Lecture 21: Mathematics and Later Composers: Babbitt, Messiaen, Boulez, Stockhausen, Xenakis,... Background By 1946 Schoenberg s students Berg and Webern were both dead, and Schoenberg himself was at the end of his career. A younger generation of composers came to prominence after World War Two, and they (mostly) rejected the practices of the `Second Viennese School. For some, the approach of organizing pitches but leaving everything else up to the composer wasn t organized enough: as we will see, the serial technique came to be applied to almost every other aspect of music composition. (For these composers, the music of Webern was closest to their ideal, because of its increased organization and rejection of lyricism in favor of crystalline texture.) For others, there was in turn a rejection of total organization in favor of letting randomness dictate certain aspects of composition and performance. In fact, some composers migrated from one camp to the other, and what both camps had in common was an approach to composition informed by mathematics. In the next couple of lectures I d like to give some thumbnail sketches of how mathematics entered into some of the music produced by these composers. Milton Babbitt (born 1916) is easily the most mathematical of 20thcentury American composers; his father and brother were mathematicians, and he wrote a PhD thesis on the combinatorial structure of twelve-tone composition. Understandably, Babbitt s early compositions were heavily influenced by Schoenberg and Webern. But in his 1947 work, Three Compositions for Piano, is the first time where serial organization is applied not just to pitches but also to rhythm and dynamics. For example, in the first movement of this piece, certain forms of the row are associated with particular rhythms and dynamic levels (see handout, and page 52 in Harkleroad). As in Schoenberg s Klavierstuck Opus 33a, the prime tone row Babbitt uses has the property that the first hexachord is, as a set of pitches,
related by a transformation to the second hexachord. (Treating subsets of the row vertically, without regard to order, is what Babbitt means by combinatorial structure.) Can you spot what the transformation is? [It is transposition by 6 semitones, so that as a set the first 6 pitches of P-6 are the same as the last 6 pitches of P-0; in Schoenberg s piece the first hexachord of I-5 is the same as the second hexachord of P-0.] In Babbitt s piece, the basic rhythmic pattern is 5 notes, then 1 note, then 4 notes, then 2 notes, where the basic unit is the 16th note. (Note that this pattern allows us to play all twelve pitches of a tone row.) This pattern is associated to the prime form and its transpositions; for retrograde forms the rhythm pattern is (naturally) 2,4,5,1, while for inversion and retrograde forms it is 1,5,2,4 and 4,2,5,1 respectively. [musical example: 3 Compositions, recorded by Robert Taub] ============= Aside #1: Technically speaking, in generating these alternate forms of the basic rhythmic pattern, Babbitt is using a map φ from the musical group of 48 transpositions into the group of permutations on 4 symbols. The transpositions are ignored by this map, R is mapped to the permutation that reverses the 4 symbols, and I is mapped onto the permutation (AB)(CD). This map is a group homomorphism, in the sense that it respects the product operation in both groups: φ(g h) = φ(g) φ(h). Such maps may not exist between any randomly chosen pair of groups (unless they are trivial, sending everything in one group to the identity element in another). Groups that have no non-trivial homomorphisms to another group are called simple; the classification of all finite simple groups was completed in the 1980s, and is perhaps the most complex mathematical problem ever solved, involving the work of hundreds of individual researchers and journal articles running to over a thousand pages altogether.
