical pairs are connected by dashed lines). The bal­ ized antisymmetrical to I x!. There is no simple symmetry anced organization of these pairs of matrices shows Boulez's between PxI and PxP, or PxI and Ix!. methodical plan for the structure of the piece. Example 17 As we saw in Example 10, much of Structures la pro­ provides the derivation for the antisymmetry between the R gresses through retrograde forms of the series. Retrograde X Rand RI X RI matrices. Similar calculations can be per­ order further complicates the identification of types of sym­ formed for the remaining pairs of matrices.s metry among and between pairs of matrices. Given any ma­ Furthermore, the PxlxPcube is generalized antisymmet­ trix, one or both axes can be retrograded. As we see in rical with respect to the IxPxI cube. Example 18 carries the Examples 14 and 15, for the P xIT-matrix, the retrograde calculation out. Retrograde affects the generalized antisym­ process produces related members.3 Applying various retro­ metry in the cubes in a similar fashion as it does to matrices, grades to a matrix changes its origin and/or the direction of so the details will be omitted. As shown in Example 19, some

3 The four versions of the P x I matrix shown in Example 14 are a subset of the dihedral group for a four-sided polygon. Furthermore, we can 4 This is not rotation in the Stravinskian rotational-array sense, but produce a different tesseract than the one presented in this paper with rather the entire plane of the matrix is rotated in 3D space. P, I, Rand RI axes. Depending on the axes chosen, the eight 2D sub­ S The P x III x R and I x PIP x RI generalized antisymmetry requires a matrices of other tesseracts may produce a larger subset of the dihedral single flip through 3D space, while the I xR/P x RI generalized anti­ group for a four-sided polygon. symmetry requires two flips through 3D space. RESEARCH NOTES 349

(Px 1) (Rx1)

P R R P - +- - +- Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 n !I n !I

RIl 1 RI RIl 1 RI

+- +- -P R R- P

(Px RI) (Rx RI)

P R R P - +- - +- RI! !RI RI! !RI

If 1I If 1I

+- +- -P R R- P

EXAMPLE 14. Retrograde and the PxI matrix

of the matrices from Structures la can exist on different sur­ faces of a single cube. For example, the PxI, IxR and RIx RI are the three matrices in the Piano 1 part, and all three of Mtl = (si - Sj + so) mod 12 them exist on the surface of the IxPxI cube. The PxI ma­ Mlf = (s(B _ i) - Sj + so) mod 12 trix occurs on the left face emanating from the origin, the I x PXRI M # = (si - sIB _ j) + so) mod 12 R matrix occurs on the left face emanating from the front­ RXRI bottom-left corner, while the RI x RI matrix occurs on the M ii = ( sIB _ i) - SIB _ j) + so) mod 12 top face emanating from the back-top-right corner. All of the matrices in Piano 1 exist on the surface of the I x P x I EXAMPLE 15. Retrograde formulae for the P xl matrix 350 MUSIC THEORY SPECTRUM 30 (zo08)

Part A Part B

P series in the order of I RI series in the order of RI PC Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 PJPJ PJPl " , H=>---700 , H 0<=0..- \ , I ( , \ Piano 1 , \ , \ \ , I RI series in the order of RI , I series in the order of R Duration ,. , / PJPJ PJPJ H 0<=0..- \ , H..-=>00 I \ , I \ , I{ ,\ , I I ~I , I I series in the order of P , R series in the order of R / PC , \ ,. i\I PJPJ , PJPJ I H ---7=>00 H <=0..-0 I , \ , Piano 2 \ \ , I , \ , \ R series in the order of R , P series in the order of RI Duration " PJPJ PJPJ H <=0..-0 H J =>..-00

EXAMPLE 16. Generalized symmetry (solid lines) and generalized antisymmetry (dashed lines) between the pairs 0/ matrices in Structures 1 a

cube, while all of the matrices in Piano 2 exist on the P X I X symmetries inherent within the tesseract in one, two, three P cube. Consulting Example 20, we see that the first Piano's and, in combination, in four dimensions. part is generalized antisymmetrical with respect to the sec­ In addition to the PCs and durations, the dynamics and ond Piano's part. Thus, the structure of the piece reflects the articulations are also serialized. To determine dynamics and

