Christopher Ariza The Xenakis Sieve as New York University Graduate School of Arts and Science, Music Department Object: A New Model and a 24 Waverly Place, Room 268 New York, New York 10003 USA Complete Implementation ariza@flexatone.net
In numerous publications from 1965 to 1992, com- Gibson 2001; Jones 2001; Andreatta 2003), though poser, architect, and theorist Iannis Xenakis (1922– none have integrated a complete software model. 2001) developed an elegant and powerful system for A sieve is a formula consisting of one or more creating integer-sequence generators called sieves. residual classes combined by logic operators. A Xenakis used sieves (cribles) for the generation of residual class consists of two integer values, a mod- pitch scales and rhythm sequences in many compo- ulus (M) and a shift (I). The modulus can be any pos- sitions, and he suggested their application to a vari- itive integer greater than or equal to 0; the shift, for ety of additional musical parameters. Though sieves a modulus M greater than 0, can be any integer from are best calculated with the aid of a computer, no 0 to M–1. A modulus and shift will be notated M@I, complete implementation has been widely distrib- read “modulus M at shift I.” A shift I greater than or uted. Xenakis’s published code is incomplete and equal to M is replaced by the common residue, or I insufficient for broad use. % M. (All computational examples in this article This article demonstrates a new object-oriented are given in the Python programming language, model and Python implementation of the Xenakis where the “%” is the modulus operator. For ex- sieve. This model introduces a bi-faceted represen- ample: 13 % 12 == 1.) tation of the sieve, expands Xenakis’s use of logic The residual class defines an infinite number se- operators, employs a practical notation, produces quence. Given a sequence generated by M@I, each sieve segments and transpositions, and easily inte- value in the sequence modulus M is equal to I. For grates within higher-level systems. This modular example, the residual class 3@0 defines an infinite implementation is deployed within athenaCL, a sequence consisting of all integers x where x % 3 == cross-platform, open-source, interactive command- 0. The resulting sieve sequence is [ . . . , –6, –3, 0, 3, line environment for algorithmic composition using 6, . . . ]. A residual class with the same modulus and Csound and MIDI. High-level, practical interfaces a shift of 1, notated 3@1, produces a sequence have been developed to provide athenaCL users with where, for each value x, x % 3 == 1, or [ ..., –5, –2, 1, sieve-based tools for the algorithmic generation of 4, 7, . . . ]. For any modulus M, there exist M unique pitches, rhythms, and general parameter values. shifts (the values 0 to M–1). For each shift of M, a sequence of equally spaced integers is produced, where the difference between any adjacent integers Definition of the Sieve is always M. A residual class produces a periodic se- quence with a period equal to M. Xenakis’s theory changed over the course of his Although modulus by zero, and thus zero-division, many writings on the sieve. Procedures, notation, is typically undefined for real numbers (and an error and nomenclature varied as the theory developed, for computers), the residual class 0@0 is permitted, yet often with inconsistent and unexplained usage. defining the empty sieve sequence: []. The residual To avoid the complexity of Xenakis’s presentation, class 1@0, the complement of 0@0, defines the infi- a sieve will be defined with a new model and nota- nite integer sequence . tion. New terms and concepts are introduced to re- Logic operators are used to combine residual place Xenakis’s sometimes-inconsistent usage. classes. Four operators are permitted: union (“or”), Alternative theoretical treatments of the sieve have intersection (“and”), symmetric difference (“xor”), been proposed by others (Riotte 1979; Squibbs 1996; and complementation (“not”). Union, intersection, and symmetric difference are binary operators; Computer Music Journal, 29:2, pp. 40–60, Summer 2005 complementation is a unary operator. The logic op- © 2005 Massachusetts Institute of Technology. erators will be notated “|” for union, “&” for inter-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 section, “^” for symmetric difference, and “-” for sieve is equal to the lowest common multiple complementation. (LCM) of all residual class moduli. In the case of For example, the sieve 3@0 | 4@0 produces the some sieves, the period is easily apparent. The sieve union of two residual classes, or the sieve sequence 3@2 | 4@1, for example, has a period of 12. In the [ . . . , 0, 3, 4, 6, 8, 9, 12, . . . ]. The intersection of the case of other sieves, however, the period can be so same residual classes, notated 3@0 & 4@0, produces large as to be practically unrecognizable. the sieve sequence [ . . . , 0, 12, 24, . . . ]. The sym- A sieve segment, or a finite contiguous section of a metric difference, or the values in each residual sieve sequence, can be extracted for practical deploy- class and not in both residual classes, notated 3@0 ^ ment. Rather than filtering the all-integer set, a sieve 4@0, produces the sieve sequence [ . . . , –3, 3, 4, 6, 8, segment filters a finite range of integers. The set of 9, 15, . . . ]. integers filtered will be called z and is specified z = Unlike union, intersection, and symmetric differ- [a,..., b], where a is the minimum, and b is the ence, unary complementation operates on a single maximum. The sieve 3@2 | 4@1 with z = [–37, ..., residual class or a group of residual classes. Binary –25], produces the sieve segment [–37, –35, –34, –31, complementation is not permitted. The sieve se- –28, –27, –25]. quence of a complemented residual class, –M@I, is The integers of a sieve segment can be thought of the sequence of all integers not in M@I. A residual as points; the remaining integers found in z but not class under complementation, –M@I, is equivalent in the sieve segment can be thought of as slots. An to the union of all residual classes of the modulus (I integer representation of a sieve segment provides from 0 to M–1) excluding the complemented resid- only the points, ignoring the slots. The internal slots ual class. For example, –3@0 is equal to the sieve between integers can be deduced; external slots be- 3@1 | 3@2, or the sieve sequence [ . . . , –7, –5, –4, –2, yond the minimum and maximum of the sieve seg- –1, 1, 2, 4, 5, 7, . . . ]. Likewise, –5@2 is equivalent to ment, however, cannot be determined without the sieve 5@0 | 5@1 | 5@3 | 5@4. knowledge of z. If external slots are not represented, Operator precedence, from most- to least-binding, for example, the segment may appear symmetrical is in the following order: unary complementation, when, in terms of its z-relative position, it is not. intersection, symmetric difference, union. By using There are four practical representations, or for- braces (“{“ and “}”) as delimiters, a sieve can employ mats, of a sieve segment. Three are slot-discarding: unlimited nesting. A delimited collection of resid- integer, width, and unit segments; one is slot- ual classes is always evaluated before residual classes retaining: binary segments. Integer segments dis- on the same hierarchical level. For example: 7@1 | card external slots and may distort z-relative position. 3@2 & 4@3 & –9@1 is evaluated 7@1 | {3@2 & 4@3 & Width segments measure the distance from one {–9@1}}. A delimited group can be complemented. point to the next, counting the point itself and the There exist two types of sieves: simple and com- intervening slots. Width segments, like integer seg- plex. A simple sieve uses at most a two-level, or- ments, discard external slots and may distort z- dered grouping of residual classes, in which the relative position. Unit segments map z to the unit inner level uses intersection, the outer level uses interval and translate segment points into real num- union, and complemented residual classes are never bers between 0 and 1. Because unit segments do not intersected. A maximally simple sieve is made of treat points as integers, both internal and external any number of single residual classes combined slots are discarded, and z, normally discrete, becomes only by union. A complex sieve uses residual a continuous range. A unit segment contains no in- classes with any combination of logic operators at formation on the number of slots between points, yet any hierarchical level, with hierarchical levels of accurate proportional spacing, including z-relative unlimited depth. position, is retained. Binary segments retain all A sieve filters the set of all integers to produce slots, both internal and external, and thus provide an infinite sieve sequence. As all residual classes the must complete representation. Figure 1 summa- are periodic, all sieves are periodic. The period of a rizes the features of sieve segment representations.
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 1. Sieve segment for- mats for sieve 3@2 | 4@1, where z = [7, ..., 16].
