Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

GGGE♭ 3
Figure 2. Thema of ’in the Mood’
But it will surprise many readers to realise that much more ancient Western music features autosimilarity:
it can be found in Beethoven’s Fifth Symphony, though the ratio 3 autosimilarity is not blatant at all (fig.
4). The ubiquitous Alberti Bass (fig. 3), well known from (for instance) the beginning of Mozart’s Sonata
in C major K. 545, is an excellent example, with autosimilarities at ratios 3, 5 and generally every odd
number. In Mozart’s first bar, left hand is exactly autosimilar while right hand significantly plays the same
three notes. Ratio 4 autosimilarity is even explicit in the first two bars of Scarlatti’s Sonata Kirk. 9 in D
minor, as pointed by an anonymous reviewer.
1.3 Mathematical generation.
Figure 3. Alberti Bass with augmentation by 3
Theorem 1.2 Any autosimilar melody with ratio a and period n is built from orbits of the affine map
x ↦→ a x mod n: denoting the orbit of x as
Ox = {a k x mod n, k ∈ Z} = a Z .x,
for each note p of the melody, the subset of indexes M −1 (p) = {i ∈ Zn, Mi = p} is one such orbit, or a
union of several ones.
Proof It is sufficient to prove that if Mx = p then Mk = p for all k ∈ Ox, hence every orbit will come in
toto for a given note. But this is obvious from the definition, as
Mk = Ma m x = Ma m−1 x = . . . Mx = p.
A basic fact about orbits is worth recalling here, namely that x ∈ Oy ⇐⇒ y ∈ Ox.
In group theory, these orbits are the classes of the action of the cyclic subgroup generated by the map
f : x ↦→ a x mod n.
At this point it seems a good idea to demand that a (or f) should generate a subgroup, which means
that a is coprime with n, or equivalently a ∈ Z ∗ n. As will be seen below, interesting situations arise when
this condition is dropped. But until section V,
The ratio a of an autosimilar melody is assumed to be coprime with the period n.
before WW II. Quoting Henry Cowell in his preface to [13], “The idea behind the Schillinger System is simple and inevitable: it undertakes
the application of mathematical logic to all the materials of music and to their functions, so that the student may know the unifying
principles behind these functions, may grasp the method of analyzing and synthesizing any musical materials that he may find anywhere
or may discover for himself, and may perceive how to develop new materials as he feels the need for them.”