= (S(B _ i) + S(B _ j) - so) mod 12, so P1P M/jR eijk = (Si - Sj + sk) mod 12, so MflxRl = (-S(B _ j) - S(B _ i) + 350) mod 12 CJfI = (-sk +.s;- Si + 250) mod 12 = (-(S(B _ i) + S(B _ j) - so) + 25 ) mod 12 0 = (- (si - Sj + Sk) + 250) mod 12 = T2 (-Mi{R) 50 Y = T2so (-err)

EXAMPLE I7- Derivation 0/ the generalized antisymmetry EXAMPLE 18. Generalized antisymmetry between the P xl X P between the R xR and RIxRI matrices and IxPxI cubes RESEARCH NOTES 351

RI with both dynamics and articulations; and the entries within ~ ____--,-+--,.\.RI the matrices that are not used in the piece are left blank. In (RIx RI) the tesseract, a diagonal can cut across two, three, or four di­

mensions, but all of the diagonals used in this piece occur Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 '" 1-+ Ip across two dimensions. Some of the diagonals used in the (Px I) ~ piece are symmetrical, while others are not. In Example 23, (IxR) the symmetrical diagonals are indicated with bidirectional arrows, and the nonsymmetrical diagonals are indicated with unidirectional arrows. The unidirectional arrows point in the direction that the musical elements progress in Structures 1 a. Since the musical elements may progress in either direction for the bidirectional arrows, for consistency, we will start at EXAMPLE 19. The PxI, IxR and RIxRI matrices the origin and progress counterclockwise around the matri­ on the surface of the IxPxI cube ces. The diagonals are combined to create four dynamic series and four articulation series, each oflength twelve. Example 24 articulations, Boulez uses diagonals of the P X P and I x I shows these series as they occur in the tesseract (the ijkl coor­ matrices, which are highlighted in Example 21 (Ligeti 1960, dinates in the tesseract are given in square brackets: [iOkO] 40, 44). These same diagonals are shown in Example 22 but for the PxP matrix, and [OjOi) for the IxI matrix). in conventional musical notation. The matrices are populated

I Part A Part B

PC P series in the order of I RI series in the order of RI P1P1 H,PIPI H=>---700 0<=0<- Piano 1 IxPxI RI series in the order of RI I series in the order of R H,PIPI P1P1 Duration 0<=0<- H<-=>00 I

PC I series in the order of P R series in the order of R PIP1 P1P1 H---7=>00 H<=0(-0 Piano 2 PxlxP R series in the order of R P series in the order of RI P1P1 P1P1 Duration H<=0(-0 H=><-00

EXAMPLE 20. Generalized antisymmetry between the PxlxP and IxPxI cubes in Structures 1a 352 MUSIC THEORY SPECTRUM 30 (2008)

(Px P) (Ix!) 3 2 9 8 7 6~il 0 A 5$ 349ABO

2 1 8 7 6 5 3 (}Jt9.4A 4 5 ABO 1 3 Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 9 8 3 2 lOA 7.6.<4B 5 9 A 3 4 5 6 8 B 8 7 2 lOB 9 6..>·3·A 4 AB 4 5 6 790 7 6 lOB A 854 2 93 B05678A124W3 Intensity 6 5 0 B A 9 '1:4 3 1 8 1 6 7 8 9B 2 3 5 A4' Diagonals :4i.)A 98.'15 2 1 B 60 89AB145706 10; 54 2 B A 8 3 9 1'2478A39 o B 4 3 1 A 9 728 235 8 9 B 4 A A 9 1 B 8 7 5 0 6 57AB160 5 '~ B A 8 6 3 2 0 7 1 o 3 4 6 B 5 iliA 5 4 3 )~O 9 8 6 1 7 6 9 A 0 5 B

