Emmanuel Amiot Modèles algébriques et algorithmes pour la ...

Thèse de Doctorat de l'Université Pierre et Marie Curie
EDITE
Présentée par
Emmanuel Amiot
Pour obtenir le grade de
Docteur de l'Université Pierre et Marie Curie
Modèles algébriques et algorithmes pour
la formalisation mathématique de structures musicales
Soutenue le 5 mai 2010
devant le jury, composé de
M. Carlos Agon, directeur de thèse
M. Moreno Andreatta, co-directeur
M. David Clampitt, rapporteur
M. Jean-Paul Allouche, rapporteur
M. Thomas Noll, examinateur.
M. Emmanuel Saint-James, examinateur
Université Pierre & Marie Curie — Paris 6
Bureau d’accueil, inscription des doctorants et base de données
Esc G, 2 ème étage
15 rue de l’école de médecine
75270-PARIS CEDEX 06 Tél. Secrétariat : 01 42 34 68 35
Fax : 01 42 34 68 40
Tél. pour les étudiants de A à EL : 01 42 34 69 54
Tél. pour les étudiants de EM à MON : 01 42 34 68 41
Tél. pour les étudiants de MOO à Z : 01 42 34 68 51
E-mail : scolarite.doctorat@upmc.fr
p. 1

p. 2

Résumé :
Cette thèse sur travaux est fondée sur cinq articles, sélectionnés tant pour leur inté-
rêt que pour leur représentativité. La synthèse ci-jointe vise à replacer et à expliciter le rôle de
ces travaux dans le contexte de la recherche contemporaine en Mathématiques et Musique. L'
auteur a été amené à utiliser des outils algébriques élaborés afin de mieux modéliser trois pro-
blèmes d'origine musicale: les canons rythmiques, les gammes, et les mélodies autosimilaires.
Sa démarche s'avère ainsi indissociable des sciences de l'informatique, sous leur double facette
d'expérimentation et d'implémentation de logiciels dédiés à l'analyse et à la composition musi-
cale.
Mots clefs :
Mathémusique, pavage, mosaïque, canons rythmiques, conjecture spectrale, Fugle-
de, canons de Vuza, gammes, Maximally Even Sets, transformée de Fourier discrète, tempéra-
ments, autosimilarité, mélodies, affine, modulaire.
p. 3

English title:
Algebraic models and algorithms for the mathematical formali-
Abstract:
zation of musical structures.
This PhD is based on a selection of five previously published papers. They have
been singled out for their scientific interest. The synthesis document aims at making clear the
role and purpose of these papers in the field of contemporary research across mathematics and
music. Their author makes use of sophisticated algebraic notions the better to study three
main topics of musical lineage: rhythmic canons, musical scales, and autosimilar melodies. His
approach intertwines with information sciences, on the one hand as a means of experimenta-
tion and on the other hand in producing software for musical analysis and composition.
Keywords:
Mathematics, music, tiling, mosaic, rythmiques canons, spectral conjecture, Fugle-
de, Vuza canons, musical scales, Maximally Even Sets, discrete Fourier transform, tempera-
ments, tuning, autosimilarity, melody, affine, modular.
p. 4

Table des matières
Introduction 6
Bref historique 6
Mes différents champs de recherche 8
Produits de mes recherches 14
Canons rythmiques 14
Gammes et transformée de Fourier discrète 22
Mélodies Autosimilaires 30
Conclusion et perspectives 37
Remerciements 42
Liste de travaux 45
p. 5

Bref historique
Introduction
Avant de présenter ma recherche, il me semble important de retracer le parcours
intellectuel et personnel qui m'y a conduit.
Le cursus traditionnel de mes études (ENS, agrégation) me prédisposait naturelle-
ment à des recherches en mathématiques «pures». Cependant, mes incursions dans ces domai-
nes (j’ai suivi un DEA sur les groupes et algèbres de Lie à l'université de Jussieu en 1983) m’ont
convaincu que je ne m'y épanouirai pas. En particulier, ma passion pour la musique 1, structurée
par mon cursus au Conservatoire de Nice, n'y trouvait pas sa place.
C’est en rencontrant André Riotte, compositeur, ingénieur et enseignant novateur à
Paris VIII d’un module intitulé Informatique et structures musicales, que j’ai trouvé ma voie. La
formalisation mathématique de structures musicales me permettait d’utiliser des outils et des
concepts à la fois puissants, élaborés et subtils, et simultanément de contrôler la validité de ces
spéculations abstraites en les appliquant immédiatement à la réalité musicale. Ces structures,
invoquées initialement pour des raisons mathématiques, s’incarnaient tout naturellement par
des implémentations; ce qui explique tant l’importance quantitative de mes contributions au
développement de logiciels d’assistance à la composition, comme OpenMusic 2, que le nombre de
mes participations à des colloques plus informatiques que mathématiques, tel l’International
Computer Music Conference (1986, 2002, 2005, 2006, 2007) entre autres.
Pour citer des exemples plus concrets, mes tous premiers travaux, comme stagiaire
à l’Ircam en 1985, faisaient la part belle aux arborescences, qu’il s’agisse des hiérarchies tonales
dans l’analyse Schenkérienne ou des arbres d’opérateurs pour l’élaboration de cribles à la Xéna-
1 Notamment contemporaine.
2 Langage de programmation et interface graphique pour l'analyse et la composition musicales, développé à l'IR-
CAM.
p. 6

kis. De ce fait, j'ai tout naturellement abordé ces problèmes sous l’angle de leur implémenta-
tion en Lisp, et, simultanément, du côté théorique, en recourant à des chaînes de Markov.
Nous étions relativement nombreux dans les années 80 à multiplier les imbrications
majestueuses de parenthèses en Lisp. Elles ont d'ailleurs laissé leur empreinte dans des logiciels
bien plus élaborés: les arborescences qu'elles modélisent ont naturellement perduré dans
Patchwork, puis surtout dans OpenMusic, logiciels développés par Carlos Agon à l'IRCAM afin
d'intégrer de façon modulaire, et dans une interface graphique (GUI), les structures utiles aux
compositeurs comme aux analystes. Il est donc logique que Lisp soit resté sous-jacent à ces en-
vironnements 3. C'est l'un des aspects de la nécessité d'une solide formalisation algébrique des
concepts et des outils de la théorie musicale.
En ce sens, une contribution décisive de Moreno Andreatta à OpenMusic fut d'y intègrer, via
l'environnement MathTools, les structures algébriques avec lesquelles nous jouions dans les an-
nées 80: groupes cycliques et diédraux, opérations modulo n, ou encore algèbres de Boole avec
tous les outils ensemblistes qui rendent accessible la Set Theory américaine des émules d'Allen
Forte. Ce sont là des progrès matériels indubitables, mettant à disposition du plus grand nom-
bre, et de manière intuitive, des concepts que leur abstraction rendait trop abscons il y a deux
décennies. En ce sens, il est donc normal que de nouveaux objets théoriques soient apparus,
puis à leur tour aient trouvé place dans ces environnements propres à démocratiser leur utilisa-
tion. Comme on le verra dans certains de mes articles, notamment dans ceux que je présente
en annexe de cette thèse, j'ai pu contribuer à cette double évolution, tant par l'élaboration de
nouveaux concepts ou de nouveaux modèles que par leur mise en œuvre sous forme d'implé-
mentations dans différents environnements — ces deux vantaux étant organiquement indisso-
ciables.
3 Il faut souligner les formalisations de Guérino Mazzola [ToM], développées parallèlement et indépendamment.
Elles ont rebuté plus d'un lecteur par leur formidable abstraction, mais ont néanmoins l'avantage de se prêter de
façon transparente à l'implémentation de leurs concepts (réalisée dans l'environnement Rubato): par exemple,
l'acte Grothendieckien de remplacement d'un point par une flêche se traduit immédiatement en terme de variables
et de pointeurs (ou plutôt de 'handles').
p. 7

Mes différents champs de recherche
Comme la discipline de « Mathémusique » n'existait pas — et n'existe toujours
pas dans le champ académique, même si les choses évoluent avec la création en 2007 à Berlin
de la Society for Mathematics and Computation and Music 4 et du Journal of Mathematics and Music 5,
qui est son organe de publication — mes recherches ont été souvent solitaires. Cela est attesté
par le nombre de mes travaux publiés sous ma seule signature. Néanmoins j'ai trouvé, notam-
ment à l'Ircam et ce à diverses époques, non seulement de l'intérêt pour mes recherches, mais
également des collaborations fructueuses, dans les équipes de recherches les plus orientées vers
les structures et la théorisation 6. Comme le prouvent les noms des co-auteurs de mes articles
collectifs, dont la plupart ont été écrits pour l' International Computer Music Conference, à com-
mencer par le tout premier en 1985. On distingue aisément la partie théorique de mes travaux
— résultant en général en un article signé de moi seul — de la part appliquée, constituée
par des contributions à l'élaboration collective de logiciels (ou au moins d'algorithmes) destinés
à l'analyse ou à la composition musicale. Tant il est vrai que mes sujets de recherche se prêtent
particulièrement à l'articulation entre théorie et pratique.
Les canons rythmiques
L' un des axes principaux de mes travaux concerne les canons rythmiques 7. C'est
l'enthousiasme communicatif de Moreno Andreatta, alors jeune étudiant, qui m'a conduit à
m'intéresser à ce sujet foisonnant. Il avait remarqué que le problème musical étudié par D.T.
Vuza 8 était, en fait, celui de la conjecture de Hajós, née dans les années 40, mais complètement
résolue bien plus tard, grâce aux efforts conjugués d'une génération de mathématiciens. Toute-
4 http://www.smcm-net.info/
5 http://www.informaworld.com/JMM
6 Le Projet 5 de la recherche musicale dans les années 1980, puis l'équipe "Représentations Musicales".
7 Dans le présent texte, il s'agira presque exclusivement de canons rythmiques mosaïques, dont l'étude équivaut à un
problème de pavage.
8 Vuza, D.T., « Supplementary Sets and Regular Complementary Unending Canons », en quatre articles : Canons.
Persp. of New Music, n os 29(2) pp.22-49 ; 30(1), pp. 184-207 ; 30(2), pp. 102-125 ; 31(1), pp. 270-305 (1991- 1992).
p. 8

fois ce problème de combinatoire relève, en fait, aussi bien de la géométrie que de l"analyse
harmonique: en témoigne la conjecture de Fuglede, qui relie la propriété de pavage à une con-
dition de spectre. Mon approche, par ce chemin détourné, m'a permis notamment de démon-
trer que la notion de « canon de Vuza » — musicalement pertinente, puisqu' il s'agit des ca-
nons tels qu'on les entend — permettait de faire progresser cette conjecture de mathématiques
dites "pures": en particulier, la conjecture est vraie si, et seulement si, elle l'est pour ces canons
de Vuza.
Un canon de Vuza de période 108
Ces derniers ne sont pas faciles à inventorier, et ils constituent un matériau extrê-
mement rare. Compte tenu de mon résultat les liant à la conjecture de Fuglede, on peut consi-
dérer ce fait comme une bonne nouvelle; toutefois il reste ardu de les fabriquer de manière ex-
haustive. J'ai participé à une des premières étapes de cette quête, en contribuant avec Harald
Fripertinger à l'établissement de leur liste exhaustive pour les périodes 72 et 108. C’est égale-
ment par une synthèse de considérations théoriques, souvent formalisées à partir des intuitions
de compositeurs et de ruses de programmation 9, que j’ai pu contribuer à la phase la plus ré-
cente de la même quête: avec les mathématiciens Kolountzakis et Matolcsi, nous avons énumé-
ré exhaustivement les canons de Vuza jusqu’à la période 144. Au passage, j’ai découvert, avec
9 J’ai élaboré le CanonCrawler, une bibliothèque d’outils en Mathematica® qui m’ont été indispensables aussi bien
du point de vue pratique que théorique.
p. 9

étonnement, l'existence de liens profonds — et inédits — entre des questions qui pouvaient
sembler se réduire à l’implémentation d’une exploration combinatoire (problème de Johnson,
canons de Vuza, canons modulo p), et l’abstruse théorie de Galois qui régit l’organisation des
racines des polynômes dans divers corps, notamment finis. Il existe là un carrefour étonnant
entre de multiples disciplines: mathématiques sous diverses formes (combinatoire, algèbre
commutative, analyse harmonique), informatique, et musique.
À l'occasion de la présentation ces travaux lors de colloques en Europe, puis en
Amérique du Nord, j’ai été amené à m'intéresser à de nouvelles problématiques, comme la
théorie des gammes musicales.
Les gammes.
C’est à l'initiative du musicologue américain David Clampitt, rencontré à une
séance du séminaire MaMuX 10 à l’Ircam, que j’ai été invité à parler de mes travaux devant les
membres de la Society for Music Theory lors d’une convention de l’American Mathematical So-
ciety à Evanston, près de Chicago. C'est aussi grâce à lui que j’ai découvert le foisonnement de
recherches sur les gammes de la nouvelle école américaine, héritière de célèbres pionniers, tels
les regrettés David Lewin ou John Clough.
p. 10
Ce sujet de recherche restera, pour moi, indissociablement lié aux acteurs améri-
cains de la théorie musicale: Richard Cohn s’est joint à David Clampitt (lui-même acteur de
premier plan dans le renouveau de la théorie des gammes) pour m’impliquer dans les travaux
de cette nouvelle école de chercheurs américains, dont la jeune génération — Cliff Callender,
Dmitri Tymoczko ou Ian Quinn notamment — a précisément démontré son génie en présen-
tant comme modèles continus des accords les "orbifolds" 11, lors des John Clough Memorial Days à
Chicago University en juillet 2005. Mon intérêt pour les gammes est d'ailleurs issu de la thèse
10 Mathématiques, Musique, et relations avec d'autres disciplines.
http://recherche.ircam.fr/equipes/repmus/mamux/
11 Orbivariétés en français, ici des quotients d'espaces vectoriels par des groupes finis traditionnels en théorie musicale,
comme T/I ou Sn.

de Quinn 12, qui, pour la première fois, explicitait les coefficients de Fourier d’une gamme à des
fins de comparaison et de classification. Par ailleurs, classer les gammes (ou plus précisément
les "pc-sets", sous-ensembles du total chromatique) selon la valeur absolue de leurs coefficients
de Fourier équivaut à considérer comme équivalents deux pc-sets ayant le même contenu inter-
vallique. Cette taxonomie est bien connue des cristallographes; pourtant, ce n'est que récem-
ment que les musiciens en ont pris conscience; or elle s'avère plus fine et plus subtile que la
classification traditionnelle sous l'action du groupe diédral T/I.
Bien entendu, ces coefficients de Fourier interviennent aussi dans les questions de
pavages (canons rythmiques) que j'avais déjà étudiées: ce sont les valeurs des polynômes carac-
téristiques des motifs, prises aux racines n ièmes de l'unité. J’ai commencé par généraliser les ré-
sultats de Ian Quinn, étudiant tous les cas de maximalité des coefficients de Fourier d’un sous-
ensemble d’un groupe cyclique. Ensuite, mon expérience des pavages m’a permis de revisiter la
plupart des questions traditionnelles sur les gammes (fonction intervallique, homométrie, gé-
nérateurs…) et d’en explorer de toutes nouvelles (comparaison de tempéraments), avec notam-
ment la surprenante confirmation, via un très simple algorithme de comparaison de coefficients
de Fourier, de l’hypothèse du musicologue Bradley Lehman sur le tempérament qu’aurait utilisé
J. S. Bach. Ce fut le fait du hasard, en étendant 13 ces transformées de Fourier discrètes à des
parties finies du cercle continu S 1, il m'est venu l'idée de les appliquer dans le cadre de diffé-
rents tempéraments musicaux. Toutefois ce domaine est riche de bien d'autres potentialités
inexplorées, et de connexions prometteuses, dans la mesure où, par exemple, cette notion de
transformée de Fourier d'une partie finie ordonnée d'un cercle permet de généraliser à de telles
parties la notion de "Maximal Evenness" 14.
Par ailleurs, cette généralisation à cheval entre discret et continu, appliquée au do-
maine des rythmes périodiques, nous a conduits à un nouveau paradigme de pensée, où les pa-
12 Quinn, I., « A unified theory of chord quality in equal temperaments », PhD dissertation, Univ. of Rochester (2005).
13 L'idée de cette généralisation revient à Thomas Noll, peu après que nous soyons revenus de Chicago.
14 Notons qu'on peut calculer ces coefficients de Fourier pour des parties de la plupart des orbifolds susmentionnés,
car leur valeur absolue passe au quotient par les groupes traditionnellement utilisés.
p. 11

amètres sur lesquels joue le musicien sont, non pas les notes, mais les coefficients de Fourier;
par exemple, par un seul paramètre on modifie, globalement et de façon cohérente, le «groove»
d'un rythme 15.
Je reviendrai ultérieurement sur cette taxonomie motifs par les différents profils de
leurs transformées de Fourier, qui devrait nous permettre enfin de jeter un pont vers les scien-
ces cognitives, via les protocoles expérimentaux actuellement mis au point à l'Ircam afin de
mettre en évidence la capacité de l’esprit humain à discerner certaines caractéristiques de ces
profils: leur platitude, ou au contraire leur "saillance"; cette dernière caractérisant des patterns
aussi célèbres que la gamme diatonique, ou le rythme traditionnel du tango .
Les mélodies autosimilaires.
Je terminerai cet exposé panoramique de mes recherches en abordant un domaine
peu ou pas exploré 16, qui illustre clairement le fait que mes travaux se trouvent toujours situés
au confluent des trois mêmes forces : musique, structures algébriques discrètes, et algorithmi-
que.
Le concept même de mélodie autosimilaire est dû à Tom Johnson, compositeur amé-
ricain vivant à Paris. L' acception mathématique du terme « autosimilaire » est plus restrictive
que celle utilisée par T. Johnson : pour rester conforme à la notion d'autosimilarité des objets
fractals, on dira qu'une mélodie est autosimilaire de rapport k si, en prenant seulement une
note tous les k temps, on entend la mélodie initiale (jouée k fois plus lentement). Enfin l' étude
de ces mélodies particulières est aussi bien abstraite (arithmétique, algèbre commutative) que
pratique (énumération exhaustive de catalogues de solutions, dénombrements, module dans
15 Cette idée a été exposée pour la première fois dans Agon, C., Amiot, E., Andreatta, M., Noll, T., « Oracles for
Computer-Aided. Improvisation », ICMC, New Orleans (2006). Une implémentation convaincante de ce concept de
«Fourier DJ» a été présentée par Thomas Noll au dernier colloque de la SMCM en juin 2009.
16 Je n'ai trouvé sur ce sujet qu'une page de David Feldman, dans sa recension de SelfSimilar Melodies de Tom Johnson.
Il y réfute brillament une conjecture de ce dernier concernant la conjonction de symétries par rétrogradation
et inversion des mélodies autosimilaires.
p. 12

