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

rhythmic world and the pitch domain. By means of the action of a (commutative) group $G$ on a $P H$ $G$ -set $S$ , the collection of orbits determined by the action of $G$ on the set of all subsets of $S$ , which are also called »modal classes« or »transposition classes«) has the algebraic structure of a commutative semigroup with a unit element.
Different notations have been used by Vuza in the more than 10 years that separate the original collection of papers on the mathematical aspects of Vieru’s Modal Theory (Vuza, 1982-83) from the most recent article on Supplementary Sets and Rhythmic Canons (Vuza, 1991-93). In summarising the principal results of Vuza’s rhythmic model we shall use concepts and definitions given in Vuza (1985), Vuza (1988) and Vuza (1991-93), and attempt to make uniform the notation of the latter.
By definition, a periodic rhythm is a periodic and locally finite subset $R$ of $|Q|$ . This means:

$E t (- Q|| +$ such that $t+ R = R$ .
$A a,b (- Q|,|a < b,#{r (- R : a < r < b}< oo$ .

This second property differs slightly from that of Vuza (1985) and Vuza (1986). The least positive rational number satisfying the first condition is called the period of $R$ whereas the greatest positive rational number dividing all differences $r - r 1 2$ with $r (- R i$ is called the minimal division of $R$ .
As pointed out in Vuza (1986) this definition implies that the collection of rhythms is a ring of sets. Moreover, one may consider the translation class of a given rhythm with respect to the group $||Q$ . More formally one can consider the action of $||Q$ on the set of periodic rhythms as defined by the map $(t,R) '---> t+ R$ and call a »rhythmic class« an orbit of $R$ under this action. Following Vuza’s notation I shall indicate with $[R]$ the rhythmic class of a given rhythm $R$ and with $Rhyth$ the collection of all rhythmic classes. More generally, I shall indicate with $T(G)$ the set of translation classes of a given group $G$ , i.e. the set of equivalence classes determined by the relation $~$ between subsets of $G$ which are the translate of each other through an element $x$ in $G$ (see Vuza, 1991-93). The sum of subsets $M$ and $N$ of $G$ are defined in the following way:

M + N = {x+ y,x (- M,y (- N }

It has been shown that $T(G)$ is a commutative semigroup with unit element under the following law (called »composition« and indicated with +):

M + N = [M + N]

where $[M ]$ signifies the translation class of a given subset $M$ of $G$ .
Taking $/ G =Z12$ we obtain the 352 »modal classes« (including the null class) which have been originally studied by Zalewski in Zalewski (1972). Figure 2 shows the symmetric distribution of transposition classes of chords in $/ Z12$ and $/ Z24$ .

Figure 2:

Number of transposition chords for the Twelve-Tone and quarter-tone temperament

Note that, using dihedral symmetry, as usually applied in American Set-Theory, the number of equivalent classes of chords in a given well-tempered division of the