Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

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 $0 =$ C $4 = 261.6Hz)$ and the unit distance is a given well-tempered interval (usually the semitone). Using the operations of union ( $U$ ), intersection ( $/~\$ ) and complementation ( $C$ ), it is also possible to formalise the diatonic collection in such a way:

(C3n+2 /~\ 4n) U (C3n+1 /~\ 4n+1) U (3n+2 /~\ 4n+2) U (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

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).

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

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).

Another music-theoretically important sort of groups that we have to mention here⁵

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).

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 almost 50 years later!) and particularly in Babbitt (1962) where he introduced the