OpenMusic permettant, entre autres choses, la construction de mélodies autosimilaires ayant un
groupe de symétries affines données), et même philosophique (car ce sont des «objets univer-
sels», i.e. des attracteurs limites d’itérations affines).
Une mélodie autosimilaire familière
Il s'avère que cette notion fort peu connue est pourtant profondément ancrée dans
la culture musicale, fût-ce inconsciemment : on trouve des mélodies autosimilaires aussi bien
chez D. Scarlatti, Mozart, dans la cinquième symphonie de Beethoven, que dans In the Mood de
Glen Miller par exemple. Or mes calculs, en établissant qu'une mélodie donnée possède une
probabilité infinitésimale d'être autosimilaire, montrent que l'existence de mélodies autosimi-
laires, même rares, dans l'œuvre d'un compositeur, ne peut être interprétée comme le fait du
hasard.
Ces trois domaines de recherche sont présentés plus en détail dans le développe-
ment qui suit, à travers cinq articles, choisis comme représentatifs de mon travail. On retrou-
vera, dans leur apparente diversité, la profonde unité des concepts qui les sous-tendent.
p. 13

Produits de mes recherches
Dans les pages qui suivent, je me propose de présenter et de détailler le contenu des
cinq articles figurant dans le dossier de mes travaux, tout en les replaçant dans le double con-
texte de mes recherches personnelles et du champ de la recherche en général. En effet mes tra-
vaux individuels sont souvent liés à des réalisations collectives. En témoignent les divers arti-
cles co-écrits avec d'autres chercheurs (cf. la bibliographie). Les cinq articles retenus comme
représentatifs de ma production sont:
✦ « À propos des canons rythmiques », Gazette des Mathématiciens, 106 (2005).
✦ « David Lewin and Maximally Even Sets », Journal of Mathematics and Music (2007)
vol. 3.
✦ « Autosimilar Melodies », JMM (2008) vol. 3.
✦ « Discrete Fourier Transform and Bach’s Good Temperament »,
Music Theory Online (2009) 15, 2.
✦« New Perspectives on rhythmic canons and the Spectral Conjecture
», JMM (2009).
Ils touchent aux trois domaines d'étude définis dans l' Introduction: canons ryth-
miques, gammes musicales, et mélodies autosimilaires.
Canons rythmiques
Parmi les nombreux articles que j'ai consacrés aux canons rythmiques, le dossier
joint proose les deux études suivantes:
1. « À propos des canons rythmiques », Gazette des Mathématiciens, 106 (2005)
2. « New Perspectives on rhythmic canons and the Spectral Conjecture », JMM (2009).
Dans la suite du texte, ces articles seront référencés par [GdM] et [AmiotJMM].
p. 14

Le premier article offre une synthèse de mes premières années de recherche sur les
canons ; il présente, notamment, le résultat séminal qui permet de limiter l'étude des conjectu-
res sur les pavages de la ligne aux canons de Vuza.
La formalisation musicale d'un canon rythmique (mosaïque), c'est à dire d'un pa-
vage parfait périodique par un unique motif A, répété à différents intervalles de temps, se ra-
mène à l'équation suivante:
équation
A ⊕ B = Z/nZ.
(A est le motif, B représente les différents départs de ce motif) qui équivaut à l'
A(X) × B(X) = 1 + X + X 2+ … X n-1 modulo X n -1 (1)
où A(X), polynôme caractéristique de la partie A, est la somme des X k quand k
décrit l'ensemble A. Le problème de la recherche de tous les canons rythmiques de période n
donnée, se traduit donc par une question de factorisation 17 du polynôme 1 + X + X 2+ … X n-1 ,
étant entendu qu'on cherche deux facteurs dont les coefficients soient exclusivement des 0 ou
des 1 (appelés polynômes 0-1 dans [GdM]), et dont le produit est calculé modulo X n -1, ce qui
signifie que tout monôme X k où k>n est remplacé (itérativement) par X k-n, ce que j'ai implé-
menté naturellement par pattern-matching. Je mentionne ce détail apparemment secondaire,
parce qu'il est significatif de la prégnance des idées musicales à tous les stades de ces recher-
ches. En effet, cette règle de programmation présente une signification perceptive forte, à
savoir que le k ème temps du canon sera occupé par une note, qui peut fort bien appartenir à une
copie du motif ayant commencé plus de n notes avant 18.
17 La difficulté vient, bien évidemment, de ce que l'anneau quotient Z[X]/(X n-1) n'est pas factoriel: les factorisations
n'y sont pas uniques, et en particulier il y en a d'autres que celles que l'on trouve dans les polynômes habituels.
J'ai exploré une autre piste (cf. [GdM]), cherchant des factorisations modulo p premier de l'équation (1). À
ma grande surprise, il se trouve que tout motif pave (pour un période suffisamment grande), au sens où tout polynôme
0-1 A(x) dans Fp[X] admet un complément B(x), lui aussi 0-1, tel que (1) soit vérifiée — modulo X p - 1.
18 L' auditeur perçoit des pavages de la ligne temporelle et non pas du cercle, structure quotient. Mais mathématiquement
les deux notions sont identiques, tout pavage de la ligne par translations d'un motif étant périodique.
p. 15

Nous rencontrerons d'autres exemples de cette utilité d'une culture musicale à pro-
pos de questions qui sembleraient relever des mathématiques les plus abstraites.
pavage de Z et de Z/12Z par le motif {0,1,5}
Les facteurs irréductibles dans Z[X] du polynôme 1 + X + X 2+ … X n-1 sont bien connus, ce sont
les polynômes cyclotomiques Φd où d divise n. Certains de ces polynômes sont 0-1; c'est le cas,
par exemple, quand d est premier et Φd = 1 + X +… X d-1. D'autres ne le sont pas, comme Φ12 = 1-
X 2+X 4. Cette remarque, qui date de la préhistoire de l'étude des pavages, a permis d'implémen-
ter dans OpenMusic un "patch" de production de canons rythmiques, dits « canons cyclotomi-
ques 19 » : pour obtenir des canons de période n, il suffit de sélectionner parmi les parties de
l'ensemble des diviseurs d1 … dk de n celles qui fournissent un polynôme 0-1 par le produit des
facteurs cyclotomiques Φd i correspondants, ce qui créera ipso facto un canon rythmique 20, à con-
dition que ce polynôme A(X) admette un complément B(X) — un polynôme 0-1 lui aussi, et qui vérifie
l'équation (1). Ainsi pour n=12
A(X) = Φ2 Φ4 Φ12 = (1 + X)(1 + X 2)(1 - X 2 + X 4) = 1 + X + X 6 + X 7
est un polynôme 0-1, correspondant à la partie A = {0,1,6,7} qui "pave" (i.e. constitue
un canon rythmique) avec par exemple B = {0, 2, 4} i.e. B(X) = 1 + X 2 + X 4.
Le problème posé par cette démarche tient à ce que fabriquer un polynôme 0-1
A(X), fût-il produit de polynômes cyclotomiques, ne garantit pas l'existence d'un complément
B(X). Ainsi pour :
19 Agon, C., Amiot, E., Andreatta, M., « Tiling the (musical) line with polynomials : Some theoretical and implementa-
tional aspects », Acts of ICMC 2005, Barcelona (2005).
20 Mais ce canon n'est jamais un canon de Vuza: l'algorithme utilisé, qui est emprunté à Coven et Meyerowitz,
donne toujours un complément B périodique.
p. 16

A = {0,1,2,4,5,6}, i.e. A(X) = (1 + X 4)(1 + X + X 2) = Φ8 Φ3
il n'existe pas de pavage de Z/12Z — ni d'ailleurs d'aucun Z/nZ — dont le motif
soit A. Ce fait résulte des conditions qui se trouvent exposées dans l'article séminal [CM], et
qui sont les premières conditions générales que l'on ait publiées pour que l'équation (1) soit vé-
rifiée. Ces conditions, (T1) et (T2), portent précisément sur les indices des facteurs cyclotomi-
ques du polynôme caractéristique A(X) d'une partie A de Z: si on note RA l'ensemble des indi-
ces d des Φd qui divisent A(X), et SA le sous-ensemble des éléments de RA qui sont des puissan-
ces de nombres premiers, alors ces conditions s'énoncent ainsi :
✦ (T1) : la valeur A(1) est le produit des Φpk(1) pour chaque p k dans SA.
✦ (T2) : pour chaque p k , q l … dans SA, on a p k × q l ×…. qui appartient à RA.
En 1998, Coven et Meyerowitz ont prouvé 21 que
vérifiée.
✦ Si A pave, alors (T1) est vérifiée.
✦ Si (T1) et (T2) sont vérifiées, alors A pave.
✦ Si A pave Z/n Z où n a au plus deux facteurs premiers, alors (T2) aussi est
On ignore encore si cette dernière propriété est vraie pour tout n. J'ai prouvé dans
[AmiotJMM] que tous les canons de Vuza de période 120 vérifient (T2), ce qui implique que
cette propriété est vraie pour une infinité d'autres canons. En effet, ainsi que je l'avais démon-
tré précédemment 22,
Tout canon rythmique peut se décomposer récursivement en canons plus courts, jusqu'à ce que
l'on arrive à un canon de Vuza;
Et, en conséquence 23, la condition (T2) étant conservée durant ces tribulations,
La conjecture « pave ⇒ (T2) » est vraie pour tout canon si, et seulement si, e(e est vraie
pour tous les canons de Vuza.
21 Leur article sera dorénavant cité comme [CM].
22 « Rhythmic canons and Galois Theory », actes du Colloquium on Mathematical Music Theory, H. Fripertinger
& L. Reich Eds, Grazer Math. Bericht Nr 347 (2005). Cité comme [Amiot05].
23 Les calculs polynomiaux sont un peu longs, mais restent élémentaires, cf. ibid.
p. 17

Edouard Gilbert [Gil] a parachevé la démonstration de ce théorème, que j'avais
énoncé sans prendre la peine d'en donner la preuve dans [GdM]. Il est notable que
et Meyerowitz.
(T1) + (T2) ⇒ spectral,
ce qui a été démontré par Izabella Łaba 26, peu après qu'elle ait lu l'article de Coven
Les canons de Vuza, objets musicaux par excellence, sont ainsi devenus incontour-
nables pour la résolution de conjectures fondamentales de mathématiques pures. Le second
article présenté en annexe répond à une question pointue et ésotérique: il s'agit de déterminer
tous les canons de Vuza de période n=120, complétant ainsi la recherche présentée par Ko-
lountzakis et Matolcsi 27 dans le même numéro du Journal for Mathematics and Music par la clas-
sification exhaustive de ces canons pour les périodes inférieures à 168, et ce pour mieux étudier
certaines des conjectures évoquées ci-dessus. Ce numéro spécial du Journal for Mathematics and
Music de 2009 consacré aux canons rythmiques fait la part belle à la conjecture de Fuglede:
c'est le sujet du troisième article (de Franck Jedrzejewski), elle est citée dans [KM], et démon-
trée au passage pour n=120 par [AmiotJMM] 28. Notons que cela prouve aussi la conjecture de
Fuglede 29 pour tout canon de période 240 (par exemple) qui n'est pas un canon de Vuza: en
effet, il se décompose alors en canons plus petits 30, qui vérifient donc nécessairement la conjec-
ture spectrale puisqu'elle est vraie pour n≤120.
La démarche, commune à ces deux derniers articles (avec n=144 dans [KM]) a été
proposée par Maté Matolcsi. Elle consiste à classifier les canons de Vuza par leurs ensembles
SA, en utilisant l'algorithme suivant:
✦ Choisir un SA qui permette d'espérer un canon de Vuza (des conditions sur RA
permettent de repèrer la périodicité de A, voire celle de son complément éventuel B, cf. [KM]).
26 Laba, I., « The spectral set conjecture and multiplicative properties of roots of polynomials », J. London
Math. Soc. 65, pp. 661–671 (2002).
27 Dorénavant citée comme [KM].
28 Les résultats de Coven-Meyerowitz et Laba suffisaient à établir pour n=144 que « pave ⇒ spectral ».
29 Au moins dans le sens « pave ⇒ spectral ».
30 Des canons dont la période divise strictement 240, donc est au plus égale à 120.
p. 19

✦ Fabriquer par l'algorithme de Coven et Meyerowitz un B qui complète tous les
A possibles ayant cet ensemble SA.
que (1) soit vérifiée.
✦ Rechercher tous les compléments de ce complément, i.e. tous les motifs A tels
✦ Trier par les valeurs de l'ensemble RA — incidemment, ceci permet de nouveau
d'éliminer des canons non Vuza.
✦ Sélectionner les solutions non périodiques, s'il y en a.
Cet algorithme nous a permis (avec quelques ruses de programmation pour les cas
les plus coriaces) de produire la classification suivante des canons de Vuza de période 72, 108,
120 ou 144, les deux premières périodes ayant déjà été exhaustivement détaillées par Harald
Fripertinger [Frip]:
n RA RB nombre de ≠A nombre de ≠B
72 {2, 8, 9, 18, 72} {3, 4, 6, 12, 24, 36} 6 3
108 {3, 4, 12, 27, 108} {2, 6, 9, 18, 36, 54} 252 3
120 {2, 3, 6, 8, 15, 24, 30, 120} {4, 5, 10, 12, 20, 40, 60} 20 16
120 {2, 5, 8, 10, 15, 30, 40, 120 {3, 4, 6, 12, 20, 24, 60} 18 8
144
144
144
144
{2,8,9,16,18,24,72,144}
or {2,8,9,16,18,72,144}
{2, 4, 9, 16, 18, 36, 144} or
{2, 4, 6,9, 16, 18, 36, 144} or
{2, 4, 9, 12, 16, 18, 36, 144}
{3, 4, 6, 8, 12, 24, 48, 72} or
{3, 4, 6, 8, 12, 24, 36, 48, 72}
{2, 3, 6, 8, 12, 24, 48, 72} or
{2, 3, 6, 8, 12, 18, 24, 48, 72}
{3,4,6,12,24,36,48} 36 6
{3,6,8,12,24,36,72} 8640 3
{2,9,16,18,144} or
{2,9,16,18,36,144}
156
+6
48
+12
{4, 9, 16, 18, 36, 144} 324 6
p. 20
Il n'y avait pas là de canons inédits pour n=120 (ceux qui ne provenaient pas de l'al-
gorithme de Vuza avaient été façonnés empiriquement à partir de canons de période 72), mais
l'algorithme utilisé a permis de l'établir avec certitude; en revanche de nouveaux canons ont été
trouvés pour n=144, montrant au passage (dans l'avant-dernier cas) qu'on pouvait avoir plu

sieurs ensembles RA en face de plusieurs ensembles RB distincts (même si SA et SB ne peuvent
changer, d'après [CM]).
Par ailleurs, j'ai utilisé depuis et utilise encore une version rapide de ce "ping-pong"
entre voix de canons pour tester des canons de plus grande taille et voir s'ils vérifiaient la con-
dition (T2), en utilisant la programmation linéaire pour chercher un complément de B (renon-
çant à les chercher tous, ce qui est trop coûteux en temps de calcul). L' idée m'en est venue en
travaillant sur des problèmes de décomposition linéaires de gammes, exactes ou approchées,
avec Bill Sethares [AS]. Le problème n'avait rien à voir a priori avec les pavages, mais la pro-
grammation linéaire fournissait une méthode rapide (et quasi infaillible) pour obtenir un com-
plément B d'un motif A qui pave 31. En itérant l'algorithme, on trouve un complément A' de B,
puis un complément B' de A', etc… jusqu'à retomber sur un motif (ou complément) déjà expri-
mé. Les ensembles SA et SB restent invariants tout au long de la procédure. Cette exploration
avait pour but de trouver des canons de Vuza qui ne vérifiraient pas la condition (T2), ni peut-
pas la condition spectrale; mais elle a failli à fournir un tel contre-exemple pour toutes les pé-
riodes allant jusqu'à n=1200 32.
En effet, la procédure décrite dans le numéro spécial de JMM présente deux points
faibles: le calcul de tous les compléments prend du temps, même après les diverses optimisa-
tions apportées par Matolcsi, lesquelles sont pourtant particulièrement pertinentes pour des
canons "irréguliers" comme ceux de Vuza; le calcul de l'ensemble RA est long lui aussi.
Pour réduire le temps de calcul de cette dernière tâche, j'ai suivi une suggestion de
Matolcsi et calculé certaines valeurs particulières du polynôme A(X) aux points X = e 2 i k π/n. En
effet, ces valeurs ne sont autres que la transformée de Fourier (discrète) de l'ensemble A, qui
permet de déceler en particulier ses périodicités internes. On obtient notamment
31 Qui plus est, l'algorithme du simplexe étant fortement asymétrique, il tend à donner des solutions non périodiques,
c'est à dire qu'on trouve des canons de Vuza quand il y en a pour les couples SA et SB considérés.
32 Ceci ne constitue pas une preuve de ce qu'il n'y ait pas de tels contre-exemples avec de "petites" périodes, mais
le laisse conjecturer, dans la mesure où les listes de canons de Vuza ainsi formées ont des cardinaux similaires à ce
que l'on trouve dans les cas où le catalogue exhaustif est connu. Il est probable que de tels contre-exemples n'existent
que pour d'assez grandes périodes, mais il serait dommage de se priver d'explorer les périodes qui nous sont
d'ores et déjà accessibles.
p. 21

(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:

Le cas particulier B=A a été réexaminé de manière magistrale dans sa thèse 35 par Ian
Quinn , qui voulut reconnaître, par les propriétés de leurs coefficients de Fourier, les parties les
plus «prototypales» , i.e. celles qui jouent le rôle de phares dans le paysage des accords. La dé-
couverte la plus remarquable de Quinn est que ces «phares» ou prototypes, consacrés par la cri-
tique traditionnelle, sont caractérisés par la valeur maximale de l'un de leurs coefficients de
Fourier. Or les prototypes en question ne sont autres que les Maximally Even Sets ou ME Sets,
que je traduis par «gammes bien réparties» dans mon article pour la Revue de mathématiques et
sciences humaines 36; ce sont les répartitions de notes les plus proches d'un polygone régulier: ainsi
A
A
G
la collection diatonique est, parmi les parties à sept notes de la
gamme chromatique, celle qui se rapproche le mieux
d'un heptagone régulier (cf. la figure ci-jointe).
Évidemment, il convient de préciser ce qu'on
entend précisément par «se rapprocher
d'un polygone régulier». Douthett et Kranz
ont prouvé qu'en termes de potentiels sur un
cercle discret, toutes les fonctions potentiel
strictement convexes (comme les potentiels coulom-
biens en physique) donnaient les mêmes ME Sets 37. Or
Quinn a mis en évidence, dans sa thèse, une autre manière d'apprécier la «bonne répartition»:
Le pc-set A à d éléments est « bien réparti » si, et seulement si, la valeur |FA(d)| (ampli-
tude du d ème coefficient de Fourier) est maximale par rapport à tous les autres pc-sets à d élé-
ments.
B
G
C
C major
F
C
F
D
E
D
p. 24
35 Quinn, I., « A unified theory of chord quality in equal temperaments », PhD dissertation, Univ. of Rochester
(2005).
36 Amiot, E., « Gammes Bien Réparties », Revue de Mathématiques et Sciences Humaines, 178 (juillet 2007).
37 Douthett, J. et Krantz, R. « Energy extremes and spin configurations for the one-dimensional antiferromagnetic
Ising model with arbitrary-range interaction », Journal of Mathematical Physics 37 (1996).

