# 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.

(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 likewise destroy the value of the whole conception." (12) To circumvent such a sense of analytical frustration, one may consider Xenakis's sieve-structure perturbations as the signature of a man rather than a machine. (13) In the "provisional conclusion" to his 1990 article, Xenakis describes the theory of sieves in a telling light, emphasizing the descriptive nature of the language with which sieves are defined: sieve theory "studies the internal symmetries of a series of points either constructed intuitively, given by observation, or fabricated completely by moduli of repetition. " (14) In phrasing the definition this way, Xenakis removes the burden of proof from his own shoulders and those of the analyst.

Since the mental algorithms governing Xenakis's sense of musicality would be very difficult to reproduce, the analyst must be content to find a statistically optimal combination of RCs with which to account for a given collection. (15) Unless the collection is an unadulterated RCset (an assumption that underlies Xenakis's algorithm), some compromise will be necessary in order to arrive at a useful, believable description of the collection as a superimposition of RCs. A few basic assumptions are required concerning one's level of tolerance (either aurally or analytically) for imperfectly realized RCs. Together, these assumptions constitute an alternative algorithm that can provide an interpretation of a modified RCset with greater efficacy than Xenakis's algorithm.

The first of these assumptions dictates that an RC must be represented by at least three points within the collection as realized. In order to be able to observe some kind of periodic recurrence, at least three points of reference are required, so that the two intervals between adjacent points can be compared and judged to be equal. This three-iteration condition eliminates from consideration all RCs whose modular values are equal to or greater than 1/2 of the range of the collection, as well as any RCs with moduli equal to or greater than 1/3 of the collection's range and an index of transposition so high as to push the highest of three points beyond the collection's range. For example, within the 88-note tessitura of the piano keyboard, this condition would be satisfied by (36, 4)--since this RC gives rise to three playable pitches, namely C1, C4, and C7--but not by its isomodular cousin (36, 28). (16)

Given this three-iteration condition, it is equally reasonable to discard from consideration any RC that is borne out in the music to a degree less than 2/3. If it is less than 2/3-present, its effects would more likely be heard as a different RC of a larger modulus, or as spurious additions to the values stipulated by other RCs, rather than as an independent pattern. As implemented in the computer program that accompanies this study, my algorithm requires in fact an even greater fraction of presence in situations involving exceptionally densely saturated collections or exceptionally large modular quantities. In a collection whose members outnumber the gaps between them by 10 to 1, for example, many different RCs will be represented to a degree of 2/3 or more. Only those RCs whose percentage-presence exceeds the saturation of the collection as a whole should be considered. In addition, one could forgive fewer exceptions in the appearance of a mod-23 RC than of a mod-3 RC; the greater the modular value of an R C suggested in partial explanation of a collection, the more rigorous its realization should be. In order to pass this substantial-presence test, then, an RC must be present to a degree no less than 2/3, no less than the ambient saturation of the collection, and no less than m-1/m , where m is the modular value associated with the RC.

We may now eliminate any redundant subsets of RCs still under consideration. For example, if the RCs (3, 0) and (6, 0) had both passed the earlier requirements, then (6, 0) would be eliminated if its percentage-presence was no greater than that of (3, 0). Any other subsets of (3, 0) that had survived the earlier weedings-out (possibly including (6, 3), (9, 0), (9, 3), (9, 6), etc.) would be subjected to the same scrutiny. This culling of redundant subsets is justified by the assumption that someone who could correctly identify a whole-tone piece as such would likely do so instead of speculating about multiple interlocking cycles of tritones, minor sixths, or minor sevenths.

Every combination of the remaining, by-and-large plausible RCs must now be rated in terms of its respective "success" in accounting for the actual collection. In order to dethrone a temporarily-reigning RCset, my algorithm requires the challenger to account for a significantly higher number of points; it also disqualifies from promotion any RCset that suggests more "wrong notes," more errors, than the current favorite. However, we might wish to ignore a slight improvement in predictive success given by a very complex RCset if the cognitive effort to grasp the many layers of regularities is too great. The criteria by which we judge the "complexity" of such a construction are not obvious, but would seem to depend on both the cardinality of the RCset and the modular sizes of its component RCs. These factors are reflected by both the sum and the product of the RCset's constituent modular quantities, and either measure might be adopted as an index of "complexity." But merely to compare two RCsets by their respecti ve modular sums is to pass perhaps too harsh a judgement against higher modular quantities (ruling, for instance, in favor of an RCset of moduli 3, 5, 11, 13, and 14 over one of moduli 20 and 30), while comparing RCsets by their modular products may be too lenient towards higher modular values (ruling in favor of an RCset of moduli 23 and 31 over one of moduli 8, 9, and 10). A blending of the sum-measure and the product-measure is achieved by adding the average of an RCset's modular values to the quotient of their product and their sum--in other words, S/C + P/S, where C is the cardinality of the RCset, P is the modular product, and S is the modular sum. This formula provides a BIAS-value that falls between the sum-measure and the product-measure and discriminates satisfactorily (as will be seen shortly) between competing RCset descriptions of a collection. To select a particular combination of RCs, then, the algorithm factors the BIAS-value of each RCset into the measure of its predictive success, and bases its choice upon this tempered measure.

Following is a concise summary, in a manner similar to Xenakis's description of his algorithm (given above), of the above analytical steps; the program in BASIC that appears in an appendix to this paper duplicates these procedures.

(a) Three-iteration condition:

Discard any RC with fewer than three iterations within the realized range.

(b) Statistical remarkability:

Discard any RC borne out in the music to a degree less than 2/3 , less than the collection's saturation, or less than m-1/m, where m is the RC's modular value.

(c) Non redundancy:

Eliminate any RC that is completely included in another RC still under consideration.

(d) Weighted RCset comparison:

Compute bias-values (average + product/sum of modular values) for all combinations of the remaining RCs; pick the particular combination of RCs that is most successful, for the size of its bias-value, in accounting for the collection.

As a test of this analytical algorithm, and particularly of my BIAS formula, I will analyze a repeated rhythm that appears several times throughout Xenakis's Tetora. The rhythm in question is first heard in measures 4 to 7, and is subsequently heard several times in various forms through the piece. Example 5 illustrates the climactic triple presentation of this rhythm at the end of the piece, where its return coincides with a marked tempo change, as well as the startling resumption of rhythmic unison and a dense, homophonic texture. As marked in Example 5, the rhythm is heard twice in succession, then once in retrograde: the end of the rhythm's second iteration overlaps with the beginning of the third.

