Composing with Sieves: Structure and Indeterminacy in-Time

Introduced by Xenakis 50 years ago, sieves have proved to be a relevant and robust device for music composition. Examples of complex and symmetric sieves usage in original works are presented along with a few possible applications not explored before. The dichotomy between predetermined abstract structures such as sieves and their actualization through random procedures is discussed and it is also shown that in the hands of a innovative musician, sieves not only serve the craft aspect of composing but could also reveal as well as impact deeper levels of thinking. 1. PERIODICITY, WEIGHTS, AND SYMMETRY. Sieves were introduced during the early 1960s by Xenakis in his works but remained a rather esoteric topic until rather recently when a number of writings on the subject have appeared Ariza [1], Gibson [6], Exarchos and Jones [5], Solomos [7] to name some testifying to the relevance of this device for music composition. Since many basic aspects and in particular sieve analysis and construction have been discussed previously, only a few brief reminders are necessary here. Sieves are logical filters expressed as boolean operations on congruence modulo classes. A trivial case is that of a sieve containing equivalence classes denoted by various indices (following the notation used by Xenakis) of a single modulo: 130 U 133 U 135 U 138. (1) This formula will generate a periodic sequence of numbers with 13 the only modulo able to define elements of the sieve since it is a prime number. Messiaen's modes with limited transpositions can be generated with simple periodic sieves: 30 U 31 (second mode, an octatonic scale) (2) while the expression offered by Xenakis in Formalized Music [13] for generating the major scale contains two modulo terms and a more involved set of operations: (3n+2 ∩ 4n)U(3n+1 ∩ 4n+1)U(3n+2 ∩ 4n+2)U(3n ∩ 4n+3) (3) In this case, the period of the sieve is the lowest common multiple (LCM) of its modulo terms while the indices show the possibility of transposing the scale. Similar pursuits, albeit from a different perspective, may be found in Anatol Vieru's Book of Modes [12]. Xenakis favored “aperiodic” pitch sieves actually, sieves with a period longer than the actual range of the sound source, hence making it impossible to determine its period. There is a clear and desirable distinction between such a pitch sieve and any octaviating scales, tonal or atonal. On the other hand, oscillating in the same piece between periodic and “aperiodic” sieves, between a recognizable structure and apparent disorder offers an enticing way of organizing musical materials. Multiple modulo terms and more intricate boolean operations allow for the addition of another feature: weights establishing preferences among the elements of the sieve. Weights may be assigned to individual elements in order to establish a hierarchy and to transform a “scale” a list into a “mode”. In DISSCO, a Digital Instrument for Sound Synthesis and Composition developed at University of Illinois Computer Music Project and Argonne National Laboratory [2], weights could follow a pattern in sync with that of the sieve elements or they could have an independent cycle or no cycle at all (aperiodic). In turn, each cycle could have its own scaling factor: e.g., the mid range octaves could have more sway than the extreme ones. Although the weights may be assigned arbitrarily following the judgement of the composer, a more abstract arrangement assigns a particular weight to each module thus reinforcing the internal structure of the sieve: sieve: (30 ∩ 40) U 31 ∩ (40 U 43) U 32 ∩ (41 U 43) modulo weight: 15 1 10 1 1 5 1 1 resulting elements: {0} {4, 7} {5, 11} element weight: 16 12 each 7 each Figure 1. Modulo weights Applied to diatonic pitches, the results in this example would favor the tonic triad {0, 4, 7} over the dominant tritone {5, 11}. Sieves that produce symmetric intervals between numbers contain modulo terms that have symmetric indices. A simple nonretrogradable rhythm: attacks 0 1 9 11 19 20 durations 1 8 2 8 1 Figure 2. Nonretogradable rhythm will be produced by the sieve: (50 ∩ 40) U (51 U 54) ∩ (41 U 43) (4) where the terms 51, 54 and 41, 43 are symmetric with respect to the origins 50 and 40 since the sieve is periodic and 50 ≡ 55. The sieve generating the Dorian mode shows a similar balance: (30 ∩ 40) U 31 ∩ (42 U 43) U 32 ∩ (42 U 41) (5) An interesting case is that of rhythmic sieves that are symmetric and extended over very large areas of a piece; they can be used to create structures similar to that of the first movement of the Symphony Op. 21 by Webern or Machaut's Ma fin est mon commencement. Multiple-entry sieves, an even more elaborate construct, could be described as involving conditional probability, or as multidimensional matrices (not unlike the sequence of “screens” used by Xenakis to generate Analogique A et B) or as related to the more common sieves through the use of equivalence modulo m relations of the type: k1m1 + k2m2 + ... + kimi + n (6) This expression is helpful, for instance, when determining the position of an attack measured in the smallest time quanta available. With k1 being the measure number, k2 the number of a beat in a 3/4 measure and the sixteen the smallest duration item or the EDU (Elementary Displacement Unit in the terminology introduced by Xenakis), m2 = 4 sixteenths, m1 = 3 beats ∙ m2 = 12, and n = number of a particular sixteen in a beat. The third sixteen of the second beat in the 7th measure will then be: 6 ∙12 + 1∙ 4 + 2 = 78 (7) (first member of each group is always 0). If instead we consider m2 the number of all dynamic levels in a piece, m1 the number of all available pitches ∙ m2 , n a particular dynamic level, k2 a particular pitch, and k1 a particular instrument in a group of size ≥ k1, we can create orchestration constraints. The period of such a sieve will be m1 and its modulo numbers will have to be divisors of m1 ∙ total number of instruments.