Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

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6 title on some pages Definition 2.1 The pc-set A ⊂ Zc, with cardinality d, is a ME set, if the number |FA(d)| is maximal among the values |FX(d)| for all pc-sets X with cardinality d: ∀X ⊂ Zc, Card X = d ⇒ |FA(d)| ≥ |FX(d)| As the number of pc-sets is finite, a solution must exist. Remember that |FA(d)| = F(ICA)(d) (see section 1). Therefore, maximal evenness is also manifest in the DFT of the interval vector as a maximality condition for |F(ICA)(d)|. From the invariance of the ‘Fourier profile’ |FA| under musical operations (see theorem 1.5 and lemma 1.2 about complementation) we obtain easily Proposition 2.2 Transposition, inversion and complementation of a ME set still yield a ME set. 2.3 Notations and Maps Throughout the remainder of this section let m = gcd(d, c) denote the greatest common divisor of d and c and let d ′ = d m and c′ = c denote the associated quotients. m Let ϕd : Zc → m Zc and ϕd ′ : Zc ′ → Zc ′ denote the linear multiplication maps ϕd(l) = d · l and ϕd ′(k) = d′ · k, respectively. Further let πc ′ : Zc → Zc ′ denote the reduction of the finer residue classes mod c to the coarser residue classes mod c ′ , i.e. πc ′(l) = l mod c′ . Finally, let im : m Zc → Zc ′ denote the isomorphism, identifying the submodule m Zc of Zc with Zc ′: im(mk) := k mod c ′ . Note that the multiplication by d is a concatenation of the multiplications by m and by d ′ title on some pages . Thus, if we concatenate the maps ϕd and im into a map πd := im ◦ ϕd, we see that the map im ‘undoes’ the previous multiplication by m. Therefore im ◦ ϕd = ϕd ′ ◦ πc ′, which means that the diagram below commutes. of ‘mathemusical’ knowledge is the continued contribution of Jack Douthett gh and other partners). He is still a beacon in the field of ME sets. eviewers have been instrumental in bringing this paper up to the quality level table task for a lone writer. I would like to thank especially Dmitri Tymocsko, R oll in that respect. s A d A mod c ϕd Zc πc ✲ mZc ϕd ✲ ′ ❅ ❅ πd ❅ ❄ ❅❘ ❄ ′ ιm Zc ′ ., 2006, Une preuve élégante du théorème de Babbitt par transformée de Fourier discrète, Quadrature, ., The Different Generators of A Scale, 2008, Figure 1. Journal Notations and ofmorphisms Music Theory, to be published. M., 1955, Some Aspects of Twelve-Tone Composition, Score, 12 , 53-61. Douthett, J.., 1994, Vector products and intervallic weighting, Journal of Music Theory , 38, 2142. V., Vicinanza D., 2004, Myhill property, CV, well-formedness, winding numbers and all that, Logique lles en musique., Keynote adress to MaMuX seminar 2004 - IRCAM - Paris. ., Clampitt, D., 1989, Aspects of Well Formed Scales, Music Theory Spectrum, 11(2),187-206. ., Douthett, J., 1991, Maximally even sets, Journal of Music Theory, 35:93-173. ., Myerson, G., 1985, Variety and Multiplicity in Diatonic Systems, Journal of Music Theory,29:249-7 ., Myerson, G., 1986, Musical Scales and the Generalized Circle of Fifths, AMM, 93:9, 695-701. , J., Krantz, R., 2007, Maximally even sets and configurations: common threads in mathematics, physics, Zc ′ B' B A'
