Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

(ou une gamme) remonte à David Lewin en 1959. et est à l’origine de nombreux concepts qui ont marqué la
recherche américaine depuis 1959. Nous choisissons la même définition que lui parmi les diverses définition
possibles équivalentes :
Définition 1. La transformée de Fourier de f : Zc → C est
F(f) :t ↦→
Z/cZ qui modélise la gamme chromatique, il s'agit d'une transformée de Fourier discrète, ou
f(k)e −2ikπt/c
p. 23
k∈Zc
Plus
DFT:
particulièrement, la transformée de Fourier de A ⊂ Zc sera la transformée de Fourier de la fonction
caractéristique 1A du sous-ensemble A5 :
FA : t ↦→
e −2iπkt/c
k∈A
Quelques exemples :
(1) FZc, la transformée de Fourier de toute la gamme chromatique, est d−1
e
k=0
−2iπkt/c Le premier qui ait utilisé cette DFT à des fins d'étude structurelle en théorie de la
1 − e−2iπt
musique est sans nul doute David Lewin, dans son tout premier = . Cette
1 − e−2iπt/c fonction est nulle sur Zc sauf quand t = 0. On voit bien sur ce calcul que seule compte la classe de
l’indice k modulo d, ce qui est adéquat.
33 ainsi que dans son dernier
article34. Dans le premier cas, il n'y fait qu'une allusion in fine, s'excusant de la difficulté de no-
tions comme l'algèbre des caractères pour les lecteurs du Journal of Music Theory. Néanmoins,
4On prend le double du sinus de l’intervalle, avec un facteur d’échelle. . .
5 1 Si l’on s’intéresse aux fonctions du cercle S à valeurs dans C, on peut voir cela comme la transformée de Fourier d’une
distribution, à savoir un peigne de Dirac
k∈A δk.
toute son analyse des rapports intervalliques entre deux parties de Z/nZ (qu'on n'appelait pas
2 title on some pages
encore des pitch-class sets ou pc-sets) repose sur les relations entre leurs transformées de Fourier.
This shows that (when the Fourier transform of the characteristic function of A is non vanishing) knowledge
of A andEn ofeffet, the interval si pour function deux parties yieldsA, complete B de Z/nZ knowledge on définit of thela characteristic fonction d'intervalles function ofpar B.
Defining the interval function between A, B ⊂ Zc as
IFUNC(A, B)(t) = nombre d'intervalles de taille t entre une note de A et une note de
IF unc(A, B)(t) = Card{(a, b) ∈ A × B, b − a = t},
B,
il s'avère que cette fonction est le produit 1 if t ∈de X
the characteristic fuction of X as 1X(t) =
convolution , IF unc appears des fonctions immediately caractéristi- as the convolution
0 if t/∈ X
ques product de -A et ofde theB: characteristic functions of −A and B:
1−A ⋆ 1B : t ↦→
1−A(k)1B(t − k) =
2 title on some pages
This shows that (when the Fourier transform of the characteristic function of A is non vanishing) knowledge
of A and of the interval function yields complete knowledge of the characteristic function of B.
Defining the interval function between A, B ⊂ Zc as
IF unc(A, B)(t) = Card{(a, b) ∈ A × B, b − a = t},
1 if t ∈ X
the characteristic fuction of X as 1X(t) =
, IF unc appears immediately as the convolution
0 if t/∈ X
product of the characteristic functions of −A and B:
1−A ⋆ 1B : t ↦→
1A(k)1B(t + k) =IF unc(A, B)(t)
1−A(k)1B(t − k) =
k∈Zc
1A(k)1B(t + k) =IF unc(A, B)(t)
k∈Zc
k∈Zc
as 1A(k)1B(t + k) is nil except when k ∈ A and t + k ∈ B. Hence from the general formula for the Fourier
as transform 1A(k)1B(t Or of la + aDFT convolution k) isdu nilproduit except product, when de convolution k ∈ A and t est + kle ∈produit B. Hence ordinaire from thedes general DFT: formula for the Fourier
transform of a convolution product,
F(IF unc(A, B)) = F(1−A) × F(1B)
F(IF unc(A, B)) = F(1−A) × F(1B)
where F(f) stands for the discrete Fourier transform of a map f.
Cela signifie qu'il est possible de récupérer B, connaissant A et IFUNC(A, B) —
where We will F(f) not stands quotefor the the formula discrete given Fourier by Lewin transform himself, of aas map it isf. hardly understandable: his notations are
sauf
undefined
quand We will la not and
DFT quote the
de
computations
A the a formula la mauvaise given extremely
grâce by Lewin cursory.
de s'annuler, himself, Of course as ce itqui this is hardly arrive
is notunderstandable: dans
for lack
le cas
of rigor:
des his «special
as notations the following are
undefined quotation and suggests, the computations Lewin did not extremely really hope cursory. to beOf understood course this when is not making for lack useofofrigor: mathematics. as the following
cases» quotation The énumérés mathematical suggests, par Lewin reasoning (telles didby not which la really gamme I arrived hope par to at tons, be this understood result ou la is gamme notwhen communicable mélodique making use tomineure aofreader mathematics. ascen- who does not
The havemathematical considerable mathematical reasoning by training. which I arrived For those at this who result have such is not a training, communicable I append to a areader sketch who of the does proof not:
dante have (0 consider 2 3 considerable 5 7 the 9 11)). groupmathematical algebra [. . . ] [13] training. For those who have such a training, I append a sketch of the proof :
Reading consider the Lewin’s grouppaper algebra gives [. . . ] one [13] a strong feeling that he wrote as little as possible on the mathematical
tools Reading that underlay Lewin’s paper his results. gives one Indeed, a strong whatfeeling little he that mentioned he wrotedid as little rouseas some possible readers on to therighteous mathematical ire in
tools the DFT next that et issue "Maximally underlay of JMT. his results. Even Indeed, Sets" what little he mentioned did rouse some readers to righteous ire in
theNowadays next issuesuch of JMT. a ‘considerable mathematical training’ will be considered basic by many readers of this
journal; Nowadays for instance such a ‘considerable D.T. Vuza made mathematical an essential training’ use ofwill the be equation considered above basic in the by many 80’s inreaders the course of this of
journal; his seminal for work instance about D.T. rhythmic Vuza made canons an(see essential [21], lemma use of the 1.9 sqq), equation wherein above heinstressed the 80’s the inimportance the course of
his Lewin’s seminal usework of DFT about of characteristic rhythmic canons functions. (see [21], lemma 1.9 sqq), wherein he stressed the importance of
33 Lewin, Lewin’s And D., as Re: use we Intervallic ofwill DFT endeavour Relations of characteristic to between prove, functions. two thiscollections approachof enables notes, Journal to define of Music ME Theory, sets (in3:298-301 equal temperament) (1959). in
a way And perhaps as we will more endeavour suggestive to prove, and even thisintuitive, approachthan enables historical/usual to define MEdefinitions. sets (in equal temperament) in
34 Lewin, D., Special Cases of the Interval Function between Pitch-Class Sets X and Y, Journal of Music Theory,
45-129
a way
(2001).
perhaps more suggestive and even intuitive, than historical/usual definitions.
k∈Zc
1.2 A quick summary of Fourier transforms of subsets of Zc
1.2 A quick summary of Fourier transforms of subsets of Zc
1.2.1 First moves.
1.2.1 Definition First1.1 moves. Following Lewin, we will define the Fourier transform of a pc-set A ∈ Zc as the Fourier
transform of its characteristic function 1A: