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

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.