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 Lutoslawski’s Derivation of Twelve­Note Harmonies from a Periodic Pitch Class Matrix Paul E. Dworak College of Music University of North Texas Denton, TX 76203‐3887 In conversations with Kaczyński and Varga, Witold Lutoslawski indicates that he had been studying the construction and classification of twelve‐tone chords and exploring their color possibilities since 1958. I’m especially interested in the somewhat elementary chords of which the adjacent notes form a limited number of types of intervals. . . . Twelve‐note chords which are made up of one, two, or three types of intervals have for me a distinct, easily recognisable character. In contrast to these, twelve‐note chords which include all types of intervals are devoid of colour and present no distinctive features. 1 Intervals have a characteristic power to create an atmosphere. It is difficult to define what I have in mind, but I think of such rather suspect words as expression, colour, atmosphere, that is, notions that can hardly be measured objectively. After all they refer to the composer’s complex inner world and sound imagination. . . . The qualitative difference between intervals is important for my musical thinking. 2 Lutoslawski disclaims any serious interest in the use of the twelve‐tone technique in his compositions. Discussing Muzyka żalobna, a work completed in 1958 that is based on a row, he says: Tadeusz Kaczyński, Conversations with Witold Lutoslawski, trans. Yolanta May (London: J. & W. Chester/Edition Wilhelm Hansen London Ltd., 1984), pp. 38‐40. 1
[The row] has completely different purposes as with Schönberg and his followers. . . . Schönberg’s principles were among other things intended to replace functional harmony. I have never been interested in that goal. The use of a row had to serve a different purpose: to create a special kind of harmony. 3 He feels instead a kinship with Debussy: An important feature of Debussy’s world of music is his sensitivity to vertical aggregations, and also the independence of functional thinking in determining the logical sequence of musical events. Schönberg’s twelve‐tone system was in my opinion a natural consequence of the functional system, and was born to replace it. Debussy’s system of organizing sound shows that he was indifferent to functions–that is what I have in common with him. 4 Stucky presents a comprehensive list of twelve‐note chords found in Lutoslawski’s compositions. He suggests that Lutoslawski principally uses three types: those emphasizing interval classes 1, 5, and 6; those based on interval class 2; and those consisting of interval classes 3 and 4 (see figure 1). Stucky also notes that the vertical distribution of pitch classes in such chords creates various patterns of symmetry. 5 Fig. 1. Twelve‐note chords based on various intervals. Bàlint Andràs Varga, Lutoslawski Profile, trans. directed by Stephen Walsh (London: J. & W. Chester/Edition Wilhelm Hansen London, Ltd., 1976), p. 21. 3 Varga, p. 11. 4 Varga, pp. 16‐17. 5 Steven Stucky, Lutoslawski and His Music (Cambridge: Cambridge University Press, 1981), pp. 116‐117. 2
Lutoslawski acknowledges being less interested in his own harmonic system than he is in its musical consequences: “As for the harmonic system itself it is perhaps overdone to call it ‘system.’ It is rather a collection of procedures still growing in number and undergoing some changes.” 6 This paper discusses Les espaces du sommeil and demonstrates that contrasting twelve‐note chords used throughout the work are all derived from the same source–a pitch class matrix with periodic properties. In his discussion of this work, Stucky indicates that the opening is based on chords using ic4 and ic1. He states that part 2, beginning at number 24, is loosely based on an informal twelve‐tone series that emphasizes ic2 and ic5. The third part, from two measures before number 83, employs chords emphasizing ic3 and ic5. The coda consists of symmetrical chord structures based on ic2 and ic3. 7 Chordal material used throughout the work actually is derived from two pitch class cells organized as follows: a = _a+1 a+3 _ b = _a+3 a+2_ _a+0 a+2 _ _a+1 a+0_ such that a is any pitch class, b = m4(a), and a = m1(b), where m4 is a 90° rotation of the cell counterclockwise and m1 is a 90° clockwise rotation. In Lutoslawski’s usage, bn = T4(m4(an)) and an+1 = T4(m1(bn)). If an and bn occur alternately in this fashion, all twelve pcs will occur after three cells, and the pattern repeats after six cells: a0 b0 a1 b1 a2 b2 a0 . . . The pattern will also repeat in this fashion if the cells are stacked vertically, or if the cells are arranged in three columns, and in each column bn is above an: 6
