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 ....