Mon article du Journal of Mathematics and Music, joint dans le dossier annexe 38, dé-
montre rigoureusement tout en la généralisant, cette découverte de Quinn. En effet, il m'a paru
nécessaire de prouver précisément 39 le fondement géométrique de cette propriété, qui est un
lemme trigonométrique (appelé « Huddling Lemma » dans [MESets]), notamment pour cla-
rifier le cas plus délicat des multi-ensembles qui apparaissent dans le cas des ME Sets dégéné-
rés — comme la gamme octatonique par exemple.
Au passage, j'ai démontré une conjecture de Quinn concernant certains ME sets
particuliers (ceux de type III dans sa nomenclature — cette propriété ne figure pas dans l'arti-
cle du JMM, mais on peut la trouver dans celui de la Revue de Mathématiques et Sciences Humai-
nes); j'ai également complètement décrit tous les cas de maximalité de tous les coefficients de
Fourier des pc-sets, et retrouvé alors de manière élégante le théorème de l'hexacorde de Babbit
(avec diverses généralisations, en particulier à tout groupe abélien compact) 40. Ce dernier
point, vu l'argument utilisé, mérite d'être approfondi ici : quand on prend B=A dans les équa-
tions ci-dessus, on obtient le contenu intervallique de A, IC(A) en lieu et place de IFUNC(A,
B) (c'est l'histogramme des intervalles présents entre les notes de A, ou, comme l'exprimait jo-
liment Lewin dans un de ses derniers articles, la probabilité que tel intervalle soit ouï si l'on
joue des notes de A au hasard). De plus, la DFT de cet histogramme donne le module alternative au title carré
(i.e. l'amplitude) de la DFT de A:
Proof If A ∈ Zc has c/2 elements, then as mentioned above, F Zc\A = −FA
F(ICA) =|FA| 2 = |F Zc\A| 2 = F(IC Zc\A) Hence (by inverse D
Cette formule permet de comparer As far astrès I know, facilement this short le contenu proof was intervallique first published de A in et [1] after I men
memorial days in july 2005. But considering the coincidence in time of Lew
celui de son complémentaire, ce qui est with précisément Babbitt, it is l'énoncé almost certain du théorème that hede was l'hexacorde. aware of it. Plus Perhaps the har
in his first paper explain why he did not publish it. It is left to the read
exercise, to prove in the same way the Generalized Hexachord Theorem,
and many others.
38 « David Lewin and Maximally Even Sets », Journal of Mathematics and Music (2007) vol. 3.
Ci-après cité comme [MESets].
2 Maximally Even Sets and their Fourier Transforms
The attribute ‘maximally even’ applies to pitch class sets, which — in com
the same cardinality — are as evenly as possible distributed within Zc. Thi
regular sets, which exist only for cardinalities d dividing the number c of pi
case — where d and c are mutually coprime — was well studied in [8].
extensive study of the general case in [7] is an explicit construction of gen
formula for this construction was later termed J-function. It departs from
numbers 0, c
c
, ..., (d − 1)
d d and ‘digitizes’ them within Zc in terms of the re
of these ratios mod c: 0, 39 Les évidences sont bien souvent trompeuses. J'ai récemment élucidé le nombre de générateurs possibles d'une
gamme monogène (comme la gamme majeure ou la gamme pentatonique, engendrées par des quintes), qui porte
bien mal son nom puisque (contrairement aux parties monogènes, alias séquences arithmétiques, dans R par
exemple) ce nombre peut être arbitrairement grand — mais n'est jamais égal à 14 par exemple, cf. mon article
« On the number of generators of a musical scale », http://arxiv.org/abs/0909.0039.
40 Ce point a été publié indépendamment dans la revue de vulgarisation mathématique Quadrature. Par ailleurs j'ai
récemment étendu le théorème de l'hexacorde à des groupes compacts non nécessairement commutatifs (non publié).
c c
, ..., (d − 1) mod c. The J-function includ
d
d
p. 25
J α c,d : k ↦→ kc + α
,k =0. . . d − 1.
d

généralement, on voit sur cette dernière formule que la connaissance de l'amplitude de la
DFT équivaut très exactement à la connaissance de IC(A), i.e. du contenu intervallique de
A.
Or l'étude des vecteurs d'intervalle, dans les années 60, a mis en lumière le fait
troublant que certains pc-sets ont même distribution intervallique, sans être pour autant con-
gruents 41. Ces exceptions, rares, ont été baptisées « Z-relation » par Allen Forte. La formule
secrète utilisée par Lewin rend donc compte très simplement de cette relation, qui exprime
l'égalité des longueurs des coefficients de Fourier.
Or c'est seulement récemment que l'on s'est aperçu que cette notion était bien
connue depuis la fin des années 40 par les cristallographes, qui l'avaient nommée homométrie 42.
Le problème de trouver des figures homométriques dans un groupe cyclique demeure, encore
aujourd'hui, le sujet d'actives recherches, au centre desquelles je me suis subitement trouvé im-
pliqué, par le biais de ces questions de ME sets. L' une de mes dernières contributions, toute
récente, est nettement plus technique que la démonstration élégante du théorème de l'hexa-
corde susmentionnée. Elle porte sur certaines unités spectrales, objets mystérieux qui permettent
de générer des classes d'objets homométriques — qui, malheureusement, ne sont pas en géné-
ral de vrais ensembles, mais des multi-ensembles 43. C'est en travaillant avec Bill Sethares sur
une autre présentation des relations entre pc-sets 44 que j'ai trouvé un théorème pointu, qui
énumère les unités spectrales rationnelles d'ordre fini. Notre projet était de décrire des rela-
tions de combinaisons linéaires entre gammes (ou accords, plus exactement). Par exemple
Do mineur = Do majeur - Fa majeur + Mib majeur.
p. 26
Pour ce faire, nous avons introduit un formalisme matriciel, qui n'est autre qu'une
représentation (au sens vulgaire comme au sens de la théorie des représentations de groupes)
41 i.e. isomorphes au groupe T/I près : il est évident que transposer, ou inverser, un pc-set A ne change pas ICA.
42 En effet, l'observation de la figure de diffraction donnée par un solide éclairé par des rayons X est essentiellement
équivalente à l'amplitude de sa transformée de Fourier. Les solides homométriques mais non congruents
sont donc des obstacles à une reconnaissance taxonomique via l'observation par ces techniques.
43 « On the Group of Rational Spectral Units with Finite Order », http://arxiv.org/abs/0907.0857.
44 « An Algebra for periodic rhythms and scales ». À paraître chez Springer.

et de considérer les DFT des applications k→ uk définies d'un groupe cyclique Z/n
Z dans le corps des nombres complexes. Une élégante propriété établie par Noll est que, dans
le cas d'une telle gamme monogène, les coefficients de Fourier sont tous alignés en tant que
nombres complexes dans le plan d'Argan-Cauchy, comme on le voit sur la figure suivante 47.
6
1
4
560123
3
Une gamme et ses coefficients de Fourier
Pour aller plus loin, j'ai plus généralement examiné une partie finie, fixée de S 1 (on
peut considérer de manière typique douzes notes obtenues par itération de la quinte pythago-
ricienne à partir d'une fondamentale) qu'on peut voir comme une gamme chromatique non né-
cessairement tempérée, et les gammes majeures, définies classiquement comme les séquences
de notes indexées par (0,2,4,5,7,9,11) ou les translatés modulo 12 de cet ensemble d'indices, dans
la gamme chromatique ambiante.
Une variante du résultat de Quinn dans ce contexte élargi, mais de démonstration
tout à fait similaire, donne que parmi toutes les gammes à 7 notes, ce sont les gammes majeures
qui ont la plus grande valeur de leur premier coefficient de Fourier a1 — ce sont les plus pro-
47 J'ai bien entendu étudié la réciproque, qui se trouve être largement fausse: il existe des familles de gammes à
plusieurs degrés de liberté dont les coefficients de Fourier sont alignés sans être pour autant monogènes.
2
5
4
0
p. 28

1
0.5
ches d'une progression géométrique par-
faite, i.e. d'un heptagone régulier (cf. figure
ci-jointe,). Ce résultat est, en fait, parfaite-
ment pur si la gamme chromatique ambiante
est également tempérée (i.e. si tous les inter-
valles entre deux notes consécutives sont
égaux), et il reste vrai pour des tempéra-
ments raisonnables (proches du tempéra-
ment égal), ce par continuité de la trans-
formée de Fourier. Il s'agit donc d'un résultat dont le domaine de validité est musical, et non
pas mathématique: il n'a pas de sens pour un contexte chromatique arbitraire, mais il est par-
faitement vrai pour tous les tempéraments 48 qui ont été utilisés effectivement par des musi-
ciens.
En préparant un exposé pour l'atelier Klang und Ton à l'institut Helmholtz de Berlin
en mai 2007 (à l'instigation de T. Noll), j'ai donc eu la curiosité de calculer les valeurs de ces
coefficients de Fourier a1 pour les 12 gammes majeures dans différents tempéraments classi-
ques, et constaté avec amusement que certains tempéraments permettaient d'obtenir une va-
leur plus grande (pour certaines gammes majeures) que le tempérament égal. Puis j'ai été frappé
par l'éventail des valeurs obtenues. Cet éventail est certes réduit 49, mais sa largeur varie consi-
dérablement en fonction du tempérament considéré. Par exemple :
tique;
1 2 3 4 5 6
Coefficients de Fourier de la gamme majeure
et d'une gamme quelconque.
• pour le tempérament égal toutes les gammes ont un coefficient de Fourier a1 iden-
• pour le tempérament pythagoricien, engendré par onze quintes justes, le coeffi-
cient varie entre 0.9856 et 0.9927.
48 Tempéraments à douze notes; j'ai exclu de cette discussion des objets étranges comme par exemple le tempérament
à 31 notes de Nicola Vicentino. Cependant, comme ce tempérament est quasiment égal, les résultats trouvés
dans le contexte des douze notes pourraient s'y étendre mutatis mutandis.
p. 29
49 Typiquement, les coefficients de Fourier des gammes majeures varient entre 0.975 et 0.995, le maximum théorique
étant de 1 pour une gamme qui réaliserait un heptagone régulier parfait.

Pour rendre compte de cette disparité de qualité entre les diverses gammes majeu-
res, j'ai créé un indicateur simple, la Major Scale Similarity ou MSS: l'inverse de l'écart maximum
entre les coefficients de Fourier des douze gammes majeures. J'ai testé, entre autres, le tempé-
rament proposé par Bradley Lehman comme celui qu'utilisait J.S. Bach, à partir d'une interpré-
tation audacieuse (et contestée !) des volutes qui ornent la première page de l'édition originale
du Wohltemperierte Klavier, publié en 1722 50:
Lehman y lit l'indication (de gauche à droite sur la figure) d'accorder cinq quintes
diminuées de deux douzièmes de comma, puis trois quintes justes, puis trois diminuées d'un
douzième de comma. J'ai alors constaté avec surprise que l' écart entre les coefficients de Fou-
rier, calculé pour ce tempérament, était plus petit que celui que l'on obtient pour tous ses con-
currents 51.
Bien sûr, des tempéraments plus récents (postérieurs à la publication du Wohltem-
perierte Klavier) donnent un écart encore plus réduit; en particulier, le tempérament égal qui
prédomine de nos jours donne un écart nul. C'est néanmoins un indicateur significatif, dans la
mesure où Bach recherchait explicitement un tempérament qui permît de faire sonner «bien»
(c'est une traduction plausible du mot wohl) toutes les gammes majeures. J'étais particulière-
ment heureux de trouver une retombée aussi concrète à des recherches à ce point abstraites.
Mélodies Autosimilaires
p. 30
50 C'est Lehman qui renverse le dessin. Ce point est discuté dans mon article, où il est expliqué pourquoi la DFT
ne donne pas de préférence entre un tempérament et son renversement.
51 Une amélioration possible du tempérament proposé par Lehman — d'ailleurs proposée par d'autres pour des
raisons musicologiques — consiste à prendre pour unité de modification des quintes un treizième, au lieu d' un
douzième, de comma pythagoricien. Cette valeur optimise la MSS, si on garde le schéma proposé par Lehman.

Certains mathématiciens du groupe Bourbaki, peut-être extrémistes, sont allés jus-
qu'à prétendre que la géométrie classique n'avait plus lieu d'être depuis qu'on avait découvert
qu'elle ne faisait que traduire l'action de certains groupes algébriques (groupes affine, orthogo-
nal, projectif, conforme…), et n'était plus qu'un codicille à l'algèbre linéaire. Cependant, la con-
sidération de figures, fussent-elles les choix partiaux de représentants d'orbites de ces groupes,
peut toujours suggérer des idées nouvelles, que l'on ne saurait découvrir aux altitudes éthérées
de l'algèbre pure. Ainsi, mon étude des mélodies autosimilaires 52 pourrait être vue comme un
codicille à propos de l'action du groupe affine modulo n. Néanmoins, son origine musicale, et
l'angle original qui en résulte, ont abouti à une quantité notable de résultats nouveaux et non
triviaux, comme les Thms. 2.8 ou 6.1. On y voit, encore mieux à l'œuvre que dans mes autres
domaines de recherche, l'interaction dialectique fructueuse entre algébrisation et implémenta-
tion.
Pour ne prendre qu'un exemple de la complémentarité des processus, le Thm. 6.1
énonce que toute mélodie périodique devient autosimilaire après un certain nombre d'itéra-
tions de "l'extraction de la k ème note". Or, sans la formalisation en termes d'action d'applica-
tions affines sur Z/nZ, l'itération à l'infini d'une telle application n'était guère concevable; mais
d'autre part, sans expérimentation informatique, je n'aurais sans doute jamais découvert que
cette itération convergeait — et il m'a fallu mettre en jeu des résultats assez profonds d'algèbre
commutative 53 pour prouver que cette convergence avait toujours lieu.
Dans le point de vue Bourbakiste évoqué plus haut, le groupe affine sur Z/nZ est
représenté fidèlement par un simple sous-groupe des matrices GL(2, n). Mais pour le musicien,
ces transformations sont fascinantes — au point qu'on peut s'interroger sur le peu de résultats
les concernant — car elles préservent, selon le contexte, nombre de notions capitales: le conte-
nu intervallique (à permutation près), les parties tous-intervalles, les modes à transpositions
52 Autosimilar Melodies, JMM, vol. 3 (2008), abrégé en [autoSimJMM].
53 Précisément il s'agit du Lemme de Fitting. Pour être parfaitement honnête, il arrive que l'attracteur autosimilaire
final soit trivial, i.e. consiste en une mélodie faite d'une seule note répétée.
p. 31

limitées 54, les pavages, divers types de séries dodécaphoniques comme les séries tous-interval-
les, etc… Pour ces raisons, je considère ces travaux comme ma contribution la plus importante
et la plus originale à la recherche mathématico-musicale.
Les mélodies autosimilaires sont des mélodies périodiques (qu'on imagine infini-
ment longues) jouissant de la propriété suivante: si on a convenu d'une unité de temps telle que
toutes les notes se produisent à des abscisses entières, alors en extrayant une note tous les a
temps on obtient une copie conforme (jouée a fois plus lentement) de la mélodie originale.
Comme on le voit ci-dessous, la cellule célèbrissime qui fonde le premier mouvement de la 5 ème
symphonie de Beethoven se retrouve quand on ne joue qu'une note sur trois — et donc, a for-
tiori, quand on joue une note sur neuf, etc.
Ce motif célèbre est autosimilaire de rapport 3
Une mélodie périodique est modélisée par une suite d'événements musicaux 55, i.e.
une suite Mk où k décrit un groupe cyclique Z/nZ. Dans l'acception originelle, la mélodie est
auto-similaire de rapport a si
54 Moreno Andreatta m'a fait remarquer ce résultat élégant.
pour tout k∈ Z/nZ , Mk = M a k (mod n)
55 … qui peuvent être absence d'événement, c'est à dire silence ou prolongation d'une note. Tom Johnson insiste
d'ailleurs pour appeler les silences "rests", de manière plus positive, et s'en sert fréquemment dans ses compositions
autosimilaires.
p. 32

Tom Johnson 56 a rapidement réalisé que cette définition manquait de généralité,
qu'il pouvait exister des décalages (a k + b au lieu de a k), ou encore, plusieurs rapports distincts
pour une mélodie. En outre, il a constaté que
Si a et a' sont deux rapports d'autosimilarité d'une mélodie M de période n alors a×a' (mod n)
est aussi un rapport d'autosimilarité de M.
ce qui exprimait en fait que l'ensemble des transformations affines, sur les indices
temporels des événements de la mélodie qui la laissent invariante, forment un groupe. D'où ma
définition plus générale:
on a
M est autosimilaire de groupe G ⊂ An (groupe affine modulo n) si pour tout élément g de G
pour tout k∈ Z/nZ , Mk = Mg(k).
Ainsi la fameuse basse d'Alberti 57, si populaire au XVIII e siècle, possède un groupe
de 8 symétries, les x → (2k+1) x + 4 ou x → (2k+1) x comme on le voit sur la figure suivante :
56 On trouvera dans [AutosimJMM] deux extraits de partitions de Johnson entièrement construites sur des mélo-
dies autosimilaires.
57 Si on la considère comme une mélodie périodique de 8 notes; on peut donc considérer la basse d'Alberti comme
très symétrique puisque son groupe est d'indice 4 seulement dans le groupe affine complet A8 qui contient 32 éléments.
p. 33