Format Sieve segment z Internal External z-relative slots slots position integer [8, 9, 11, 13, 14] discrete retain discard discard width [1, 2, 2, 1] discrete retain discard discard unit [0.11, 0.22, 0.44, 0.66, 0.77] continuous discard discard retain binary [0, 1, 1, 0, 1, 0, 1, 1, 0, 0] discrete retain retain retain
A sieve can be transposed by any integer. A sieve ment. Compression by segment requires generating transposition is created by adding the transposition a sieve segment from a complex sieve, then re- value to the shift of each residual class in the sieve. sampling the values within this set to produce a For example, the sieve 5@2 & 2@0 produces the maximally simple sieve. Compression by segment sieve sequence [ . . . , 2, 12, 22, 32, 42, 52, . . . ]. If can be lossy. The compressed and expanded sieves this sieve is transposed by a value of 4, the sieve will generate identical sieve segments only for the z 5@1 & 2@0 results (modulus reduction of 5@6 & provided during compression. Segments generated 2@4), producing the sieve sequence [ . . . , 6, 16, 26, with the compressed sieve beyond this z may devi- 36, 46, 56, . . . ]. A transposition value will be ate from segments generated with the expanded called n. sieve. Increasing the size of z, and thus the size of When z is discrete (integer, width, and binary seg- the segment sampled, improves the quality of com- ments), the integer step can be mapped to any value. pression. Compression by segment with a z-length The integer step will be called the elementary dis- equal to the sieve period, or even multiple periods, placement, or ELD. When z is continuous (unit seg- may not result in a compressed sieve that, when ments), points can function as floating-point scalars compared to the expanded sieve, produces identical of any value. Thus, what a sieve creates is a sequence segments for all z. of points on a line, a sequence of proportions between Sieve compression is defined by two theorems pro- these points, or a distribution of points within a vided by Xenakis. First, any two non-complemented range. This sequence can be treated as an ordered or residual classes under intersection can be reduced unordered collection. to a single residual class. It follows that any number Simple and complex sieves can undergo compres- of residual classes, if intersected, can be reduced to sion. There are two forms of compression: by inter- a single residual class. This is compression by inter- section and by segment. Compression always section. Second, any finite integer set can be ex- results in the production of a maximally simple pressed as a maximally simple sieve. This is sieve. A maximally simple sieve cannot be further compression by segment. compressed. A sieve that has not been compressed The Xenakis sieve is a set theory of infinite peri- is an expanded sieve. odicities. This sieve is a unique structure. Cer- A simple sieve can be compressed by intersection. tainly, the concept of selecting a collection of Compression by intersection is a process of combin- numbers by removing elements from a set is com- ing all residual classes within inner intersection mon in mathematics (Hawkins 1958). Few mathe- groups into a single residual class. A maximally matical sieves, however, have the sole goal of simple sieve, a collection of single residual classes creating geometrically and aesthetically pleasing combined by union, results. Compression by inter- structures for sonic deployment. As Xenakis states, section is non-lossy: sieve segments produced with “the image of a line with points on it, which is close any z will be identical for both compressed and ex- to the musician and to the tradition of music, is panded sieves. very useful” (Xenakis 1996, p. 147). Such an image A complex sieve can be compressed only by seg- is provided by the sieve.
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 History and Foundations of the Sieve sents a moment in time with a two-dimensional plane of frequency and intensity; this plane can be In 1963, Aaron Copland invited Xenakis to teach at populated with events, or what Xenakis calls grains. the Berkshire Music Center at Tanglewood. Xenakis’s Xenakis employs the logic operators union, inter- notes and sketches from that summer contain his section, and complementation to “envisage in all its first experiments with sieves (Barthel-Calvet 2001). generality the manner of combining and juxtaposing As a result of a Ford Foundation grant, Xenakis screens” (1963; 1992, p. 57). Analogique A (1958) lived in Berlin from the fall of 1963 to the spring of and Analogique B (1959) are offered by Xenakis as 1964. During this time, he developed sieve theory demonstrations of this technique (1963; 1992, further (Barthel-Calvet 2001). Xenakis’s theory of pp. 98–108). The sieve might be seen as a one- sieves can be seen in the context of his interest in dimensional screen. In chapter 3, “Symbolic Mu- number sequences, his use of logic operators with sic,” logic operators are further used by Xenakis to screens and groups, and his desire to develop combine groups of musical materials. The pitch “outside-time” musical structures. groups of Herma (1960–1961), for solo piano, are of- The sieve produces numerical sequences. Xenakis’s fered by Xenakis as a demonstration of this tech- interest in the musical deployment of numerical nique (Xenakis 1963; 1992, p. 175). sequences has been frequently demonstrated. An The sieve is a generator of outside-time struc- example can be seen in his prominent use of the Fi- tures. To Xenakis, a structure that is outside-time is bonacci series in his early compositions Zyia (1952) unique regardless of order or temporal deployment. and Sacrific (1953; see Solomos 2002, pp. 26–29). This contrasts with an “in-time” structure, which Xenakis may have had a similar interest in the se- only retains identity in an ordered or temporal de- ries of prime numbers. The sieve of Eratosthenes of ployment (1992, p. 207). Using the standard trans- Cyrene (c. 276–c. 194 BCE), a well-known method formations of a twelve-tone row as an example, a of generating prime numbers (Horsely 1772), may retrograde transformation is an “in-time” operation, have inspired Xenakis’s sieve (Flint 1989). whereas inversion, in producing new intervals, is an The sieve of Eratosthenes can be explained through “outside-time” operation: “of the four forms of the the following steps: (1) list a range of ordered inte- series, only the inversion of the intervals is related gers starting from 1; (2) let M be 2; (3) then select M: to an outside-time structure” (Xenakis 1992, p. 193). this value is prime; (4) eliminate all values that are Xenakis, taking an historic view, often lamented multiples of M (i.e., M × 2, M × 3, M × 4, . . . ); (5) set the decline of outside-time structures to in-time M to the next available (i.e., not eliminated) value; structures in Western music. He offered the sieve as (6) if integers remain, return to step 3. After M ex- an antidote. hausts all available values, only primes will remain. Xenakis wrote frequently about sieves. There are This process has features in common with the Xe- at least five unique discussions of the sieve; each is nakis sieve. The collected multiples of M are simi- found in multiple translations and editions. (Through- lar to the periodic integers defined by a residual out this article, each discussion will be referenced class; combining these prime-number multiples is by the date of the earliest document, followed, similar to the union of multiple residual classes. when necessary, by the date of the actual citation.) Multiples are used here, however, to generate slots, Sieves were first introduced in “La voie de la not points, and they exclude their first multiple (the recherche et de la question” (Xenakis 1965), pub- prime). lished in Preuve and later reprinted in Kéleütha The sieve combines structures with logic opera- (1994). Xenakis further demonstrated the sieve in tors. Xenakis’s writings collected in the 1963 edi- “Vers une philosophie de la Musique,” published tion of Musiques Formelles demonstrate similar in French in Gravesaner Blätter (1966), Revue combinations with different materials. Chapter 2, d’Esthétique (1968), and as chapter 6 in Musique “Markovian Stochastic Music,” introduces the con- Architecture (Xenakis 1976). This text, detailing the cept of screens (1963; 1990, p. 51). A screen repre- application of sieves in Nomos Alpha (1965–1966),
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 was later translated to English by John and Amber Two Models of the Sieve Challifour and included in the 1971 edition of For- malized Music as chapter VIII. A detailed presenta- Xenakis offers two sieve models. The first model, tion of sieves occurs in the 1967 article “Vers une the complex sieve, is described in his earliest writ- métamusique,” published first in La Nef, and sub- ings (Xenakis 1965, 1966, 1967, 1988). The second sequently republished as chapter five of Musique model, the simple sieve, is found only in his last Architecture (Xenakis 1976). The first English trans- treatment of the topic and its accompanying soft- lation by G. W. Hopkins was published in Tempo in ware implementation (Xenakis 1990). Interestingly, 1970 and was included in the 1971 English edition of this second model fails to incorporate aspects of the Formalized Music as chapter VII, “Towards a Meta- original, and Xenakis provides no explanation for music.” Sieves are also discussed in “Redécouvrir le this difference. The models differ in their allowed Temps” (Xenakis 1988). In 1989, excerpts from this logic operators and their levels of residual class article, translated into English by Roberta Brown, nesting. appeared in Perspectives of New Music as “Con- Xenakis’s first model was based on the manual cerning Time.” The complete text, titled “Concern- calculation of sieves. Xenakis (1966; 1992, p. 234) ing Time, Space and Music,” was included as provides an image of a hand-written diagram on chapter X in the 1992 edition of Formalized Music. graph paper labeled “Nomos alpha Sieves.” Each The article “Sieves” (Xenakis 1990), translated column of the graph is treated as an integer