7 Varga, p. 34. Stucky, pp. 186‐188. b0 b1 b2 b0 . . . a0 a1 a2 a0 . . . In Les espaces du sommeil, Lutoslawski’s use of these cells is a variant of this last arrangement. He stacks the lower row of bn above an but places the higher row below an+1 in the next column. This organization forms a matrix of pcs that forms the opening series of chords: M0 = 5 1 0 B 4 3 2 A 1 9 8 7 0 B A 6 9 5 4 3 8 7 6 2 5 1 0 B 4 3 2 A 1 9 8 7 0 B A 6 ... ... ... ... Fig. 2. Opening chord series in Les espaces du sommeil. As Stucky points out, this opening progression emphasizes ic1 and ic4. Fig 3. Notation of opening chords. The second chord in this series, B = <4, 3, 2, A>, is derived from the first chord, A = <5, 1, 0, B>, by the relationship B = T5(RI(A)). Likewise, every subsequent pair of chords is similarly related. Since there are eight pcs, the equivalent of two cells, in every pair of chords, the third chord, C = <1, 9, 8, 7>, is derived from the first by the relationship C = T8(A). As figure 4 shows, the columns of the matrix generate ic1 and ic4, the outer rows generate ic1 and ic3, and the inner rows generate ic2 and ic6. row ics column ics 4 1 1 1 2 2 1 1 1 4 3 6 6 3 4 1 1 1 2 2 1 1 1 4 3 6 6 3 4 1 1 1 2 2 1 1 1 4 Fig. 4. Interval classes generated by the matrix. In this figure, odd numbered rows represent intervals between adjacent row members in the matrix, and odd numbered columns represent intervals between column members. (The numbers in even‐numbered rows and columns have no meaning). Lutoslawski generally derives chords from the columns when he wishes to emphasize ic1 and ic4. He uses the rows when he wishes to emphasize ic1 and ic3. He also “borrows” pcs from adjacent rows or columns to add other intervals to the sets and segments that he uses. The intervals between borrowed pcs and other set members can be determined by summing the actual intervals separating any two members of the matrix. Figure 5 shows the actual intervals in M0, assuming that the matrix is read from the upper left hand corner to the lower right hand corner. 8 B B Fig. 5. Intervals in M0. B 2 2 B B B 4 9 6 6 9 8 B 1 B 2 2 B B B 4 9 6 6 9 8 B 1 B 2 2 B B B 4 The interval between M0<0, 0> (5) and M0<2,2> (8) is B+9+8+B = 3 (mod 12). The design of the matrix also enables all ics to occur within limited areas. One example follows, but all others are similar: 7 6 5 1 0 B 4 3 2 A 9 8 Fig. 6. Sample of interval class orientations within the matrix. The design of the pc matrix also creates symmetrical, interlocking sets of identities. The middle two pcs of the first chord in any pair become the top pcs of the next chord pair. Likewise, the middle two pcs of the second chord in any pair are the bottom two pcs of the previous chord pair: – – – d – – – c – b a – – d c – b – – – a – – – Fig. 7. Identities in the pitch class matrix. Four unique transpositions of this matrix exist, M0 ­ M3. M0: 5 4 1 0 9 8 1 3 9 B 5 7 0 2 8 A 4 6 B A 7 6 3 2 M1: 6 5 2 1 A 9 2 4 A 0 6 8 1 3 9 B 5 7 0 B 8 7 4 3 Fig. 8. Transpositions of the matrix. M2: 7 6 3 2 B A 3 5 B 1 7 9 2 4 A 0 6 8 1 0 9 8 5 4 M3: 8 7 4 3 0 B 4 6 0 2 8 A 3 5 B 1 7 9 2 1 A 9 6 5 Note that odd numbered transpositions do not alter the set content of the inner rows of M0, but instead exchange the set content between these rows: { 0 2 4 6 8 A } { 1 3 5 7 9 B } M1 does create new hexachords for the top and bottom rows, when compared with M0, but this transformation leaves the trichords { 0 4 8 }, { 1 5 9 }, { 2 6 A }, and {37B } invariant. Even numbered transpositions exchange the set content of the top and bottom rows and rotate the pc content of the outer rows forward by two columns. The remaining transpositions introduce neither new sets nor new segments, but merely reorder pairs of columns within the matrix. The remainder of this paper shows how Lutoslawski uses these matrices and their inherent symmetries to generate the varied chordal colors that occur throughout Les espaces du sommeil. As a result, all the chordal material originates from a common source in an orderly fashion. He uses the matrix as a source both for ordered segments of pcs and also for unordered sets, depending on the needs of each section of the composition. Lutoslawski uses M0 from the beginning of the work until number 3 in the score. At number 5, he uses both the original matrix and M1. Number 5 begins with three measures and ends with two measures derived from M1. The middle two measures are derived from M0 . Chords are derived by moving both forward and backward through both matrices. The goal of both motions through M1 is <6,2,1,0>. 6 5 2 1 2 4 A 0 1 3 9 B 0 B 8 7 Fig. 9. Motions through M1 at number 5. A 6 5 4 9 8 7 3 6 2 1 0 ... ... ... ... Sometimes the matrix also is the source of material for aleatoric counterpoint. Lutoslawski describes this as follows (see figure 10): The individual parts have no identical time organisation, no common pulsation which would be identical in fractions of seconds. The parts are independent of each other in time. . . . Within an aleatoric section, pitch can be strictly fixed. That may appear strange if you think of the loosening of time relations between sounds. . . . if you take a twelve‐tone chord which we regard as twelve different sounds that we hear simultaneously, we can write down a passage that is based on the notes belonging to this twelve tone chord. The different parts can play very complicated rhythms, even sound sequences, and yet they only play notes of that chord. . . . It may occur that the chord never actually sounds in its entirety–it is supplemented by our memory and imagination. 8 Fig. 10. Example of aleatoric counterpoint. 8
Varga, pp. 24‐25. At number 3, Lutoslawski writes a string chord accompanied by aleatoric counterpoint in the winds and brass. The vertical structure of the string chord includes three trichords derived from the columns of M0 , such that two members of each trichord occur in any column and the third is borrowed from an adjacent column. These trichords form symmetrical partitions of M4. Fig. 11. Deriving trichords at number 3 by partitioning M4. The aleatoric counterpoint at number 3 in the winds and trumpets employs sets of notes derived from the columns of M0, e. g., { 8 7 2 } and { 9 5 4 3 }. Beginning at number 6 in the score, Lutoslawski derives harmonies from the rows of the matrix. Throughout the remainder of Les espaces du sommeil, the segments <9, 8, 5, 4>, <9, 0, 1, 4>, and <7, 6, 3, 2> become prominent. These segments are actually derived from the hexachords that constitute the outer rows of M0: {014589} and {2367AB}. Sometimes harmonies are derived exclusively from one set or the other, sometimes by combining a member of one hexachord with several from the other. 9 Since the matrix is periodic in both directions, it can be represented by a torus. Since each four note cell recurs every third cell, but is rotated, the torus can be viewed as folded back on itself and twisted. Such a model, unfortunately, contributes little to the understanding of this composition. Curiously, if the cells are arranged as diamonds and stacked diagonally to the northeast or southeast, the sets that are prominent in both the matrix and the composition emerge. 9
At number 22, e. g., the segments used are <9, 0, 1, 7>, <6,3,2,B>, and <8,5,A,4>. Figure 12 shows how these segments are derived from symmetrical partitions of the outer rows of M8. Fig. 12. Deriving tetrachords at number 22 by partitioning M8. At number 23, Lutoslawski again uses { 9 8 5 4 }, this time as a set and as one tetrachord of a twelve‐note chord. Since the others tetrachords are { A 6 2 1 } and {B730}, however, these sets result from symmetrical partitions of the outer rows of M3. M3: 8 2 7 1 4 A 3 9 0 6 B 5 ... ... Fig. 13. Deriving tetrachords at number 23 by partitioning M3. 1
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At this point in the score, this twelve note chord is distributed among nine instruments and played eight times. Six presentations of the twelve‐note chord (612 = 72) sound during the eight instrumental chords (98=72). Each time, the set members are reorganized. By the time the fifth chord is sounded, M0 returns with its sets {014589} and {2367AB}. Some mixing of set members occurs in these chords, but by the end of the section, pcs from the latter set predominate. 0 5 4 9 8 6 1 2 A 1 9 6 5 2 A 7 3 B 2 B A 8 7 6 3 0 4 B 8 7 4 3 0 9 5 1 8 5 4 1 0 9 6 2 A 7 3 0 B 9 8 4 1 5 6 2 B A 3 0 7 4 8 6 1 A 9 3 2 6 B 7 Fig. 14. Tetrachords at number 23, read from bottom to top. In the first three chords of this section, Lutoslawski also employs an interesting exchange of pcs among the cells that constitute the matrix. Members of cells that are written as follows may be identified by their relative compass direction (N = north, E = east, S = south, W = west): Cell 0: 0 1 2 3 Cell 1: 4 5 6 7 Cell 2: 8 9 A B W N S E Pcs then can be identified by the position they hold in the cell and their cell number. Pc 5 is N1, pc A is S2. Using this notation, the first six tetrachords at number 23 exhibit the following interchange of cell members: A 2 1 6 A 2 S2 S0 N0 S1 S2 S0 8 9 4 5 9 1 W2 N2 W1 N1 N2 N0 0 B 3 7 3 6 E0 E1 E0 S1 W0 E2 Fig 15. Pcs derived from exchanges of cell members. 5 N1 6 S1 0 W0 4 W1 7 E1 8 W2 Note that in the first two tetrachord pairs, only the cell from which a pc is derived changes, but the same number of relative cell positions is preserved. In the final pair, S1 is also substituted for E2, which begins the process of chordal transformation that continues in subsequent chords. Beginning at number 24 and continuing through 82, the pitch classes fall into hexachordal groupings and account for the sustained string passages in part 2 of Les espaces du sommeil. Lutoslawski refers to this section as one of his most successful attempts to organize pitches in simple textures. 10 The hexachords he uses are {024579} and {6 8 A B 1 3}. These are related at the tritone and project the symmetrical interval structure [2 2 1 2 2]. These hexachords are derived from the following partition of M4: 10
Kaczyński, pp. 74‐75.