OpenMusic réalise les transformations affines d'une mélodie.
L'étude d'une propriété de certaines mélodies isolées est donc ramenée à celle de
l'action de sous-groupes du groupe affine modulo n sur les indices temporels des événements
de cette mélodie. Cette démarche n'est pas neuve en soi (cf. par exemple la classification des
pc-sets par Forte, qui est, de fait, une nomenclature des orbites de l'action du groupe T/I sur
l'ensemble Z/nZ); et si elle est devenue classique dans nombre de domaines scientifiques 58,
c'est qu'elle permet une taxonomie pertinente qui réduit une combinatoire considérable (tou-
tes les mélodies de période n) à un petit nombre de classes, tout en conservant le sens musical
à l'intérieur d'une même classe : ainsi, toutes les mélodies qui y appartiennent vont être inva-
riantes par les mêmes augmentations/extractions. Bien entendu, la faculté de reconnaître le
58 On peut dater son origine au fameux Programme d'Erlangen de Felix Klein: Vergleichende Betrachtungen über neuere
geometrische Forschungen, 1872.
p. 34

sous-groupe des symétries affines d'une mélodie (autosimilaire) donnée est ausi fondamentale
théoriquement que pratiquement, et elle a été dûment implémentée dans OpenMusic 59.
Une fois obtenue la description algébrique complète (cf. [autosimJMM] qui se veut
exhaustif sur la question), il devenait possible de renverser la perspective et de créer une mélo-
die autosimilaire jouissant de symétries don-
nées. Le "patch" que j'ai développé avec Car-
los Agon pour OpenMusic 60 permet de don-
Fig. 9. A palindrome with period 14
ner en entrée une ou plusieurs "symétries
affines", c'est à dire diverses manières d'ex-
traire des notes de la mélodie, ainsi qu'une
séquence de notes (suffisamment longue), et
d'en déduire une mélodie qui soit autosimi-
laire modulo toutes ces symétries. Par exem-
Fig. 10. Autosimilar melody with period 15 and its palindromic
deformation
ple, en prenant les trois notes do, sol, mi, et
en imposant les applications affines
x → x + 4, x → 3x, x → 3x + 4, x → 5x + 4, x → 5x
modulo 8, on retrouve la basse d'Alberti61. sequences
OpenMusic crée une mélodie ayant
Fig. 8. This patch shows how to produce an autosimilar melody (the
Alberti Bass) un groupe starting with donné a set of symmetries, d'autosimilarités.
a period and a collection
of pitches.
2) Find a melody with some given symmetries. One
builds the orbits first as explained below, from there it
is the composer’s choice to associate notes to each orbit
with a standard Tout OpenMusic ce travail procedure. sur les structures algébriques permet ainsi d'atteindre l'expression la
The user inputs a collection of affine maps. Starting from
plus an parfaite element xde in la Zn, créativité a set is initialized du compositeur, with x as sole qui peut jouer sur tous les paramètres qui restent
element. Then all the maps in the collection are applied
libres repeatedly après toqu'il that ait set fixé untilles it no symétries longer changes. qu'il désirait. All Un utilisateur de ce programme, suffisamelements
of this set, now an orbit, are set aside and
the algorithm carries on with the next element in Zn
that has not been reached yet, until Zn is exhausted
(see Fig. 8). This is the dual approach from the last
one, providing the composer with the simplest structure
admitting autosimilar copies with the desired ratios and
offsets.
D. Palindromes
The above concept enables to clarify which autosimilar
melodies will be palindromes, as it is only a question
of whether x ↦→ −x (or some more general inversion
x ↦→ c − x) is present in the stabilizer of the melody.
The algorithms allow the straighforward construction of
palindromic melodies (among other symmetries), and the
theory reaches interesting result, as (simplifying a little)
Theorem 6: An autosimilar melody with ratio a will
be palindromic iff there is some power of a equal to -1
1 under reference A126949.
Besides of course, it is always possible to build a palindromic
(autosimilar) melody from any autosimilar melody
by just collapsing together notes belonging to orbits that
are symmetrical (the map f : x ↦→ −x exchanges orbits of
a primitive autosimilar melody with ratio a). For example
see the original and the ‘palindromized’ on Fig. 10
obtained by collapsing together the two inversionallyrelated
orbits (1, 2, 4, 8) and (7, 11, 13, 14). 2
A nice theoretical property of autosimilar melodies
appears when one tries to iterate an affine map with
a ratio that is not invertible modulo n: the map being
no longer one-to-one, there is no reversibility and some
information is lost at each iteration, but (this is related
to the very deep Fitting Lemma of commutative algebra
that already appeared in a musical context in Anatol
Vieru’s sequences, [3]) after a time an autosimilar melody
59 L' algorithme choisi cherche le coefficient de corrélation entre emerges la mélodie (see Fig. originale 11). et les mélodies extraites à
divers rapports, en les considérant comme des circlists, cf. [autosimJMM] Thus chaos 5.1. hides inside itself the deepest harmony.
60 cf. Agon, C., Amiot, E., Andreatta, M., « Autosimilar melodies and their implementation in OpenMusic »,
Proceedings SMC 07, Le1ada, Grèce (2007).
61 Il est suffisant de donner quelques symétries, qui engendreront tout un sous-groupe qu'il n'est pas nécessaire
d'expliciter, non plus que son nombre d'éléments — l'algorithme calcule tout pour l'utilisateur, en appliquant toutes
les symétries données jusqu'à ce que rien de nouveau ne puisse être engendré: Se vogliamo che tutto rimanga come
è, bisogna che tutto cambi (G.T. di Lampedusa, Il Gattopardo).
Fig. 11. An autosimilar melody from a random one
IV. CONCLUSION
We presented some theoretical and implementational
aspects of melodic autosimilarity. After describing a gen-
p. 35

ment versé en mathématiques, pourrait prévoir exactement le nombre d'occurrences des diffé-
rentes notes, ou le nombre maximal de notes différentes, la possibilité ou non que la mélodie
soit palindromique, ou faire en sorte que la mélodie s'exprime comme un pavage par augmenta-
tions: toutes ces questions, et d'autres, sont élucidées dans [autosimJMM]; mais en pratique,
même sans aucune connaissance mathématique, en expérimentant avec les paramètres du
patch développé sous OpenMusic on arrive rapidement aux mêmes résultats.
p. 36

Mémoire de thèse Amiot 37
Conclusion et perspectives
L’ élaboration de cette synthèse de recherches initiées il y a 25 ans m’a permis de
comprendre pourquoi, et en quoi, l’informatique a toujours joué un rôle central dans chacun de
mes travaux. Mes deux pôles d’intérêt ont toujours été musique et mathématiques (non trivia-
les); or, refusant toute séparation schizophrénique entre elles, je me suis tout naturellement
placé au carrefour de l’informatique. Cette discipline touche à la première, notamment par la
validation des théories élaborées via des implémentations concrètes (et bien sûr, plus tradition-
nellement par tout l’aspect expérimental, analyse de données ou formulation de conjectures), et
aux dernières via l’organisation rigoureuse des données, qui fait de l’écriture d’un programme le
reflet fidèle (pour ne pas dire platonicien) des structures algébriques, qu’en retour il aura par-
fois suggérées.
Il est fréquent de considérer les mathématiques comme des prestataires d'outils et
de concepts pour d'autres sciences; certes il en est ainsi, dans ma démarche comme dans d'au-
tres: les structures de groupe, d'anneau, d'algèbre, la transformée de Fourier et autres mor-
phismes montrent leur « déraisonnable » efficacité. Cependant, comme parfois avec la physi-
que, il arrive que le champ d'étude lui-même suscite des retombées mathématiques, que l'on
aurait obtenues bien plus tard, si la perspective musicale n'avait suggéré des questionnements
d'une nature bien différente.
En premier lieu, je peux citer mes résultats sur les pavages de Z par translation. La
notion de canon de Vuza est «évidente» pour un musicien, et la réduction récursive à un canon
de Vuza est exactement ce que l'on perçoit d'un canon musical (on n'entend jamais que des ca-
nons de Vuza…). Et par ailleurs, cette réduction apporte des lumières nouvelles sur certaines
des plus fascinantes conjectures (mathématiques) sur les pavages.

Mémoire de thèse Amiot 38
Un signe manifeste de l'injuste obscurité des canons de Vuza est l'absence, dans
l'encyclopédie des suites d'entiers en ligne de Sloane, de la liste des cardinaux des «mauvais
groupes» (ceux qui ne sont pas de Hajós: 72, 108, 120, 144…). Cette omission a été réparée de-
puis; j'y ai fait rajouter en plus une suite originale, celle des entiers n tels que -1 soit une puis-
sance 62 modulo n. Ces entiers sont liés aux propriétés palindromiques 63 de certaines mélodies
autosimilaires.
D'autres propriétés intéressantes, et à ma connaissance inédites, ont surgi de l'étude
de ces mélodies: ordre maximal d'une application affine, définition des mélodies autosimilaires
commme limites asymptotiques d'extraction de toute mélodie périodique. D'autres propriétés
déjà connues par ailleurs, mais demeurées obscures, ont pris un sens nouveau grâce à l'orienta-
tion que leur ont donnée mes recherches. Ainsi en va-t-il du lemme ésotérique qui assure que
tout polynôme à coefficients modulo p, n'ayant pas 0 comme racine, divise un X n-1 pour n as-
sez grand. Lorsque je l'ai retrouvé, ce lemme m'a permis de prouver que tout motif fini «pave
modulo p».
Enfin, pour conclure provisoirement cette liste (qui, je l'espère, continuera de s'en-
richir), il me semble que seul un musicien pouvait s'interroger sur le nombre d'intervalles géné-
rateurs d'une gamme, même s'il a fallu requérir des compétences de mathématicien pour établir
que ce nombre ne peut jamais être 14,par exemple 64. C'est en étudiant les diverses opérations
musicales sur les gammes (transpositions, inversions) et les rapports algébriques entre elles que
j'ai été amené à classer toutes les spectral units rationnelles d'ordre fini, qui sont (entre autres
représentations) des matrices de passage entre gammes homométriques. Recherche qui m'a in-
cidemment fait découvrir le fait beaucoup plus simple, que la différence entre deux nombres
inversibles modulo n décrit le sous-groupe d'indice 2 de Z/nZ 65.
62 Les premières valeurs sont 5, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33…
63 Mon intérêt pour les palindromes est largement né de la passion que leur voue Moreno Andreatta, qui avait remarqué
que nombre de canons de Vuza exhibent une certaine "palindromicité", quand ce n'est pas une palindromicité
certaine.
64 Puisque j'ai prouvé que ce nombre est toujours de la forme Φ(n) où Φ est la fonction d'Euler.
65 Et le groupe tout entier quand n est impair.

Mémoire de thèse Amiot 39
L' approche musicienne est, pour moi, une source d'émerveillement continuel, car
elle ne cesse d'engendrer des découvertes nouvelles, dues à ses angles de pensers originaux. Ce-
pendant — et ce n'est pas une opposition mais une consolidation dialectique — il s'agit en
vérité de l'unité profonde qui relie naturellement l'esprit d'un musicien et celui d'un mathéma-
ticien. Si l'on en croit un vieux poncif pythagoricien, cette unité va tellement de soi qu'elle n'a
nul besoin d'être prouvée. Récemment encore, ce constat se fondait sur l'abondance du nombre
dans la musique (mètre, hauteurs, combinatoire), abondance qui n'offre pourtant rien de parti-
culièrement remarquable, ou de spécifique à la musique! Car, pour citer le mot trop célèbre de
Leibniz,
La musique est un exercice d'arithmétique secrète, et celui qui s'y livre, ignore qu'il ma-
nie des nombres.
Si le secret perdure sans doute, il est clair dorénavant, et particulièrement je l'espère
pour mes lecteurs, que cette harmonie dépasse de loin la notion de nombre et donc la perspec-
tive Leibnizienne 66, embrassant tout particulièrement les structures algébriques qui modélisent
au plus près les concepts du musicien, concepts simples de son point de vue (autosimilarité,
contenu intervallique, etc… ) mais qui nécessitent des outils relativement complexes pour leur
formalisation. Comme le dit fort justement Guérino Mazzola,
On ne peut prétendre que Bach, Haydn, Mozart ou Beethoven — pour ne nommer que
quelques uns des plus grands compositeurs, sont des génies exceptionnels qui ont élaboré des
chefs-d'œuvres éternels, sans se donner, pour essayer de comprendre leurs créations uniques, des
outils appropriés, c'est à dire suffisamment puissants et profonds 67.
66 La dépassant et même la renversant, comme tente de le montrer M. Andreatta dans « Mathematica est exercitium
musicae : la recherche "mathémusicale" et ses interactions avec les autres disciplines », HDR (soutenance
prévue pour octobre 2010).
67 Mazzola & alii, Topos of Music, Birkhaüser, 2002, p. vii. (ma traduction).

Mémoire de thèse Amiot 40
Or l' ordinateur apporte un interfaçage bienvenu entre l'aridité de l'algèbre et l'im-
médiateté de la perception musicale; il est donc naturel que mes recherches se soient si souvent
traduites en implémentations (comme d'ailleurs nombre de celles de Mazzola). Cependant il
existe une autre piste qui confirmerait ce renouveau du poncif — la musique comme algèbre
secrète ? — qui est celle de l'expérimentation en sciences cognitives.
5
4
3
2
1
Un FLID: distribution plate des
intervalles et des coefficients de Fourier
DFT of 0,2,3,4,8 mod 11
2 4 6 8 10
Avec Isabelle Viaud-Delmon, Carlos Agon et Moreno Andreatta, nous avons lancé à
l'IRCAM un projet destiné à mesurer la perception par les sujets testés du caractère uniforme,
ou pas, des coefficients de Fourier d'un rythme. Ce projet vise à établir si l'on discerne le carac-
tère saillant (comme pour les ME sets) ou au contraire plat (cf. l'illustration ci-dessus, avec une
platitude maximale des coefficients de Fourier) de structures discrètes périodiques. L'étude du
caractère perceptif des coefficients de la DFT fait partie du projet plus large "Mathématiques/
Musique et Cognition 68" cherchant à relier les approches formelles en musicologie computa-
tionnelle et les sciences cognitives (cf. figure infra). Le projet "Mathématiques/Musique et
Cognition", qui avait reçu initialement le soutien de l'AFIM (Association Française d'Informa-
tique Musicale) est désormais intégré en tant qu' un axe de recherche, à laquelle je participe, au
sein de l'équipe Représentations Musicales de l'IRCAM.
68 http://recherche.ircam.fr/equipes/repmus/mamux/Cognition.html

Mémoire de thèse Amiot 41

Mémoire de thèse Amiot 42
Remerciements
Avant tout, je remercie André Riotte, pionnier courageux grâce à qui j'ai pu trouver
ma voie, ainsi que nombre de mes complices dans cette belle aventure. Il a eu le courage d'aller
de l'avant dans une entreprise que presque tous jugeaient impossible. Il avait raison.
Je remercie chaleureusement mon directeur de thèse, Carlos Agon, qui lui aussi a
rendu possible ce qui semblait ne pas l'être. Sans lui jamais je ne me serai lancé dans cette
thèse. Par ailleurs, je garde un souvenir ému des nombreuses heures que nous avons passées en-
semble à implémenter mes idées les plus étranges. Son inégalable sens pratique a permis de
concrétiser nombre de mes abstractions.
Moreno Andreatta m'a convaincu de l'intérêt des canons rythmiques mosaïque, et
m'a ainsi remis en selle dans la recherche mathémusicale après une longue parenthèse. Sans lui,
je n'aurais sans doute pas infligé tous ces articles à la communauté scientifique. Son enthou-
siasme communicatif a impulsé un grand nombre d'événements et de recherches novatrices, et
en particulier des miennes.
David Clampitt porte une responsabilité non moins lourde, puisque c'est lui qui m'a
fait connaître aux membres actifs de la théorie musicale aux USA. J'ai ainsi pu, avec quelques
siècles de retard, découvrir l'Amérique, ce qui a eu des conséquences extrêmement fécondes
sur ma recherche. Sa générosité, et son hospitalité chaleureuse, m'ont grandement aidé à trou-
ver mon chemin en ces terres inconnues, et son travail visionnaire sur les We( Formed Scales
(avec Norman Carey que je salue au passage) m' ont ouvert un nouvel univers.
Thomas Noll a été un des chercheurs qui m'ont le plus stimulé, par nos discussions
informelles comme par nos innombrables échanges d'emails. Ses idées visionnaires permettent
d'aller plus loin qu'on ne s'en serait cru capable. Par ailleurs, il m'a lui aussi permis de rencon-
trer nombre de chercheurs fascinants, comme William Sethares dont je tiens à dire combien
j'ai apprécié la collaboration, l'incurable optimisme et l'inaltérable bonne humeur. C'est un réel
plaisir de faire de la recherche avec lui.

