Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

GGGE♭ 25
as a c − b = b + c by hypothesis.
From there we get immediately S(f k (x)) = f k (S(x)) and hence S(Ox) ⊂ O S(x). By symmetry (reasoning
on S −1 which is none other than S itself !) the inclusion is an equality.
These examples open a new alley for future research, combining inner symmetries (the autosimilarity)
of a melody with outer symmetries (e.g. inversion), using some structural features of the space of musical
events.
8 Online Supplementary II: about general affine maps
Here we consider what happens when iterating an affine map that is not bijective.
8.1 Universal property
Theorem 6.1, establishes autosimilar melodies as universal objects. Musically this means that one hears
an autosimilar melody after several augmentations of any periodic melody.
The iteration of any affine map f modulo n (not one to one) eventually reduces to iterating an affine trans-
formation on some subset of Zn.
Mathematically, it means that the submelody M = f p (M) =
Mf p (k) is autosimilar by some power of
k∈Z
f: f q [ M] = M for some p, q > 0.
Proof The set (algebraically, a monoid) of all affine maps modulo n is finite. Thus the sequence of powers
of f will only take a limited number of different values. So there must exist two different exponents p, p + q
with f p = f p+q . Now for any r > p,
f r+q = f (p+q)+(r−p) = f p+q ◦ f r−p = f p ◦ f r−p = f p+(r−p) = f r
We have just shown that the sequence of powers of f is ultimately periodic. So is for any x ∈ Zn, the
sequence f k (x). This means that after p iterations of f, any further iteration of f q will preserve the
sequence.
8.2 A Fitting ending
We will round up this last theorem with a more detailed explanation in the simpler case of homotheties,
which links this result with the abstract Fitting Lemma already connected with several musicological
situations (Anatol Vieru’s iteration of the difference operator, [5]). 1 A connection to the general case is
that the ultimate period of f ∈ Affn : x ↦→ a x + b is a multiple of the ultimate period of its linear part
f : x ↦→ a x.
Let us consider this map x ↦→ a x mod n. First we will assume for simplicity’s sake that gcd(a, n) = p
is a prime factor. This means n = p m q where q is coprime with p. Now x ↦→ a x maps Zn into the cyclic
subgroup of index p, namely p Zn, isomorphic with Zp m−1 q. After m iterations we are working in p m Zn,
cyclic subgroup of Zn isomorphic with Zq. There x ↦→ a x is one-to-one, at long last.
Proposition 8.1 The attractors Ax = {a k x, k ≥ m} are the orbits of f : x ↦→ a x operating on Z/qZ,
identified to subgroup p m (Z/nZ).
1 It is worth noticing that orbits, for homotheties, or their difference sets, for general affine maps, are exactly the eigenvectors of
Vieru’s difference operator acting on subsets of Zn:
∆(x, a x, a 2 x . . . ) = (a x − x, a 2 x − a x, . . . ) = (a − 1)(x, a x, a 2 x . . . )