Example 6 contrasts the twenty-six points of attack of the rhythm with the points suggested by the RCset {(3, 0), (4, 0), (5, 0)}, the "answer" returned by my algorithm after the rhythm was entered as data. Only five of the twenty-six points in Xenakis's collection (points 19, 23, 29, and 31) are foreign to the RCset {(3, 0), (4, 0), (5, 0)}. It is of less consequence that four values (points 6, 21, 24, and 36) are nor represented in Xenakis's rhythm because there is, of course, no stricture in effect dictating that every point in a background sieve-collection must be touched upon. It is not my current aim to explain the deviations, nor to ascribe any sense of correctness or truth to this explanation, but merely to note the simplicity of the sieve-expression with which we have accounted for so great a proportion (21/26, over 80%) of the rhythmic attacks.

By tracing the inner workings of the algorithm, we reveal the decisions it made in order to arrive at this explanation for the rhythm. Steps (a) through (c) of the algorithm, as given in the summary above, succeeded in narrowing the field of viable RCs down to nineteen candidates, ranging in size from (3, 0) up to (20, 0). Combinations of these were then evaluated, and the algorithm's BIAS suggested {(3, 0), (4, 0), (5, 0)} as the best choice, in terms of simplicity and descriptive efficacy. In order of increasing descriptive accuracy, and increasing complexity, the seven RCsets from among which ((3, 0), (4, 0), (5, 0)} was chosen are shown in Example 7. The second column of Example 7 gives a percentage value indicating the RCset's predictive success of the rhythm's overall membership, both points of articulation and points between articulations. The third column gives the number of articulations correctly accounted for by each RCset candidate. The fourth column gives the BIAS-value for each set of moduli. It can be seen, then, that {(3, 0), (4, 0), (5, 0)} was chosen from among these seven possibilities because the ratio of its BIAS-value to that of any other RCset was always less than the ratio of their respective tallies of correctly-predicted articulations. The algorithm compared the BIAS-value of 9.00 to that of the next RCset, 34.45, and decided that a gain of 1 out of 26 points was not worth the increased complexity. (17)

Many of Tetora's adjacent collections turn out to be transpositionally related. Of the forty-seven measures between measures 38 and 85, thirty-six measures are relatively strict iterations of one clearly articulated collection, transposed up or down by small intervals (four semitones or smaller) from its original transpositional level. The first nineteen measures (measures 38-56) of this section are heard at the same level (the same pitches are heard for nineteen measures, with a few anomalous exceptions in measures 50-6) before the collection is transposed to nearby pitch levels. These transpositional changes, and most of the other collectional "modulations" throughout the piece, are characteristically abrupt. The music in Example 8a passes through three well-established scalar collections (as marked), occurring respectively from measure 59 to the middle of measure 62, from there to near the end of measure 65, and from the last note of measure 65 to the middle of measure 68. Scalar representations of the thr ee collections are given in Example 8b. Each of these three collections are transpositions of nearly identical subsets of the firmly-established collection in measures 38-56; compared to measures 38-56, the first of the three collections has been transposed down two semitones, the second up three semitones, and the third down one semitone. (18)

An intriguing and unexpected connection between the dimensions of pitch and rhythm can be observed in this passage. Whether or not the recurring rhythm discussed previously originated as the RCset {(3,0), (4,0), (5,0)}, it is manipulated here in a way that would seem to confirm its origin in Xenakis's mind as a sieve. In measures 59 and 65, the rhythm undergoes what Xenakis calls "metabolae" at points exactly coincident with the changes in pitch collection. A metabola is any kind of modification in the structure of a sieve, whether a change in the moduli, the transposition indices, the unit of elementary displacement, or the logical operations that connect them. At measure 59, each rhythmic value--either one, two, or three sixteenth-notes in length--is augmented by one sixteenth-value, and taken "mod 3-sixteenths" (in other words, a sixteenth-note becomes an eighth, an eighth becomes a dotted eighth, and a dotted eighth becomes a sixteenth). When the rhythm is completed, partway through measure 65, the same o peration is performed on the new rhythm. Example 9 documents the metabolae involved.

Before submitting any data to the algorithm, our goal must be to relate as many of the discrete pitch collections to one another as possible, so as to consolidate groups of related collections into individual, fundamentally different scales. This is accomplished by observing congruences at different transpositional levels, as well as the embedding of certain collections within other larger collections. Example 10 is a table of pitch collections throughout the piece, and explanations of their interrelations by inclusion and/or transposition; some measure numbers are qualified by eighth-note numbers. Example 11 provides scalar representations of the six main collections; measure numbers are provided to distinguish between multiple presentations of a particular collection.

The application of my analytical algorithm to the six main collections (as given in Example 11) reveals certain RCsets in evidence. Example 12 provides suggested RCset explanations for the six collections; in each case, the indices of transposition are measured relative to the lowest pitch in each collection. The third column gives a percentage value indicating the RCset's predictive success of the collection's overall membership--pitches both present and absent. The fourth column gives the number of pitches correctly accounted for by each RCset candidate.

While the percentages given in the third column of Example 12 are all "passing grades," not all of the fractions in the fourth column--representing the number of notes in the collection with which the suggested RCset agrees--seem very encouraging. Either of the two columns can be taken to indicate the degree of perversion from a basic RCset framework. Such perversion might take the form of saturating a sparse RCset with extraneous elements, excising certain elements of the RCset, or (as can be seen in the case of the recurring rhythm in Example 6) the slight shifting of certain elements in one direction or another without any great change in the degree of saturation. In the numerical representations of our RCsets' success rates, then, we have a measure of the severity of those alterations.

If, in an earlier stage of the analysis, two realizations of the same collection had been misinterpreted as two different collections, an analyst might hope for the algorithm to reveal underlying similarities. Indeed, Collections V and VI elicit exactly the same RCset suggestion from the algorithm, {(2, 0),(3, 2)}. The two collections, however, sport huge differences in their pitch content, as seen in Example 11. The regularities that the algorithm observed in Collection V (the even whole-tone scale and the diminished-seventh chord built on D) are not in evidence at the same pitch level in Collection VI, in which the entire span from E5 up to F6 is simply a long chromatic "smear." Collection V recurs without any alterations ten measures after its initial presentation. The divergence between Collections V and VI must preclude an assumption of their equivalence, despite the resemblance of the algorithm's RCset descriptions.