Mémoire de thèse Amiot 43
Guerino Mazzola est un chercheur d'une envergure immense, dont les travaux ont
plusieurs décennies d'avance sur le reste de la communauté. Je suis toujours pantois qu'il ait, si
tôt, cru en moi (m'invitant à un colloque à Zürich dès 2003 et ne cessant jamais de m'encoura-
ger) et cela m'a été d'un grand secours dans les moments de doute ou de découragement. Ses
idées fascinantes m'ont aussi grandement influencé, qu'il s'agisse des objets de nos études ou,
plus vertigineusement, de ses théories sur que(es théories utiliser pour les étudier.
Ma mère a eu le courage admirable de relire un texte dont elle ne pouvait que rare-
ment suivre le sens, pour me conseiller une écriture mieux articulée, plus riche en virgules, et
généralement plus limpide. S'il reste des obscurités dans ce mémoire, ce n'est certes pas de sa
faute, mais de la mienne.
Tom Johnson a toujours été un formidable réservoir d'idées et de questions. Je lui
suis, entre autres, redevable de la merveilleuse question des mélodies autosimilaires, mais aussi
de la découverte d'un bon nombre de belles propriétés — qu'il ne restait plus qu'à prouver…
C'est néanmoins un vieux complice, Gérard Assayag, qui m'a signalé ce dernier problème. Je lui
dois cela et bien plus, car il n'a pas peu contribué à m' inspirer, notamment grâce à ses belles
réalisations informatiques, dont le fleuron est sans doute OMax.
Jean-Paul Allouche m'a ouvert les colonnes de sa revue pour divulguer au grand pu-
blic les mystères des canons rythmiques et je l'en remercie.
J'ai toujours un grand plaisir à échanger des idées avec Jon Wild, un des chercheurs
les plus fertiles que je connaisse. Nos discussions sont toujours extraordinairement stimulantes,
et j'attends les prochaines avec impatience.
Aux USA, j'ai été très sensible à l'accueil chaleureux de Rick Cohn, qui m'a notam-
ment invité (avec David Clampitt) à être présent lors de la préentation mémorable des Orbi-
folds par Cliff Callender, Ian Quinn et Dmitri Tymosczko. J'ai dit plus haut à quel point la thèse
de Ian m'a profondément influencé, et je répète ici à quel point je lui en suis reconnaissant. J'ai
eu par ailleurs des échanges d'une grande fécondité et profondeur avec Dmitri, qui m'a très
gentiment invité à un séminaire mémorable — et fécond — à la Barbade, dont je garde un
souvenir ému.

Mémoire de thèse Amiot 44
Je tiens à rappeler la mémoire de Marcel Mesnage, récemment disparu, qui a porté
à son époque, et quasiment seul, la qualité de l'implémentation des théories musicales à un ni-
veau longtemps inégalé. Son Morphoscope avait des années d'avance sur son temps. C'était un
ami et une source d'inspiration, nous ne l'oublierons pas.
Last but not least (by far), mes remerciements émus à mon épouse Pascale, qui a pa-
tiemment relu avec soin, non seulement ce mémoire, mais aussi la plupart de mes articles dont
elle a aidé à polir l'anglais. Son mérité est d'autant plus immense que les sujets évoqués étaient
bien éloignés de ses connaissances, et qu'elle dû, et su, supporter de nombreux moments de dé-
couragement, mais aussi — et ce sont sans doute les pires — ceux d'intense créativité.

Mémoire de thèse Amiot 45
Liste de travaux
ARTICLES FIGURANT DANS LE DOSSIER DES TRAVAUX
✦ Amiot E., « À propos des canons rythmiques », Gazette des Mathématiciens, 106 (2005). Consultable en ligne.
✦ Amiot E., « David Lewin and Maximally Even Sets », Journal of Mathematics and Music (2007) vol. 3.
✦ Amiot E., « Autosimilar Melodies », Journal of Mathematics and Music (2008) vol. 3.
✦ Amiot E., « Discrete Fourier Transform and Bach’s Good Temperament », Music Theory Online (2009) 15, 2.
✦Amiot E., « New Perspectives on rhythmic canons and the Spectral Conjecture », Journal of Mathematics and
Music (2009) special issue on rhythmic canons, C. Agon et M. Andreatta dir..
AUTRES ARTICLES PUBLIÉS DANS DES REVUES avec comité de lecture
✦ Amiot E., « Pour en finir avec le Désir », Revue Analyse Musicale, 22 (1991).
✦ Amiot E.,« Mathématiques et analyse musicale: une fécondation réciproque », R.A.M., 28 (1992).
✦ Amiot E.,« La série dodécaphonique et ses symétries », Quadrature, 19 (1994).
✦ Amiot E.,« Chopin, virtuose de la théorie des groupes ?» Quadrature, 24 (2000).
✦ Amiot E.,« Une preuve élégante du théorème de l’hexacorde de Babbitt », Quadrature, 28 (2006).
✦ Amiot E.,« Gammes Bien Réparties », Revue de Mathématiques et Sciences Humaines, 178 (2007).
ACTES DE COLLOQUES avec comité de lecture
✦ Riotte, A., Amiot, E., Assayag, G., Malherbe, C., « Duration Structure Generation and Recognition in Musical
Writing », Proceedings, ICMC (International Computer Music Conference), la Haye (1986).
✦ Agon, C., Amiot, E., Andreatta, M., « Tiling problems in music composition : Theory and Implementation »,
Voices of Nature Proceedings, ICMC, Göteborg (2002).
✦ Amiot, E., « Rhythmic canons and Galois Theory », Actes du Colloquium on Mathematical Music Theory, H.
Fripertinger & L. Reich Eds, Grazer Math. Bericht Nr 347 (2005).
✦ Agon, C., Amiot, E., Andreatta, M., « Tiling the (musical) line with polynomials : Some theoretical and im-
plementational aspects », Acts of ICMC 2005, Barcelona (2005).
✦ Agon, C., Amiot, E., Andreatta, M., Noll, T., « Towards Pedagogability of Mathematical Music Theory: Alge-
braic Models and Tiling Problems in Computer-Aided Composition », Bridges Conferences, Proceedings, Lon-
don (2006).
✦ Agon, C., Amiot, E., Andreatta, M., Noll, T., « Oracles for Computer-Aided. Improvisation », ICMC, New
Orleans (2006).
✦ Agon, C., Amiot, E., Andreatta, M., « Autosimilar melodies and their implementation in OpenMusic », Pro-
ceedings SMC 07, Lefkada, Grèce (2007).

Mémoire de thèse Amiot 46
✦ Amiot, E., « Eine kleine Fourier Nachtmusik », Actes du colloque de la Society for Mathematics and Computa-
tion in Music, Berlin (2007), Springer (à paraître).
✦ Amiot, E., Sethares, W., « An Algebra for periodic rhythms and scales », Actes du Helmholtz Workshop "Klang
und Ton", Berlin (2007), Springer (à paraître).
OUVRAGE COLLECTIF
✦ « Why rhythmic canons are interesting », Perspectives in Mathematical and Computational Music Theory,
Mazzola, Noll, Lluis Puebla Ed, Epos, 190-209, Univ. Onasbrück (2004).
CONFÉRENCES
✦ Un bon nombre de ces conférences ou communications ont été données dans le cadre du séminaire MaMuX à
l'IRCAM, Paris.
✦ « Why rhythmic canons are interesting », colloque MaMuTh de Zürich (octobre 2002).
✦ « Outils pour les canons rythmiques », MaMuX (janvier 2003).
✦ « Sur des canons de Vuza que son algorithme ne permet pas d'obtenir », MaMuX (janvier 2004).
✦« Rhythmic canons and Galois Theory », conférence au colloque MaMuTh de Graz (mai 2004).
✦ « Les Canons rythmiques », conférence de vulgarisation, Perpignan (octobre 2004).
✦ « Rhythmic Canons, Galois Theory, Spectral Conjecture », American Mathematical Society’s Fall Session,
Evanston, IL, USA (octobre 2004).
✦ « Canons rythmiques pour les musiciens », « canons rythmiques pour les mathématiciens », MaMuX (juin
2005).
✦ « Canons that worked » à McMaster University, et « Mathematical Properties of Rhythmic Canons », St Cathe-
rine University, Ontario, Canada (juillet 2005).
✦ « Sur les canons rythmiques », à l’invitation du Club Maths de l’université Pierre et Marie Curie, Paris VI (oc-
tobre 2006).
✦ « Why Fourier ? », journées à la mémoire de John Clough à l’Université de Chicago, IL, USA (7-10 juillet
2005).
✦ « Gammes et transformée de Fourier Discrète », MaMuX (mai 2006).
✦ « Mélodies Autosimilaires » (avec Tom Johnson), Colloque Mélodie de la Société Française d'Analyse Musi-
cale, IRCAM (oct. 2006).
✦ « L’action du groupe affine modulo n et les mélodies autosimilaires », Séminaire MaMuX (oct. 2006).
✦ « Scales and Fourier in the non tempered universe », Helmholtz Workshop "Klang und Ton", Berlin (mai
2007).
✦ « Eine kleine Fourier Nachtmusik », colloque de la SMCM, Berlin (mai 2007).
✦ « DFT vs JSB: about some fine tuning », MaMuX, Paris (mai 2008).
✦ « Canone ritmici: come, quanto ? », université de Pise (octobre 2008).

Mémoire de thèse Amiot 47
✦ « Fourier, Music, Mathematics and the Brain », colloque de la Society for Music Theory, Nashville (novembre
2008).
✦ « La trasformata di Fourier discreta & applicazione musicale », cours de Mastère à l’université de Pise (janvier
2009).
✦ « Aspects cognitifs de la DFT dans la perception de structures musicales », MaMuX (avril 2009).
✦ « Promenades musicales dans Z/12 Z »: une concerférence de vulgarisation sur certaines structures algébriques
discrètes en musique; donnée à Perpignan, Toulouse, et Leucate. Une partie du concert (La cage aux chiffres
d'André Riotte) est disponible en ligne.
AUTRES RÉFÉRENCES BIBLIOGRAPHIQUES
✦ Actes du colloque sur la Set Theory, Ed. Delatour France / IRCAM Centre Pompidou, Paris (2008).
✦ Amiot, E., « On the Group of Rational Spectral Units with Finite Order », http://arxiv.org/abs/0907.0857.
✦ Amiot, E., « About the Number of Generators of a Musical Scale », http://arxiv.org/abs/0909.0039.
✦ Andreatta, M., Méthodes algébriques en musique et musicologie du XXe siècle : aspects théoriques, analytiques
et compositionnels, thèse, École des hautes études en sciences sociales / Ircam, Paris (2003).
✦ Coven, A., Meyerowitz, E., « Tiling the integers with translates of one finite set », J. Algebra 212 , pp 161--174
(1999).
✦ Douthett, J. et Krantz, R. « Energy extremes and spin configurations for the one-dimensional antiferromagnetic
Ising model with arbitrary-range interaction », Journal of Mathematical Physics 37 , pp. 3334-3353 (1996).
✦ Fidanza, G., « Canoni ritmici a mosaico », tesi di laurea, Università degli Studi di Pisa, Facoltà di
SSMMFFNN, Corso di laurea in Matematica (2008).
✦ Fripertinger, H., « Tiling problems in music theory », Perspectives of Mathematical and Computational Music
Theory, (G. Mazzola, E. Puebla et T. Noll eds.) EpOs, Universität Osnabrück, 153-168 (2004).
✦ Gilbert, E., « Canons mosaïques, polynômes cyclotomiques, rythmes k-asymétriques », mémoire ATIAM, M.
Andreatta dir. (2007).
✦ Kolountzakis, M., et Matolcsi, M. « Algorithms for Translational Tiling », special issue on rhythmic canons, C.
Agon et M. Andreatta dir, JMM (2009).
✦ Kolountzakis, M., Matolcsi, M., « Tiles with no spectra », Forum Math., to appear.
✦ Łaba, I., « The spectral set conjecture and multiplicative properties of roots of polynomials », J. London Math.
Soc. 65, pp. 661–671 (2002).
✦ Lewin, D., Re: Intervallic Relations between two collections of notes, Journal of Music Theory, 3:298-301
(1959).
✦ Lewin, D., Special Cases of the Interval Function between Pitch-Class Sets X and Y, Journal of Music Theory,
45-129 (2001).
✦ Quinn, I., « A unified theory of chord quality in equal temperaments », PhD dissertation, Univ. of Rochester
(2005).
✦ Riotte, A., « Formalisation de structures musicales », cours Paris VIII (1978-1990).

Annexe I
Articles figurant dans le dossier des travaux

À propos des canons rythmiques
Emmanuel Amiot
Résumé
Cet article fait le point sur une notion riche, issue de préoccupations musicales mais qui s’est
avérée féconde en problèmes mathématiques fascinants.
En particulier l’étude des canons rythmiques a permis de découvrir des résultats inédits sur
les corps finis et de démontrer de nouveaux cas de la conjecture spectrale. En retour, bien sûr,
les outils mathématiques performants utilisés ont donné de nouvelles dimensions à explorer aux
compositeurs. La théorie de GALOIS a donc fait une apparition inattendue dans certains logiciels
d’aide à la Composition Assistée par Ordinateur !
Des illustrations musicales (ou en tout cas, sonores. . .) de cet article peuvent être trouvées sur le
site http ://canonsrythmiques.free.fr/Midi, sous forme de fichiers MIDI.
Notations
Je note Zn le groupe (parfois l’anneau) quotient Z/nZ, Fq est le corps à q éléments.
Dans tout anneau, a | b signifie que a divise b.
Φn désigne le n ème polynôme cyclotomique dans Z[X].
Enfin [[ a, b ]] désigne l’intervalle d’entiers [a, b] ∩ Z.
1 Canons musicaux et canons rythmiques
1.1 Canon musical
Le principe du canon musical est probablement bien connu du lecteur ; l’exemple le plus connu
des francophones est sans doute « Frère Jacques », qui se chante de préférence à quatre, chaque
chanteur reprenant exactement la même comptine mais décalé d’une mesure par rapport au chanteur
précédent.
Frère Jacques, frère jacques Dormez-vous ? dormez-vous ? Sonnez les matines, sonnez les matines Ding deng dong, ding deng dong
qui devient, en régime de croisière,
Frère Jacques, frère jacques Dormez-vous ? dormez-vous ? Sonnez les matines, sonnez les matines
Frère Jacques, frère jacques Dormez-vous ? dormez-vous ?
Frère Jacques, frère jacques
Frère Jacques, frère jacques Dormez-vous ? dormez-vous ? Sonnez les matines, sonnez les matines Ding deng dong, ding deng dong
Ding deng dong, ding deng dong Frère Jacques, frère jacques Dormez-vous ? dormez-vous ? Sonnez les matines, sonnez les matines
Sonnez les matines, sonnez les matines Ding deng dong, ding deng dong Frère Jacques, frère jacques Dormez-vous ? dormez-vous ?
Dormez-vous ? dormez-vous ? Sonnez les matines, sonnez les matines Ding deng dong, ding deng dong Frère Jacques, frère jacques
Ce principe de jouer une même mélodie (ou une forme légèrement déformée de la même mélodie)
le long de diverses voix est aussi celui de la fugue, dont le plus célèbre spécialiste est J.S. BACH.
C’est tout un art (de la fugue !) que de faire coïncider harmonieusement des notes diverses avec un
décalage. J.S. BACH, justement, a par exemple montré sa virtuosité dans les Variations Goldberg où
il fait des canons décalés dans le temps et dans l’espace des hauteurs, successivement d’un unisson,
d’une seconde, d’une tierce, etc. . .
Pour modéliser de façon constructive les canons, nous allons nous montrer moins ambitieux,
en nous concentrant exclusivement sur le domaine rythmique, et plus exigeants, en posant une
condition rigoureuse :
✞
☎
Sur chaque temps, on doit entendre une seule note
✝
✆
Sans cette contrainte, on pourrait (on peut !) faire un canon fondé sur n’importe quel motif. Cela
n’a pas grand intérêt, sauf peut-être combinatoire.
Le canon rythmique « canonique », si j’ose dire, est donc fondé sur un pattern rythmique discret,
qu’on peut imaginer joué par un instrument de percussion (on néglige la question de la durée des
notes), pattern qui est répété à l’identique par d’autres voix.
On peut alors modéliser ce pattern très simplifié par une série d’entiers, qui repèrent les moments
où une note est jouée. Sans perte de généralité, on peut fixer l’origine des temps à la première note

du pattern, qui va donc commencer par le nombre 0. Le principe même du canon signifie que les
diverses entrées du motif sont obtenues par des translations, égales aux décalages avec la première
entrée : les diverses voix seront A, A + b1, A + b2 . . .
En mettant dans un même vecteur tous ces décalages, on obtient une première formalisation,
provisoire :
DÉFINITION 1. Soit A = {0, a1, . . . ak−1} un sous-ensemble de N ;
≀ A sera le motif (inner rhythm) d’un canon rythmique s’il existe un pattern des voix (outer rhythm)
≀ B = {0, b1, . . . bℓ−1} tel que A × B ∋ (a, b) ↦→ a + b est injective.
≀ A est le motif du canon, B la séquence des entrées (les moments où chaque instrument commence sa
≀ partie).
≀ Cette condition s’écrit aussi A + B = A ⊕ B.
Exemple : Le motif A = {0, 1, 3, 6} donne un canon à quatre voix avec B = {0, 4, 8, 12}. En effet,
A ⊕ B = {0, 1, 3, 4, 5, 6, 7, 10}. Une représentation simplifiée de partition en est donnée figure 1.
FIG. 1 – Un canon rythmique
REMARQUE 1. Chaque note est jouée sur un multiple entier de l’unité de temps ; ceci peut paraître une
≀ contrainte artificielle et forte, mais en fait aussi bien DAN TUDOR VUZA [20], qui est le pionnier des re-
≀
cherches sur les canons rythmiques, que LAGARIAS [15] dans un article purement mathématique, ont
≀ montré essentiellement que ce cas est le seul possible pour un motif fini.
Un corollaire élémentaire de la définition :
PROPOSITION (DUALITÉ). Si A ⊕ B est un canon rythmique, il en est de même de B ⊕ A : on peut échanger les
≀ rôles des inner et outer rhythms.
La commutativité de l’addition permet donc de fabriquer un canon à p voix de q notes en partant d’un
canon à q voix de p notes. Avec 4 × 5 ou 5 × 4 notes on par exemple la figure 2.
Un canon et son dual
FIG. 2 – Deux canons duaux
1.2 Canons périodiques
On remarque, sur l’exemple ci-dessus, qu’il y a des trous dans le canon – que les musiciens
appellentdes silences – mais que ces trous se trouveraient naturellement bouchés par d’autres copies
du motif. En fait on peut obtenir un canon infini avec une note et une seule par temps, soit avec un
nombre infini de voix, soit de façon plus réaliste en prolongeant le motif par périodicité (ici la période
8) comme on le constate sur la figure 3, qui reprend le motif de la figure 1.
....
Variations sur uncanon
....
ou
FIG. 3 – Canon prolongé à l’infini
.... ....
On obtient ainsi un pavage périodique de Z par le motif A. Dorénavant, je parlerai donc indifféremment
de pavages (pavages de Z) ou de canons rythmiques. Par passage au quotient, cela
2

evient à dire que l’on a un pavage du groupe cyclique Zn = Z/nZ. D’où une nouvelle définition, plus
restrictive, qui est celle que nous considérerons désormais :
DÉFINITION 2. On a un canon rythmique de motif A = {a0, . . . ak−1} et de période n s’il existe B ⊂ N tel que
≀
A ⊕ B = Zn
≀
La condition de somme directe (⊕) exprime que sur chaque temps on a exactement une note et une
seule.
Exemple : Le motif A = {0, 1, 3, 6} donne un canon de période 8 avec B = {0, 4}. En effet, A ⊕
B = {0, 1, 3, 4, 5, 6, 7, 10} = {0, 1, 2, 3, 4, 5, 6, 7} si on travaille modulo 8. C’est l’effet obtenu si on reprend
périodiquement ce canon (comme Frère Jacques).
FIG. 4 – La forme d’un motif est définie à une rotation près
REMARQUE 2. Comme Zn est cyclique, la notion de pavage ou de canon rythmique est indépendante d’un choix
≀ d’origine de ce cercle comme on le voit sur les figures 4 et 5. C’est pourquoi en pratique on convient, sans
≀
perte de généralité, que A, B commencent par 0. Formellement cela est lié à l’action du groupe Zn sur ses
≀ parties par translation.
FIG. 5 – Un canon de Vuza, de période 108
Cette condition de périodicité que nous avons apparemment imposée semble très forte. Mais on
connait depuis 1950 le théorème suivant :
THÉORÈME 1. ([de Bruijn)]
≀ Si A est une partie finie de N telle qu’il existe C ⊂ Z avec A ⊕ C = Z, alors il existe un entier n et une
≀ partie (finie) B tels que C = B ⊕ nZ. Donc le pavage est périodique, et A ⊕ B = Zn.
La démonstration de ce théorème repose sur l’incontournable principe des tiroirs, je laisse le lecteur
intéressé se référer à [2] et à la figure suivante.
Mentionnons à ce propos un premier problème ouvert : Si ℓ(A) = Max A − Min A est la largeur du
3