============ Aside #2: Another way one could approach rhythm in a way that s analogous to how twelve-tone composers approach pitch is to apply more basic transformations to rhythms: reflection, translation and dilation in time. One setting in which the translational approach to varying rhythm appears in its purest form is Steve Reich s Clapping Music (1972), in which two players or groups of players start by clapping the same 12-beat rhythm, made up of a sequence of short and long notes, SSLSLLSL, where the longs are two beats and the shorts are one. This is repeated 12 times, then one part continues the rhythm while the other translates the rhythm back by one beat, to SLSLLSLS, so that two groups of clappers are playing slightly different 12-beat rhythms. The shifts continue after every 12 repeats, until finally both parts finish by clapping out identical rhythms again. [ animated excerpt: http://www.youtube.com/watch?v=eu-trxgordg ] Olivier Messiaen (1908-1992). We ve already heard the story of Messiaen s landmark piece Quartour pour le fin de temps (written and performed in 1940 in a POW camp), and the use of isorhythmic techniques in the movement Liturgie de Cristale in that piece. Messiaen s music is rhythmic and harmonically complex; he actually published a two-volume treatise in which he explains his compositional techniques. Messiaen was deeply influenced by his study of Indian rhythms. In Indian music a tala is a repeated rhythm; a catalog of 120 talas was compiled by the 13th century Indian musicologist Sharngadeva. Rhythms from this catalog show up in Messiaen s music, but he also combined these with another technique, the use of compositional materials that have symmetry. Messiaen assembled his own more complex talas which he frequently used in his music: [ Tala 1, Tala 2 from Johnson, page 37 ] These rhythmic patterns are composed of smaller cells, most of which are symmetric about their midpoints (Messiaen calls these non-
retrogradable rhythms). For example, the 17-beat Tala #1 is used in the piano part in the Liturgie de Cristale. [play excerpt] Tala #2 is allegedly used in the 6th movement of Messiaen s 1944 cycle of piano pieces, Vingt Regards sur l Enfant Jesus; the following excerpt gives you some idea of the texture of Messiaen s music in fast mode : rhythmic complexity coupled with harmonic complexity. The main harmonic device Messiaen introduced were modes of limited transposition. These are proper subsets of the 12-note chromatic scale that are symmetric under a transposition. For example, the following scale C D E F F# G# A# B C has the property that transposing it up 6 semitones gives back exactly the same subset of the chromatic scale. There are exactly 7 such modes in all, containing between 6 and 10 notes; the above example is Messiaen s 6th mode. This mode is used in the theme of the 6th movement of the Quartour pour le fin du temps, entitled Dance of fury, for the seven trumpets. Given Messiaen s penchant for using complex, repeated rhythmic patterns in his compositions, it s not surprising that he briefly experimented with using series to control the durations of notes. In his 1950 piano composition Modes de valeurs et d intensités, Messiaen used three series of 12 notes, wherein each note had a specified duration, register (in which octave they are played on the keyboard) and intensity (loudness, and mode of attack). To avoid complete repetition, Messiaen permutes the order in which these notes are played, using patterns like 1-12-2-11-3-10-4-9-5-8-6-7. Aspects of a note like duration, loudness, attack, etc., are often called its parameters. In this piece, the parameters of each note are fixed, so that it is played exactly the same way (and for the same length of time) every time. For this reason, making chords is
impossible, and Messiaen never attempted this kind of multiparameter serialism again. [listen to excerpt at Alex Ross s blog http://www.therestisnoise.com/2007/01/chapter-11-brav.html ] It was Messiaen s most famous student, Pierre Boulez (born 1925), who first used independent series for pitch, duration, loudness, and attack in his 1950 composition Structures Ia. Boulez began with the first series used by Messiaen in Modes de valeurs et d intensités, and using just the pitch classes, writes it as a row of numbers modulo 12: P=3, 2, 9, 8, 7, 6, 4, 1, 0, 10, 5, 11 with the convention that 0 is C, 1 is C#, and so on. The inversion of this row about its starting point generates the series of numbers I=3, 4, 9, 10, 11, 0, 2, 5, 6, 8, 1, 7 It was Schoenberg who first hit upon the idea of listing the prime row and its inversion along two edges of a matrix, so that each row of the matrix (read left-to-right) is a transposition of the prime and each column is a transposition of the inversion. 3 2 9 8 7 6 4 1 0 10 5 11 4 3 10 9 8 7 5 2 1 11 6 0 9 8 3 2 1 0 10 7 6 4 11 5 10 9 4 3 2 1 11 8 7 5 0 6 11 10 5 4 3 2 0 9 8 6 1 7 0 11 6 5 4 3 1 10 9 7 2 8 2 1 8 7 6 5 3 0 11 9 4 10 5 4 11 10 9 8 6 3 2 0 7 1 6 5 0 11 10 9 7 4 3 1 8 2 8 7 2 1 0 11 9 6 5 3 10 4 1 0 7 6 5 4 2 11 10 8 3 9 7 6 1 0 11 10 8 5 4 2 9 3 (The retrograde and retrograde-inversion forms of the row are obtained by reading right-to-left and bottom-to-top respectively.)