The inverse case would involve two collections that are believed to be slightly different versions of each other, but which are assigned different RCsets by the algorithm. Collection VI reappears at its original pitch level in measures 128-37; no other collection has been heard since measure 112, and its transpositions have diverged only minimally from the original collection. The similarity of the two versions of Collection VI is obvious from a comparison of Examples 13a and 13b, yet instead of suggesting {(2, 0), (3, 2)} as it did for measures 101-8, the algorithm suggests the RCset {(3, 2), (4, 3)}. Out of the 35 pitches in measures 101-8, 9 do not reappear in measures 128-37, and 6 out of 32 pitches in measures 128-37 are not found in the original version of the collection; these differences must have disturbed the fine balance struck between the 33 different RC candidates that the algorithm had considered in measures 101-8. It should be recalled that the algorithm does not take into account such perceptu al clues as pitches in common, tessituras in common, or the shared chromatic smear noted above; it is merely a tool with which to measure the interior periodicities of a collection as a whole, and to formulate a suitable combination of several RC candidates.

From Example 12 one can observe the following: (1) a recurring sense of diminished-seventh frameworks (mod-3 RCs playing eminently perceivable roles in Collections I, IV, V and VI), (2) the apparent decrease throughout the piece of extraneous pitches (notice the increasing fractional representations of predictive success in the fourth column), and (3) an encouraging correspondence between the RCset suggested for Collection III and Xenakis's known sieve tendencies. The modular values suggested for Collection III, 11 and 13, are in fact the same two modular values with which Xenakis begins Nomos Alpha, his landmark 1966 piece for solo cello, which he himself analyzes at many levels in Formalized Music. The suggestion by my algorithm of modular values 11 and 13 in explanation of Collection III might be a clue as to its analytical prowess.

According to Xenakis, the arrangement in time of sieves in a piece of music may engender its dramatic form, defining the structural demarcations of the piece. In a discussion of the processes that determine his pitch collections, Xenakis identifies the sieves used throughout Nomos Alpha, which circulate among prime-number modular values 1, 5, 7, 11, 13, and 17. (19) These are the only integers less than 18 that share no common factors with 18; thus they form a commutative group of elements whose products, taken mod 18, yield only other members of this group. For example: (5 x 11) mod 18 = 1; (7 x 17) mod 18 = 11; (11 x 13) mod 18 = 17. A 24-item cycle is generated from an initial pairing of modular values 11 and 13, comprising the vocabulary of modular values in Nomos Alpha: 11-13-17-5-13-11-17-7-11-5-1-5-5-7-17-11-7-5-17-13-5-11-1-11. To generate this cycle, the product of two adjacent prime numbers (taken mod 18) is notated to their immediate right, and then itself multiplied by the number to its left. Cont inuing this recursive multiplication pattern, one finds that after its 24th value the cycle would repeat from its beginning (continuing on from the end of the above sequence with 11, 13, 17,...), but Xenakis ends the piece just in time to prevent the recycling of the cycle. (20) Thus, Xenakis's in-time program of a "cinematic" progression leading towards what he calls "closure" is realized. (21)

The goal of this section of my paper is to arrive at an intervallic expression that could map the perceptual distances traversed between sieve collections throughout a piece. It would be unrealistic to hope that a strictly numerical expression of the arithmetical differences between sieve parameters could ever match the elegance of the prime-number scheme of the Nomos Alpha moduli. Nevertheless, a listener who fails to perceive the prime-number cycle at the deepest level of Nomos Alpha's sieve-structure deserves an analytical expression that accurately models the sensations experienced in the audience, instead of a compositional scheme virtually imperceptible even to a sieve theory initiate. More importantly, such an intervallic expression would be equally useful in the analysis of later works, like Tetora, in which Xenakis has abandoned (by his own testimony) the kind of long-term arithmetic connections that he identifies in Nomos Alpha. (22) In the absence of an imposed organizational framework like the pri me-number cycle, such an intervallic measure can provide perhaps the only model of the large-scale relationships between adjacent or non-contiguous sieve collections.

For these purposes, any consideration of transposition indices of RCs is irrelevant, as Example 14 suggests. It is more appropriate, and perceptually more relevant, to consider all isomodular RCs together, envisioning an endlessly-repeating pattern of binary code. A realization in semitones of the RCset in Example 14a is inevitably heard not as five interlocking, superimposed chains of major sevenths, but as a series of diatonic fragments ("do-re-mi-fa-sol") recurring consistently at the interval of a major seventh. Since small intervals and conjunct patterns are in general more readily noticeable and identifiable than large and disjunct ones, the listener would automatically concatenate the clumps of pitches, rather than considering the five RCs as separate entities. Similarly, Example 14b is more readily classified as recurring chromatic pentachords than as five interlaced chains of major sixths. If these two pentachordal patterns were combined, as in Example 14c, the highly complex result would be understo od in terms of the interaction between the two pentachords. Since every possible transpositional superimposition of the two pentachords will occur at some point within any given 99-semitone span (the product of the modular values 9 and 11), the particular pattern of interference at a certain place--though it might be distinctive in some way--tells us less than the actual modular values themselves. Without an arbitrarily-assumed zero-point, the question of relative transpositional levels of the repeating modular patterns is moot.

What may prove more wrenching an aspect of the "modulation" that occurs between two RCsets is the change in their respective periodicities. As shown in Example 15, however, identifying RCsets solely by their periodicity does not by any means afford a unique representation of their modular values. Even though both sieves shown in Example 15 have a periodicity of 210 units, the patterns of interference on a local level are noticeably different between the two RCsets. Since 14 is very close to 15, the RCset in Example 15a will only gradually show divergence between its mod-14 component and its mod-15 component (no matter how many mod-14 or mod-15 RCs are in fact present). Conversely, the mod-10 and mod-21 patterns in Example 15b will exhibit a "two-voice" (or "second-species") pattern of divergence. Since 10 is very nearly half of 21, every other iteration of the mod-10 residue classes will intersect with the mod-21 RCs in similar ways.

In measuring the interval between two RCsets, then, it is insufficient to refer to their locations in RCset-space by their periodicities alone. To describe the particular pattern of interference among the members of a set's RCs, it would be useful to have a tool with which to refer to the intervals between an RCset's modular quantities. The tool that leaps most naturally to mind is Allen Forte's interval vector. (23) Notated within square brackets, Forte's interval vector provides a tally of a set's total unordered interval-class content. In contradistinction to Forte's, however, the interval vector that we will use for our collections' modular values must be of indeterminate length (since a sieve whose moduli are very widely separated in value is conceivable), and its elements must be of indeterminate size (because we might have a very large number of mod-n RCs for which to account). Our new construction, the moduli interval vector (or MIV), would express an RCset of moduli 3, 5, and 8 as [01101], since ther e is one interval of 2 units (between 3 and 5), one interval of 3 units (between 5 and 8), and one interval of 5 units (between 3 and 8).