Fig. 16. Deriving hexachords at number 24 by partitioning M4. Beginning at number 83, the texture is contrapuntal with several chordal interruptions. The chords used three measures before number 84 are derived from symmetrical partitions of the outer rows of M5. Fig. 17. Partitions of M5 before number 84. The chord at number 85 consists of only eleven notes. As a result, it lacks the symmetry of other chords in its partition of M5, but it still displays the pc borrowing characteristic of other sets and segments in this composition. Fig. 18. Partitions of M5 at number 85. Between numbers 92 and 94, Lutoslawski begins exploring the inner row space of M8. The following figure shows the chords at number 92 and one measure before number 94. Fig. 19. Partitions of the inner rows of M8. Lutoslawski returns to the outer rows of M0 and M8 for the chords between numbers 94 and 96. The three chords in the three measures before 95 and the chord at 96 create the following partitions. Fig. 20. Partitions of the outer rows of M0 and M8. Probably the most calm and lyrical section of Les espaces du sommeil, beginning at number 98, also uses material from the opening pitch class matrix. The chord at that point, organized into pentachords read from top to bottom, is {0}{97531} {A8642} {B9753} {0}. This chord is composed of pitch classes separated by ic2 and pentachords separated by ic3. These sets are derived directly from the inner rows of M0. The pentachords in the celesta, harp, and piano are pentatonic and possess the symmetrical interval structure [32223]: {B2468B}, {035790}, {1468A1}, {2579B2} These sets are derived from those in the strings merely by substituting the pc that will form ic3 as follows: 0 B, 1 0, 2 1, 3 2 At number 105, the final chord that precipitates quickly to silence, the hexachords used after number 24 are broken into trichords and recombined. The new chord is {7 5 2} {B 8 3} {0 9 4} {A 6 1}. Note that when the two dyads that form each hexachord are broken (cf. figure 16), the left member of each joins with the right member of the other to form two notes of the trichord. These two notes combine with the remaining monads to form the symmetries shown in figure 21. Fig. 21. Deriving trichords at number 105 from hexachordal partitions of M4. These trichords then recombine to form the tetrachords {7 5 2 B} {8 3 0 9} {4 A 6 1}. These tetrachords form the basis for the aleatoric counterpoint that ends the composition. In summary, Lutoslawski constructed the chordal material for Les espaces du sommeil by partitioning and recombining sets from the opening matrix of pitch classes. He designed this matrix to display periodic patterns of pitch classes and to sound all the interval classes. He explores this space throughout the composition, much as the work itself explores sleep’s spaces. As a result, he has developed a tool for unifying the composition, while exploring the many and varied colors of the chords he used throughout this work.
References Kaczyński, Tadeusz. Conversations with Witold Lutoslawski. Translated by Yolanta May. London: Chester Music, 1972. Lutoslawski, Witold. “Rhythm and the Organization of Pitch in Composing Techniques Employing a Limited Element of Chance.” Polish Musicological Studies 2 (1986) : 37‐
53. Morris, Robert. Composition with Pitch Classes: A Theory of Compositional Design. New Haven: Yale University Press, 1987. Stucky, Steven. Lutoslawski and His Music. Cambridge: Cambridge University Press, 1981. Varga, Bàlint Andràs. Lutoslawski Profile. Translation directed by Stephen Walsh. London: Chester Music, 1976.