Note that, as we move to the left along a row, the differences between successive entries are always the same, mod 12, and the same is true for the columns. These differences are, of course, the same as the steps in the prime row and its inversion, respectively. Boulez added two more dimensions to this picture, effectively constructing a 4-dimensional array of numbers. Suppose the coordinates on this hypercube are x,y,z, and w; then moving within the cube in the positive z-direction, the differences are the same as along P, and in the positive w-direction the differences are the same as along I. The consequence of this convention is that the number at position (x,y,z,w) of the hypercube is P x - P y + P z - P w + P 0, where P 0 is the first number in the prime series P, and P x is x places along in P. Effectively, the matrix we have above is just the slice of the hypercube where z=w=0. To get rows for durations, Boulez sets z or w to nonzero values, then proceeds in the usual P, R, I, or RI directions (i.e., in positive or negative x- or y-directions). The numbers obtained are mapped to durations by composing with the inverse of the prime row as a function from 1-12 to 0-11. For example, the durations for the top voice in the example come from setting z=0 and w=11, x=0, and counting off y from 11 down to 0 (i.e., move in the RI direction), giving the numbers 11, 5, 0, 10, 9, 6, 4, 3, 2, 1, 8, 7. The pitch 11 is the 12th note of the row, so the first note is 12 beats long; the pitch 5 is the 7th note of the row, so the next note is 7 beats long, and so on (a beat being a 32nd note) For the durations in the lower lower voice, z=11, w=0 and we take the R direction, giving the numbers 7, 1, 6, 8, 9, 0, 2, 3, 4, 5, 10, 11 which are mapped to durations 5 beats, 8 beats, 6 beats, and so on. Boulez selects the pitch rows for successive blocks in the upper voice by going in the P direction with y=0, then y=1, y=2, y=3 and so on (or, in the lower voice, in the I direction with x=0, x=1, x=2,..), while selecting duration rows in the RI direction with z=0, and w=11, w=10, w=9, and so on (in the lower voice, in the R direction with w=0 and z=11, z=10, z=9,...). In effect, Boulez is taking a several
simultaneous walks through the hypercube. He also uses rows derived from the diagonals of these matrices to control the attack and dynamic levels, which remain constant while a pitch/dynamic row is worked through. [listen to opening of Structures 1a at Alex Ross s blog http://www.therestisnoise.com/2007/01/chapter-11-brav.html ] The approach Boulez used in Structures Ia, of total serialism, was in fashion for only a short time in the 1950s, used by Boulez in a few compositions and by other composers such as Ernest Krenek and Karlheinz Stockhausen. By the late 1950s these composers had developed a more complex approach to composition, involving both determined elements and randomness. For example, Stockhausen s 1957 Klavierstuck XI is printed on single large of paper. On it are printed 19 fragments of music, distributed unevenly over the page. The performer is to look at a random place on the sheet, begin playing the music there, choosing his own tempo and loudness. At the end of the fragment he reads the directions for the tempo and loudness of the next fragment, and chooses another one at random to play; and so on, until all fragments have been played and the piece is over. This means that there are 19! = 1.21 10 17 possible ways to play this piece. (Think of this as like a musical dice games, but without dice and without a regular rhythm.) [listen to this at http://www.youtube.com/watch?v=rhsvkejxy7c ] Another composer who employed randomness in composing music, but in a much more sophisticated way, was Yannis Xenakis (1922-2001). Xenakis studied architecture as well as music, and his apprenticeship with the architectural theorist Le Corbusier influenced Xenakis thinking about structure. Xenakis took a top-down approach to music---often designing the overall structure, but using randomness (and computers) to fill in some of the details in his scores. This stochastic approach to music---using randomness to achieve some overall goal that is not precisely, but only statistically, defined---first
appeared in Xenakis landmark 1954 composition Metastaseis. (The title means beyond immobility.) Xenakis began this composition for orchestra (mostly strings, with some brass and percussion) with an overall diagram of the structure, in which the players fill in the detail. The dominant sonority is the string glissando (where for example a violin player bows while changing the length, and therefore the pitch, of the string). A short excerpt from the score is given on a handout; here you can see that Xenakis conception is primarily geometrical. [musical example, with follow-along diagram: http://www.youtube.com/watch?v=32oy43ptoxe&feature=channel ] Again, the effect is an overall statistical impression of the music, with individual details almost lost in a cloud of notes. As this example shows, Xenakis was known for introducing new textures into music, and this extends to his compositions for percussion. One such piece is Pleiades, written for an ensemble of 6 percussion players playing all metal instruments in one movement, all keyboards in another, and all drums in another. Here is an excerpt from the drum movement: [ Peaux: http://www.youtube.com/watch?v=9oqtyklwcla ] In the keyboard movement, players start playing at a common speed, but then begin to play at different speeds, going in and out of phase and branching into independent layers. [ Claviers: http://www.youtube.com/watch?v=vnvfuiuhmpm ] [ Metaux: http://www.youtube.com/watch?v=41ecwtjcuma ] References: L. Dallin, Twentieth Century Composition, W.C.Brown, 1974. P. Griffiths, Messiaen and the Music of Time, Cornell, 1985. J. Harley, Xenakis, Routledge, 2004. R. Lombardi, M. Wester, A Tesseract in Boulez's Structures 1a, Music Theory Spectrum 30 #2 (2008), 339--359. R. Sherlaw Johnson, Messiaen, University of California Press, 1980. G. Perle, Serial Composition and Atonality, University of California Press, 1962.