Sieve Theory

Dimitris Exarchos

A sieve is an abstract generalization of the concept of the scale. In Xenakis’s own words, a sieve is “every well-ordered set [that] can be represented as points on a line, as long as a reference point is given for the origin and a length u for the unit distance” (1992, 268). As an abstract construct, a sieve represents a class of scales; that is, a sieve includes all the possible transformations applied to a scale that do not change its internal intervallic structure (the sequence of intervals between its successive elements). These transformations comprise cyclic transposition (a shift that causes the intervallic structure to start at a different interval) and inversion (where the intervallic structure is inverted). For example, the white-key modes are all cyclic transpositions of one single sieve (see Xenakis 1992, 268).

Xenakis first conceived of sieve theory (based on related mathematical theories) in 1963-64, during a residency in Berlin, on a grant awarded by the Ford Foundation and the West-Berlin Senate (Xenakis 1992, 371; see also Group Theory). The first published material on sieve theory is found in Xenakis’s article “La voie de la recherche et de la question” of late 1965 (reprinted in Xenakis 1994, 67-74). The first extended reference was published in 1967 in “Vers un métamusique” (reprinted in Xenakis 1992, 180-200) whose manuscript also dates back to late 1965 (see Solomos 2001, 233). This was followed by the ultimate publication of “Sieves” in 1990 (reprinted in Xenakis 1992, 268-88).

Sieves were applied in two distinct ways by Xenakis, during his middle and late periods (see Exarchos 2019). The former concerns the works of the mid-sixties such as Akrata (1964-65) and Nomos Alpha (1965-66). During this period Xenakis’s sieves had the form either of pitch scales (frequently based on quarter-tones) or of rhythmic sequences, in works such as Persephassa (1969). Back then, Xenakis would apply sieve theory using mathematical formulas and manual calculations, in order to produce scales and their transformations (metabolae was Xenakis’s term), while he would represent these as points on a straight line, using graph paper. In the late period Xenakis applied sieves “in a new way” (1977), starting with Jonchaies (1977), Aïs (1980), and Nekuïa (1981), and extending to the works of the early 1990s. During this period he composed sieves invariably as pitch scales (based exclusively on semitones), perhaps with the aid of a computer and without the level of mathematical rigour of the earlier middle period. All sieves of the late work share similar features with regard to the range and average size of intervals (see Exarchos 2008). The aforementioned publications of the 1960s correspond to the middle period, while the most recent one (“Sieves” of 1990), which also included two computer programs for the construction and analysis of sieves, corresponds to the late period.

The most characteristic aspect of the middle period with regard to sieve application, is Xenakis’s double discovery of decomposition and commutativity. Decomposition stands for the process whereby an element is broken down into its component parts. Given a law of composition, elementary components can be combined in order to make up a composite entity (just like atoms make up a molecule). Elementary components are those that cannot be further decomposed into smaller entities. A common such process in arithmetic is factorization: the dividing of a number into smaller numbers whose product yields the original. Those numbers that cannot be divided exactly by another number are called primes and comprise the building blocks of all other numbers. Commutativity is also common in arithmetic: when adding or multiplying numbers, the order of their appearance does not count. Both of these properties are assumed to be valid when one mathematically notates and analyses a sieve (note that due to commutativity, sieves consist structures outside of time). What underlies the application of the above is Xenakis’s celebratory remark on the neutrality of equal temperament, which entails that musical intervals are constructed by addition (based on a “unit distance”), rather than by proportion of lengths.

[The tempered chromatic scale] is for music what the invention of natural numbers is for mathematics and it permits the most fertile generalisation and abstraction. Without being conscious of its universal theoretical value, Bach with his Well-Tempered Clavier was already showing the neutrality of this scale. […] Today, we can affirm with the twenty-five centuries of musical evolution, that we arrive at a universal formulation concerning the perception of pitch, which is the following:

The totality of melodic intervals is equipped with a group structure with addition as the law of composition.