2.4 Pich Class Sets and related Multisets alternative title 7 Our goal is to translate the maximality condition for the absolute value |FA(d)| of the d-th Fouriercoefficient for pc-sets A into an equivalent maximality condition for the absolute value |FA ′(1)| for associated pc-multisets A ′ . To that end we investigate the image of a pc-set A ⊂ Zc under the map πd in a refined way. The refinement of the image πd(A) is a multiset which controls the multiplicity of each single image l = πd(k) ∈ Zc ′ for k ∈ A, i.e. the cardinality of the pre-image π−1 d (l) ∩ A. A suitable definition of the concept of a multiset is given in terms a generalized concept of characteristic function. Recall that the ordinary characteristic function 1A : Zc → {0, 1} ⊂ C serves as an alternative representation of the set A. In this way, the set of subsets of Zc appear as the subspace of complex-valued functions on Zc, with the condition that the values are only 0, 1. The Fourier transform is an automorphism of this last algebra. In extrapolation of this idea, we consider the function νd A : Zc ′ → {0, ..., m} ⊂ C with ν d A(l) := Card(π −1 d (l) ∩ A) = Card({k ∈ A | q · k = l}). The multiset associated with A consists of the elements of πd(A), each being repeated with the multiplicity νd A (l). For the non-elements of πd(A), i.e. for all l ∈ Zc ′\πd(A), the multiplicity vanishes: νd A (l) = 0. In order to manipulate this multiset like an ordinary set, we attach the multiplicity of each element as a superscript: A ′ := { νd A (l) l | l ∈ πd(A)}. For instance, the multiset associated with c = 12, d = 3 and the regular set (augmented fifth) A = {0, 4, 8} ⊂ Z12 is the multi-singleton set A ′ = { 3 0} (with 0 ∈ Z4). The multiset associated with c = 12, d = 8 and the octatonic set A = {0, 1, 3, 4, 6, 7, 9, 10} ⊂ Z12 is A ′ = { 4 0, 4 2} (with 0, 2 ∈ Z3). The straightforward following lemma relates the d-th Fourier coeffient of the set A to the first Fourier coefficient of the function ν d A . When the meaning of A′ is clear, we may adopt the notation from pc-sets and write: FA ′ := F(νd A ) and call this the Fourier transform of the multiset A′ . Lemma 2.3 With the notations above we have: FA(d) = FA ′(1). Proof We need to re-interpret a Fourier coefficient defined over Zc as a Fourier coefficient over Zc ′: FA(d) = e −2πikd/c = e −2πikd′ /c ′ = k∈A k∈A l∈Z c ′ k∈A∩π −1 d (l) e −2πil/c′ = ν d A(l)e −2πil/c′ = FA ′(1). In order to faithfully translate the maximality conditions from sets in Zc to multisets in Zc ′, we need to determine the correct collection of multisets involved. The following definition and lemma clarify this issue. Definition 2.4 A m|d-multiset in Zc ′ is a function ξ : Zc ′ → {0, ..., m} satisfying k∈Z ξ(k) = d. c ′ Lemma 2.5 m|d-multisets are exactly the multisets associated with a subset A with cardinality d. Proof We represent Zc as a disjoint union of the pre-images π −1 d (l) of single residue classes l ∈ Zc ′ under the surjective map πd, i.e.: Zc = π −1 d (0) ⊔ π−1 d (1) ⊔ ... ⊔ π−1 d (c′ − 1) and we list the m elements of each of these pre-images in some arbitrary way: π −1 d (l) := {kl,1, ..., kl,m} for each l ∈ Zc ′. Now for A = {k0,1, ..., k0,ξ(0)} ⊔ {k1,1, ..., k1,ξ(1)} ⊔ ... ⊔ {kc ′ −1,1, ..., kc ′ −1,ξ(c ′ )}, we easily see that νd A = ξ. Conversely, the kernel of ϕd is the subgroup c ′ Zc, with m elements, so the multiplicity of any element of πd(A) is at most d. And of course the sum of multiplicities is Card A = d. Corollary 2.6 The absolute value |FA(d)| of the d-th Fourier coefficient of a pc set A ⊂ Zc is maximal among the values |FX(d)| for all d-element subsets X ⊂ Zc iff the absolute value |FA ′(1)| = |F(νd A )(1)| of the 1-st Fourier coefficient of the associated multiset A ′ is maximal among the values |F(ξ)(1)| for all m|d-multisets ξ in Zc ′. l∈Z c ′
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Thèse de Doctorat de l'Université
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Résumé : Cette thèse sur travaux
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Table des matières Introduction 6
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kis. De ce fait, j'ai tout naturell