unit and into English by John Rahn, was first published in is clearly labeled “ELD = 1/4 tone.” Numerous par- Perspectives of New Music. Nearly the same article allel, horizontal lines are used to calculate the sieve. is included as chapter XI in the 1992 edition of For- Each line illustrates a residual class segment (with malized Music. “Sieves” (Xenakis 1990) includes points marked as vertical tick-marks) and is labeled the only published software implementation of the with a logic operation such as “∩ (5,2).” By applying sieve, written in the C language. Chapter XI of For- the logic operator of each line to the previous line, malized Music, titled “Sieves: A User’s Guide,” in- the final sieve segment is realized. Xenakis men- cludes nearly the same software implementation. tions this technique elsewhere: when introducing Xenakis frequently mentioned composing with the sieve in a later document, he describes setting sieve structures. Xenakis describes sieve-based the ELD to both a “semitone or a millimeter,” and pitch structures in Akrata (1964–1965; Xenakis notating a sieve on graph paper (Xenakis 1990; 1992, 1966), Nomos Alpha (1965–1966; Xenakis 1966), p. 269). This tedious process has been duplicated by Jonchaies (1977; Varga 1996, p. 164), Pléïades (1978; analysts of Xenakis’s music (Vriend 1981, pp. 55–57; Varga 1996, p. 180), and Aïs (1979; Varga 1996, Harley 2004, p. 43). p. 165). Xenakis describes sieve-based time struc- Intersection, union, and unary complementa- tures in Psappha (1975; Emmerson 1976, p. 24) and tion operators are employed in the first model (Xe- Komboï (1981; Varga 1996, p. 71). In addition to Xe- nakis 1965, 1966, 1967, 1988). In the second nakis’s explicit statements, analysts have found model (Xenakis 1990), however, both in prose and sieve structures in numerous compositions includ- in code, there is no mention or example of com- ing Eonta (1963; Barthel-Calvet 2000), Anaktoria plementation. Xenakis says nothing about this (1969; Solomos 1996 p. 93), Persephassa (1969; omission. Harley 2004, p. 64), Pléïdes (1978; Harley 2004, In the first model, logic operators are permitted in p. 121) Pour la Paix (1981; Solomos 1996, p. 93), any combination, within any hierarchical level. The Embelllie (1981; Solomos 1996, p. 93), Mists (1981; following complex sieve is an example: Squibbs 1996, 2002), Neküia (1981; Gibson 2001), {–M@I & M@I & {–M@I | M@I}} | M@I & M@I Serment-Orkos (1981; Solomos 1996, p. 93), À R. (Hommage à Maurice Ravel) (1987; Squibbs 1996, The second model exclusively uses two-level or- p. 61), Tetora (1990; Jones 2001), and Paille in the dered groups. The inner level consists only of inter- Wind (1992; Solomos 1996, p. 93). sections, the outer level consists only of unions, and
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 complementation is not used. The following simple all common logic operators, including symmetric sieve is an example: difference (after Amiot et al. 1986), are permitted. This model reformulates Xenakis’s reduction algo- M@I & M@I & M@I | M@I | M@I & M@I rithms into the concept of compression, a method It can be suggested that Xenakis ordered group- of moving from a complex to a maximally simple ings and reduced the use of logic operators because sieve. Though these features are not formally neces- it became apparent that these features were not for- sary to produce all sets, the expressive power of the mally necessary to produce all segments. Further, sieve formula is expanded by their use. This nota- software implementation may have encouraged the tional convenience is, in fact, an essential feature of definition of a simpler, easily computable model. the model. Two proofs, introduced with the second model (Xe- nakis 1990), support this claim. Xenakis (1990; 1992, p. 275) provides an algo- The Notation of the Sieve rithm for sieve compression by segment: the deriva- tion of a sieve from an arbitrary integer set. As The notation of a sieve is important. To a user, the mentioned earlier, this proof demonstrates that any logic formula is the primary interface, and its nota- finite set of integers can be generated by a maxi- tion directly affects the utility of the model. A sieve mally simple sieve: a sieve created exclusively from notation must include a means of specifying a resid- the union of residual classes. Thus, intersection, ual class as a modulus and a shift, symbols for the complementation, and multiple levels of nesting are logic operators, and symbols to delimit residual not required to construct a sieve of any complexity class groups. and may be deemed superfluous. Union is the only Xenakis uses one notation, with slight variation, necessary logic operator. for complex sieves (1965, 1966, 1967, 1988). Shifts In Xenakis’s second model, complementation is re- are represented as subscripts of the modulus. (In moved, yet intersection is retained. This peculiarity some instances, this notation is reversed, with the is explained by Xenakis’s algorithm for compression modulus represented as a subscript of the shift; see by intersection. This proof demonstrates that any Xenakis 1965.) Traditional logic symbols are used: number of residual classes, upon intersection, can be ∪ for union, ∩ for intersection, and an over-score reduced to a single residual class (1990; 1992, p. 271). line for complementation. (Xenakis 1963 uses the Intersection, in the second model, is used not out same logic symbols for intersection and union; of necessity, but as a notational convenience. For complementation, as a binary operator, is notated example, the following simple sieve (Xenakis 1992, with a “–”.) Transposition is represented by the p. 274) contains four groups of intersections joined variable n and is included with every subscript as by union: “n+shift” (or “n” when shift is equal to zero). Groups are notated with parentheses. The following example 3@2 & 4@7 & 6@11 & 8@7 | 6@9 & 15@18 | 13@5 & (Xenakis 1992, p. 197) demonstrates this notation: 8@6 & 4@2 | 6@9 & 15@19 (3¯ ∩ 4 ) ∪ (3¯ ∩ 4 ) ∪ (3 ∩ 4 ) ∪ (3¯ ∩ 4 ) Applying compression by intersection, this sieve n+2 n n+1 n+1 n+2 n+2 n n+3 is reduced to a maximally simple sieve. The last Xenakis uses two notations for simple sieves. The pair of residual classes, having no points in com- first is introduced in the two proofs presented in mon, reduces to the null residual class 0@0: “Sieves” (Xenakis 1990). Residual classes are repre- sented as number pairs delimited by parentheses 24@23 | 30@3 | 104@70 | 0@0 and are given with either integers or variables The new model presented here combines both (where M represents modulus, and I represents Xenakis’s first and second models. Complex and shift). Logic operators are notated as before. Trans- simple sieves are incorporated into a single, bi- position, as the variable n or otherwise, is no longer faceted object. All forms of hierarchical nesting and notated. Groups are notated with braces. The fol-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 lowing example (Xenakis 1992, p. 274) demon- complexity. Finally, the sieve contains only ASCII strates this notation: characters, no commas, and (optionally) no spaces, allowing for easy parsing and isolation when passed ∩ ∩ ∩ ∪ ∩ ∪ {(3,2) (4,7) (6,11) (8,7)} {(6,9) (15,18)} as an argument or included with complex data. This {(15,5) ∩ (8,6) ∩ (4,2)} ∪ {(6,9) ∩ (15,19)} notation, for example, could easily be sent as an Open Sound Control (OSC) string; if stored in Hy- The second notation for simple sieves, part of the perText Markup Language (HTML) or eXtensible C-language implementation accompanying Xenakis Markup Language (XML), one character used in this (1990), uses an American Standard Code for Infor- notation, the ampersand (“&”), must be converted mation Interchange (ASCII) symbol representation. to a character entity (“&”). The use of ASCII ensures the availability of every character on a computer keyboard. Modulus and shift are represented as a number pair. Intersection Implementations of the Sieve is represented by the symbol “*,” and union is rep- resented with the symbol “+.” Incidentally, Xenakis A software model of the sieve must treat the logic had used these symbols earlier in “Symbolic Mu- formula as the primary object, not one of the for- sic” (1963) and “La voie de la recherche et de la mula’s possible segments. The logic formula should question” (1965). Complementation is not speci- be retained as a reusable object such that it can be fied, and transposition is not notated. Groups are represented as conceived, multiple transpositions notated with square brackets, for example: and segments can be extracted, and the formula it- [(3,2) * (4,7)] + [(6,9) * (15,18)] self can be logically combined. Simply combining fixed sets with logic operators reduces the function- The new notation presented here is a “logic ality of a sieve to simply an input notation: the in- string.” Similar to the notation presented in Xe- formation contained in the design of the formula is nakis (1990), ASCII characters are used to represent lost only to one of its many output potentials. Of a sieve formula. To avoid using commas or paren- the handful of implementations available to date, theses, a residual class is given as a modulus and a none has treated the logic formula of the sieve as a shift separated by the “@” symbol. This symbol is reusable object, allowing the input, maintenance, chosen primarily for its infrequent use in other no- representation, and processing of a complex sieve tations. (Representing a residual class with two fig- from a single argument. ures separated by a symbol has a precedent in Xenakis (1990) provides the first published soft- Xenakis 1966, where shift and modulus—in that or- ware implementation of a sieve. This software con- der—are separated by the character “M”.) Logic op- sists of two main procedures: the generation of a erators are notated using the pipe (“|”) for union, the sieve segment from a logic formula, and the genera- ampersand (“&”) for intersection, the circumflex tion of a logic formula from an arbitrary set of inte- (“^”) for symmetric difference, and the dash (“–”) gers. Xenakis, in the 1992 edition of Formalized for unary complementation. All four logic operators Music, credits the C code to Gérard Marino, stating are permitted, and complementation can be applied that it is a port of Xenakis’s original BASIC code. to a single residual class or a group. A group is no- For the version published in Perspectives of New tated with braces. Music, no authorship is given. Functionally, the two This notation has many advantages. It is more versions are nearly identical. The version in For- compact and requires fewer characters than Xenakis’s malized Music (1992), however, contains numerous notations. The use of bitwise logic operator symbols typographic omissions. Ronald Squibbs has listed (pipe, ampersand, and circumflex) are transparent in these errors and provides corrected code (1996, meaning to those familiar with programming lan- pp. 292–303). guages such as C, Java, or Python. The use of a single Unfortunately, the implementation Xenakis of- set of characters for grouping (braces) reduces visual fered in 1990 cannot process the many complex
