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On group-theoretical methods applied to music : some compositional and implementational aspects
| Content Provider | Semantic Scholar |
|---|---|
| Author | Andreatta, Moreno |
| Abstract | This paper focuses on the group-theoretical approach to music theory and composition. In particular we concentrate on a family of groups which seem to be very interesting for a ›mathemusical‹ research: the non-Hajós groups. This family of groups will be considered in relationships with Anatol Vieru’s »Theory of modes« as it has been formalised and generalised to the rhythmic domain by the Roumanian mathematician Dan Tudor Vuza. They represent the general framework where one can formalize the construction of a special family of tiling canons called the »Regular Unending Complementary Canons of Maximal Category« (RCMC-canons). This model has been implemented in Ircam’s visual programming language OpenMusic. Canons which are constructible through the Vuza’s algorithm are called Vuza Canons. The implementation of Vuza’s model in OpenMusic enables to give the complete list of such canons and offers to composers an useful tool to manipulate complex global musical structures. The implementation shows many interesting mathematical properties of the compositional process which could be taken as a point of departure for a computational-oriented musicological discussion. 1 Introductory remarks on the role of group theory in music »The question can be asked: is there any sense talking about symmetry in music? The answer is yes« (Varga (1996), p. 86). By paraphrasing Iannis Xenakis previous statement, one could pose a similar question about groups and music: is there any sense talking about mathematical groups in music? With the assumption of the relevance of symmetry in music the answer follows as a logical consequence of this universal sentence: »Wherever symmetry occurs groups describe it« (Budden (1972)). As Guerino Mazzola’s Mathematical Music Theory suggests, there are many reasons for trying to generalise some questions about symmetry in music. But the Perspectives in Mathematical and Computer-Aided Music Theory 2 question needs to be asked as to whether new results could be musically relevant, or whether they represent purely mathematical speculations. A concept of »musical relevance« in a mathematical theory of music is one of the most difficult to define precisely. Inevitably there is a »tension« between mathematics and music which has, as a practical consequence, the »mystical aura of pure form« (Roeder (1993)) of some mathematical theorems in contrast to the »mundanity« of their application to music . Criticism could be levelled against the potential competence of a mathematician expressing »in a very general way relations that only have musical meaning when highly constrained« (Roeder (1993), p. 307). This essay is an attempt to discuss some general abstract group-theoretical properties of a compositional process based on a double preliminary assumption: the algebraic formalization of the equal-tempered division of the octave and the isomorphism between pitch space and musical time. Historically there have been different approaches from Zalewski’s »Theory of Structures« (Zalewski (1972)) and Vieru’s »Modal Theory« (Vieru (1980)), to the American Set-Theory (Forte (1973), Rahn (1980), Morris (1987)), whose special case is the so-called diatonic theory, an algebraic-oriented ramification of Set-Theory which is usually associated with the so-called »Buffalo School« at New York (Cf. (Clough J. (1986)) and (Clough (1994))). See (Agmon (1996)) for a recent summary in the theory of diatonicism.1 The common starting point is that every tempered division of the octave in a given number n of equal parts is completely described by the algebraic structure of the cyclic group / Zn of order n which is usually represented by the so-called ’musical clock’. Three theorists/composers are responsable for this crucial achievement: Iannis Xenakis, Milton Babbitt and Anatol Vieru. They form what we could call a »Trinity« of composers for they all share the interest towards the concept of group in music.2 In Babbitt’s words, »the totality of twelve transposed sets associated with a given [twelve-tone set] S constitutes a permutational group of order 12« (Babbitt (1960), p. 249). In other words, the Twelve-Tone pitch-class system is a mathematical structure i.e. a collection of »elements, relations between them and operations upon them« (Babbitt (1946), p. viii). Iannis Xenakis is sometimes more emphatic, as in the following sentence: »Today, we can state that after the Twenty-five centuries of musical evolution, we have reached the universal formulation for what concerns pitch perception: the set of melodic intervals has a group structure with respect to the law of addition« (Xenakis (1965), p. 69-70). But unlike Babbitt’s and Vieru’s theoretical preference for the division of the octave in 12 parts, Xenakis’ approach to the formalisation of musical scales uses a 1 A detailed bibliography on Set-Theory, diatonic theory and Neo-Riemannian theory is available online