Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

GGGE♭ 19
Proposition 7.2 Let M be an autosimilar melody with ratio a and any offset; put r = −a −1 (mod n);
then picking one note every r yields the retrograde −M (up to some offset).
n − 1
This is particularily audible when r = 2, i.e. when a = (for odd n).
2
In musical terms, this means that any autosimilar melody has an augmentation equal to one of its
retrogradations.
7.1.2 Inverse-retrograde symmetry. Johnson’s conjecture mentions melodies whose inverse IS the retrograde.
These can be build from an autosimilar melody with a symmetry between its orbits, setting
opposite notes on symmetric orbits: then the retrograde of the melody will be its inversion. It is uncommon
to find autosimilar (primitive) structures without such a symmetry (this is related to Johnson’s
conjecture): it is mandatory for instance when b = 0. But 4x + 1 (mod 21) does the trick, as its orbits (0,
1, 5), (2, 9, 16), (3, 11, 13), (4, 6, 17), (7, 8, 12), (10, 18, 20), (14, 15, 19) exhibit no inversional symmetry
whatsoever. We have a condition ensuring that such retrogradation symmetries between orbits do exist:
Theorem 7.3 Assume a − 1 divides 2b, 2b = c(a − 1); then all orbits are permuted by the symmetry
x ↦→ −c − x, i.e.
∀x ∈ Z/nZ O−x−c = −c − Ox
(some orbits may be self-invariant under this symmetry).
Proof to be found in the Online Supplementary I.
7.2 Other spaces
Further perspectives include the use of spaces other than pitch and time. The full group Aff12 would
provide, on the one hand, sequences of series derived from a seminal one (not unlike Jean Barraqué’s
Séries proliférantes) and on the other hand, series with interesting symmetries when the sequence turns
out to be shorter than expected. This has been studied, notably for f : x ↦→ −x + b (e.g. retrogradations),
for instance in the ancient [2], with enumeration and construction of such series. But more general affine
transforms may be of interest for composers, especially with n = 12.
Acknowledgements
First and foremost I must thank composer Tom Johnson for his pioneering work on the subject and the
wonderful music he managed to create out of this basically simple idea. I am also grateful to Gerard
Assayag who introduced me to the notion on an informal occasion, and later raised the fine question of
detecting approximate autosimilarity. Carlos Agon implemented all this in OpenMusic T M , while Moreno
Andreatta helped clarify a number of delicate points. I have received precious and learned advice from
anonymous reviewers. I owe them several interesting additional references.
References
[1] Amiot, E., Why Rhythmic Canons are Interesting, in: E. Lluis-Puebla, G. Mazzola et T. Noll (eds.), Perspectives of Mathematical
and Computer-Aided Music Theory, EpOs, 190–209, Universität Osnabrück, 2004.
[2] Amiot, E., La série dodécaphonique et ses symétries (1994), Quadrature, 19, Ed. du Choix, Marseille. Online version:
http://pagesperso-orange.fr/chuckydoo/SeriesSym/index.html
[3] Amiot, E., Agon, C., Andreatta, M., Implementation of autosimilar melodies in OpenMusic (2007), ICMC acts.
[4] Andreatta, M., Méthodes algébriques en musique et musicologie du XXe sicle : aspects théoriques, analytiques et compositionnels
(2003), Ph.D. dissertation, EHESS.
[5] Andreatta, M., Agon, C., Vuza, D.T., Analyse et implémentation de certaines techniques compositionnelles chez Anatol Vieru, in
Actes des Journées dInformatique Musicale, Marseille, (2002), pp. 167-176
[6] Batstone, Philip. Multiple Order Functions in Twelve-Tone Music. (1972) Perspectives of New Music 10(2); 11(1).
[7] Feldman, D., Self-Similar melodies, in Leonardo Music Journal, (1998) 8:80-84.