(Px P) (Ix!) 9 8]Y~4 1 OA5 B 2 5 6 8 1 7 .65 3 0 B 9 4 A 1 3 6 792 8 lOA 7 6 4 B 5 n'..... ;. 5 6 8 B 0 2 7 1 B 9 6 5 3 A 4 6 790 1 3 8 2 8 5 4 2 9 3 B 56;1'8A12493 Articulation 4 3 1 8 2 .~;'.;. 1 6 7 8 ...9';. B 2 3 5 A 4 Diagonals 1 BO. 2 3 8 9 A BIl; 4 5 7.06 9 5 6 B 0 1 2 47 8 A3.9 8 6701235 8!111E~4A 6 8 9 2 3 4 5 7 AB;1; 6 0 1 2 7 8 9 A 03' 4 6:Bi5 781234:.69A051

EXAMPLE 21. Intensity and articulation diagonals

Every onset in Structures 1 a is set to a single PC and a and 6 in Part B. We define a block as a section of the piece single duration from the tesseract, and every form of the PC comprised of complete forms of the PC and duration series. and duration series is set to a single dynamic and articula­ The forms' of the PC and duration series do not overlap, but tion. So, the dynamics and articulations change every twelve rather begin simultaneously with each new block. In onsets for each line of polyphony. The entire precomposi­ Example 25, the measure numbers and the tempo changes tion as it progresses through the piece and tesseract is given (if changing) are given at the beginning of each block. For in Example 25. Part A is given in the first column; Part B, in both pianos, the serial content is given for each voice of the second. The entire piece comprises 14 blocks, 8 in Part A polyphony. Both pianos always have between 0-3 voices of RESEARCH NOTES 353

(PxP) (Ix!)

mf quasi .. ffff 0; p

'1ft mf Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 A .. f 0; mp ppp

0; mp mf f ppp '1ft quasI ~ A fff pppp f PPPP quasI pp 0; norm. pp .. fff f quasi ppp .. p 0; ffff mf mf quasi 0; ppp '1ft p ffff A .. quasi f norm. fff mp f mp fff norm. f quasI f '1ft mf PPPP norm. .,- PPPP A

mf pp norm. ppp pp > quaSI ffff ppp 0; norm. p mf ..

EXAMPLE 22. Intensity and articulation diagonals with musical nomenclature

polyphony, combined to create a total density of 1-6 voices of polyphony. The serial content is given in square brackets in four parts, one for each serialized element: PC, duration, (PxP) (Ix!) intensity, and articulation. For each serialized element, a co­ ordinate or series in the tesseract is provided. Example 10 shows that each piano simultaneously pro­ gresses through different portions of the tesseract for both the PCs and durations. Examples 16 and 20 further elabo­ rate on Example 10 by showing the various symmetries be­ tween the forms of the series as they are grouped into matri­ ces and cubes. These examples quantize large portions of the precomposition according to PC and duration across Parts A and B for both pianos. Example 25 gives more precise detail about the progression through the tesseract in that it incor­ porates the PC and duration forms of the series with the EXAMPLE 23. Symmetry in the intensity and articulation diagonals 354 MUSIC THEORY SPECTRUM 30 (2008)

Intensity

Piano 1 [BOOO] [AOlO] [9020] [8030] [7040] [6050] [5060] [4070] [3080] J2090] [lOAO] [OOBO] Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 quasi quasi Part A f.f.tf mJ' mf f.tf f.tf p p f.tf f.tf mJ' mJ' f.f.tf

Piano 1 [B050] [A040] [9030] [8020] [7010] [6000] [0060] [1070] [2080] [3090] [40AO] [50BO]

Part B ppp pp pppp mp f mf mf f mp pppp pp ppp

Piano 2 [OBOO] [OA01] [0902] [0803] [0704] [0605] [0506] [0407] [0308] [0209] [010A] [OOOB] quasi quas, quas, quasi quasi quas, Part A p ppp ppp f f f.f.tf f.f.tf f f ppp ppp p

Piano 2 [OB05] [OA04] [0903] [0802] [0701] [0600] [0006] [0107] [0208] [0309] [040A] [050B]

Part B mJ' pp pppp f mp ppp ppp mp f pppp pp mJ'

Articulation

Piano 1 [OBOB] [OAOA] [0909] [0808] [0707] [0606] [0505] [0404] [0303] [0202] [0101] [0000]

ifz -;- :> -;- :> ~ ~ ifz Part A A norm. A norm.

