Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

18 title on some pages
Proof For transposition and inversion it is theorem 1.5. For complementation we see that
|F Zc\A(c − d)| = |F Zc\A(−d)| = | − FA(d)| = |FA(d)|
holds for any subset A. So the one value is maximal whenever the other is, e.g. A is a ME set iff Zc \ A is
maximally even.
Proof of remark 1, linking he Fourier coefficients of A and its reduction B mod c ′ :
FA(d) =
e −2iπdk/c =
k∈A
m−1
k ′′ ∈B ℓ=0
e −2iπd(k′′ +ℓ c ′ )/c =
k ′′ ∈B
e −2iπd k′′ m−1
/c
Supplementary III: about other maximums of Fourier coefficients
e −2iπℓ = m
ℓ=0
k ′′ ∈B
e −2iπd′ k ′′ /c ′
= m FB(d ′ )
When d is coprime with c, generated 〈c, d〉 ME sets (the Clough-Myerson type) get their maximum Fourier
coefficient value in d: FA = FA∗ sin(π, d/c)
= |FA(d)| = µ(c, d) =
sin(π/c) .
Any generated scale with a generator coprime with c will share the same value of FA∗ , as
• any ME set A is in affine bijection with any such generated scale, both being affine images of the cluster
{0, 1, 2 . . . d − 1}.
• If f : x ↦→ a x + b, a ∈ Zc ′, is a bijective affine map then F f(A) ∗ = FA ∗ , as ∀t ∈ Zc |F f(A)(t)| =
|FA(at)| (the Fourier coefficients are permuted by affine maps).
We can reformulate prop. 3.2 in more detail:
Proposition 5.3 Fix a cardinality d coprime with c. For all d-element subsets of A ⊂ Zc. we find that
FA ∗ ≤ µ(c, d). With regard to equality the following three conditions are equivalent:
(i) FA ∗ = µ(c, d).
(ii) A = r · M(c, d) + s for suitable r ∈ Z ∗ c and s ∈ Zc and M(c, d) as in Definition 3.1. above.
(iii) A = a0 + {0, f, ..., (d − 1)f} is generated by a residue f ∈ Z ∗ c coprime with c.
Proof Choose t ∈ Z ∗ c such that FA ∗ = |FA(t)|. Then we have |FA(t)| = |Fd −1 t·A(d)| ≤ µ(c, d). To prove
(i) ⇔ (ii) we argue that the equality FA ∗ = µ(c, d) holds iff d −1 t · A = M(c, d) + s ′ or equivalently iff
A = t −1 d · M(c, d) + t −1 ds ′ . To prove (ii) ⇔ (iii) we remember that M(c, d) = k0 + {0, d −1 , ..., (d − 1)d −1 },
hence A = r · M(c, d) + s = (rk0 + s) + {0, d −1 r, ..., (d − 1)d −1 r}.
When c, d are no longer coprime this is not true anymore. The value of µ(c, d) = |FA(d)| for a 〈c, d〉 ME set
is now m sin(d ′ π/c ′ )/ sin(π/c ′ ) (this comes from thm. 2.8), which is larger than ρ(c, d) = sin(dπ/c)/ sin(π/c)
because (by concavity) sin π
c ′
m π π
= sin ≤ m sin .
c
c
But in that more general case, and with this value, we can characterize scales generated by some invertible
generator (among which the chromatic clusters): this is prop. 3.3, whose proof follows.
Proof Choose t0 ∈ Z ∗ c such that FA ∗ = |FA(t0)|. Then we have |FA(t0)| = |Ft0A(1)| ≤ µ(c, d) by the
huddling lemma in the simple case m = 1. The maximal case is that of a cluster, i.e. t0A = τ +{0, 1 . . . d−1}
is a cluster. Multiplying by f = t −1
0 we get A = a0 + f{0, 1 . . . d − 1}.
We do not find a characterization of all generated scales, i.e. also for generators not coprime with c. This
is because, for instance, the chunck of whole tone scale A = {0, 2, 4} ⊂ Z12, generated though certainly
not Maximally Even, realizes FA(6) = 3, clearly an unbeatable value (notice |FA(3)| = 1 < 3).
In order to understand better the maximality condition for FA, it is is useful to inspect the subgroup
of Zc which is generated by the intervals of a pitch class set A.