Two such expressions are indeed comparable by superimposing them and comparing the binary values in corresponding places. More concisely, however, Example 16 illustrates a procedure by which the MIV may be expressed as one number, so that we may compare MIVs simply by integer division. The second row of numbers in Example 16 is the MIV just mentioned, [01101]; the third row lists consecutive prime numbers beginning with 2. The numbers in the fourth row are calculated by applying the values in the second row as exponents to the corresponding elements in the third row. Multiplied together, these fourth-row values provide a unique single-integer representation of any MTV or indeed of any ordered one-line array of positive integers. This product will henceforth be referred to as MIVPOP, in abbreviation of "product of primes." MTVPOP is a unique representation of any MIV precisely because all its factors are prime; Euclid's Unique Factorization Theorem dictates that for any positive number there can be only one se t of prime factors whose product is the number. MIV is easily derivable from this product of primes, MTVPOP, simply by checking how many times MIVPOP is divisible by 2, 3, 5, et cetera. In addition, two MIVPOPs are directly comparable to each other by taking their quotient, since each prime factor will affect only its corresponding prime in the other MIVPOP. In a single rational number, this quotient informs the analyst of the differences in interval content of the set of modular values of two RCsets.

Just as the periodicity of an RCset cannot afford it a unique representation of the modular quantities involved, however, the MIV of the RCset by itself is also insufficient because, for example, the same MIV would apply equally to an RCset with moduli 3, 5, and 8 and to an utterly different one with moduli 31, 34, and 36. We might hope that listing the periodicity of an RCset together in an ordered pair with the corresponding MIV would guarantee a unique expression. The modular content of many RCsets is in fact described uniquely by such an ordered pair, but in certain cases two different RCsets of the same periodicity can have the same MIV. The periodicity of the repeating-rhythm RCset examined earlier, {(3, 0), (4, 0), (5, 0)}, for example, is 60, the lowest common multiple of its modular quantities. Unfortunately the lowest common multiple of the numbers 4, 5, and 6 is also 60, and an RCset of moduli 4, 5, and 6 would have a MIV identical to that of the 3-4-5 RCset.

Recognizing this, we will simply use the product of the modular quantities (which I abbreviate to PROM) instead of their lowest common multiple, and append to it our intervallic descriptor, MIVPOP, in an ordered pair: (PROM, M1VPOP). The perceptual salience of this measure is admittedly somewhat compromised in the case of an RCset of an observable periodicity that differs from the product of the modular quantities--that is, an RCset whose moduli are not coprime. Nevertheless, if two RCsets are described by the same modular product and by the same modular interval vector (and thus by the same MIVPOP), the sets of their respective modular values will be identical. (24) Moreover, two RCsets, S1 and S2, may now be compared by comparing the two elements of the respective ordered pairs (PROM1, MTVPOP1) and (PROM2, MIVPOP2). The interval between S1 and S2 is expressed as (PROM2 / PROM1, MIVPOP2 / MIVPOP1). Since this interval locates a unique, specific set of modular values a certain "distance" away from those of a departure RCset, intervallic "roadmaps" or networks of any configuration may be constructed to link together all the collections in a piece. Because the analytical method for deriving the modular values is applicable to any collection, regardless of whether or not it originated as an RCset, such a network can be used to represent the changes throughout a piece of music in the degree of interior interval-repetition or modularity of any musical parameter.

It would be highly desirable to compare the large-scale picture given by a knowledge of the prime-number cycle in Nomos Alpha with a map of intervallic connections separating the sieves involved. We have restricted ourselves, however, to the consideration of musical works throughout which the unit of elementary displacement (the smallest unit of parametric division, such as the semitone) remains constant. Similarly, Xenakis's 1990 algorithm assumes a single unit of elementary displacement, and his 1990 string quartet, Tetora, remains strictly within semitonal space. It is impossible to try out this new tool on Nomos Alpha, however, because the unit of elementary displacement in Nomos Alpha is not the same for the whle piece. Similar to the binary oppositions in Xenakis's compositions Duel and Strategie (both 1959-62) in which two small orchestras "compete" according to game-theoretical principles, this division of Nomos Alpha destroys the equal footing of a uniform unit of elementary displacement necessary fo r the comparison of different collections.

Since Tetora, on the other hand, maintains a solely semitonal basis throughout, we can compare collections and thus model the changes between pitch-material totalities. In its third column, Example 17 provides such an ordered-pair description of the six main collections in Tetora (as identified in Example 11); the collections are listed in score order, in order to reflect the reprise of Collection I, for example, before the first introduction of Collection W In the fourth column, the interval is calculated for each pair of adjacent collections.

My agenda for this paper concludes with the suggestion of a longer-term application of this intervallic function. Without any attempt to justify or rationalize the hierarchical subjugation of some collections and not others, an interpretation at which a listener or analyst might conceivably arrive is advanced in which the sequence of collectional changes in Example 17 is condensed into an hierarchical structure featuring several levels of "departure and return." (25) The largest level involves Collection I departing to and returning from Collection IV, as in Example 18. Both of these collections may themselves be heard as ensconcing other collections' appearances: for example, the interjections of Collection V in measures 58 and 68--transpositionally equivalent scales of 14 sixteenth-notes ascending from A#4 and B4, respectively--are, as seen in Example 19, notably brief, high in pitch, and texturally sparse in comparison to the surrounding harmonic density of Collection IV. An independent final section is in itiated with the introduction of Collection VI in measure 101, and the unprecedented level of rhythmic complexity that accompanies it (as seen in Example 20). Collection VI remains unchallenged throughout the last 26 measures, perhaps reflecting the first 26 measures of the piece, which featured continuous exposure to Collection I. (26)

Any such succession of non-contiguous collections--either the largescale interpretation just advanced or any other that seems subjectively more satisfactory--can be described with the same intervallic function as previously introduced to gauge the degree of disjunction between adjacent collections. Example 18 documents the long-range intervallic connections between the collections that I have described. Pleasingly, the ordered-pair expressions of Collections I and IV are exactiy the same, so Collection IV does not disturb the surrounding sense of the mod-3 basis.

The goal of this paper has been to provide a tool with which to assess and quantify the degrees of interior interval-repetition within a piece's pitch collections, or other collections of elements in any parameter defined by a unit of elementary displacement. Such an analytical approach may be applied not only to the compositions of Xenakis but also to many other areas of compositional practice. Fruitful inquiry may be made into the oeuvres of composers especially concerned with interval cycles, recurring pitch patterns or rhythmic figures, or other periodic iterations of a basic musical seed. Underlying intervallic regularities in some parameter--scalar structure, metric or hypermetric periodicities, or largescale schemes of tonal progression--may bring insight into the structure of any piece of music in new, yet-unexplored ways.

An analyst who accepts Xenakis's tenet on which his theory of sieves is built--that the perception of a basic intervallic unit, and multiples thereof, is fundamental to all musical perception--must concede the necessity of a tool with which to measure modular repetition such as the algorithm I have outlined. Further, the temporal evolution of a piece to which such a tool is applicable can be described with the intervallic expression introduced in this paper. Thus equipped, we have at our fingertips a powerful analytical resource capable of revealing both intentional and incidental compositional structure.

[MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]
```APPENDIX

A Program in BASIC with which to reach an RCset approximation of a given
collection.

10 REM *** This program suggests a
residue-class set to describe a given ***
20 REM *** collection. Written by
Evan Jones, December 7-8, 1994, in
the ***
Facility of the Eastman School of
Music. ***
40 REM
99 REM *** Input collection ***
100 INPUT "Number of elements in
collection";ELEM
110 IF ELEM<1 OR ELEM<ABS(INT
(ELEM)) THEN 100
120 DIM MBRS (ELEM)
individually,in any order but
without"
140 PRINT "duplication. No negative
or fractional values can be
accommodated."
150 FOR N=1 TO EMEM
160 PRINT"No.";N;" ";;INPUT MBRS(N)
170 IF MBRS(N)<>ABS(INT(MBRS(N))
THEN 150
180 IF MBRS(N)>MAX THEN MAX=MBRS(N)
190 NEXT N
200 PRINT "Original collection:";:
FOR N=1 TO ELEM:PRINT MBRS(N);:
NEXT N:PRINT
207 REM
208 REM *** Calculate percentage-
presence of all RCs, ***
209 REM *** and eleminate those <
50% or < saturation ***
210 SATN=100*ELEM/MAX:MODMAX=INT
(MAX/2):DIM BINAR(MAX),RC(MODMAX,
MODMAX)
220 FOR N=1 TO ELEM:BINAR(MBRS(N))=
1:NEXT N
230 FOR M=2TO MODMAX
240 FOR 1=0 TO M-1:IF 1>MODMAX-MTHE
310
250 X=1; PRES=0:ABST=0
260 IF BINAR(X)=1 THEN PRES=PRES+1
ELSE ABST=ABST+1
270 X=X+M:IF X<=MAXTHEN 260
280 PCTG+100*PRES/PRES+ABST)
290 IF PCTG>=100*(M-1)/MAND
PCTG>=200/3 AND PCTG>=SATN THEN
RC(M,1)=PCTG
300 NEXT 1
310 NEXT M
317 REM
318 REM *** For each modular level,
eliminate all multiples thereof
whose ***
319 REM *** percentage-presence is
less than that of the original ***
320 FOR M=1 TO INT(MODMAX/2):FOR
1=0 TO M-1
330 IF RC(M,1)=0THEN 380
340 FOR MX-M*2 TO M*INT(MODMAX/M)
STEP M:FOR 1X=1TO MX-M+1 STEP M
350 IF RC(MX,1X)>=RC(M,1)THEN 370
360 RC(MX,1X)=0
370 NEXT 1X:NEXT MX
380 NEXT 1:NEXT M
388 REM
389 REM *** Count all remaining RCs
***
390 FOR M+1 TO MODMAX;FOR 1=0
TO M-1
400 IF RC(M,1)>0THEN RCTOT=RCTOT+
1:REMAIN+1
410 NEXT 1:NEXT M:1F REMAIN=0THEN 0
420 DIM MLIST(RCTOT),1LIST(RCTOT),
COMB(RCTOT),TOPMOD(MODMAX),
TOPIND(MODMAX)
430 DIM SIVPTS(MAX):X=0:TOPCARD=
RCTOT:TOPBIAS=MAX*2
440 FOR M=I TO MODMAX:FOR I=0 TO
M-1
450 IF RC(M,I)>0 THEN X=X+I:MLIST
(X)=M:LIST(X)=I
460 NEXT 1:NEXT M
467 REM
468 REM *** Increase by one the
cardinality ***
469 REM *** of RCsets under
Consideration ***
470 CARD=CARD+1:IF CARD>RCTOT THEN
810
480 FOR X=1 TO CARD:COMB(X)=X:NEXT
X:COMB(CARD)=COMB(CARD)-1
487 REM
488 REM *** Examine every RCset at
current ***
489 REM *** cardinality in order of
modular size ***
500 FOR X=CARD TO 1 STEP - 1
520 IF COMB(X)<X+RCTOT-CARD THEN
530 OF CP,N(X-1)>=RCTOT+X-CARD THEN
550
540 COMB(X)=COMB(X-1)+2:FOR N=X+1
TO CARD:COMB(N)=COMB(X)-X:NEXT N
550 NEXT X:IF ADD=1 THEN 470
557 REM
558 REM *** Check to see if the
RCset under consideration
suggests even ***
559 REM *** as may points as the
temporary-favorite RCset explains
***
560 TOTAL=0:FOR X=1 TO CARD:TOTAL=
TOTAL+(MAX+1)/MLIST(COMB(X):NEXT X
570 IF TOTAL<TOPPTS THEN 490
580 FOR N=1 TO CARD:PRINT COMB(N);
CHR\$(20);:NEXT N:PRINT:PRINT CHR\$
(30);
587 REM
588 REM *** Compile all RCs under
consideration into one collection
***
589 REM *** so that it may be
compared to the original
collection ***
590 FOR WIPE=0 TO MAX:SIVPTS(WIPE)=
0:NEXT WIPE
600 FOR M=1 TO CARD:FOR PASS=0 TO
MAX
610 IF PASS MOD MLIST(COMB(M))=
ILIST(COMB(M)) THEN SIVPTS(PASS)=1
620 NEXT PASS:NEXT M
627 REM
628 REM *** Gauge overall
descriptive success-rate of
curent ***
629 REM *** RCset compared to that
of the reigning favorite ***
630 SCORE=0:PTS=0
640 FOR COMPARE=0 TO MAX
650 IF BINAR (COMPARE)=SIVPTS
(COMPARE) THEN SCORE=SCORE+1:PTS=
PTS+BINAR(COMPARE)
660 NEXT COMPARE
670 IF SCORE<=TOPSCORE FOR PTS<=
TOPPTS THEN 490
677 REM
678 REM *** Evaluate current RCset
in light of its BIAS-value ***
679 REM *** compared to that of the
reigning favorite  ***
680 MPROD=1:MSUM=0:MNUM=0
690 FOR M=1 TO CARD:IF MLIST(COMB
(M)=MLIST(COMB(M-1)) THEN 710
700 MPROD=MPROD*MLIST(COMB(M):MSUM=
MSUM+MLIST(COMB(M)):MNUM=MNUM+1
710 NEXT M:BIAS=MSUM/MNUM+MPROD/
MSUM
720 IF PTS*TOPBIAS/BIAS>=TOPPTS
THEN 490
730 IF (ELEM-PTS)*BIAS/TOPBIAS>=
ELEM-TOPPTS THEN 490
738 REM
739 REM *** Confirm current RCset
as regning favorite ***
740 TOPCARD=CARD:TOPSCORE=SCORE:
TOPPTS=PTS:TOPBIAS=BIAS
750 FOR M=1 TO CARD
760 TOPMOD(M)=MLIST(COMB(M):TOPIND
(M)=ILIST(COMB(M))
770 PRINT TOPMOD(M)"."TOPIND(M)" ":
780 NEXT M
790 PRINT SCORE/(MAX+1)" "PTS"/"
ELEM
800 IF SCORE<MAX+1 THEN 490
808 REM
809 REM *** Announce results of
algorithm ***
810 PRINT"This algorithm proposes
the following residue-class set:"
820 FOR M=1 TO TOPCARD
830 PRINT"(":TOPMOD(M):",";TOPIND
(M):")",
840 NEXT M:PRINT
850 PRINT 100*TOPSCORE/(MAX+1)"%
truth;:TOPPTS"out of":
860 PRINT ELEM"points accounted for
":END
870 PRINT"This algorithm cannot
recommend a residue-class set."
EXAMPLE 7

COMPETING RCSET DESCRIPTIONS OF THE RECURRING RHYTHM IN TETORA

RCset                          Overall %-score  Articulations predicted

{(3,0)}                             53.33            10 out of 26
{(3,0), (4,0)}                      66.67            17 out of 26
((3,0), (4,0), (5,0)}               75.56            21 out of 26
{(3,0), (4,0), (5,0), (11,5)}       77.78            22 oct of 26
{(3,0), (4,0), (5,0), (19,0)}       80.00            23 out of 26
((3,0), (4,0), (5,0), {11,9),       82.22            24 out of 26
(19,0)}
{(3,0), (4,0), (5,0), (11,9),       84.44            25 out of 26
(13,3), (19,0)}

RCset                          BIAS-value

{(3,0)}                         4.00
{(3,0), (4,0)}                  5.21
((3,0), (4,0), (5,0)}           9.00