Piano 1 [060B] [070A] [0809] [0908] [OA07] [OB06] [0500] [0401] [0302] [0203] [0104] [0005]

Part B . . :> ~ ~ :> .. 0;- 0;- 0;- 0;-

Piano 2 [BOBO] [AOAO] [9090] [8080] [7070] [6060] [5050] [4040] [3030] [2020] [1010] [0000]

-;- ~ -;- ~ ifz :> ifz :> Part A norm. norm. A A

Piano 2 [5000] [4010] [3020] [2030] [1040] [0050] [60BO] [70AO] [8090] [9080] [A070] [B060]

Part B . . . . 0;- 0;- . . . . 0;- :> norm. norm . :> 0;-

EXAMPLE 24. Intensity and articulation series RESEARCH NOTES 355

Part A Part B

(1) Measure 1, Tres Modere (J'l = 120) (9) Measure 65, Lent ()! = 120) Piano 1 Voice 1 [[ ~OOO], [O~OB], [BOOO], [OBOB]] Piano 1 Voice 1 [[O~OB], [B~OO], [B050], [060B]] Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 Piano 2 Voice 1 [[O~OO], [~OBO], [OBOO], [BOBO]] Voice 2 [[O~OA], [A~OO], [A040], [070A]] Voice 3 [[0~09], [9~00], [9030], [0809]] (2) Measure 8, Modere, presque vif (J'l = 144) Piano 2 Voice 1 [[ ~OBO], [~BOO], [OB05], [5000]] Piano 1 Voice 1 [[~100], [O~OA], [A010], [OADA]] Voice 2 [[ ~OAO], hAOO], [OA04], [4010]] Voice 2 [[~200], [0~09], [9020], [0909]] Piano 2 Voice 1 [[1~00], [~OAO], [DA01], [AOAO]] (10) Measure 73, Modere, presque vif (J'l = 144) Voice 2 [[2~00], [~090], [0902], [9090]] Piano 1 Voice 1 [[0~08], [8~00], [8020], [0908]] Piano 2 Voice 1 [[~080], h800], [0802], [2030]] (3) Measure 16 Voice 2 [[ ~090], h900], [0903], [3020]] Piano 1 Voice 1 [[ ~300], [0~08], [8030], [0808]] Voice 2 [[~400], [0~07], [7040], [0707]] (11) Measure 82, Tres Modere (J'l = 120) Piano 2 Voice 1 [[3~00], [~080], [0803], [8080]] Piano 1 Voice 1 [[0~06], [6~00], [6000], [OB06]] Voice 2 [[0~07], [7~00], [7010], [OA07]] (4) Measure 24 Piano 2 Voice 1 [[~060], h600], [0600], [0050]] Piano 1 T acet Voice 2 [[ ~070], h700], [0701], [1040]] Piano 2 Voice 1 [[ 4~00], [~070], [0704], [7070]] (12) Measure 90, Modere, presque vif (J'l = 144) (5) Measure 32, Lent ()! = 120) Piano 1 Voice 1 [[0~05], [5~00], [0060], [0500]] Piano 1 Voice 1 [[ ~500], [0~06], [6050], [0606]] Voice 2 [[0~04], [4~00], [1070], [0401]] Voice 2 [[ ~600], [0~04], [5060], [0505]] Piano 2 Voice 1 [[ ~040], [~400], [0107], [70AO]] Voice3 [h700], [0~03], [4070], [0404]] Voice 2 [[~050], h500], [0006], [60BO]] Piano 2 Voice 1 [[7~00], [~040], [0407], [4040]] Voice 2 [[6~00], [~050], [0506], [6060]] (13) Measure 98, Lent ()! = 120) Voice 3 [[5~00], [~060], [0605], [5050]] Piano 1 Voice 1 [[0~03], [3~00], [2080], [0302]] Piano 2 Voice 1 [[ ~030], [~300], [0208], [8090]] (6) Measure 40, Modere, presque vif (J'l = 144) Piano 1 Voice 1 [h800], [0~03], [3080], [0303]] (14) Measure 106, Tres Modere (J'l= 120) Piano 2 Voice 1 [[8~00], [~030], [0308], [3030]] Piano 1 Voice 1 [[0~02], [2~00], [3090], [0203]] Voice 2 [[O~OO], [O~OO], [5080], [0005]] (7) Measure 48 Voice 3 [[0~01], [1~00], [40AO], [0104]] Piano 1 Voice 1 [h900], [0~02], [2090], [0202]] Piano 2 Voice 1 [[~010], h100], [040A], [A070]] Voice 2 [[ ~AOO], [0~01], [10AO], [0101]] Voice 2 [[ ~OOO], [~OOO], [050B], [B060]] Piano 2 Voice 1 [[9~00], [~020], [0209], [2020]] Voice 3 [[ ~020], [~200], [0309], [9080]] Voice 2 [[A~OO], [~010], [010A], [1010]] Voice 3 [[B~OO], [~OOO], [OOOB], [0000]]