Un musicien attend
pour rentrer ICI
FIG. 6 – Pourquoi tout canon de motif fini est périodique
motif A, la démonstration de DE BRUIJN montre que n 2 ℓ(A) majore la période du canon, mais tous
les exemples connus vérifient n 2 × ℓ(A). . . Est-ce général ?
EXERCICE 1. Donner un motif de largeur ℓ pour lequel la plus petite période possible du canon est effectivement
≀ 2ℓ (solution en fin d’article).
En revanche un motif infini peut très bien donner un canon apériodique (par exemple les nombres
dont l’écriture binaire n’a que des bits d’ordre impairs), ainsi que des pavages plus « tordus »[2].
2 Modélisation polynomiale et facteurs cyclotomiques
Pour travailler dans une structure plus riche, on fait comme SOPHUS LIE passant d’un groupe de
Lie à son algèbre : par exponentiation.
2.1 Polynôme associé à un motif rythmique
On va enrichir la structure algébrique ambiante, remplaçant les sommes d’ensembles par des
produits de polynômes :
DÉFINITION 3. Soit A ⊂ N un sous-ensemble fini non vide. Alors on pose A(X) =
≀
X
k∈A
k .
PROPOSITION. La somme A + B est directe (i.e. A × B ∋ (a, b) ↦→ a + b est injective) ssi
≀
A(X) × B(X) = (A ⊕ B)(X)
≀
Sinon on trouverait des coefficients > 1. Et donc la définition des canons rythmiques est la condition
suivante, notée (T0) :
THÉORÈME 2. A est le motif d’un canon rythmique avec « motif des entrées » B et période n ssi
≀
(T0) A(X) × B(X) = 1 + X + X
≀
2 + . . . X n−1
(mod X n − 1)
Par exemple {0, 1, 3, 6} ⊕ {0, 8, 12, 20} donne les polynômes (1 + X + X 3 + X 6 ) × (1 + X 8 + X 12 + X 20 ) dont le
produit est
1 + X + X 3 + X 6 + X 8 + X 9 + X 11 + X 12 + X 13 + X 14 + X 15 + X 18 + X 20 + X 21 + X 23 + X 26
qui se réduit modulo X 16 − 1 à 1 + X + . . . X 15 – concrètement on applique la règle X k → X k mod n .
NB : c’est en découvrant cette formalisation appliquée par ANDRANIK TANGIAN [16] à un problème
de TOM JOHNSON que je me suis enthousiasmé pour la la cause des canons rythmiques ; mais ce
procédé remonte à REDEI dans les années 1950.
2.2 Les ‘perfect square tilings’ de Tom Johnson
Le but de paragraphe est de montrer sur un exemple assez simple que l’introduction des polynômes
n’est pas qu’une simple commodité d’écriture : si on sort l’artillerie lourde, c’est qu’elle
s’impose pour l’étude des canons ! On sait plus de choses dans une algèbre que dans le monoïde
P(Zn). . .
T. JOHNSON, compositeur minimaliste américain vivant à Paris, s’est posé récemment la question
de réaliser une forme très particulière de canon (canon avec augmentations) avec le motif très simple
T1 =(0 1 2) mais avec les contraintes suivantes :
– Utilisez des augmentations de ce motif, T2 = 2 × T1, . . . Tk (au sens musical : comprenez des
multiples, translatés, comme (5 9 13)= 5 + T4 = 5+(0 4 8)= 5 + 4 × T1 ),
– les multiplicandes sont tous distincts,
– et on pave de façon ‘compacte’, i.e. sans faire de réduction modulo n.
4

Ce problème a été exposé dans la rubrique de J.P. DELAHAYE dans Pour la science (novembre 2004)
ce qui a donné l’occasion à plusieurs lecteurs, bons programmeurs, de trouver des solutions avec le
motif initial (0 1 2 3) – à cette heure la question de l’existence d’un pavage parfait pour un motif de
5 notes est ouverte.
5
4
3
2
1
0
0 2 4 6 8 10 12 14
FIG. 7 – Plus petit ‘perfect square tiling’.
Quant Tom s’est ouvert de cette nouvelle question, il nous a présenté la plus petite solution (cf.
figure, où l’on voit T1 aux échelles 1,2,4,5,7), et une question troublante : pourquoi était-il impossible
de trouver de ‘pavage en carré parfait’ avec seulement un ou deux des motifs T3, T6, T9 ? Son
programme lui donnait soit les trois à la fois (figure suivante), soit aucun.
7
6
5
4
3
2
1
0
0 5 10 15 20
FIG. 8 – Plus petit ‘perfect square tiling’ avec T3, T6, T9.
Cette petite question admet une solution très simple, si l’on exprime le motif de base par le polynôme
Φ3(X) = 1 + X + X 2 , ses augmentations sont de la forme X k (1 + X i + X 2i ) = X k Φ3(X i ), et la
question de Tom revient à trouver une expression algébrique de la forme
X kiΦ3(X i ) = 1 + X + . . . =
i∈I
3n−1
j=0
X j = X3n − 1
X − 1
EXERCICE 2. Montrer que dans une relation comme la précédente, le nombre d’indices i qui sont multiples de
≀ 3 est lui-même un multiple de 3.
2.3 Polynômes cyclotomiques
L’intérêt de ce changement d’espace, de Z à une partie de Z[X], est que l’on sait plusieurs choses
sur les polynômes qui apparaissent dans notre problème (voyez n’importe quel bon livre d’algèbre
commutative pour une preuve du théorème suivant) :
THÉORÈME 3. Les facteurs irréductibles (dans Q[X] ou Z[X]) de 1 + X + X
≀
2 + . . . Xn−1 sont les célèbres polynômes
cyclotomiques Φd, avec d | n (et d > 1 ici). Φd est le produit (dans C[X]) des X − ξ où ξ décrit l’ensemble des
racines de l’unité d’ordre exactement d.
On peut les calculer par récurrence par la formule
Φd(X) = X
d|n
n − 1 ou, par inversion de MÖBIUS
Φn(X) =
(X
d|n
d − 1) µ(d) .
Leurs coefficients sont entiers (relatifs). On a par exemple Φp(X) = 1 + X + X2 + . . . Xp−1 quand (ssi) p est
premier.
On a tout de suite un critère très utile qui résulte du théorème précédent :
COROLLAIRE 1. Pour un canon de période n, chaque polynôme cyclotomique Φd, 1 < d | n, divise A(X) ou B(X).
Ceci résulte du théorème du GAUSS dans l’anneau principal Q[X], appliqué à la relation (T0) (cf. Thm
2). Ce phénomène explique une remarque d’ANDREATTA, faite sur les canons de VUZA (cf. infra), observant
que beaucoup de canons sont « presque » (i.e. à peu de termes près, voire exactement) des
palindromes. En effet, tous les polynômes cyclotomiques sont autoréciproques, i.e. palindromiques,
5

ainsi que leurs produits. Comme ce sont (presque) les seuls facteurs de A(x), B(x) cela explique que
ces derniers sont (presque) palindromiques.
Nous verrons que la répartition de ces facteurs cyclotomiques entre A(x) et B(x) est cruciale pour
permettre l’existence d’un canon rythmique.
Le cas de Φp d’indice premier admet une généralisation utile quoique élémentaire (récurrence) :
LEMME 1. On a Φd(1) = 1 ssi d est une puissance d’un nombre premier.
≀ Si d = p
≀
α on a alors Φd(1) = p, car
Par exemple, Φ9 = 1 + X 3 + X 6 .
3 Les conditions de Coven-Meyerowitz
Φpα(X) = 1 + Xpα−1 + X 2pα−1
+ . . . + X (p−1)pα−1
Avant 1998, on ne connaissait quasiment aucune condition générale pour déterminer si le motif
A était capable d’engendrer un canon rythmique, i.e. de paver. (à l’exception du cas où |A| était une
puissance d’un nombre premier).
EXERCICE 3. Sauriez-vous dire par exemple si (1, 4, 9, 16) forme un canon rythmique ?
Des considérations précédentes, les deux mathématiciens AARON MEYEROWITZ ET ETAN COHEN
ont déduit (cf. [5]) les critères suivants :
THÉORÈME DE COVEN-MEYEROWITZ. Soit RA l’ensemble des d ∈ N où Φd divise A(x), et SA le sous-ensemble
≀ des puissances de nombres premiers éléments de RA. On définit alors les conditions
≀ (T1) : A(1) =
≀
pα p et
∈SA
(T2) : si pα , qβ , · · · ∈ SA alors pα .qβ · · · ∈ RA.
Alors
(Thm A1) Si A pave, alors (T1) est vérifiée.
(Thm A2) Si (T1) et (T2)sont vérifiées, alors A pave.
(Thm B) Si |A| = A(1) n’a que deux facteurs premiers et si A pave, alors (T2) est vérifiée.
Donnons un exemple : le motif A = {0, 1, 8, 9, 17, 28} a un polynôme associé qui se factorise en
(1 + X) 1 − X + X 2 1 + X + X 2 1 − X 2 + X 4 1 − X 3 + X 6 1 + X 3 − X 4 − X 7 + X 8 − X 9 + X 11 − X 12 + X 13
On reconnaît 1 les facteurs cyclotomiques d’indices 2, 3, 6, 12, 18, plus un outsider qui n’est pas cyclotomique
du tout. On a donc RA = {2, 3, 6, 12, 18}, SA = {2, 3} ; or 2 × 3 = 6, ce qui prouve à la fois (T1) (car
A(1) = 6) et (T2) (car 6 ∈ RA). Effectivement, A pave :
FIG. 9 – Canon vérifiant (T1)&(T2)
Seul le troisième de ces résultats (thm B) est véritablement difficile ; il repose sur un lemme de
SANDS qui prouve que A ou B est inclus dans pZ (où p est l’un des deux facteurs premiers), ce qui est
faux dans le cas général, et ce en utilisant un résultat on ne peut plus Galoisien :
LEMME 2. Si n est premier avec m alors Φn est encore irréductible dans le corps cyclotomique Q[e 2iπ/m ].
Qu’on me pardonne de mentionner ce résultat technique : dans une partie ultérieure où l’on travaille
dans Fq[X], les Φn cessent généralement d’être irréductibles et le contraste avec la situation en caractéristique
0 méritait, je pense, d’être mentionnée.
Cet article serait interminable si toutes les démonstrations y figuraient, mais je vais tout de même
reproduire brièvement ici la démonstration du (Thm A1), qui illustre bien l’intérêt d’avoir élargi le
contexte de parties de Zn à une algèbre de polynômes.
1 j’ai dû implémenter pour cela une procédure qui marie harmonieusement théorie de GALOIS et mathématiques
numériques, utilisant notamment que si toutes les racines d’un polynôme unitaire irréductible à coefficients entiers sont
de module 1, alors ce sont des racines de l’unité. La précision du calcul a dû être adaptée au degré du polynôme passé en
variable !
6

Démonstration. La preuve repose sur le lemme 1. Observons que si A ⊕ B = Zn, on a en termes de
polynômes A(1)B(1) = n. Mais dans la décomposition en facteurs premiers de A(1)B(1) figurent tous
les Φd(1), qui valent 1 ou p (ce dernier cas ssi d est une puissance de p). Le nombre premier p figure
un nombre de fois égal au nombre de puissances de p qui divisent n, i.e. sa multiplicité dans n.
Donc les facteurs premiers de n apparaissent dans A(1)B(1) sous la forme Φpα(1) et seulement
sous cette forme. Les autres facteurs (cyclotomiques ou pas) de A(X) (ou B(X)) contribuent seulement
pour la valeur 1 quand X = 1, puisque tous les facteurs de n sont recensés.
La valeur de A(1) est donc égale au seul produit des facteurs premiers p tels que Φpα soit un
facteur de A(X) : c’est bien la condition (T1).
Notons sans insister, pour l’instant, qu’on ignore toujours aujourd’hui si la condition (T2) est
nécessaire dans tous les cas pour paver.
Ces conditions ne sont pas dénuées d’applications pratiques : en septembre, nous avons présenté
à Barcelone pour l’ICMC 2 une nouvelle fonctionnalité du logiciel d’aide à la composition OpenMusic,
développé à l’IRCAM notamment par CARLOS AGON et MORENO ANDREATTA, et qui permet de fabriquer
des « canons cyclotomiques », de période donnée, en utilisant les conditions ci-dessus.
4 Canons et corps finis
Nous allons faire un détour instructif en généralisant de façon naturelle la notion de canon rythmique
à celle de canon modulo p. Il s’agit alors d’avoir sur chaque temps un nombre de notes égal à
1 modulo p, condition plus généreuse que « une note et une seule ». On se retrouve naturellement
à factoriser des polynômes dans l’anneau k[X], où k est un corps fini. En effet, comme on l’a vu la
condition définissant un canon rythmique est
(T0) A(X) × B(X) = 1 + X + X 2 + . . . X n−1 mod (X n − 1)
La question se pose alors de considérer les facteurs irréductibles de cette identité polynômiale. Dans
le cas de Z[X] on avait affaire aux polynômes cyclotomiques ; dans l’exposé qui suit la situation est
plus compliquée, notamment du fait de la multiplicité > 1 des racines (de l’unité).
4.1 Le problème de Johnson
Je résume ici le problème qui m’a poussé à considérer des factorisations dans des corps finis en
lieu et place de Z[X].
TOM JOHNSON a présenté en 2001 aux Journées d’Informatique Musicale [10] le problème suivant
de canon par augmentations :
Faire un canon (compact) avec le motif (0 1 4) et (certaines de) ses augmentations (0, 2, 8) (ainsi
que (0, 4, 16) etc). Le compositeur et mathématicien A. TANGIAN, de l’université de Hanovre, rédigea
aussitôt [16] un programme en Fortran pour calculer toutes les solutions de taille bornée par un
N donné ; il s’avéra que toutes ces solutions avaient une longueur multiple de 15. La plus petite se
trouve sur la figure suivante.
5
4
3
2
1
0
0 2 4 6 8 10 12 14
FIG. 10 – Le plus petit pavage avec (0 1 4) et augmentations
Qu’elles soient multiples de 3 n’avait rien de surprenant, mais pourquoi de 15 ?
Si l’on pose J(X) = 1 + X + X 4 , le problème de JOHNSON revient à trouver des facteurs 0-1 A, B . . .
tels que
A(X)J(X) + B(X)J(X 2 ) . . . = 1 + X + X 2 + . . . X n−1
En effet, une augmentation comme (0, 2, 8) a pour polynôme associé J(X 2 ) = 1 + X 2 + X 8 .
Dans la plus petite solution, on a A(X) = 1 + X 2 + X 8 + X 10 et B(X) = X 5 pour n = 15.
Je me suis demandé s’il y avait moyen de trouver une racine commune de tous ces facteurs. Pour
cela il s’est avéré nécessaire de changer de corps. En effet, dans tout corps de caractéristique 2 on a
2 International Computer Music Conference
J(X 2 ) = 1 + X 2 + X 8 = (1 + X + X 4 ) 2 = J(X) 2
7