Xenakis 1994, 69 (my translation)

During the middle period of sieve application Xenakis made use of set-theoretical operations, which he had originally used in the piano work Herma (1961). Herma‘s symbolic music (Xenakis’s term) unfolds a predetermined logical process of combining three pitch sets, A, B, and C, via the logical operations of union (+), meaning “either/or,” intersection (*), meaning “both/and,” and complementation (–) (see Xenakis 1963, 206 & 1992, 170-77). For the composition of Akrata three years later, in the place of sets, Xenakis used residue classes (RC) instead: these are series of multiples of a unit, generated by numbers equivalent under a modulo relation. Two numbers are said to be congruent modulo M if they give the same residue (remainder) when divided by M. For example, 4, 7, and 10 all give residue 1 when divided by 3. This relation is symbolized by “≡”:

1 ≡ 4(mod3) ≡ 7(mod3) ≡ 10(mod3)

Each RC is notated by a pair M_I that indicates a modulus M (a natural number) and a residue I (a number smaller than M). The above RC notated as 3₁. A single RC is the most elementary sieve. If middle C is set to 0 and the unit is the semitone, the example above produces the following pitches:

3₁ = 1 4 7 10 … = C# E G B♭…

The purpose of working with RCs is two-fold: to either construct scales by combining RCs with the aforementioned logical operations; or to analyse a given scale by decomposing its overall modulus into RCs (e.g. the octave into a major 3^rd and a minor 3^rd, as 12 = 4 x 3). As Schaub 2005 has shown, Akrata employs scales produced by RCs with moduli A = 2, B = 3, and C = 5. Other notable works using sieves in this period include Nomos Alpha and Persephassa, composed in the following years. In Nomos Alpha several quarter-tone pitch scales were produced via an intricate mechanism of sieve transformations (metabolae), with the following “departure formula”:

which produces the scale of the example below.

Persephassa, a work for unptiched percussion, comprises rhythmic sequences based on transformations of a sieve of a modulus (period) of 60 sixteenth notes (see Gibson 2011, 103-14).

For the construction of sieves in the middle period, Xenakis relied on the decomposition of the overall modulus; in his late period however, he abandoned this approach. In fact, the pitch scales of the later music are of a different kind, and analysis in terms of decomposition tends to not apply to these, as there is no clear overall period to be decomposed. The last work to use a sieve with a clear period is Mists (1981, for piano), whose structure however relies on cyclic transpositions, and not on decomposition (see Squibbs 1996, 180-202 & Exarchos 2008, 144-48). Subsequent works make use of a different kind of sieve, itself comprising a class of pitch scales of a particular character.

In Jonchaies (1977) and other works of the following two years, Xenakis used a scale made up of intervals between a minor 2^nd and a major 3^rd; its period is one octave and a perfect 4^th, which is a prime (17 semitones) and thus cannot be decomposed (see Exarchos 2008, 142-44). That characteristic scale was partly inspired by his “study of Javanese music, and of the scale called the pelog in particular, which is based on a very powerful interlocking of two fourths” (Varga 1996, 144). Analysis of periodicity in such sieves can only be applied on the internal structure of the sieve (see Exarchos 2008, 101-76). There are approximately forty works that make use of around one hundred pitch scales, all of which (with very few exceptions) exhibit the same general profile, based on the following particular characteristics:

• there are no repeating patterns and no perceivable periodicity
• the size of consecutive intervals ranges from the minor 2^nd to the major 3^rd, with no use of quarter-tones
• the longest string of semitones in the intevallic structure is two
• the density of the sieve, defined as the ratio of its range over the number of its pitches, is around 0.5
• finally, due to the above, the complement of the scale retains the same characteristics.

We can term the class of the above scales collectively as the Xenakis sieve.

In such approach, Xenakis was researching a way to produce timbres, rather than melodic patterns (see Exarchos 2007). As he said,

The structure of the melodic scale is very important […]. If you take a given range, and if the structure of the scale is rich enough, you can stay there without having to resort to melodic patterns – the interchange of the sounds themselves in a rather free rhythmic movement produces a melodic flow which is neither chords nor melodic patterns. […] They give a kind of overall timbre in a particular domain.