(8) Measure 57, Tres Modere (J'l = 120) Piano 1 Voice 1 [hBOO], [O~OO], [OOBO], [0000]] Piano 2 Tacet

EXAMPLE 25. The series forms for all rifStructures la MUSIC THEORY SPECTRUM 30 (2008)

Voice 1 [[0 ~OB], [B~OO], [060B], [BOSO]] Voice 2 [[0 ~OA], [A~OO], [070A], [A040]] Voice 3 [[0 ~09], [9 ~OO], [0809], [9030]] Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021 65 Lent () = 120) Al

92 BO pp 74 2 > 5 Pno.l 8 16 pppp

BO pppp ~~>~ Lent () = 120)

7~

83 3 mJ' 5 Pno.2 8~ 16 . BO .~

pp pp pp 83 .' pp pp pp Voice 1 [[ ~OBO], [-BOO], [5000], [OBOS]] Voice 2 [[ ~OAO], [~AOO], [4010], [OA04]]

EXAMPLE 26. Structures la, Part B, Block 9, measures 65-72

intensity and articulation diagonals. Furthermore, Example and in addition to these mistakes, Structures la has other 25 shows that the two pianos progress through the tesseract miscellaneous notation errors. at different rates. Because each voice is set to its own form of The nomenclature in Example 25 is combined with the the series, the pianos progress through the tesseract faster score in Example 26 to illustrate an analysis of one block of when the polyphony is denser (Ligeti 1960,49-51). For ex­ the piece. For both pianos, the voices are labeled from top ample, in Block 3, Piano 1 simultaneously plays two forms to bottom. Every onset has two order positions attached to of the series while Piano 2 only plays one, so here Piano 1 is it: the first is for the PCs and the second is for the dura­ progressing quicker through the tesseract. The polyphonic tions. The order positions for the dynamics and articula­ density is quantized to each block. Example 25 combines tions are not labeled because each voice has one dynamic these forms of the series and accounts for every onset in the and one articulation per block. For the PCs, Piano 1 has entire piece. The serial mistakes are given in Appendix 2, three forms of the RI series in the order of RI ([Of-OB], RESEARCH NOTES 357

69 ~ppp Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021

Pna.1

pppp

EXAMPLE 26. [Continued]