par l’automorphisme de FROBENIUS 3 on a plus généralement
LEMME 3. pour tout polynôme A(X) à coefficients dans Fp, on a A(X) p = A(X p ).
De même J(X 4 ) = J(X) 4 et ainsi de suite.
Donc J(X) doit diviser 1 + X + X 2 + . . . X n−1 à condition de calculer modulo 2. Or J se décompose
dans le corps F16 à 16 elements (résultat élémentaire de GALOIS : J est irréductible sur F2 et donc
F16 = F 2 d ◦ J ≈ F2[X]/(J)), de plus dans ce corps toutes ses racines 4 sont d’ordre (multiplicatif) 15, i.e.
vérifient α n = 1 ⇐⇒ 15 | n.
Maintenant on a donc, si α est racine de J dans F16, 1 + α + α 2 + . . . α n−1 = J(α) × (. . . ) = 0. On en
déduit en multipliant par α − 1 que α n − 1 = 0, donc α n = 1, ce qui impose bien que 15 | n, cqfd.
4.2 « Der verfluchte Ring »
La condition (T0) a un sens dans tous les anneaux 5 k[X], et même A[X] pour tout anneau A contenant
0, 1. En effet c’est une identité entre polynômes 0-1, c’est-à-dire entre éléments de l’ensemble
{0, 1}[X]. Je dis bien ensemble : car il n’est fermé ni pour + ni pour × (par exemple développer (1+X 2 ) 3
fait intervenir des opérations autres que 0+0 ou 0+1).
Il est bien clair que Z[X] est beaucoup trop vaste (il y a une écrasante majorité de polynômes NON
0-1), et il est bien difficile de donner des conditions nécessaires et suffisantes sur un polynôme 0-
1∈ Z[X] pour déterminer s’il va paver ou non (cela reste un problème ouvert), sinon en exhibant un
tel pavage 6 .
Il est bien plus tentant de travailler dans F2[X] = Z2[X] = Z/2Z[X] cet anneau ne contient que des
polynômes 0-1, et il les contient même tous une fois et une seule.
Malheureusement bien que l’application canonique
Z[X] → F2[X]
P = akX k ↦→ P (mod 2) = akX k
envoie bijectivement {0, 1}[X] ⊂ Z[X] sur F2[X], il n’y a pas de bonne application réciproque vers Z[X],
faute de structure algébrique sur {0, 1}[X].
Par exemple on ne peut pas remonter dans Z[X] l’équation suivante :
(1 + X)(1 + X + X 2 ) = 1 + X 3 ∈ F2[X]
en termes de polynômes 0 − 1.
J’ai donc recherché des conditions équivalentes au fait que « (T0) soit vérifiée dans Z[X] ». La plus
convaincante est :
THÉORÈME 5. (T0) est vraie dans Z[X] ⇐⇒ elle est vérifiée dans tous les Fp[X]
C’est un lointain cousin du théorème chinois, qu’il est bien plus facile[2] de démontrer musicalement
que mathématiquement : il signifie que le nombre de notes sur chaque temps est exactement 1, si et
seulement si il vaut 1 modulo tous les p premiers (on peut affaiblir les hypothèses d’ailleurs).
REMARQUE 3. Il est capital de souligner que l’énoncé ci-dessus est différent de ce qui suit :
≀ « Il existe B(X) ∈ {0, 1}[X] ⊂ Z[X], n tel que A(X) × B(X) ≡ 1 + x + . . . x
≀
n−1 (mod Xn − 1) dans Z[X]
⇐⇒
Pour tout p premier, il existe Bp, np tel que A(X) × Bp(X) ≡ 1 + x + . . . xnp−1 (mod Xnp − 1, p) »
Bien sûr, ⇒ est vraie. J’ai essayé assez longtemps de prouver la réciproque, conformément à la philosophie
de YONEDA ou aux théorèmes (local ⇒ global) du genre de ceux de HASSE sur les formes
quadratiques. Le résultat fut surprenant. . .
4.3 Canons modulo p : todos locos !
Nous avons posé la question de l’étude locale, i.e modulo p, de la relation (T0). Quelles conditions
a-t-on pour qu’un motif donné pave modulo p ? De façon stupéfiante, il n’y en a aucune ! (cf. [3])
3 Pour p premier, q une puissance de p, F : x ↦→ x p est un automorphisme du corps Fq (et il engendre le groupe de GALOIS
de Fq sur Fp). L’ensemble de ses points fixes est le sous-corps premier Fp. Cela résulte de l’identité (x + y) p = x p + y p en
caractéristique p.
4 Ceci résulte du théorème de LAGRANGE : tout élément de F ∗ 16 a un ordre qui divise 15, et du fait que des éléments d’ordre
inférieur seraient racines d’autres polynômes (ex. 1 + X + X 2 pour les éléments d’ordre 3) qui sont premiers avec J.
5 Sauf peut-être celui des Nibelungen auquel réfère bien sûr le titre de cette section.
6 Ce serait un problème NP, si l’on en croit [12].
8

THÉORÈME 6. (Amiot, avril 2004)
≀ Pour toute partie finie non vide A ⊂ N (contenant 0), pour tout p premier, il existe B ⊂ N, n ∈ N
≀
∗ tel que
(T0,p) A(X) × B(X) ≡ 1 + X + X 2 + . . . X n−1
(mod X n − 1, p)
c’est-à-dire qu’on a cette congruence dans Fp[X].
En termes musicaux, cela signifie que sur chaque temps, le nombre de notes est 1, mais à un multiple
de p près.
J’ai d’abord trouvé ce théorème avec le modulo 2, toujours très particulier (dans ce cas on a même
un pavage compact, c’est-à-dire que la réduction modulo X n − 1 est superflue).
Ainsi avec le motif (0 1 4), qui ne pave certes pas Z, on a un canon avec des notes isolées ou des
accords de trois sons :
FIG. 11 – Un pavage dans F2
EXERCICE 4. Le lecteur est invité à chercher un tel pavage avec le motif (0,1,3) modulo 2.
L’argument clef est un théorème assez simple, mais frappant, de la théorie de GALOIS des corps finis :
THÉORÈME 7. Dans tout Fp[X], tout polynôme A(X) non nul en 0 divise X n − 1, pour n assez grand.
Démonstration. Je ne donne que les grandes lignes (cf. [3]). Toute racine de A(X) ∈ Fp[X] est dans un
corps Fq, extension de Fp de degré fini. Or dans ce corps, Fq, tout élément non nul vérifie xq−1 = 1
(à cause du théorème de LAGRANGE). Avec quelques points techniques (à cause de la multiplicité des
racines), on en déduit un n tel que A(X) | Xn − 1 (un multiple de tous ces q − 1).
Si A(X) représente le motif rythmique, il existe donc n tel que A(X) × (X − 1) | Xn − 1, et donc en
X
posant C(X) =
n − 1
X
on a bien A(X) × C(X) =
A(X)(X − 1) n − 1
A(X)(X − 1) .
Certes, C(X) n’est pas en général un polynôme 0-1 ; mais en caractéristique finie on construit
facilement un polynôme 0-1 B(X), qui soit congru à C(X) modulo Xn − 1 : il suffit de remplacer tout
terme αXk par (α − 1)Xk + Xn+k−1 , α > 1, et d’itérer cette transformation jusqu’à ce qu’il ne reste plus
que des coefficients 0 ou 1, ce qui achève la preuve du théorème.
Ce procédé est constructif : la figure précédente est obtenue par un algorithme qui implémente
précisément cette méthode (en optimisant n, en plus).
Des références sur ce sujet trop peu étudié sont [21] et, plus récent (avec des applications à la
cryptographie), [1].
En conclusion, il n’y a PAS de condition locale (modulo p) pour qu’un motif donné A pave : todos
locos ! il nous faut donc revenir à Z[X]. Au moins, les facteurs de Xn − 1
∈ Z[X] seront simples. . . Pour
l’amour de l’art, observons en effet que la situation est bien moins claire dans les corps finis :
• X n − 1 peut avoir un discriminant nul (i.e. des racines multiples) quand p | n.
• Des polynômes jadis irréductibles (ex. Φ8 = X 4 + 1) sont factorisables dans TOUT corps fini.
• Le produit des X − ξ où ξ parcourt l’ensemble des racines d’ordre multiplicatif donné dans F ∗ q
n’est plus en général un polynôme irréductible dans Fp[X] (cf. [1]).
EXERCICE 5. Trouvez les facteurs irréductibles du produit
≀
(X−ξ) quand ξ décrit l’ensemble des huit générateurs
du groupe (F∗ 16 , ×), (les éléments d’ordre 15). Vous pouvez vous référer utilement au paragraphe sur le
problème de JOHNSON)
J’ai découvert incidemment une propriété étrange et mystérieuse, bien que sa preuve ne soit pas très
difficile, qui permet de calculer la multiplicité de 1 comme racine de A(X) donné :
9
X − 1

PROPOSITION.
≀ Soit A(X) un polynôme 0-1 qui pave avec période n et soit p un facteur premier de A. Alors
≀
• La multiplicité de 1 comme racine de A dans Fp, s’exprime en base p comme un nombre
≀
dont les chiffres sont exclusivement 0 ou p − 1.
≀
• le nombre des chiffres non nuls dans ce nombre en base p n’est autre que la multiplicité
≀
de p comme facteur de A(1) dans N, A(1) étant le nombre de notes du motif A (ceci est
≀
quasiment la condition (T1) de Coven-Meyerowitz).
Curieusement, ce nombre de bits non nuls apparaît pour le calcul de complexité de l’exponentiation
rapide, comme dans le difficile théorème de SMALE sur l’immersion d’une variété sans singularité
dans R n .
5 Retour dans Z[X]
Ce bref passage en caractéristique p nous a permis de toucher à d’autres formes de canons. Il
en existe autant d’espèces que de compositeurs et je ne puis les énumérer toutes ; mentionnons
seulement le résultat remarquable du canadien Jon WILD (2000) : « tout motif de trois notes pave
avec son rétrogradé », qui renvoie à une pratique courante à l’âge baroque.
On revient en caractéristique nulle, avec l’intention de pousser l’utilisation des polynômes 0-1
aussi loin que possible. On parviendra au fait remarquable que la conjecture spectrale en dimension
1 repose sur le cas particulier des canons de VUZA, dont l’intérêt dépasse donc de loin les simples
applications musicales.
Dans le cas de Z[X] les facteurs irréductibles de l’identité (T0) polynomiale sont les polynômes
cyclotomiques Φd, d | n ; les conditions trouvées en 1998 par COVEN - MEYEROWITZ sont cruciales
pour la suite de cet exposé.
J’ajoute ici un lemme qu’ils jugent capital :
Les automorphismes du groupe Zn sont bien connus, ce sont les x ↦→ αx où α ∈ Z ∗ n est un des
éléments inversibles de l’anneau Zn. Il est donc clair que A ⊕ B = Zn ⇐⇒ αA ⊕ αB = Zn. Il est
beaucoup moins évident que l’on a
LEMME 4. A ⊕ B = Zn ⇐⇒ αA ⊕ B = Zn pour tout α inversible modulo n.
COVEN & MEYEROWITZ ([5]) donnent une démonstration de ce lemme « capital » dans un anneau de
polynômes, apparemment insatisfaits de la démonstration originale (combinatoire) de [17].
Mais en fait sa première preuve est dûe à D.T. VUZA 6 ans auparavant. Il en a senti d’ailleurs
la raison profonde, qui est que ψα : z ↦→ z α est un automorphisme du groupe des racines n èmes
de l’unité ; et l’utilise pour une démonstration encore assez compliquée à base de transformée de
FOURIER et convolution.
Ma version consiste à remarquer que changer A en αA revient à changer A(X) en A(X α ), qui est
une bijection sur l’ensemble des polynômes 0-1 considérés modulo X n − 1 (cf. exo), et cela applique
(l’inverse de) ψα aux racines de A(X), et donc les ensembles des racines n èmes de l’unité qui sont
racines de A(X) ne sont pas changées, ce qui signifie que les facteurs cyclotomiques de A(X) sont
invariants dans cette transformation.
Comme ceux de B(X) n’ont pas bougé, on a encore entre A(X α ) et B(X) tous les facteurs cyclotomiques
de (X n − 1)/(X − 1), qui doit donc diviser A(X α ).B(X). Un argument de degré permet de conclure
(on connaît les n − 1 racines de A(X α ).B(X) modulo X n − 1).
EXERCICE 6. Vérifier que changer A(X) en A(X α ) (modulo X n − 1) conserve sa nature de polynôme 0-1.
Je tiens à souligner que ce procédé est musical 7 : par exemple TOM JOHNSON l’a redécouvert tout seul,
expérimentalement !
Bien sûr cette action de groupe permet une description plus économique des canons : par exemple
pour l’ensemble des canons de VUZA de période 72, qui consiste de 3 formes pour A et 6 pour B 8 , il
ne reste qu’une orbite pour chacun et on peut donc déduire facilement tous les canons de VUZA 72
du couple
A = (0, 3, 6, 12, 23, 27, 36, 42, 47, 48, 51, 71) B = (0, 8, 10, 18, 26, 64)
6 Les groupes de Hajós, Dan Tudor Vuza, et les canons rythmiques
irréductibles
7 Alban Berg par exemple l’a utilisé pour transformer des séries dodécaphoniques dans son opéra Lulu.
8 À rotation près.
10

Le mathématicien et musicien roumain DAN TUDOR VUZA a passé près de dix ans, de 1980 à 1990,
à étudier la question des canons rythmiques en long et en large. En particulier, il s’est intéressé aux
canons pour lesquels ni A ni B n’ont de période propre.
Précisons : bien sûr, d’après le théorème de DE BRUIJN (tout canon de motif fini est périodique),
on travaille avec une période globale, n, du canon. Mais il arrive très souvent (presque toujours, en
fait) que l’un des termes de la somme directe A ⊕ B = Zn ait une sous-période. Ainsi pour l’exemple
simple de A = {0, 1, 4, 5} qui pave avec période 8 : c’est en fait le sous-motif {0, 1} répété avec période 4
qui constitue A = {0, 1} ⊕ 4.
MORENO ANDREATTA s’est aperçu que VUZA avait redécouvert des résultats sur les factorisations
des groupes cycliques issus de la conjecture de HAJÒS : un groupe cyclique Zn est de HAJÒS, ou
encore est un « bon groupe », si dans toute factorisation Zn = A ⊕ B, on a A + p = A pour un
certain p < n (ou la même chose pour B). VUZA a caractérisé tous les groupes de HAJÓS cycliques en
utilisant la théorie de FOURIER, suivant une remarque déjà ancienne du grand théoricien LEWIN qui
remarquait qu’une somme directe de parties revient à un produit de convolution de leurs fonctions
caractéristiques.
Cela est décrit en détail dans [4]. Le plus petit « mauvais groupe » est Z72.
On connaît des algorithmes pour fabriquer des canons de VUZA (i.e. des factorisations de « mauvais
» groupes), mais aucun procédé qui assure de les trouver tous. La formule la plus simple est
dûe[11] à FRANK JEDRZEJEWSKI :
PROPOSITION (2003). Si on considère p1, p2 premiers et ni, i = 1..3 tels que n1p1 soit premier avec n2 (et
≀ réciproquement) alors en posant [[ a, b ]] = {a, a + 1, . . . b} on a pour n = p1p2n1n2n3 le canon de VUZA
≀ suivant :
≀
A = n2n3 × ([[ 0, p2 − 1 ]] ⊕ p2n1 × [[ 0, p1 − 1 ]]) B = n1n3 × ([[ 0, p1 − 1 ]] ⊕ p1n2 × [[ 0, p2 − 1 ]])
≀
S = n3(p2n2 × [[ 0, n1 − 1 ]] ⊕ p1n1 × [[ 0, n2 − 1 ]]) R =
≀
[[ 1, n3 − 1 ]] ⊕ B ∪ A
R ⊕ S = Zn
La factorisation de n est générale : si on ne peut ainsi écrire n c’est que Zn est un bon groupe ([18]).
Une autre façon de l’écrire[18] consiste à énumérer les cardinaux des « bons » groupes :
THÉORÈME (HAJÒS, REDEI, DE BRUIJN, SANDS,. . .).
≀ Les « bons groupes » cycliques sont les Zn tels que n s’écrive de l’une des façons suivantes (où p, q, r, s
≀ sont des nombres premiers distincts) :
≀
n = p
≀
α
n = p α q n = p 2 q 2
n = p 2 qr n = pqrs
Remarque : si Zn a un sous-groupe qui est « mauvais », alors on montre que Zn est aussi « mauvais ».
Ces canons sont d’un grand intérêt pour les compositeurs, car ils introduisent une non-répétitivité
dans la régularité (du phénomène globalement périodique), un peu comme la rime en poésie. La notion
a beau être relativement récente, une liste d’outils sur les canons de VUZA est présente dans divers
logiciels d’aide à la composition, comme Open Music développé à l’Ircam, et plusieurs compositeurs
contemporains (GEORGE BLOCH, FABIEN LÉVY) s’en servent dans leurs œuvres. J ’ai par exemple sur
mon piano un morceau très simple de G. BLOCH qui a servi de musique pour une version française
d’un film de Hitchcock. Il nous a expliqué de façon très convaincante pourquoi ces cellules qui se
répètent, mais à intervalles imprévisibles, excellent à faire monter la tension du spectateur/auditeur !
7 Génération de canons
Poussés par le dynamisme des compositeurs, nous avons étudié nombre de transformations sur
les canons rythmiques :
7.1 Plusieurs transformations
7.1.1 Le groupe affine
Le Lemme 4 donne l’exemple même d’une transformation non triviale, qui préserve la notion de
canon sous l’action d’un groupe. Il s’agit ici du groupe affine Aff(Zn). Souvenons-nous en effet que
l’on a convenu d’identifier un motif rythmique A à la classe de tous ses translatés A + m mod n. Le
lemme 1 rajoute les homothéties (de rapport a inversible) et on a donc affaire aux orbites sous les
actions de toutes les bijections x ↦→ ax + b dans Zn. La figure suivante montre une telle orbite et les
canons correspondants : (0, 1, 4, 5) et (0, 3, 4, 7) sont les deux formes du motif modulo 8.
Le lemme 4 prouve que les conditions (T1) et (T2) de [5] sont préservées par une telle transformation.
11

FIG. 12 – Orbite d’un motif sous l’action du groupe affine
Je me suis posé la question de généraliser ce résultat aux autres transformations utilisées par les
musiciens. Les voici :
7.1.2 Zoom/augmentation
Cette transformation revient à dilater le temps et à remplacer une note (ou un silence) par k notes
(ou silences). Illustration :
FIG. 13 – Zoom d’un canon rythmique
Du point de vue polynômial, on change B(X) en B(X k ) et A(X) en A(X k ) × (1 + X + . . . X k−1 ). En
travaillant par récurrence sur les facteurs premiers de k, j’ai pu montrer dans [3] que cette opération
préserve aussi les conditions (T1) et (T2) de [5].
L’ importance particulière de cette opération vient de ce qu’elle permet de fabriquer de nouveaux
« canons de VUZA », à partir d’anciens. On obtient ainsi des canons inédits (non fournis par l’algorithme
de VUZA). Cela a été remarqué par divers chercheurs (Carlos Agon, Thomas Noll) et notamment
par Harald FRIPER TINGER de l’université de Graz, qui a donné des formules remarquables
de dénombrement des canons rythmiques et s’est lancé à la recherche de tous les canons de VUZA
de « petite » taille. Une amicale compétition (il a gagné) nous a permis en 2003-2004 de trouver
force nouveaux canons de période 108, 120 ou 144. Au colloque de Graz[7] en mai 2004, Harald
a fait sensation en montrant que tous les canons de VUZA que nous avions trouvés pour les deux
premières périodes, 72 et 108, étaient en vérité les seuls possibles. Pour cela il s’est appuyé à la
fois sur des algorithmes habilement conçus par ses soins, des actions de groupes et dénombrements
d’orbites à coups d’équation aux classes, et de la combinatoire (théorie de POLYA). Incidemment, les
canons de VUZA apparaissent comme un matériau exceptionnellement rare (probabilité inférieure au
1 millionième), ce qui a son intérêt pour la suite.
7.1.3 Concaténation
L’opération de concaténation est très simple, elle consiste à répéter un même motif à la queueleu-leu
plusieurs fois :
FIG. 14 – Un canon rythmique répété
Je suis redevable à H. FRIPER TINGER pour m’avoir fait comprendre l’importance théorique de cette
opération si simple. Elle lui a permis [8] de donner des formules exactes pour dénombrer les canons
dans les « bons groupes », puisque par définition même un des facteurs d’une décomposition en
somme directe d’un tel groupe est concaténé {0, 1} ⊕ {0, 2} → {0, 1, 4, 5} ⊕ {0, 2}. d’un motif plus court.
Ceci permet de proche en proche de trouver tous les canons de période donnée, à condition d’éviter
les périodes fatidiques 72,108,120 , etc. . .
12