{(3,0), (4,0), (5,0), (11,5)}   34.45
{(3,0), (4,0), (5,0), (19,0)}   44.52
((3,0), (4,0), (5,0), {11,9),   306.97
(19,0)}
{(3,0), (4,0), (5,0), (11,9),   2973.17
(13,3), (19,0)}
EXAMPLE 10

INTERRELATIONS OF THE PITCH COLLECTIONS IN TETORA

Measures/eighth-bears    Collection reference-numeral or
relationship to other collection

1-26                     Collection I (D2-G6)
27-30                    Collection II (C2-B6)
31/7 (all)-36/2 (vns)    Collection III (D#2-G#6)
34 (va, vc) -38/3 (all)  subset of Collection I,
untransposed
38/4-39                  Collection IV (E * 2-A#6)
40-42                    subset of Collection IV,
untransposed
43-50/7                  Collection IV, untransposed
50/8-56/2                Collection IV, untransposed;
anamalous notes in lyrical
56/3-57/2                lines
57/3-58/1                subset of Collection IV, transposed
down 1 semitone
58/2-58/8                subset of Collection IV, transposed
up 2 semitones
59-62/4                  Collection V (A#4-A6)
62/5- 65/7               subset of Collection IV, transposed
down 2 semitones
65/8- 68/4               subset of Collection IV, transposed
up 3 semitones
68/5- 69/3               subset of Collection IV, transposed
down 1 semitone
69/4-71/1                Collection V, transposed up 1
semitone
71/2-73/5                subset of Collection IV, transposed
up 1 semitone
73/6-76/5                subset of Collection IV, transposed
down 3 semitones
76/6-82/3                subset of Collection IV, transposed
up 2 semitones
82/4-83/2 (vns & Va)     Collection II, untransposed
82/3-82/6 (vc)           subset of Collection I,
untransposed
82/6-83/2 (vc)           subset of Collection II, transposed
up 1 semitone
83/2-85                  subset of Collection II,
untransposed
86-88/2                  subset of Collection IV, transposed
down 4 semitones
88/3-89/4                subset of Collection I, transposed
down 1 semitone
89/5-91/5                subset of Collection I, transposed
down 2 semitones
91/6-101/1               subset of Collection I, transposed
down 1 semitone a dense saturation
of pitch-space, stepwise chromatic
lines
101/2-108                hard to parse or relate to another
collection
109-111                  Collection VI (D2-G6)
slow chorale featuring parallel
dyads and between vns and between
va and vc; no pitch-recurrence and
no obvious relation to another
collection
112-116/5                Collection VI, untransposed
116/6-119/3              subset of Collection VI, transposed
up 1 semitone
119/4-122/3              subset of Collection VI, transposed
down 1 semitone
122/4-125/1              subset of Collection VI, transposed
up 3 semitones
125/2-128/4              subset of Collection VI, transposed
down 2 semitones
128/5-137                Collection VI, untransposed

* [MUSICAL NOTES NOT REPRODUCIBLE IN ASCII]
EXAMPLE 12

THE ALGORITHM'S EXPLANATIONS FOR THE SIX MAIN COLLECTIONS

Collection  RCset suggested  Overall %-score  Pitches predicted

I            {(3,2)}           61.11         12 out of 27
II           {(5,0)}           51.67         10 out of 37
III      {(11,0),(13,0)}       72.22          9 out of 24
IV           {(3,1)}           62.50         13 out of 28
V        {(2,0), (3,2))        75.00         12 out of 14
VI        {(2,0),(3,2)}        61.11         25 out of 35
EXAMPLE 16

CONVERTING MIV TO MIVPOP

Inter-modulus intervals  1     2     3     4     5      6      7      8
Tally thereof (MIV)      0     1     1     0     1      0      0      0
Corresponding primes     2     3     5     7     11     13     17     19
MIVPOP =                 1  x  3  x  5  x  1  x  11  x  1   x  1   x  1

Inter-modulus intervals     9 ...
Tally thereof (MIV)         0 ...
Corresponding primes        23 ...
MIVPOP =                 x  1 x ... = 165
EXAMPLE 17

INTERVALS BETWEEN ADJACENT COLLECTIONS IN TETORA DESCRIBED AS ORDERED
PAIRINGS OF THE MODULAR PRODUCT (PROM) WITH MIVPOP

Collection (and                         RCset from       Ordered pair
first measure no.)                       Example 12      (PROM, MIVPOP)

I   (1)                              {(3, 2)}           (3, 1)
II  (27)                             {(5, 0)}           (5, 1)
III (31)                        {(11, 0), (13, 0)}     (143, 3)
I   (34)                             {(3, 2)}           (3, 1)
IV  (38)                             {(3, 1)}           (3, 1)
V   (58)                         {(2,0), (3, 2)}        (6, 2)
IV  (59)                             {(3, 1)}           (3, 1)
V   (68)                         {(2, 0), (3, 2)}       (6, 2)
IV  (69)                             {(3, 1)}           (3, 1)
II  (76)                             {(5, 0)}           (5, 1)

[I and II (both 82) superimposed in
different instruments; omitted due
to question of order]

IV  (83)                             {(3, 1)}           (3, 1)
I   (86)                             {(3, 2)}           (3, 1)

[Measures 91-101: dense pitch-
space, hard to parse or to relate
to another collection]

VI  (101)                        {(2, 0), (3, 2)}       (6, 2)

[Measures 109-11: slow chorale, no
pitch recurrence and no reference
to other collections]

VI  (101)                        {(2, 0), (3, 2)}       (6, 2)

Interval from immediately
Collection (and                          preceding collection
first measure no.)                   (PROM2/PROM1, MIVPOP2/MIVPOP1)

I   (1)                                     -----
II  (27)                                   (5/3, 1)
III (31)                                  (143/5, 3)
I   (34)                                 (3/143, 1/3)
IV  (38)                                    (1, 1)
V   (58)                                    (2, 2)
IV  (59)                                  (1/2, 1/2)
V   (68)                                    (2, 2)
IV  (69)                                  (1/2, 1/2)
II  (76)                                   (5/3, 1)

[I and II (both 82) superimposed in
different instruments; omitted due
to question of order]

IV  (83)                                   (3/5, 1)
I   (86)                                    (1, 1)

[Measures 91-101: dense pitch-
space, hard to parse or to relate
to another collection]

VI  (101)                                   (2, 2)

[Measures 109-11: slow chorale, no
pitch recurrence and no reference
to other collections]

VI  (101)                                   (1, 1)
EXAMPLE 18

LONG-RANGE INTERVALLIC MAP OF TETORA

Collection (and       RCset from       Ordered pair
first measure no.)     Example 12     (PROM, MIVPOP)

I  (1)            {(3, 2)}          (3, 1)
IV (38)           {(3, 1)}          (3, 1)
I  (86)           {(3, 2)}          (3, 1)
VI (101)      {(2, 0), (3, 2)}      (6, 2)

Interval from immediately
Collection (and         preceding collection
first measure no.)  (PROM2/PROM1, MIVPOP2/MIVPOP1)

I  (1)                    -----
IV (38)                   (1, 1)
I  (86)                   (1, 1)
VI (101)                  (2, 2)
```