[O(-OA], and [0(-09]), while Piano 2 has twa forms of the Structures 1 a by accounting for the multiple orderings of the R series in the order of R ([ (-OBO] and [(-OAO]). For the forms of the series, and reflects the various symmetries in­ durations, Piano 1 has three forms of the I series in the herent within the piece. order of R ([B~OO], [A~OO] and [9~00]), while Piano 2 The tesseract we presented here is specifically constructed has two forms of the P series in the order of RI ([ ~BOO] to capture the serial design in Structures la, but serial cubes and [~AOO]). For the intensities, the three voices of Piano and hypercubes may indeed have other applications in 1 are set to pianississimo, pianissimo, and pianissississimo music. Cubes and/or hypercubes with other cardinalities respectively, while the two voices of Piano 2 are set to may be applicable to music with series of lengths other than mezzo forte and pianissimo. For the articulations, the three twelve. The tesseract we used here has axes of P and I, where voices of Piano 1 are set to staccati underneath slurs, mar­ lis the inversion of Pbeginning on the same PC. It is possi­ cato, and legato respectively, while both voices of Piano 2 ble to make twelve-tone hypercubes where the axes are re­ are set to staccati underneath slurs. The tesseract as applied lated by Tn' T';' M n, M';' and their retrogrades as long as to Block 9, shown in Example 26, as well as to the whole the axes' origins share a common PC. Since these operations piece succinctly summarizes the technical aspects of are twelve-tone, the resulting hypercube is twelve-tone in MUSIC THEORY SPECTRUM 30 (zo08)

Suppose the generating series of the twelve-tone P xl matrix Mis

s = (so, ... , Sl1 ) , Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021

then the Oth row of M will be given by S:

M iO = si (i = 0, ... ,11)

The Oth column of M will be given by the inversion of S transposed so that Moo will be the same as before:

hence

MOj = (2so -s) moo 12 (j = 0, ... , 11).

The remainder of the rows of M are transpositions of S, with the amount of transposition determined by the corresponding element in the Oth column of M computed above, that is, the jth row of M

This leads to the general formula for M,

where i,j= 0, ... ,11.

ApPENDIX I. Proifif the generalformula for a twelve-tone matrix

that all the horizontals and verticals across any pair of sub­ hypercube such as this may be useful in examining pieces dimensions are some transformation of the generating se­ with multiple series. If these multiple series vary in cardinal­ ries. If the axes are related by a non-ITO, then the resulting ity, then the axes may be different lengths. Other reorder­ hypercube is not twelve-tone. In this case, the horizontals ings such as transposed rotations of the series, like the ones and verticals across the sub-dimensional pairs may not be used by Stravinsky, may be used for the various axes. The aggregates and may have repeated pes. A non-twelve-tone hypercubes described here all have axes placed in 90 degree RESEARCH NOTES 359

Given Corrected

Block 5, Piano 1, Voice 2 [[~600], [0~04], [5060], [0505]] [[~600], [0~05], [5060], [0505]] Downloaded from https://academic.oup.com/mts/article/30/2/339/1000085 by guest on 27 September 2021

Block 5, Piano 1, Voice 3 [[~700], [0~03], [4070], [0404]] [[~ 700], [0~04], [4070], [0404]]

2 2 m. 89, Piano 1, lower staff 8 ~:;~)1~1 8 )1:;~)1~1

3 3 m. 94, Piano 1, upper staff 8 t)1:;1 8 t:;)11

2 2 m. 113, Piano 2, Voice 3 8 ~.h.hl 8 ~V.I

m. 40, Piano 1 f.f sempre J'f.fsempre

ApPENDIX 2. Serial mistakes in Structures la orientation to each other. Other n-dimensional objects can Music Theory Spectrum, Vol. 30, Issue 2, pp. 327-359, ISSN 0195- be constructed, such as tetrahedra with axes placed at 60 de­ 6167, electronic ISSN 1533-8339. © 2008 by The Society for Music grees. All of the scenarios described here are based on PC Theory. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University structure, however, a whole new array of multidimensional of California Press's Rights and Permissions website, at http://www. ideas would emanate from structures that are not based on PC. ucpressjournals.com/reprintinfo.asp. DOl: 10.1525/mts.2008.30.2.327

REFERENCES

Brindle, Reginald Smith. 1987. The New Music: The Avant­ Garde since 1945. 2nd ed. New York: Oxford University Press. Dallin, Leon. 1974. Techniques of Twentieth-Century Composition:A Guide to the Materials ifModern Music. 3rd ed. Dubuque IA: W.e. Brown. DeYoung, Lynden. 1977-78. "Pitch Order and Duration Order in Boulez Structures la." Perspectives ifNew Music 16.2:27-34. Ligeti, Gyorgy. 1960. "Pierre Boulez: Decision and Automation in Structures la." Die Reihe 4:36-62.