Preliminaries
Applying methods from Pólya Theory it is possible to
enumerate various kinds of musical objects as intervals,
chords, scales, tone-rows, motives and so on. See for instance
[7],[15],[4],[5],[14]. Usually these results are given
for an n-scale, which means that there are exactly
n tones within one octave. Collecting all tones, which are
any number of octaves apart, into a pitch class, there are exactly
n pitch classes in an n-scale. These pitch
classes can be considered as elements of the residue class
group (Zn,+) of Z modulo
nZ. The musical operator of transposing by
one pitch class can be interpreted as a permutation of
Zn
| T: Zn→
Zn, i↦ T(i) :=
i+1... |
Inversion at pitch class 0 is the following permutation
| I: Zn→
Zn, i↦ I(i) :=
-i... |
The group of all possibilities to transpose is the
cyclic group ⟨T⟩ =Cn of order
n. The permutation group generated of T and
I is the dihedral group Dn of
order 2n (for n≥ 3). Sometimes in twelve tone
music a further operator, the so called quart circle
symmetry, is used, which is given by
| Q: Z12→
Z12, i↦ Q(i) :=
5i... |
Generalising this concept to n tone music the affine
group
| Aff (1,Zn) :=
{(a,b) | a∈ Zn*, b∈
Zn}.. |
(the set of all unit elements in the ring
Zn is indicated by
Zn*) acts on Zn by
(a,b)(i) := ai+b.
These permutation groups induce group actions on the
sets of musical objects. (For basic definitions and notions in
enumeration under finite group actions see [10].) Let me explain these group
actions by introducing the so called mosaics in
Zn (see chapters 2 and 3 of [1]). In [9] it is stated that the
enumeration of mosaics is an open research problem communicated by
Robert Morris. A partition π of
Zn is a collection of subsets of
Zn, such that the empty set is not an element of
π and such that for each i∈ Zn
there is exactly one P∈ π with i∈ P. If
π consists of exactly k subsets, then π
is called a partition of size k. Let
Πn denote the set of all partitions of
Zn, and let Πn,k be the set
of all partitions of Zn of size k. A
permutation group G of Zn induces the
following group action of G on Πn:
| G × Πn→
Πn, (g,π)↦ gπ
:= {gP | P∈ π},.. |
where gP := {gi | i∈ P}. This action can be restricted
to an action of G on Πn,k. The
G-orbits on Πn are called
G-mosaics. (This is a slight generalisation of the
definition given in [9].) Correspondingly the
G-orbits on Πn,k are called
G-mosaics of size k.
harald.fripertinger "at" uni-graz.at, May 10,
2016