on: http://www.ircam.fr/equipes/repmus/OutilsAnalyse/BiblioPCSMoreno.html 2 We could easily add some further references to the history of group-theoretical methods applied to music by also including music theorists as W. Graeser (Graeser (1924)), A.D. Fokker (Fokker (January 1947)), P. Barbaud (Barbaud (1968)), M. Philippot (Philippot (1976)), A. Riotte (Riotte (1979)), Y. Hellegouarch (Hellegouarch (1987)) ). We chose to concentrate on Babbitt, Xenakis and Vieru because of the great emphasis on compositional aspects inside of an algebraic approach. For a more general discussion on algebraic methods in XXth Century music and musicology see my thesis (Andreatta (2003)). For a detailed presentation of the algebraic concepts in music informatics see Chemillier (1989). Perspectives in Mathematical and Computer-Aided Music Theory 3 different philosophy. He considers the keyboard as a line with a referential zeropoint which is represented by a given musical pitch and a unit step which is, in general, any well-tempered interval. Algebraically, the chromatic collection of the notes of keyboard could be indicated in such a way 10 = {... − 3,−2,−1, 0, 1, 2, 3, ...} The symbol 10 means that the referential point is the 0 (usually 0 =C4 = 261.6Hz) and the unit distance is a given well-tempered interval (usually the semitone). Using the operations of union (∪), intersection (∩) and complementation (C), it is also possible to formalise the diatonic collection in such a way: (C3n+2 ∩ 4n) ∪ (C3n+1 ∩ 4n+1) ∪ (3n+2 ∩ 4n+2) ∪ (C3n ∩ 4n+3) where n = 0, 1, 2, ..., 11 and ax = an+i if x ≡ (n + i)mod(a) (cf.Orcalli (1993),p. 139). In a similar way one can formalise some other well-known music-theoretical constructions, like Messiaen limited transposition modes.3 But Sieve-Theory could also be useful to construct (and formalise) musical scales which are not restricted to a single octave or which are not necessarely applied to the pitch domain.4 Another music-theoretically important sort of groups that we have to mention here5 is the family of the dihedral groups. Historically they have been introduced by Milton Babbitt in a compositional perspective aiming at generalising Arnold Schoenberg’s Twelve-Tone System to other musical parameters than the pitch parameter. This generalisation of the Twelve-Tone technique is usually called »integral serialism« and it represents an example of a remarkable convergence of two slightly different serial strategies. We will not discuss this point from a musicological perspective, although one would be tempted to say that a critical revision of some apparently well-established historical achievements will be soon necessary. European musicologists do not seem to have been particularly interested to seriously analyse Milton Babbitt’s contribution in the field of the generalised serial technique. On the other hand, American musicologists consider M. Babbitt as the first total serialist, thanks to pieces like Three Compositions for piano (1947), Compositions for Four Instruments (1948), Compositions for Twelve Instruments (1948). Moreover M. Babbitt widely discussed this isomorphism between pitch and rhythmic domain in some crucial theoretical contributions, starting from his already quoted PhD thesis of 1946 (accepted by the Princeton Music Departement 3 The following quotation shows how the problem of expressing Messiaen’s modes of limited transpositions in sieve-theoretical way was a central concern in Xenakis’ theoretical speculation during the 60s: »I prepared a new interpretation of Messiaen’s modes of limited transpositions which was to have been published in a collection of 1966, but which has not yet appeared« (Xenakis (1991), p. 377). 4 Following Xenakis’ original idea, André Riotte gave the formalisation of Messiaen’s modes in sieve-theoretical terms (Riotte (1979)) and suggested how to use Sieve-Theory as a general tool for a computer-aided music analysis. This approach has been developed in collaboration with Marcel Mesnage in a series of articles which have been collected in a two-volumes forthcoming book (Riotte and Mesnage (2003)). 5 A forthcoming article is dedicated to the sieve-theoretical and transformational strategies underlying Xenakis’ piece Nomos Alpha involving generalized Fibonnacci sequences taking values in the group of rotations of the cube, see Agon and al. (2003). For more generalized investigations into the role of Coxeter groups in music see Andreatta (1997). Perspectives in Mathematical and Computer-Aided Music Theory 4 almost 50 years later!) and particularly in (Babbitt (1962)) where he introduced the concept of Time-Point System. We argue that there is a possibility to better understand the developments of integral serialism by »transgressing the (geographical) boundaries«, to seriously quote the title of Sokal’s famous hoax, and by analysing how some ideas could have freely moved from Europe to USA and vice-versa. The French theorist and composer Olivier Messiaen has probably played a crucial role for what concerns the European |
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| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