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 sieves he had demonstrated since 1965. This soft- ments a variety of sieve structures using union, ware, permitting only the calculation of simple intersection, and complementation, and produces sieves, has other shortcomings. The code lacks sieve segments in integer, binary, and width for- modular design, limiting both flexibility and expan- mats. The system’s sieve notation, however, signifi- sion. The program is intended exclusively for user cantly deviates from a logic-formula representation. rather than programmatic interaction; a general pur- A more sophisticated sieve model, also written in pose interface, necessary for use in larger systems, Macintosh Common Lisp, was included in Mes- is not provided. A complete sieve cannot be sup- nage’s “Criblograph,” a graphical user interface en- plied as a single argument: when entering a sieve, vironment for music composition developed from for example, the user must first declare the number 1989 until 1997, and then merged into Mesnage’s of unions, then declare the number of residual “Mélusine” system. The sieve implementation in classes in each intersection, then enter a modulus these systems offered all common logic operators, and shift one at a time. Further, there is no imple- the creation and modification of reusable sieve seg- mentation of transposition and the user cannot ments, and a unique graphical sieve segment editor. freely designate z. Some of these tools have been subsequently incor- In 1980, prior to the publication of Xenakis’s porated into Mesnage’s “Morphoscope” (1993). De- code, Sever Tipei implemented a sieve model in spite the broad functionality of these systems, the FORTRAN for use with his computer-assisted com- sieve is not given a practical notation, and a com- position program MP1 (Tipei 1975). In addition to plex sieve cannot be provided as a single argument. using sieves for parameter generation, Tipei pro- Within Paul Berg’s AC Toolbox (available online duced weighted sieve-segments. If each residual at www.koncon.nl/ACToolbox), a Lisp-based soft- class is assigned a weight, a sieve can be con- ware system for algorithmic composition running structed that produces segments encoding com- on MacOS X are four sieve-related objects: Sieve, bined weights. These techniques and others were Sieve-Union, Sieve-Filter, and Interpret- used in his compositions and described in numer- Sieve. The Sieve object here can be better thought ous articles (Tipei 1981, 1987, 1989). of as a residual class: the user provides a modulus, a Researchers at the Institut de Recherche et Coor- shift (as start value) and a z (as minimum and maxi- dination Acoustique/Musique (IRCAM) employed mum). Evaluation returns an integer segment. The the sieve in a number of publications. Malherbe et object Sieve-Union is a union operator that com- al. (1985) demonstrate a novel application of sieve bines any number of lists (generated by Sieve or structures to model spectral peaks found in ana- otherwise) and returns a new list. Sieve-Filter lyzed sound files. Amiot et al. (1986) develop upon employs a list (generated by Sieve or otherwise) to this model and provide a more complete descrip- select values from another list. Interpret-Sieve is tion. Here, sieves are employed primarily for filter- designed for the production of rhythms: a list of val- ing metric patterns. Logic operators are expanded ues, interpreted as either a binary or a width sieve to include symmetric difference, composition, and segment, is used to process a list of rhythm durations exponentiation. This sieve was implemented by into note (positive) or rest (negative) values. The user Gérard Assayag as the “Langage de Cribles” (LC) in provides a list of durations, the sieve segment, and a the now-obsolete Le Lisp language. Neither details control variable. Although providing useful tools for of the implementation nor examples of its use are the deployment of combined residual class segments, available. Assayag’s Lisp-based ScoreBoard applica- this model does not allow the input or maintenance tion (1993) may offer a related model. of a complex sieve as a complete logic formula. In 1990, Marcel Mesnage programmed a text- Jones (2001), following the limits and notation of based system in Macintosh Common Lisp called Xenakis (1990), models only simple sieves (called “Partitions d’Ensembles de Classes de Résidus” RCsets) as residual classes combined under union. (PCR; Riotte 1992; Mesnage and Riotte 1993). This Jones is primarily interested in deriving a maxi- system, based on a model by André Riotte, imple- mally simple sieve from an arbitrary set for the pur-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 pose of analysis. A new method of compression by presented above. This implementation is portable, segment is proposed. He rejects Xenakis’s algorithm modular, and offers the easy creation and deployment for only accepting residual classes with perfect cor- of sieve segments and transpositions. The model respondence to the source set. Because Xenakis of- combines Xenakis’s two models into a bi-faceted ten varied the realization of a sieve in its musical object: a sieve can contain both complex and simple deployment, Jones proposes an algorithm that pro- representations simultaneously. The file sieve.py, vides a “statistically optimal” (2001, p. 225) match part of the athenaCL library libATH, implements to the source set, weighted toward fewer residual this model as the Residual and Sieve objects. classes with lower moduli, even if this requires pro- ducing sieve segments that do not match the source. Candidate residual classes, for example, must match The Residual Object at least three points within the source set. Although this model suggests an interesting method of lossy The Residual object is a representation of the sieve compression, the deviation Jones allows be- residual class. A Residual contains data attributes tween source set and resulting sieve is so high as to for modulus (m), shift (shift), complement (neg), suggest a transformation beyond mere compression. and integer range (z). Modulus and shift are integer In his analysis of Tetora, Jones offers simple sieves values as defined above. As complementation of a to replace source sets that, in one case, cover as few single residual object is a unary operation, comple- as 10 out of 37 original points. Jones (2001) includes mentation is a Boolean value (neg) stored as an at- an implementation, programmed in BASIC, provid- tribute of the Residual object. Each Residual ing an interface similar to Xenakis (1990). instance contains a reference to a finite list of con- Some, perhaps owing to the absence of a complete tiguous integers (z) from which sieve segments are software implementation, have calculated sieves filtered. The attribute segFmt determines segment with a numeric table (Squibbs 1996, pp. 304–306; output format, where format strings for integer, bi- Gibson 2001). If a sieve uses only two moduli, a nary, unit, and width are notated int, bin, unit, table can be constructed with each modulus as- and wid, respectively. The default segFmt is int. signed to an axis. Each axis has rows or columns for Object initialization requires a modulus (m) value. every shift in the modulus (0 to M–1), thus repre- Optional arguments can be provided for shift, neg, senting every residual class of each modulus. Values and z. If no z is given, a default range is provided. in the table are filled by wrapping a z (from zero to The segment() method of a Residual instance one less than the product of the moduli) diagonally returns a transposed sieve segment in one of four through the table. The resulting values, for any two formats. This method has three optional arguments: residual classes, provide the intersection point an integer value for transposition (n), by default found within a single-period sieve segment. Though zero; a list of integers (z), by default the z set at ini- Squibbs has demonstrated the use of multiple tables tialization; and a format string. Arguments passed to handle sieves employing more than two moduli, to the segment method do not change attributes of this technique is very limited. the object: they are used only in the calculation of The predecessor of the sieve model presented here the desired segment. If changes to these attributes (Ariza 2004) introduced the logic string notation and are desired, the method zAssign() can be used to modeled the logic formula as a reusable object. Sieve set z, and the method segFmtSet() can be used to objects, however, were limited to simple sieves. set segFmt. The method period() provides the pe- riod of the residual, which in all cases is equal to the modulus. The repr() method provides a string Object-Oriented Implementation representations of the residual class, and the copy() method returns a new object with identical An object-oriented Python implementation has been m, shift, neg, and z attributes. developed based on the new model and notation Additional functionality is provided through op-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 2. Residual object class diagram.
