Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

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Definition 2.1 Let M be a periodic melody with period n. M is autosimilar with ratio a and offset b iff
∀k ∈ Zn Ma k+b = Mk.
An example is the following common rhythmic beat, which is autosimilar with ratio 3 and offset 1:
Figure 7. Autosimilarity with offset
Theorem 2.2 Any autosimilar melody of ratio a, period n, and offset b is built up from orbits of the affine
map x ↦→ a x + b (mod n).
The proof is identical to that of Thm. 1.2. 1
Remark 1 This new, more general setting, includes the case a = 1 with melodies invariant under x ↦→ x+τ,
i.e. maps with a period smaller than n. Each orbit, and hence each preimage
M −1 (p) = {k, Mk = p}
is then a Limited Transposition Subset of Zn. We will henceforth exclude this case and assume a = 1. 2
For instance, the Kientsy Loops 3 melody G F E D E F G D G F E D E F G D G F E. . . can be viewed
as generated by x ↦→ 3x + 6 (mod 8) if we set origin at G=0.
It can be viewed more simply as generated by x ↦→ 3 x if we decide that the consecutive F, E are labelled
0, 1 instead of 1, 2. This ambivalence will be elucidated below.
2.1 Orbit lengths
Lemma 2.3 The order of map f : x ↦→ a x + b (mod n) (e.g. the size of the subgroup gr(f) generated by
f in Affn) is:
Proof Easily from formula
o(f) = min k > 0 such that gcd(a − 1, b) (1 + a + . . . a k−1 ) = 0 (mod n) (2)
f k (x) = f ◦ f ◦ . . . f(x) = a k x + b (1 + a + . . . a k−1 ) (3)
In practice, compute the ‘missing factor’ mf = n/ gcd(a − 1, b, n), and look up the first number 1 + a +
. . . a k−1 that is a multiple of mf.
1 The mathematical standpoint would be here to define an action of Affn on the set of maps M : Zn → (pcs) by f • M = M ◦ f.
2 The setting of affine maps modulo n might be unfamiliar to many readers, and a few reminders may be useful. The main point is to
distinguish between the monoid of general affine maps, and the group of affine transformations, which are one-to-one maps; these
last are exactly the x ↦→ a x + b (mod n) with gcd(a, n) = 1. Their group Affn is a semi-direct product of its translation subgroup (all
x ↦→ x + b), isomorphic to the group Zn, and its homotheties subgroup (all x ↦→ a x, gcd(a, n) = 1) which is isomorphic to Z ∗ n; Affn is
not abelian and several open problems remain about its structure. [10] is justified in demanding that the exact sequence
0 → (Zn, +) → (Affn, ◦) → (Z ∗ n , ×) → 1 (ES) (1)
(which is another way of expressing that Affn = Z ∗ n ⋉ Zn) be taken into account; but it does not explain all that follows.
3 CD Pogus productions, P21033-2, 2004.