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

Idempotent chords

Partitioning chords

In the present context we will concentrate basically on families which are central to the formalisation of tiling rhythmic canons. According to Zalewski’s and Vieru’s original idea, a translation class $M$ will be indicated by the intervallic structure $(m1,m2, ...,mi)$ which counts the number of unit steps between successive notes. $#M$ is called the order of the translation class (i.e. of the intervallic structure or, more simply, of the structure). It follows that $sum ij=1mj = n$ where $n$ is the order of the cyclic group $Z/n$ .
The three families which are particularly relevant in the context of the present paper are:

Idempotent classes.
Given two classes $A,B (- T (/Zn)$ we introduced, following Vuza (1982-83), a law of composition ( + ) which is by definition: $A + B = [M + N ]$ where $M (- A,N (- B$ . This operation is formally equivalent to what the American music-theoretical tradition calls »transpositional combination« between chords. It represents a generalisation of Boulez’ technique of »multiplication d’accords«, as it has been initially introduced in Boulez (1963) and Boulez (1966). For a discussion of this compositional technique from an american music-theoretical perspective see Cohn (1986).
In the case of $Z/12$ there are exactly 6 special classes $A$ for which $A+ A = A$ . These are called idempotent classes and their collection is indicated by $Tid$ . All of these are well known to musicians, from the unison $U = (0)$ to the total chromatic $TC$ . They correspond to Zalewski’s »monomorphic structures«, and mathematically speaking there are simply all subgroups of a given cyclic group.
Limited transposition classes.
A modal class $A (- T(Z/n)$ is a limited transposition structure if $#A < n$ i.e. its subgroup of stability $SA := {t (- Z/n : t+ M = M, M (- A}$ is not trivial. I shall indicate with $Ttl$ the family of all limited transposition structures. There is a strong connection between transposition limited modes and idempotent classes thanks to the previous concept of »composition« between intervallic structures. It is easy to see that a class belongs to $Ttl$ iff it is the transpositional combination of two classes, where at least one of these classes must be idempotent. This enables to calculate very easily the transposition limited classes for any well-tempered division of the octave in a given number $n$ of equal parts.
Partitioning classes
The family $Tp$ of »partitioning classes« has been introduced in Vuza (1982-83)