erator overloading. The object’s __call__() Residual method is mapped to the segment() method, and the __str__() method is mapped to the repr() m method. (In Python, all objects can overload opera- shift tors by defining specially named methods. The neg __call__() method is called when an object x is z evaluated as x(). The __str__() method is called whenever a string representation of an object is segFmt needed, for example print x.) zAssign() The __and__() method, called with the binary & segFmtSet() operator, applies compression by intersection to two Residual operands and returns the Residual __call__(), segment() of the reduced intersection. The z of this new object period() is formed from the union of each operand’s z. Com- copy() pression by intersection is implemented after the al- __str__(), repr() gorithm presented in Xenakis (1990). If intersection of a complemented residual class is attempted, an __and__() error is raised. The __or__() method, evoked with __neg__() the | operator, is not implemented: the union of __eq__() two residual classes can only be represented by two __ne__() residual classes. The __xor__() method, evoked with the ^ operator, is likewise not implemented. The __neg__() method, called with the unary - operator, changes the neg attribute to its Boolean or simple) is stored as the expType attribute and is opposite. To compare two Residual objects, the used to determine the form of compression. Each equal and not-equal operator methods, __eq__() state (expanded and compressed) has unique attri- and __ne__(), are defined by comparing m, shift, butes for logic string, residual classes, and period. and neg attributes. Each state generates independent segments. The z A Unified Modeling Language (UML) class dia- attribute is shared between both states. The object gram of the Residual, summarizing the public at- has an attribute for current state, set at initializa- tributes and methods of this object, is provided in tion to expanded. To change states, the compress() Figure 2. Figure 3 demonstrates the Residual ob- or expand() methods are called. ject within a Python interactive session. (The Python The Sieve shares aspects of the Residual object prompt >>> precedes user input. Comments are pre- interface. The zAssign() and segFmtSet() meth- ceded by #.) ods set z and segment formats, respectively. The segment() method, with the addition of an argu- ment for state, returns a sieve segment for an optional The Sieve Object shift, z, and format argument. This method, using the current state, is mapped to the __call__() method. The Sieve object is a bi-faceted representation of the Segment formats are the same as for Residual ob- Xenakis sieve. One representation is the expanded jects: integer, binary, unit, and width. The pe- state, which can be any sieve from simple to com- riod() method returns the period of the current plex. The other representation is the same sieve com- state, or the state designated with an optional argu- pressed; the compressed state is always a maximally ment. The period of each state is calculated by find- simple sieve. Each state is an independent sieve. ing the lowest common multiple of all residual The logic string provided at initialization becomes periods. The repr() method, also mapped to the the expanded sieve. The type of this sieve (complex __str__() method, returns a string representation
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 3. Python session demonstrating the Resid- ual object.
>>> from athenaCL.libATH import sieve >>> # the built-in Python function "range" can be used to create a z >>> z = range(0,25) # creates a list of contiguous integers from 0 to 24 >>> a = sieve.Residual(3,2,0,z) # a new Residual object >>> print a # return a string representation 3@2 >>> a() # calling the Residual returns a sieve segment [2, 5, 8, 11, 14, 17, 20, 23] >>> a(2) # the first argument is a transposition [1, 4, 7, 10, 13, 16, 19, 22] >>> a(2, range(-20,-10)) # the second argument is z [-20, -17, -14, -11] >>> a(0, range(0,13), 'bin') # the third argument is segment format [0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0] >>> a(0, range(0,13), 'unit') # a unit segment [0.16666666666666666, 0.41666666666666669, 0.66666666666666663, 0.91666666666666663] >>> a(0, range(0,13), 'wid') # a width segment [3, 3, 3] >>> a.period() # period, modulus, and width are equal 3 >>> b = sieve.Residual(8,0,1) # a complemented Residual, with a default z >>> print b -8@0 >>> b = -b # unary complementation >>> print b 8@0 >>> b() # calculating a segment with default transposition, z, and format [0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96] >>> c = a & b # the intersection of two Residuals >>> print c 24@8 >>> c() [8, 32, 56, 80] >>> c == a # Residuals tested for equality 0 >>> -a != a # Residuals tested for inequality 1
of the logic formula; the copy() method returns a mula is identified and replaced by a key, in the form new Sieve instance with identical attributes. of “
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 in the appropriate tree string is replaced by the string If compression by segment is mandated by exp- necessary to instantiate a Set; this Set contains the Type, a segment is created at the current z and is integer segment from the corresponding Residual processed. Compression by segment cannot be per- object. If a residual class is complemented, the seg- formed if, owing to the logic formula or the size of ment returned already reflects this complementation. z, an empty segment is retuned. Compression re- After all keys are replaced, the entire string is evalu- turns a list of Residual objects that, when com- ated, causing the instantiation and evaluation of all bined under union, will create a sieve that returns Set objects. Evaluation processes Set objects with the source segment for the provided z. The Resid- the logic operators specified in the tree string. A ual objects are stored in resLib, and a new tree single Set results and is returned by the Sieve as a string (cmpTree) is created. list. If a string representation is requested from the A variation of Xenakis’s algorithm (1990; 1992, Sieve, a similar operation is performed: all keys in p. 274) for compression by segment is implemented the tree string are replaced with the string represen- as follows: (1) Two copies of the source integer set tation of the corresponding Residual object. are stored as src and match lists. (2) A z list, if not Sieve objects forbid binary complementation, provided, is constructed by taking the range of inte- but allow unary complementation of Residuals gers from the minimum to the maximum of the and groups. Python Set objects, however, employ match list. (3) A value in the match list is treated as only binary complementation. This difference is a shift value. (4) A Residual object is created handled in two ways: (1) A single Residual class, with this shift, a modulus of 1, and z. (5) A sieve under complementation, internalizes its comple- segment is created by calling the Residual in- mented state. A Set is then instantiated from an stance. (6a) If this sieve segment is a subset of src, already-complemented Residual segment; the re- the object is appended to the list residuals and the sulting Set thus does not require complementation. shift value, as well as any points on the segment (2) In the case that a group of Residual objects is also found in the match list, are removed from complemented outside of a delimiter, the comple- match. (6b) Otherwise, the shift is retained, the mentation operator, at evaluation, is preceded by a modulus is incremented, and the process is repeated Set object corresponding to a segment of 1@1 for until a subset sieve segment is found. (7) Remaining the current z. Because binary complementation is values in match are each treated as a shift (repeat- evaluated before intersection, symmetric difference, ing from step 3), until match is empty. (Note that, and union, this effectively converts unary negation because the segment is always compared to src and into binary negation at the time of Set evaluation. not match, found Residual objects may cover re- During initialization, and after the expanded dundant points; this addresses a shortcoming of Xe- sieve is parsed, compression is performed. Two nakis’s algorithm mentioned in Jones 2001, p. 233.) methods of compression are available: by intersec- If an integer in a source set is treated as a residual tion and by segment. The expanded sieve type class shift, a modulus can always be found that, (expType) determines which compression is per- with this shift, produces a segment that is both a formed. If a sieve is complex, compression by seg- subset of the source and matches at least the shift. ment must be performed. If a sieve is simple, This segment will always be found before the mod- compression by intersection is performed. ulus is incremented past the number of points in z. If compression by intersection is mandated by At an extreme, a residual class can be created for expType, the expTree is divided into intersection each point in the source set, each residual, for the groups. The keys for each group of Residual ob- current z, contributing only one point to a union. It jects are collected, and the corresponding objects are follows that any arbitrary set can be represented as a intersected. A new tree string (cmpTree) is con- maximally simple sieve. structed, joining by union keys for each of the re- Compression occurs at initialization with the pro- sulting Residual objects. Each Residual is stored vided or default z. After initialization, compression in resLib. can be re-performed if a new z is provided. This new
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 4. Sieve object class diagram.