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fois ce problème de combinatoire r
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de Quinn 12, qui, pour la première
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OpenMusic permettant, entre autres
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Le premier article offre une synth
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A = {0,1,2,4,5,6}, i.e. A(X) = (1 +
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✦ Fabriquer par l'algorithme de C
Page 23 and 24: (ou une gamme) remonte à David Lew
Page 25 and 26: Mon article du Journal of Mathemati
Page 28 and 29: et de considérer les DFT des appli
Page 30 and 31: Pour rendre compte de cette dispari
Page 32 and 33: limitées 54, les pavages, divers t
Page 34 and 35: OpenMusic réalise les transformati
Page 36 and 37: ment versé en mathématiques, pour
Page 38 and 39: Mémoire de thèse Amiot 38 Un sign
Page 40 and 41: Mémoire de thèse Amiot 40 Or l' o
Page 42 and 43: Mémoire de thèse Amiot 42 Remerci
Page 44 and 45: Mémoire de thèse Amiot 44 Je tien
Page 46 and 47: Mémoire de thèse Amiot 46 ✦ Ami
Page 49: Annexe I Articles figurant dans le
Page 52 and 53: du pattern, qui va donc commencer p
Page 54 and 55: Un musicien attend pour rentrer ICI
Page 56 and 57: ainsi que leurs produits. Comme ce
Page 58 and 59: par l’automorphisme de FROBENIUS
Page 60 and 61: PROPOSITION. ≀ Soit A(X) un polyn
Page 62 and 63: FIG. 12 - Orbite d’un motif sous
Page 64 and 65: . Un domaine K (compact d’intéri
Page 66 and 67: période 900 : A = (0, 36, 72, 100,
Page 69 and 70: Journal of Mathematics and Music Vo
Page 71 and 72: alternative title 3 Lemma 1.2 The F
Page 73: alternative title 5 Proof If A ∈
Page 77 and 78: in so doing, the sum S = FA ′(1)
Page 79 and 80: 12 10 8 6 4 2 4 0 ME7,3 4 alternati
Page 81 and 82: alternative title 13 For instance,
Page 83 and 84: alternative title 15 To state it wi
Page 85 and 86: Supplementary II: pictures and proo
Page 87 and 88: 3 chunk of whole tone 1 2 3 4 5 6 7
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Page 92 and 93: 2 autosimilar melodies flake, Sierp
Page 94 and 95: 4 autosimilar melodies Example 1.3
Page 96 and 97: 6 autosimilar melodies Proposition
Page 98 and 99: 8 autosimilar melodies Proposition
Page 100 and 101: 10 autosimilar melodies homothety h
Page 102 and 103: 12 autosimilar melodies These compu
Page 104 and 105: 14 autosimilar melodies Theorem 3.6
Page 106 and 107: 16 autosimilar melodies 5.2 Example
Page 108 and 109: 18 autosimilar melodies at most q
Page 110 and 111: 20 autosimilar melodies [8] Fripert
Page 112 and 113: 22 autosimilar melodies Now we woul
Page 114 and 115: 24 autosimilar melodies But in Z15
Page 116 and 117: 26 autosimilar melodies At this jun
Page 118 and 119: 2 circle of fifths begins with five
Page 120 and 121: 4 (it is the Fourier Transform of t
Page 122 and 123: 6 Definition 2. A temperament, or t
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8 MSS A Bb B C C D Eb E F F G G Pyt
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10 sequence of 3 (and preferably mo
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12 i.e. an arithmetic progression w
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2 EMMANUEL AMIOT If we visualize Zn
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4 EMMANUEL AMIOT Theorem 3. (1) Til
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NEW PERSPECTIVES ON RHYTHMIC CANONS
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NEW PERSPECTIVES ON RHYTHMIC CANONS
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