Residue-class sets in the music of Iannis Xenakis: an analytical algorithm and a general intervallic expression.

Publication: Perspectives of New Music

Publication Date: 22-JUN-01

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COPYRIGHT 2001 University of Washington

IN A 1990 ARTICLE entitled "Sieves," IANNIS XENAKIS elaborated upon the description of sieve theory he gave twenty years before in "Towards a Metamusic," which was also included in his monograph, Formalized Music. (1) XENAKIS'S "sieves" are custom-designed collections constructed from interwoven chains of elements (pitches, rhythms, timbres, textures, or iterations in some other parameter) separated by congruent intervals. The "sieve" label is a metaphor for the set-theoretical filtration process involved in restricting the compositional material to members of carefully chosen sets that exhibit internal intervallic repetition at one or more levels. Constituting an open invitation to achieve informed analysis of his own works, Xenakis's thorough formalization of sieve theory has been largely neglected as an analytical tool. (2) A complete description of a multi-sieve composition, or of any piece in which reiterative structures are evident to some degree, would explore relationships both within a given sieve an d between different sieves heard at different times. The twofold purpose of this study, then, is to suggest an algorithm with which to reach a description of any collection as a sieve and to formulate a general expression of the intervallic distance between two collections in terms of their structure as sieves.

A sieve may be constructed out of one or more chains of repeating intervals in some parameter. In Example 1, three such chains are postulated; each is identified by an ordered pair containing the size of the repeated interval (measured as some integral multiple of an elementary unit of displacement, such as a semitone) followed by the "starting value," the transpositional level with respect to some zero value. (3) Such an ordered pair defines a residue class, a set of values all of which yield the same remainder, or residue, when subjected to division by some number. The elements of the residue class in Example la are those values which yield a remainder of 3 when divided by 7. Envisioning these values as the rungs of a ladder, it can be seen that each rung is exactly 7 units away from its nearest neighbors. Example 1b features a different mod-7 residue class, the elements of which yield 5 under reduction mod 7. The residue class in Example 1c features a different unit of modular repetition; the rungs of this ladder are separated by 8 units, and the index of transposition is 2.

By combining individual residue classes (henceforth RCs), Xenakis transforms their pedantic monotony into an astonishingly powerful and varied vocabulary of collections featuring complex patterns of interference between the various units of modular repetition. (4) Example 2a features a superimposition of the three RCs from Example 1, to which I will refer as a "residue-class set," or RCset. The periodicities of the three RCs, easily perceivable in isolation, are masked in combination, as a listener will attest for whom the resulting RCset is clapped at a fast tempo, or realized (as in Example 2b) in a semitonal space. Notice that the leftmost sequence of values recurs at 58 (that is, 56 units later than its first appearance); this is because the periodicity of an RCset (in this case, 56) is equal to the lowest common multiple of the modular values of its constituent RCs (in this case, 7 and 8).

The character of the scale given in Example 2b strongly evokes much of the scalar material of Xenakis's 1990 string quartet, Tetora. Published in the same year as his "Sieves" article, Tetora is a continuous seventeenminute work dedicated to the Arditti String Quartet, its first interpreters.5 Generally speaking, Tetora falls into short sections demarcated by sudden and audible changes of pitch collection. Because the rhythmic character of the piece is for the most part fairly mundane and the tempi fairly slow, these pitch-collection changes are easily heard by even a novice listener. Within a given section, Xenakis circulates within one collection in an uncomplicated way, with much stepwise motion, with the effect that the whole collection is relatively easy to perceive aurally. The opening collection, for example, lasts for 26 measures, just less than one fifth of the quartet's 137-measure total. In Example 3 (measures 16-21), only eight pitches--discounting the final sixteenth note--are heard, in a quasi-o ctatonic scalar fragment: A4, B4, C#5, E5, F5, G5, A#5, and C#5.