z will be set as the current z and will be used to cre- Sieve ate the segment sampled for compression. Changing the z value, in the case of compression by segment, expType can result in different compressed representations. state If, rather than a logic string, a list of integers is z entered at initialization, compression by segment is used to create the necessary Residual objects; segFmt these objects are stored in resLib, and a tree string resLib (expTree) is created. The expanded sieve, in this expTree case, will always be a maximally simple sieve: fur- cmpTree ther compression is not possible. The compressed sieve representation of the object will be identical compress() to the expanded sieve. expand() Sieve objects themselves, through operator over- zAssign() loading, can be combined with logic operators to pro- segFmtSet() duce new Sieve objects. The methods __and__(), __or__(), __xor__(), and __neg__() are defined __call__(), segment() to overload operators &, |, ^, and –. A new sieve is period() created by instantiating an object with a new logic copy() string and the union of each operand’s z. This new __str__(), repr() logic string encodes the operation on each operand’s expanded sieve, producing a new object that models __and__() the operation. __or__() A UML class diagram, summarizing the public at- __xor__() tributes and methods of this object, is provided in __neg__() Figure 4. The Python session provided in Figure 5 demon- strates, for a complex sieve, the internal representa- tion of tree strings for both the expanded and ding to the major scale for all z. Using the conver- compressed states. Combining Sieve objects with sion function psToNoteName() from the athenaCL logic operators is also demonstrated. Practical ex- libATH module pitchTools, each integer can be amples of the Sieve object are provided next. converted to a pitch name to verify the accuracy of the scale over four octaves. (The Python map func- tion applies a function to each value in a list and re- Demonstrating the Sieve turns a new list.) The compressed sieve, after compression by segment, is a maximally simple The Sieve object can realize all of Xenakis’s sieves, sieve. It also has a period of 12 and produces the ma- both complex and simple. One of Xenakis’s earliest jor scale over four octaves. examples is a complex sieve for the generation of a Figure 6 hides a potential confusion. Because a major scale. This elegant formula is, however, in- complex sieve is supplied at initialization, compres- compatible with both his second model and its soft- sion must occur by segment. As no z is supplied, a ware implementation (Xenakis 1990). This sieve default z of 0 to 99 is provided. For some smaller z provides an excellent example of the utility of a ranges, however, compression by segment will re- bi-faceted representation of the sieve. sult in a different compressed sieve. A Sieve object can be created from Xenakis’s for- If a Sieve object is created from an integer set of mula for the major scale. This expanded sieve has a a one-octave major scale (where z is automatically period of 12, and it produces a segment correspon- determined by the minimum and maximum of the
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 5. Python session demonstrating internal representation of a logic formula and the combina- tion of Sieve objects with logic operators.
>>> from athenaCL.libATH import sieve >>> a = sieve.Sieve('7@0|{-5@2&-4@3}') # create a complex sieve >>> a.expType # the expanded type is automatically identified 'complex' >>> a.expTree # the expanded tree string '
segment), a logic formula is found. This formula, is finally found, having the necessary period of 12 however, is only valid for the points covered by the and producing correct segments for all z. Xenakis, z of one octave. Beyond this, pitches deviate from aware of this aspect of compression by segment, the major scale. Figure 7 creates four Sieve objects. states that “one should take into account as many Sieve “a” is initiated with a one octave set of the points as possible in order to secure a more precise major scale; the resulting sieve is shown to have a logical expression” (1990, p. 275). Gibson (2001), period of 210 and deviate from the major scale at likewise using segments of the major scale, has also values beyond the z set at initialization. Sieve “b” demonstrated this constraint. extends the source set to two octaves. A different Jan Vriend, in his examination of Nomos Alpha, sieve is found with the same period. Again, z values demonstrates the complex system Xenakis em- beyond the z set at initialization result in a scale ployed for creating a succession of sieves, all based that deviates from the desired major scale. This pro- on a common formulation (1981, p. 55). The sieve is cess is repeated with a three-octave set. Given a provided as an algebraic logic formula (Xenakis four-octave set, the desired maximally simple sieve 1992, p. 230):
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 6. Python session demonstrating Xenakis’s sieve for the major scale and its compressed form.
>>> from athenaCL.libATH import sieve, pitchTools >>> # Xenakis's logical formula for the major scale >>> a = sieve.Sieve('{-3@2&4}|{-3@1&4@1}|{3@2&4@2}|{-3@0&4@3}') >>> print a {-3@2&4@0}|{-3@1&4@1}|{3@2&4@2}|{-3@0&4@3} >>> a.period() # the period of the major scale is 12 12 >>> a(0, range(0,13)) # one octave segment as pitch class [0, 2, 4, 5, 7, 9, 11, 12] >>> # four octave segment as note names >>> map(pitchTools.psToNoteName, a(0, range(0,49))) ['C4', 'D4', 'E4', 'F4', 'G4', 'A4', 'B4', 'C5', 'D5', 'E5', 'F5', 'G5', 'A5', 'B5', 'C6', 'D6', 'E6', 'F6', 'G6', 'A6', 'B6', 'C7', 'D7', 'E7', 'F7', 'G7', 'A7', 'B7', 'C8'] >>> a.compress() # toggle the current sieve state >>> print a # display the compressed sieve 6@5|12@0|12@2|12@4|12@7|12@9 >>> a.period() # return the compressed period 12 >>> # four octave segment from the compressed sieve >>> map(pitchTools.psToNoteName, a(0, range(0,49))) ['C4', 'D4', 'E4', 'F4', 'G4', 'A4', 'B4', 'C5', 'D5', 'E5', 'F5', 'G5', 'A5', 'B5', 'C6', 'D6', 'E6', 'F6', 'G6', 'A6', 'B6', 'C7', 'D7', 'E7', 'F7', 'G7', 'A7', 'B7', 'C8']
L (M,N) = {–{N@i | N@j | N@k | N@l} & M@p} | {–{M@q {8@3 & {11@1 | 11@2 | 11@3 | 11@4 | 11@10}} | | M@r} & N@s} | {N@t | N@u | N@v} {8@4 & {11@0 | 11@4 | 11@8}} | By algorithmically generating variables for modu- {8@5 & {11@0 | 11@2 | 11@3 | 11@7 | 11@9 | 11@10}} | lus and shift, this structure is used to produce six unique sieves and their resulting segments. A sieve, {8@6 & {11@1 | 11@3 | 11@5 | 11@7 | 11@8 | 11@9}} | using this formulation and employed in Nomos Al- {8@7 & {11@1 | 11@3 | 11@6 | 11@7 | 11@8 | 11@10}} pha, is given below (Xenakis 1966; 1992, p. 230): Two segments from this sieve are given, at trans- {–{13@3 | 13@5 | 13@7 | 13@9} & 11@2} | {–{11@4 | positions 0 and 10. These are provided in Squibbs’s 11@8} & 13@9} | {13@0 | 13@1 | 13@6} notation (1996, p. 61–62): Vriend (1981, p. 57), after demonstrating the man- K = {0, 2, 3, 4, 7, 9, 10, 13, 14, 16, 17, 21, 23, 25, 29, ual calculation of a segment from this sieve, pro- 30, 32, 34, 35, 38, 39, 43, 44, 47, 48, 52, 53, 57, 58, vides the resulting sieve segment mapped onto a 59, 62, 63, 66, 67, 69, 72, 73, 77, 78, 82, 86, 87} quarter-tone ELD pitch scale. This scale is notated
in Figure 8. Figure 9, using the Sieve object to model T10(K)(mod 88) = {0, 4, 8, 9, 10, 12, 13, 14, 17, 19, 20, Xenakis’s sieve, confirms Vriend’s realization of the 23, 24, 26, 27, 31, 33, 35, 39, 40, 42, 44, 45, 48, 49, segment. 53, 54, 57, 58, 62, 63, 67, 68, 69, 72, 73, 76, 77, 79, Squibbs (1996, p. 61) provides the logic formula 82, 83, 87} for a sieve from Xenakis’s À R. (Hommage à Mau- Figure 9 demonstrates that the creation of a Sieve rice Ravel): object with this logic formula accurately produces {8@0 & {11@0 | 11@4 | 11@5 | 11@6 | 11@10}} | the desired segments. Ellen Rennie Flint, in her analysis of Psappha, {8@1 & {11@2 | 11@3 | 11@6 | 11@7 | 11@9}} | demonstrates Xenakis’s use of sieves to generate {8@2 & {11@0 | 11@1 | 11@2 | 11@3 | 11@5 | 11@10}} | rhythms. Flint (1989, p. 228) provides the logic for-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 7. Python session Figure 8. Sieve generated demonstrating compres- scale from Nomos Alpha, sion by segment of a with ELD equal to a quar- major scale. ter tone (Vriend 1981).
>>> from athenaCL.libATH import sieve, pitchTools >>> # one octave of the major scale, entered as a list >>> a = sieve.Sieve([0,2,4,5,7,9,11]) >>> print a 5@2|5@4|6@5|7@0 >>> # pitches deviate from the major scale, excluding E5 >>> map(pitchTools.psToNoteName, a(0, range(0,25))) ['C4', 'D4', 'E4', 'F4', 'G4', 'A4', 'B4', 'C5', 'D5', 'F5', 'G5', 'A5', 'A#5', 'B5', 'C6'] >>> a.period() # expected period of 12 not found 210 >>> # two octaves of the major scale, entered as a list >>> b = sieve.Sieve([0,2,4,5,7,9,11,12,14,16,17,19,21,23]) >>> print b 5@4|6@5|7@0|7@2|7@5 >>> # pitches deviate from the major scale at F#6 >>> map(pitchTools.psToNoteName, b(0, range(0,37))) ['C4', 'D4', 'E4', 'F4', 'G4', 'A4', 'B4', 'C5', 'D5', 'E5', 'F5', 'G5', 'A5', 'B5', 'C6', 'D6', 'E6', 'F6', 'F#6', 'A6', 'A#6', 'B6'] >>> b.period() # expected period of 12 not found 210 >>> # three octaves of the major scale, entered as a list >>> c = sieve.Sieve([0,2,4,5,7,9,11,12,14,16,17,19,21,23,24,26,28,29,31,33,35]) >>> print c 6@5|7@0|7@5|10@9|12@0|12@2|12@4|12@7 >>> # pitches deviate from the major scale at D#7 >>> map(pitchTools.psToNoteName, c(0, range(0,48))) ['C4', 'D4', 'E4', 'F4', 'G4', 'A4', 'B4', 'C5', 'D5', 'E5', 'F5', 'G5', 'A5', 'B5', 'C6', 'D6', 'E6', 'F6', 'G6', 'A6', 'B6', 'C7', 'D7', 'D#7', 'E7', 'F7', 'F#7', 'G7', 'B7'] >>> c.period() # expected period of 12 not found 420 >>> # four octaves of the major scale, entered as a list >>> d = sieve.Sieve([0,2,4,5,7,9,11,12,14,16,17,19,21,23,24,26,28,29,31,33,35,36,38,40, 41,43,45,47]) >>> print d 6@5|12@0|12@2|12@4|12@7|12@9 >>> d.period() # expected period of 12 found 12
mula of the sieve used to create the rhythmic articu- lations of the first 39 time units of the composition: {{8@0 | 8@1 | 8@7} & {5@1 | 5@3}} | {{8@0 | 8@1 | 8@2} & 5@0} | {8@3 & {5@0 | 5@1 | 5@2 | 5@3 | 5@4}} | {8@4 & {5@0 | of group B provides a clear statement of this sieve 5@1 | 5@2 | 5@3 | 5@4}} | (Harley 2004, p. 96). These points, notated and expressed as an integer segment, are provided in {{8@5 | 8@6} & {5@2 | 5@3 | 5@4}} | {8@1&5@2} | Figure 10. {8@6&5@1} Using the Sieve object, this sieve can be recre- The resulting sieve segment, as attack points, ated from its logic formula. In Figure 9 integer and can be transcribed from the score; the second voice binary segments, over a z from 0 to 38, are created.
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 9. Python session demonstrating Xenakis’s sieves from Nomos Alpha, À R. (Hommage à Maurice Ravel) and Psappha.
>>> from athenaCL.libATH import sieve >>> # sieve from Xenakis’s Nomos Alpha >>> x = sieve.Sieve('{-{13@3|13@5|13@7|13@9}&11@2}|{- {11@4|11@8}&13@9}|{13@0|13@1|13@6}') >>> print x {-{13@3|13@5|13@7|13@9}&11@2}|{-{11@4|11@8}&13@9}|{13@0|13@1|13@6} >>> x() # segment notated in Vriend (1981) [0, 1, 2, 6, 9, 13, 14, 19, 22, 24, 26, 27, 32, 35, 39, 40, 45, 52, 53, 58, 61, 65, 66, 71, 78, 79, 84, 87, 90, 91, 92, 97]
>>> # sieve from Xenakis’s À R. (Hommage à Maurice Ravel) >>> a = sieve.Sieve('{8@0&{11@0|11@4|11@5|11@6|11@10}}|{8@1&{11@2|11@3|11@6|11@7|11@9}} |{8@2&{11@0|11@1|11@2|11@3|11@5|11@10}}|{8@3&{11@1|11@2|11@3|11@4|11@10}}|{8@4& {11@0|11@4|11@8}}|{8@5&{11@0|11@2|11@3|11@7|11@9|11@10}}|{8@6&{11@1|11@3|11@5|1 1@7|11@8|11@9}}|{8@7&{11@1|11@3|11@6|11@7|11@8|11@10}}') >>> a.period() # as modulus values of only 8 and 11 are used, the period is 88 88 >>> a(0, range(0,88)) # segments that correspond with Squibbs (1996) [0, 2, 3, 4, 7, 9, 10, 13, 14, 16, 17, 21, 23, 25, 29, 30, 32, 34, 35, 38, 39, 43, 44, 47, 48, 52, 53, 57, 58, 59, 62, 63, 66, 67, 69, 72, 73, 77, 78, 82, 86, 87] >>> a(10, range(0,88)) # transposition at 10 [0, 4, 8, 9, 10, 12, 13, 14, 17, 19, 20, 23, 24, 26, 27, 31, 33, 35, 39, 40, 42, 44, 45, 48, 49, 53, 54, 57, 58, 62, 63, 67, 68, 69, 72, 73, 76, 77, 79, 82, 83, 87]
>>> # sieve from Xenakis’s Psappha (Flint 1989) >>> x = sieve.Sieve('{{8@0|8@1|8@7}&{5@1|5@3}}|{{8@0|8@1|8@2}&5@0}|{8@3&{5@0|5@1|5@2|5@ 3|5@4}}|{8@4&{5@0|5@1|5@2|5@3|5@4}}|{{8@5|8@6}&{5@2|5@3|5@4}}|{8@1&5@2}|{8@6&5@ 1}') >>> segment corresponds to opening rhythm in group B, voice 2 >>> x(0, range(0,39)) [0, 1, 3, 4, 6, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 38] >>> x.segFmtSet('bin') # change the segment format >>> x(0, range(0,39)) [1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1]
The binary segment provides a clear representa- programmed in Python by using dynamically gener- tion of a sieve as a sequence of points (notes) and ated Sieve objects. slots (rests). Xenakis discussed many methods of transforming sieve structures. One method, demonstrated above Integration in athenaCL with the sieve system from Nomos Alpha, includes the algorithmic generation of sieve moduli and The objects provided in sieve.py can be used in iso- shifts; other methods employ permutations of shift lation or in cooperation with other software. Within values or logic operators, transpositions, or ELD athenaCL (Ariza 2002, 2003, 2004), high-level, prac- transformations. Xenakis called these operations tical object interfaces have been developed to pro- métabolae. Although space does not permit demon- vide access to sieve functionality. These interfaces stration here, métabolae of any complexity can be provide sieve-based tools for the algorithmic com-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Figure 10. Psappha, open- Figure 11. Three sieve- ing passage, group B, voice generated simultaneities: 2. Complex sieve with a (1) 5@1 | 7@1,C2,C5,C4; (2) period of 40. 6@2 | 7@3,C4,C7; and (3) 10@4 | 7@4,C3,C6,C4.
position of pitches, rhythms, and general parameter values. Complete demonstration is beyond the :: PIn a 5@1|7@1,C2,C5,C4 6@2|7@3,C4,C7 10@4|7@4,C3,C6,C4 scope of this article; basic functionality of a selec- tion of these tools, however, will be summarized. Each sieve has a cycle of fifths (a residual class with modulus 7), and all share a common origin (C4). This Path, notated in Figure 11 as a succession of Sieve Pitch Generation un-transposed, sustained simultaneities, demon- strates the ease with which sieves can be used to The athenaCL system features specialized objects create diverse pitch structures. A Texture in for pitch structures, including Pitch, Multiset athenaCL can employ these pitch structures in a va- (collections of Pitch objects), and Path (ordered riety of fashions: as a scale from which melodies are Multiset objects). When using athenaCL for com- created, as a collection from which subset chords position, a Path defines reusable pitch collections. are derived, or as a pointillistic cloud of pitches. The actual use of a Path is dependent on interpreta- tion by a Texture, to be described below. The Mul- tiset objects of a Path can be provided by the user Sieve Rhythm Generation in many representations, including pitch-class (e.g., 0, 5, 6), pitch names (e.g., C2, F6, F#4), and Forte A Texture in athenaCL is a model of an algorithmic (1973) set-class names (e.g., 3–5B). The Sieve object music layer employing two levels of algorithmic de- enables users to enter a Multiset as a sieve. sign. The lower level of algorithmic design is con- When using a sieve in athenaCL to create a Path, trolled by ParameterObject objects, objects that the sieve segment is constructed upon a chromatic provide values for every parameter of an event. The pitch range. When entering a sieve, the logic string number of parameters in a Texture, as well as their can be followed by two comma-delimited argu- function and interaction, is determined by the parent ments for lower- and upper-bounding pitches. These type and instrument model of the Texture. The up- pitches are notated with pitch names, where middle per level of algorithmic design is controlled by the C is denoted by C4. Rather than defining z with TextureModule, the parent class of each Texture integers, z is derived from the range given by the instance. The TextureModule manages the deploy- bounding pitches. An optional third argument is ment and interaction of lower-level ParameterOb- the pitch of origin, or the location where the zero ject objects, as well as offering linear or non-linear of the sieve is placed within the pitch scale. This event processing. TextureModule objects can gener- value, if treated as an integer, is the same as trans- ate either monophonic or polyphonic musical parts, position. If no argument is provided, the origin is and they are capable of algorithmic procedures not automatically set to the lower-bounding pitch. possible by parameter generation alone, such as al- Commands in athenaCL can be executed with gorithmically generated ornamentation (Ariza 2003). single command-line arguments or through a series A ParameterObject is entered by the user as a of interactive, text-based prompts. The following list of arguments. Arguments can be strings, num- command-line argument, using the PathInstance bers, lists, or arguments for nested ParameterOb- New command (PIn), creates a new Path named “a” ject objects. The first argument is always the from three simple sieves: name of the ParameterObject.
