Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

2 autosimilar melodies
flake, Sierpinski sponge). Diverse musical renderings are possible; in the simplest, one melody plays within
itself contrapuntally (see fig. 2), something the Kantor of Leipzig might have dreamed of. Several instances
of such melodies have been identified in classical music.
1 First definitions, historical examples
We begin with the simplest case, when all augmentations begin on the same note. This is historically the
case studied by Tom Johnson in [9], though he came across the more general case, with different starting
points, which will be studied in section III; further generalizations will occur in the last sections.
1.1 Autosimilar melody with ratio a
Definition 1.1 Let M be a periodic melody with period n: M0, . . . Mn = M0, Mn+1 = M1, . . . wherein the
values Mk are musical events (key strokes, for instance) and k is some measure of time. M is autosimilar 1
with ratio a iff
∀k ∈ Zn Ma k = Mk.
This means that taking one note every a beats yields the same melody, only slower; or equivalently that
some augmentation of the melody is part of the melody itself, as is obvious on the score below (fig. 1, with
a = 3). This explains why the melody has to be infinite. Non-periodic solutions are possible, but this is
another subject.
1.2 Musical examples
Figure 1. First bars of ’La Vie Est Si Courte’ by Tom Johnson
Of course, the use of augmentation is quite ancient. J.S. Bach is probably the best known exponent
of melodies played simultaneously with their augmentations in numerous fugas; he is also famous for
contriving several voices inside one monody (the Suites for solo strings spring to mind). Tom Johnson has
discovered this possibility around 1980 (cf. [9]), and is probably the first composer who made use of it so
systematically, as in La Vie Est Si Courte (fig. 1: one can see that the left hand voice is the right hand one
played thrice slower, and that each note of the former falls in with the same note in the latter), Kientsy
Loops, Rational Melodies, or Loops for Orchestra (fig. 13), though – as he acknowledges – a few other
contemporary American musicians (David Feldman, Paul Epstein) toyed with it at times.
There are some earlier American examples: consider Glen Miller’s famous In the Mood. Ratio 4 autosimilarity
is perfectly audible (if one understands it as it is written, i.e. with regular eighth notes, and not
as it is played), since one note out of four emerges on each strong beats, probably quite voluntarily on
Miller’s part as he studied with the mathematically minded (some say ‘obsessed’) Joseph Schillinger. 2
1 We decided to change Tom Johnson’s ‘selfRep’ to ‘autosimilar’, which he himself used in the broad sense of ‘result of some iterated
process’, because this is the traditional mathematical meaning, for instance with classical fractals.
2 This was pointed out by T. Johnson. Almost forgotten nowadays, Schillinger taught Miller, Gerschwin and other prominent composers