Varga 1996, 145

This is occasionally achieved via techniques like heterophony (“artificial reverberation” – Xenakis 1977, or “halo sonores” – Solomos 1996, 84), sieve clusters, cellular automata, and others. As Hoffman put it, “it is as if a collection of tuned chimes [are] rung by a complex mechanism” (2002, 126). Quite a few instances of the Xenakis sieve were used in more than one works, thus providing a strong sense of continuity between them. One such instance, the sieve of Nekuïa (1981), was used in more than fifteen works in the six years to follow (see Exarchos 2008, 151-62).

During the late period, sieves were probably produced by computerized algorithmic processes. Further, it is likely that Xenakis was inspired by a mathematical model of a random sieve (itself based on the mathematical prime number sieve, also known as the Sieve of Eratosthenes – see Hawkins 1958). This hypothesis fits with the features of the aforementioned sieve profile, with regard to its average expected interval (major 2^nd) and its larger expected interval (perfect 4^th). Although at first it might seem that Xenakis abandoned the high degree of mathematical formalization of the early phase, the new computational sophistication available during the later phase allowed him to further advance his early theoretical conceptions.

References

Exarchos, Dimitris. 2007. “Injecting Periodicities: Sieves as Timbres.” In Proceedings of the 4th Sound and Music Computing Conference (SMC07), edited by C. Spyridis, A. Georgaki, G. Kouroupetroglou, and C. Anagnostopoulou, 68–74. Lefkada, Greece. http://smc07.uoa.gr/SMC07%20Proceedings/SMC07%20Paper%2011.pdf.

Exarchos, Dimitris. 2008. “Iannis Xenakis and Sieve Theory: An Analysis of the Late Music (1984-1993).” PhD diss. Goldsmiths, University of London.

Exarchos, Dimitris. 2019. “The Berlin Sketches and Xenakis’s Middle Period Style.” In Exploring Xenakis: Performance, Practice, Philosophy, edited by Alfia Nakipbekova, 21–36. Wilmington, Delaware: Vernon Press.

Gibson, Benoît. 2003. “Théorie et pratique dans la musique de Iannis Xenakis : À propos du montage.” PhD diss., École des hautes études en sciences sociales.

Gibson, Benoît. 2011. The Instrumental Music of Iannis Xenakis: Theory, Practice, Self-Borrowing. Hillsdale, N.Y. : London: Pendragon Press.

Hawkins, David. 1958. “Mathematical Sieves.” Scientific American, December, 105–12.

Hoffmann, Peter. 2002. “Towards an ‘Automated Art’: Algorithmic Processes in Xenakis’ Compositions.” Contemporary Music Review 21 (2–3): 121–31. https://doi.org/10.1080/07494460216650.

Solomos, Makis. 2001. “Bibliographie commentée des écrits de/sur Iannis Xenakis.” In Presences of Iannis Xenakis [Présences de Iannis Xenakis], edited by Makis Solomos, 231-65. Paris: Cdmc.

Solomos, Makis. 2004. Iannis Xenakis. Revised edition. Mercuès: P.O. Editions. https://hal.archives-ouvertes.fr/hal-01202402/document.

Squibbs, Ron. 1996. “An Analytical Approach to the Music of Iannis Xenakis: Studies of Recent Works.” PhD diss. Yale University.

Xenakis, Iannis. 1965. “La voie de la recherche et de la question.” Preuves 177, 33-36.

Xenakis, Iannis. 1967. “Vers un métamusique.” La Nef 29, 38-70

Xenakis, Iannis. 1977. Jonchaies. Paris: Éditions Salabert.

Xenakis, Iannis. 1994. Kéleütha (Ecrits). Paris: L’Arche.

Xenakis, Iannis. 1990. “Sieves.” Perspectives of New Music 28 (1), 58-78.

Xenakis, Iannis. 1992. Formalized Music: Thought and Mathematics in Composition. Rev. ed. Stuyvesant, NY: Pendragon.

How to cite

EXARCHOS, Dimitris. 2023. “Sieve Theory.” In A Xenakis Dictionary, edited by Dimitris Exarchos. Association Les Amis de Xenakis. https://www.iannis-xenakis.org/en/sieve-theory