In analyzing these collections individually, it would be valuable to have at one's disposal an algorithm that was sensitive to the modular repetition of intervallic quantities within the collection--in other words, a tool with which to gauge the presence or absence of entire RCs. Xenakis indeed provides such an algorithm in his 1990 article, which also includes a computer program in C that implements the algorithm. He describes his algorithm as follows:

(a) Each point is considered as a point of departure (= [I.sub.n]) of a modulus.

(b) To find the modulus corresponding to this point of departure we begin by applying a modulus of value Q = 2 unities. If each one of its multiples meets a point which has not already been encountered and which belongs to the given sieve, we keep the modulus and it forms the pair ([M.sub.n], [I.sub.n]). But if any one of its multiples happens not to correspond to one of the points of the series, we abandon it and pass on to Q + 1. We proceed so until each one of the points in the given series has been taken into account.

(c) If for a given Q we garner all its points (Q,[I.sub.k]) under another pair (M, I), that is if the set (Q,[I.sub.k]) is included in (M, I), then we ignore (Q,[I.sub.k]) and pass on to the following point [I.sub.(k+1)].

(d) Similarly, we ignore all the (Q, I) which, while producing some of the not-yet-encountered points of the given series, also produce, upstream of the index I, some parasitical points other than those of the given series. (6)

Xenakis's algorithm searches for "perfect" occurrences of interior interval-repetition within a given collection, and then discounts them from future consideration as successively larger modular quantities are evaluated. This algorithm is flawed in two respects, one trivial and one more serious. (7) First, discounting points as soon as they have been accounted for puts any sieve with at least one point of intersection between its RCs outside the bounds of this algorithm's effectiveness. The RCset in Example 2a would be scanned for component RCs of increasing modular size. Example 4 shows what would be left after the two mod-7 RCs had been discounted. Xenakis's algorithm would have to account for Example 4 with an awkward conglomeration of five mod-56 RCs, rather than a single mod-8 RC. This problem would be easy to fix.

More seriously, however, Xenakis's algorithm might give a misleading result if the collection under examination had been altered at all from it original status as an RCset, or if it had been intuitively or randomly constructed rather than calculated. This obstacle is alarming because Xenakis's own sieve-realizations in his compositions are far from normative. Xenakis has conceded that he sometimes requires certain adjustments, above and beyond the structure that emerges from combining the RCs, to conform to his own personal criteria of interest. (8) Not only are hi RCs selected according to taste, so as "to be not too symmetric (regular) nor too empty, interesting from the point of view of the scale, that is" (9) but his intuitive criteria of interest demand further alterations that distort the original structure of the sieve. In other words, after the desired RCs coalesce into an RCset, certain musical decisions are then made about the end result by the composer.

Jan Vriend, in a footnote to his 1981 analysis of Nomos Alpha, Xenakis's 1966 piece for solo cello, concedes the near impossibility of working backward from score to sieve (Vriend calls sieves "grids"). (10) Most intriguingly, Vriend cites personal communication with Xenakis in which the composer acknowledges the pluralistic, if not entirely arbitrary, nature of the analyst's decisions.

I had a hard time trying to find only the first grid L (11, 13) from the different sources I had at my disposal. I produced 6 different grids out of them, i.e. none was conform the other [sic]. I then produced a seventh to satisfy myself that this was the only good one. In a letter Xenakis acknowledges the confusion and touchingly sends me an eighth, strikingly different again..., adding to my collection to be sure, but not to the truth, I'm afraid. And as if he cannot be stopped, he concludes with the refreshing remark that "here too are errors and/or adjustments..." True, checking the score with each of the possible grids produces a favorite one on statistical grounds, but definitely produces more adrenaline in the researcher's blood... (11)

It may be difficult for the analyst to come to terms with such fundamental tampering, especially when faced with the daunting task of reconstructing the original sieve expression. Vriend thinks that "to conceive of a machinery as sophisticated as the one we have been constructing so far, with all its philosophical pretensions and historical ambitions..., and then to destroy it with purposely planned deviations and confusions, would...

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