NOTES

Earlier versions of this paper were read at the 1996 meeting of the Music Theory Society of New York State (at SUNY Stony Brook) and at the 1997 Joint AMS/SMT Conference (in Phoenix, Arizona). I thank Nell Minturn and Robert Morris for their helpful comments on earlier versions of this paper.

(1.) Iannis Xenakis, "Sieves," trans. John Rahn, Perspectives of New Music 28/1 (Winter 1990): 58-78; "Vers une metamusique" La Nef 29 (1967); "Towards a Metamusic," trans. G. W. Hopkins, Tempo 93 (Summer 1970): 2-19; Formalized Music: Thought and Mathematics in Music (Bloomington: Indiana University, 1971; New York: Pendragon, 1991 (expanded edition)). The "Sieves" article is included in the revised edition of Formalized Music as Chapters 11 and 12.

(2.) Xenakis's own analysis of his 1966 solo violoncello piece Nomos Alpha (in Formalized Music, 218-36) has been expanded upon by a number of scholars. Both Fernand Vandenbogaerde (in his "Analyse de Nomos Alpha de Iannis Xenakis," Mathematiques et Sciences Humaines 24 (Paris: Centre de Mathematique Sociale, 1968): 35-50) and Jan Vriend (in "Nomos Alpha for Violoncello Solo: Analysis and Comments," Interface 10 (1981): 15-82) reiterate and comment upon the composer's sieve scheme for the piece. Thomas DeLio briefly alludes to the existence of sieves without explanation in "Iannis Xenakis' Nomos Alpha: The Dialectics of Structure and Materials," Journal of Music Theory 24/1 (Spring 1980): 63-95. Sieve theory is not addressed by Gilles Naud in "Apercus d'une analyse semiologique de Nomos Alpha" (Musique en Jeu 17 (January 1975): 63-72).

A few analyses of Xenakis's other works have made some use of sieve theory. Ellen Rennie Flint, in "An Investigation of Real Time as Evidenced by the Structural and Formal Multipilcities in Iannis Xenakis's Psappha" (Ph.D. diss., University of Maryland, 1989) cites the sieve expressions for the opening passages of Psappha as obtained from the composer's sketches. In his "Sonic and Parametrical Entities in Tetras An Analytical Approach to the Music of Iannis Xenakis" (Canadian University Music Review 16/2 (1996): 72-99), James Harley gives a general account of sieve theory, but does not analyze any of his examples as sieves. Ronald Squibbs, in "An Analytical Approach to the Music of Iannis Xenakis" (Ph.D. diss, Yale

University, 1996), gives sieve theory a very thorough exposition but does not attempt to achieve sieve expressions for any given excerpt of music.

(3.) This ordered-pair notation follows the notational convention of Xenakis's 1990 article. In earlier writings, Xenakis notated the transpositional index of a residue class as an exponent applied to its modulus. Thus (7,3) would have been represented instead as 73.