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 Rhythms are generated using specialized Param- for the rhythmSieve ParameterObject are as fol- eterObject objects. Pulse objects, the rhythm lows: name, sieve logic string, length, selection primitive in athenaCL, are measures of duration rel- string, and rhythm ParameterObject. The rhythm ative to a beat. These objects are notated with three ParameterObject is given as a list of arguments values: a divisor, a multiplier, and an accent. The di- for the desired rhythm generator. visor designates a fraction of the beat; the multiplier All sieves, when interpreted as rhythms, can be scales this fractional division, and the accent can thought of as composite polyrhythms. As each code for notes (value of 1) or rests (a value of 0). Residual object produces a periodic stream of Thus, at a quarter-note beat equal to 120 BPM, a pulses, so the sieve is a periodic stream of composite Pulse object of (2,1,1) would be an eighth-note du- pulses. The sieve, used as such, provides a compact ration equal to 0.25 seconds. A Pulse object of and precise notation for composite polyrhythms. In (4,3,1) would produce a dotted eighth-note duration athenaCL, for example, numerous independent equal to 0.375 seconds. Texture objects, each with related sieves, could be Two sieve-based rhythm ParameterObject ob- used to create dense polyrhythmic structures. jects, pulseSieve and rhythmSieve, are deployed within athenaCL. Xenakis (1990; 1992, p. 269) demonstrates converting a binary sieve segment to Sieve Parameter Generation a rhythm by setting the ELD to a duration, segment points to notes, and segment slots to rests. The With the exclusion of rhythm, all event values in athenaCL pulseSieve object extends this applica- athenaCL can be controlled by generator Parame- tion. The arguments for pulseSieve are as follows: terObject objects. Event values include tempo, name; sieve logic string; length; pulse; a selection amplitude, panning, microtonal pitch transposition, string randomChoice, randomWalk, randomPer- and, depending on instrument model, values for mutate, orderedCyclic, or orderedOscillate; synthesis parameters. Xenakis describes the utility and an articulation string attack or sustain. The of controlling many attributes of a note event with length value creates a z from 0 to length–1; pulse a sieve: “[S]ieve theory is very general and conse- defines the durational ELD as a Pulse object; the quently is applicable to any other sound characteris- selection string determines the method of read- tic that may be provided with a totally ordered ing values from the sieve segment; and articula- structure, such as intensity, instants, density, de- tion selects whether the rhythm is created from a grees of order, speed, etc.” (1966; 1992, p. 199). binary segment (an attack articulation, in which The athenaCL ParameterObject called value- intervening slots become rests) or from a width seg- Sieve facilitates flexible generation of masked, ment (a sustain articulation, in which intervening floating point values applicable to a wide range of slots are sustained). parameters. The arguments for valueSieve are as The pulseSieve object can be thought of as fil- follows: name, sieve logic string, length, minimum, tering a z of equal-valued pulses. For more rhythmic maximum, and selection string. Sieve logic strings, variety, the ParameterObject called rhythm- length values, and selection strings function as in Sieve allows z to be supplied by any non-rest pulse the pulseSieve and rhythmSieve objects. How- list. This pulse list is then filtered by the sieve: ever, where pulseSieve and rhythmSieve use bi- points on the sieve segment remain notes, and slots nary or width sieve segments, valueSieve uses a on the sieve segment become rests. The pulse list, unit segment. Segment points, as floating-point val- or z, can be dynamically generated by any other ues within the unit interval, are read from the unit rhythm ParameterObject. A ParameterObject segment (as dictated by the selection string) and for generating rhythmic variants with genetic algo- then scaled within dynamic minimum and maxi- rithms (Ariza 2002), for instance, could be used to mum values. Varying the maximum and minimum generate the z filtered by the sieve. The arguments with nested ParameterObject objects provides dy-
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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/0148926054094396 by guest on 24 September 2021 namic scaling of the sieve segment’s proportions. References The sieve is thus bound by what G. M. Koenig called a tendency mask (1970). Although with such a mask Amiot, E., et al. 1986. “Duration Structure Generation absolute values of the sieve are not maintained, the and Recognition in Musical Writing.” Proceedings of proportional distribution of values between points the International Computer Music Conference. San remains intact. Within athenaCL, such a generator Francisco, California: International Computer Music could be used to control tempo values, panning po- Association, pp. 185–192. Andreatta, M. 2003. “Méthodes algébriques en musique sitions, amplitudes, or microtonal transpositions; et musicologie du XXe siècle: aspects théoriques, ana- in combination with a Csound instrument, sieve- lytiques et compositionnels.” Dissertation. École des derived values could be used for selecting filter cut- Hautes Etudes en Sciences Sociales. off frequencies or start times of an audio file. Ariza, C. 2002. “Prokaryotic Groove: Rhythmic Cycles as Real-Value Encoded Genetic Algorithms.” Proceedings of the International Computer Music Conference. San Conclusion Francisco, California: International Computer Music Association, pp. 561–567. With the sieve, Xenakis sought to “add to our arse- Ariza, C. 2003. “Ornament as Data Structure: An Algo- nal sharper tools”—tools of “trenchant axiomatics rithmic Model based on Micro-Rhythms of Csángó and formalization” (1967; 1992, p. 194). The sieve Laments and Funeral Music.” Proceedings of the Inter- national Computer Music Conference. San Francisco, model presented here shares the elegance and power California: International Computer Music Association, of Xenakis’s original theory while providing an open- pp. 187–193. source, portable, modular, and complete implemen- Ariza, C. 2004. “An Object-Oriented Model of the Xe- tation. With this new model, rigorous exploration of nakis Sieve for Algorithmic Pitch, Rhythm, and Para- the sieve is possible, including applications in algo- meter Generation.” Proceedings of the International rithmic composition, algorithmic synthesis, music Computer Music Conference. San Francisco, Califor- analysis, and music theory. In 1996, Xenakis stated nia: International Computer Music Association, “the basic problem for the originator of computer pp. 63–70. music is how to distribute points on a line” (1996, Assayag, Gérard. 1993. “Cao: vers la partition potentielle.” p. 150); the sieve is one solution. The Python mod- Les Cahiers de l’Ircam: La composition assistée par or- ule sieve.py is distributed under the General Public dinateur 1(3). Available at mediatheque.ircam.fr/arti- cles/textes/Assayag93b/. License (GPL) as part of athenaCL, and it can be Barthel-Calvet, A. S. 2000. “Le Rythme dans l’Oeuvre et downloaded online at www.athenacl.org. la Pensée de Iannis Xenakis,” Dissertation. L’Ecole des Hautes Etudes en Sciences Sociales. Barthel-Calvet, A. S. 2001. “Chronologie.” In F. B. Mache, Acknowledgments ed. Portrait(s) de Iannis Xenakis. Paris: Bibliotheque nationale de France, pp. 133–142. I particularly thank Elizabeth Hoffman, who, over Emmerson, S. 1976. “Xenakis Talks to Simon Emmer- the course of this project, provided advice and guid- son.” Music and Musicians 24:24–26. ance. I am grateful to Paul Berg, Sean H. Carson, Flint, E. R. 1989. “An Investigation of Real Time as Evi- Robert Rowe, and the Editors and anonymous re- denced by the Structural and Formal Multiplicities in viewers for providing valuable comments and sug- Iannis Xenakis’ Psappha.” Dissertation. Graduate School of the University of Maryland. gestions on earlier versions of this article. Thanks Forte, A. 1973. The Structure of Atonal Music. New also to the following people for discussing and shar- Haven, Connecticut: Yale University Press. ing their research: Emmanuel Amiot, Gerard As- Gibson, B. 2001. “Théorie des cribles.” In M. Solomos, ed. sayag, Anne-Sylvie Barthel-Calvet, Ellen Flint, Presences of / Présences de Iannis Xenakis. Paris: Cen- Benoît Gibson, James Harley, Evan Jones, Marcel tre de Documentation de la Musique Contemporaine, Mesnage, André Riotte, and Sever Tipei. pp. 85–92.
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