(4.) George Perle's "cyclic sets" (combinations of two forms of an interval cycle) and "derived sets" (combinations of interval cycles of different moduli) are the pitch-class cousins of Xenakis's sieves. Unlike Perle, Xenakis does not presuppose a mod-12 compositional space. See George Perle, Twelve-Tone Tonality (Berkeley: University of California Press, 1977; 2d ed. 1995). John Roeder's theory of pulse streams is also closely akin to sieve theory in that they both consider the juxtaposition of regularly reiterative event-classes. See his "Interacting Pulse Streams in Schoenberg's Atonal Polyphony," Music Theory Spectrum 16/2 (Fall 1994): 231-49.

(5.) Tetora is available from Editions Salabert (S. A. 22, rue Chauchat 75009 Paris, France) and has been recorded by the Arditti Quartet as part of a two-disc compilation entitled "Iannis Xenakis 1: Chamber Music (1955-1990) for strings, piano, strings and piano" (Montaigne Records MO 782005). The title of the piece (meaning "four" in the Dorian language) refers to the number of players, as do the titles of his two earlier string quartets, ST/4 (1955-62) and Tetras (1983; the title means "four" in Greek).

(6.) Xenakis, "Sieves," 64-5.

(7.) Aside from these flaws, there are numerous typographical errors in Xenakis's program listings (Formalized Music, chapter 12), of which Ronald Squibbs provides correction in Appendix I(288-300) of his dissertation, "An Analytical Approach to the Music of Iannis Xenakis."

(8.) Vriend, "Nomos Alpha for Violoncello Solo: Analysis and Comments," 64.

(9.) Vriend cites personal communication with Xenakis in "Nomos Alpha," 78 n. 14.

(10.) Vriend, "Nomos Alpha," 35-50.

(11.) Vriend, "Nomos Alpha," 79 n. 15.

(12.) Vriend, "Nomos Alpha," 54.

(13.) In the early 1960s, Xenakis generated a series of compositions (ST/4, ST/10; ST/48; Atrees; Morsima-Amorsima, and Amorsima-Morsima) on an IBM-7090 computer at the headquarters of IBM-France. His program, which is reproduced in Chapter 5 of Formalized Music, applies chance processes (controlled by predetermined biases) to compositional decisions. The absence of any mechanizable algorithm controlling Xenakis's modifications "to taste" in Nomos Alpha and other pieces distinguishes them aesthetically from the products of his computer program.

(14.) Xenakis, "Sieves," 66.

(15.) See David Lewin, "Some Investigations into Foreground Rhythmic and Metric Patterning," in Richmond Browne, ed., Music Theory: Special Topics (New York: Academic Press, 1981), 101-37. Lewin examines Schoenberg's op. 19 no. 6 and formalizes a model of the perception of ametric rhythm; this is analogous to the quest for observable periodicities within Xenakis's seemingly aperiodic collections, except for the ostensible origin of Xenakis's collections as a combination of regularities.

(16.) In Formalized Music (197-8), Xenakis describes three different pitch-class sets--a C-major scale, a mixed Byzantine scale, and the Raga Bhairavi of the Audava-Sampurna type ("Audava" is misspelt "Andara")--as the union of various RCs' intersections. Prior to his 1990 article, in fact, Xenakis's theory of sieves relied equally upon the three set-theoretical operations of union, intersection, and complementation, acting in consort upon his chosen RCs. Since the intersection of two RCs is just another RC, however, and since the complement of an RC is an RCset consisting of all other RCs of the same modular value, the union operation is sufficient to describe any RCset. Xenakis's algorithm (in his 1990 article) analyzes collections only as the union of RCs, and this study follows that precedent.

(17.) The reader can judge the degree to which the choice of ((3, 0), (4, 0), (5, 0)) over all the other possibilities matches one's intuition. If, in this case or any other, a discrepancy is found between one's intuition and my algorithm's BIAS, the reader is encouraged to formulate a BIAS that models analytical intuition and/or perceptual impression more closely. For the sake of uniformity in analysis, I shall continue to use this BIAS-formula in analyzing the pitch material of the quartet, but I would like to emphasize the truism that one's

intuition (either analytical or perceptual) is often heavily context dependent.

(18.) In Example 8b, the first pitch in the second collection (measures 62-5) is parenthesized because it does not appear in the score; it is provided merely to document the transpositional relation that obtains, with only a few exceptions, between the notes of this collection and those of the collections immediately before and after. The scales' uniformity is especially noticeable in comparing their first few notes. I supply the parenthesized f#2 (which is implied by the shared intervallic succession at the lowermost end of each of the three scales) only to point out their mutual affiliation. Also, the six highest notes in the first collection--the entire treble-clef portion of which is octatonic--recur faithfully under their respective transpositions in the two subsequent collections, without exception or interpolation.

The odds that three random scalar collections of this tessitura (52 semitones) would match as closely as these three do are over two hundred and sixty quintillion to one. Thanks to Damon Scott for helping me determine this.

(19.) Xenakis, Formalized Music, 218-36.

(20.) Vriend, "Nomos Alpha," 54-5.

(21.) Xenakis, Formalized Music, 199.

(22.) In an interview with Simon Emmerson ("Xenakis talks to Simon Emmerson," Music and Musicians 24 (May 1976): 24-6), Xenakis contrasts his employment of sieves in Nomos Alpha with the nonstructural role that sieves play in his percussion piece Psappha (1975): "It [sieve theory] is not used in a formal way [in Psappha] because I didn't want to repeat myself. Based on my experience I used a more immediate and intuitive method." (24) The admission of the incidental nature of Psappha's sieves (as opposed to the structural role that they play in Nomos Alpha) precludes any presumption of long-term sieve structure in his later works.

(23.) Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973).

(24.) Dave Rusin, a professor of topology at Northern Illinois University, has informed me that in fact not every set of moduli may be represented uniquely by this ordered-pair measure. Among integral modular quantities, however, the cases in which uniqueness is not achieved may involve integers so large as to diminish our concern.

For example, an RCset of moduli 6655, 11730, and 12584 will have the same modular product as an RCset of moduli 7986, 8840, and 13915; this product is 982,346,679,600. These two RCsets also have the same MIV.

(25.) For an important examination of issues relevant to the proposition of such a structure, see Joseph Straus, "The Problem of Prolongation in Post-Tonal Music," Journal of Music Theory 31/1 (Spring 1987): 1-21.

(26.) One other element of symmetry around the central measure 69--central, that is, discounting tempo changes--is the introduction of Collection IV in measure 38 and its "reflection" through to the end of measure 100, after which the duple subdivisions are abandoned and Collection VI is introduced.

EVAN JONES is an Assistant Professor of Music Theory at the Florida State School of Music. He participated in the North American premiere of lannis Xenakis's Tetora in 1993.
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