Cette opération consiste, polynômialement, à multiplier A(X) par ce que j’appelle un métronome :
DÉFINITION 4. Un métronome est un motif de la forme
≀
A = (0, k, 2k, 3k, . . . (m − 1)k) i.e. A(X) = 1 + X
≀
k + X 2k + . . . X (m−1)k = Xmk − 1
Xk
=
− 1
Il en résulte assez facilement (cf. [3]), travaillant avec m premier sans perte de généralité, le
d|k et d|k
THÉORÈME 9. Si A s’obtient par concaténation de Ã, pavant avec B, alors l’un vérifie (T2) si et seulement si
≀ l’autre vérifie aussi (T2).
7.1.4 Entrelacement et équirépartition
Ce procédé n’est pas (encore) connu des musiciens mais je ne doute pas qu’ils en fassent bientôt
leurs choux gras. Je l’ai découvert dans un article récent et très général (de KOLOUNTZAKIS et MA-
TOLCSI, [12]) (dans un contexte d’algèbre commutative), mais il s’avère que vu sous un autre angle,
c’est un outil essentiel pour le dernier théorème de [5], le plus difficile. Je reformule ensemble ces
différents résultats. La démonstration n’en est pas très difficile (le lecteur courageux pourra s’y essayer)
:
THÉORÈME 10. Soient A1, . . . Ak des motifs
≀
qui pavent un canon rythmique avec un MÊME B. Alors on obtient
≀ un canon rythmique en posant A = (i + kAi), qui pave avec kB.
≀
i=0...k−1
≀
Réciproquement, un tel canon est caractérisé par le fait que l’« outer rhythm » B est multiple d’une
≀ constante k > 1 : B ⊂ kZ, ou, de manière équivalente, par le fait que A est équiréparti modulo k : les
≀ ensembles
≀
Ai ≀
= A ∩ (i + kZ)
≀ ont tous même cardinal, et pavent avec un même B = B/k.
On peut ainsi fabriquer de nouveaux canons de VUZA, par exemple en prenant pour les Ai (une partie
de) l’orbite d’un motif sous le groupe affine. De façon bien plus remarquable, tous les canons VUZA
recensés (ceux obtenus par algorithme et les autres) peuvent être fabriqués par ce procédé à partir
de canons plus petits, qu’ils soient de VUZA ou pas.
La figure ci-dessous illustre la genèse d’un canon de période 72. On y reconnaît (cf. la voix
supérieure) les deux motifs génériques qui sont dilatés et s’entrelacent pour donner le motif final.
FIG. 15 – Un canon de Vuza 72 comme entrelacement de deux canons 36
7.2 Réduction des canons rythmiques
Les canons de VUZA sont finalement assez similaires aux nombres premiers, au sens où ils sont
« irréductibles », et engendrent tous les autres canons ; en effet, résulte de leur définition le :
THÉORÈME 11. On peut réduire récursivement tout canon par déconcaténation – l’opération inverse de la
≀ concaténation – appliquée à l’un des deux termes A ou B, soit à un canon de VUZA, soit au canon trivial
≀ ({0} ⊕ {0}).
Nous disposons finalement de diverses transformations, dont plusieurs (zoom, concaténation, entrelacement)
changent la période ; toutes ces opérations conservent les conditions (T1), (T2) de COVEN-
MEYEROWITZ et il est temps de mentionner le lien avec la conjecture spectrale.
8 La conjecture de Fuglede
8.1 La conjecture spectrale
Dans le cas le plus général, la conjecture publiée par FUGLEDE en 1974 ([13]) est une question qui
relie la géométrie et l’analyse harmonique :
13
Φd

. Un domaine K (compact d’intérieur non vide) pave R
≀
n ,c’est-à-dire qu’il existe B ⊂ Rn tel que
b + K = R
b∈B
n ◦ ◦
et b + K ∩ b ′ + K = ∅ pour b = b ′
si et seulement si K possède une base hilbertienne, i.e. il existe une famille Λ telle que (e2iπλ )λ∈Λ est une
famille orthonormée qui engendre une partie dense de L2 (K).
FUGLEDE a prouvé sa conjecture dans le cas où B est un réseau (B ≈ Z n ). On comprend mieux cette
conjecture dans le cas simple où K est par exemple l’ hypercube unité : il suffit alors de prendre pour
K le réseau Z n , c’est la théorie de la décomposition en série de FOURIER.
Jusqu’à l’été 2003, tous les résultats publiés sont allés dans le sens de la confirmation de cette
conjecture. Elle est vraie en particulier pour tous les K assez réguliers (les convexes plans, par
exemple). Comme il s’agit d’orthogonalité dans R n , on a assez vite établi une condition équivalente en
terme d’existence d’une matrice de HADAMARD douée de propriétés adéquates.
C’est en exhibant des matrices de HADAMARD complexes que TERENCE TAO a finalement prouvé
que cette conjecture est fausse en dimension 5. Depuis on a trouvé des contre-exemples dans
les deux sens, dont la dimension est descendue à 3 [12]. Mais cela reste un problème ouvert en
dimension 1 malgré une kyrielle de résultats partiels (cf. [13]). Un des plus récents concerne les
produits de métronomes, au sens de la définition ci-dessus ([14]). Il est en partie contenu dans le
théorème que je démontre ci-dessous.
Notons le lien trivial entre pavages de R et pavages de Z : si A pave Z alors A + [0, 1[ pave R. La
réciproque est moins triviale mais résulte des travaux de VUZA et de façon très différente de LAGARIAS
& WANG[15].
8.2 La conjecture spectrale en dimension 1
IZABELLA LABA, lisant l’article de COVEN-MEYEROWITZ, a rapidement compris qu’on pouvait en
tirer une connexion à la conjecture spectrale. Elle publia peu après (2000) [13] le résultat suivant :
PROPOSITION. (T1) + (T2) ⇒ spectral (et spectral ⇒ (T1)).
Cela utilise des calculs élémentaires, et le lemme suivant qui caractérise le caractère spectral en dimension
1 (on peut le prendre comme définition) :
LEMME 5. A pave Z si et seulement si il existe une famille 0 = λ0 < λ1 < . . . λk−1, k = |A| = A(1) telle que les e
≀
2iπλj
soient racines de A(X).
LABA prend tout simplement des λj de la forme i/p α , i = 0 . . . p−1, pour montrer que si (T2) est vérifiée
alors A est spectral.
REMARQUE 4. Il ne faut pas croire que le caractère 0-1 du polynôme A(X) oblige ses racines de module 1 a être
≀ d’ordre fini dans S
≀
1 . En d’autres termes, théoriquement un spectre peut très bien exister et être irrationnel :
EXERCICE 7. Trouver un polynôme 0-1 ayant des racines de module 1 d’ordre infini.
D’après ce résultat de LABA, si un motif rythmique pave mais qu’il n’est pas spectral, il ne peut
vérifier (T2). Mais si l’on en croit le principe de réduction énoncé au théorème 11, cela ne peut se
produire que dans un groupe non-HAJÓS. En effet, tout pavage dans un « bon groupe » se réduit
récursivement à des canons plus petits dans des sous-groupes, qui sont donc encore « bons », et
donc on peut encore réduire jusqu’à tomber sur le canon trivial à une note, (0) ⊕ (0). Qui vérifie
la condition (T2) ( !). Par conservation d’icelle dans le procédé de concaténation des canons (voir le
théorème 9), on déduit
THÉORÈME (AMIOT, JUIN 2004).
≀ Si un canon n’est pas spectral, alors il peut se réduire par déconcaténation à un canon de VUZA,
≀ lui aussi non spectral. A fortiori, si n a l’une des formes suivantes (p, q, r, s étant des nombres premiers
≀ distincts) :
≀
n = p
≀
α
n = p α q n = p 2 q 2
n = p 2 qr n = pqrs
alors tout motif d’un canon rythmique de période n est spectral.
On peut même aller plus loin en prenant au lieu de la période n la taille du motif (= le nombre de
notes = A(1)). Les trois premiers cas avec deux facteurs premiers résultent du théorème (B2) de [5],
les deux derniers sont nouveaux à ma connaissance.
14

Ce résultat s’applique aussi aux pavages d’un intervalle d’entiers, que j’appelle canons compacts
(ceux pour lesquels il est inutile de procéder à une réduction modulo n car on a A ⊕ B = [[ 0, n − 1 ]]
dans N, par exemple {0, 1, 4, 5} ⊕ {0, 2} = {0, 1, 2, 3, 4, 5, 6, 7}), car d’après un résultat ancien de DE BRUIJN
ils sont tous déconcaténables.
COROLLAIRE 2. En conséquence de l’énumération par FRIPER TINGER des canons de VUZA pour n = 72, 108 nous
≀ savons que tous les canons de période 108 (et même jusqu’à 119) sont spectraux (au sens où aussi bien
≀ A, le « inner rhythm », que le « outer rhythm » B, sont spectraux).
Ceci suggère une idée assez intuitive, à savoir que s’il existe un motif A qui pave sans vérifier (T2),
alors A, n doivent être grands. Jusqu’ici, tous les algorithmes qui fabriquent des canons de VUZA
assurent que (T2) est vérifiée, et tous les procédés d’augmentation de taille des canons vus ci-dessus
conservent cette propriété : on ne sait donc vraiment pas comment construire un éventuel canon
de VUZA qui nie la propriété (T2), ce qui ne prouve pas qu’il n’en existe pas puisqu’on ne sait pas
comment les construire tous. . .
8.3 Réduction par équirépartition
Il ne paraissait pas impossible d’espérer réduire TOUS les canons de VUZA, et donc de démontrer
le sens (pave ⇒ spectral) de la conjecture de FUGLEDE.
En effet l’algorithme de VUZA fabrique toujours un second membre multiple de n3 (cf. la formule
de JEDRZEJEWSKI). Dans ce cas on a équirépartition de l’inner rhythm modulo n3 (cf. [5], lemme
2.5). Le groupe affine préserve d’ailleurs cette condition d’équirépartition.. Mais elle signifie que l’on
peut réduire un tel canon à un canon plus petit, en préservant la condition (T2). Si donc il s’avérait
que TOUT canon de VUZA ait, à l’instar de ceux que l’on sait fabriquer, un facteur équiréparti, la
méthode de réduction (utilisant dualité, concaténation, équirépartition selon le cas) permettrait de
réduire TOUT canon au canon trivial. Ce qui démontrerait la condition (T2) pour tout canon, ce dont
on pourrait déduire (un sens au moins de) la conjecture de FUGLEDE.
8.4 Le cimetière des conjectures
Mais le champ de bataille des canons rythmiques est jonché des cadavres de nombreuses conjectures.
. .À commencer, historiquement au tout début, par la conjecture de DE BRUIJN : si un groupe
abélien fini est somme directe de A et B alors l’un des deux est périodique, tuée dans l’œuf par les canons
de VUZA et bien avant cela, par les contre-exemples de REDEI, HAJÒS, DE BRUIJN et consorts ;
La conjecture de FUGLEDE a certes tenu 31 ans avant de connaître son premier contre-exemple –
mais il est vrai qu’elle aura été prouvée dans nombre de cas particuliers ; au contraire, brèves auront
été la vie de celle de TIJDEMAN 1996 (si ppcm(A)=1 alors il existe un nombre premier tel que B ⊂ pZ,
tuée par SZABÓ), ou de celle de LAGARIAS & WANG (si T pave avec les compléments T1, . . . Tn alors ils
sont spectraux et de même spectre) qui fut victime de KOLOUNTZAKIS & MATOLCSI en juin 2004[12].
On ignore actuellement si la conjecture que COVEN ET MEYEROWITZ se sont soigneusement retenus
d’énoncer (pave ⇒ (T2)) est prouvable ; elle est logiquement plus forte que le sens (pave ⇒ spectral)
de celle de FUGLEDE, d’après LABA. Les résultats en sens inverse sont encore peu nombreux (si A est
spectral alors ?. . .), à part [14] qui utilise des métronomes c’est-à-dire des canons très simples, et il
est difficile de se faire une opinion sur cette direction.
Mais je finirai cette hécatombe par l’extermination de ma propre conjecture. En effet, la construction
mentionnée par [5] du hongrois SZABÓ dans [19] réfute au départ une conjecture de SANDS,
proche de celle de TIJDEMAN. Mais il s’avère qu’elle donne, incidemment, un canon de VUZA, qui
n’est donc par définition pas réductible par dé-concaténation, et par construction pas réductible par
équirépartition ! Signalons tout de même que le plus petit contre-exemple donné par cette méthode,
que j’ai implémenté avec Mathematica TM , est de période 30030, ce qui explique qu’il n’ait pas sauté aux
yeux. De plus la méthode de construction est particulièrement perfide, même si elle n’est pas sans
rappeler certains procédés de construction des canons de VUZA ; j’en donne ici l’idée très simplifiée :
En hommage au théorème de DE BRUIJN intitulé ’on British Number systems’, j’emprunterai une
métaphore pécuniaire. On considère un ensemble de pièces et de billets qui permettent de payer
exactement n’importe quelle somme (< n). On prend pour A la somme des « pièces jaunes », et pour
B ′ les sommes de « gros billets ». De façon encore plus imagée, A contient les unités et B ′ les dizaines,
et A ⊕ B ′ = [[ 0, n − 1 ]]. Dans l’exemple donné plus bas, B ′ = {0, 30, 60, . . . 30k, . . .}.
L’idée hongroise consiste alors à perturber B ′ en une nouvelle partie B, en modifiant certains
éléments, choisis exprès irrégulièrement, par l’ajout d’un élément variable de A, tout cela variant
circulairement ; ceci ne modifie pas le fait que A ⊕ B = Zn mais cela rend B plus irrégulier. Avec
certaines conditions techniques (cf. [19]) on montre que B (ainsi que, plus trivialement, A) engendre
Zn et en conséquence, qu’il n’est pas contenu dans un sous-groupe strict k Zn : donc pas de réduction
possible par équirépartition. De plus, la méthode employée assure qu’on a affaire à deux facteurs A, B
apériodiques, et donc à un canon de VUZA. . . Le plus petit que j’ai pu construire de cette façon est de
15

période 900 :
A = (0, 36, 72, 100, 108, 136, 144, 172, 200, 208, 225, 236, 244, 261, 272, 297, 308, 325, 333, 344, 361, 369, 397, 425, 433, 461, 469, 497, 5
B = (0, 30, 60, 90, 156, 210, 240, 250, 270, 330, 336, 360, 390, 405, 420, 510, 516, 540, 550, 570, 600, 690, 696, 720, 780, 810, 850, 855, 8
9 Coda
9.1 Stretta
Nous pouvons nous croire arrivés bien loin de Frère Jacques et de son petit canon à quatre voix.
Mais les contre-exemples obtenus par des arguments sophistiqués à ces conjectures mathématiques
pointues permettent de mettre en évidence des objets musicaux doués de propriétés intéressantes,
qui vont certainement faire leur apparition dans des partitions prochaines : cela a déjà été le cas par
le passé, particulièrement avec les canons de VUZA mais aussi dans bien des domaines – il n’est pas
nouveau que des idées mathématiques servent, consciemment ou non, l’inspiration de musiciens.
En retour, et de façon bien plus novatrice, on peut espérer que les suggestions des compositeurs
continuent, comme elles ont commencé de le faire, à éclairer la recherche mathématique de leurs
idées spécifiques. L’étonnante fécondité de cette irruption de la musique dans les mathématiques
s’explique à mon avis par la fringale des mathématiciens pour les concepts nouveaux, qui ont toujours
servi spectaculairement l’avancement de notre science : bien des outils mathématiques aujourd’hui
banals sont issus de la physique, bien sûr, mais aussi de la biologie, de l’économie, etc. . .
Prophétisons que le temps est venu de reformer et d’élargir le Quadrivium antique, la musique reprenant
avec les autres son statut de pilier de la connaissance.
9.2 Solutions des exercices
9.2.1 Canon de période 2 × ℓ(A)
Il suffit de prendre A = (0, n − 1) qui pave avec B = (0, 1, . . . , n − 1) mais pas moins.
9.2.2 Perfect square tilings
Partant de :
X
i∈I
kiΦ3(Xi ) = 1 + X + . . . = 3n−1
X
j=0
j = X3n − 1
X − 1 on pose X = j = e2iπ/3 : les indices i
multiples de 3 sont caractérisés par le fait que Φ3(ji ) = Φ3(1) = 3 = 0. Pour tout autre indice on aura
Φ3(ji ) = 0 – c’est la relation classique 1 + j + j2 = 0. Il reste donc une somme des jki , i ∈ 3N qui doit
valoir 0. Or la plus courte somme valant 0 est encore 1 + j + j2 = 0, ce qui impose qu’il y ait 0 ou 3 (ou
6, ou 9. . .) multiples de trois parmi les indices i. Cela explique pourquoi T3 n’apparaît pas sans T6 et
T9.
9.2.3 (1, 4, 9, 16) pave-t-il ?
Non. On a bien deux facteurs cyclotomiques, d’indices 2 et 10, mais la condition (T1) n’est pas
vérifiée, sans parler de (T2).
X 16 + X 9 + X 4 + X = X × (1 + X) × (1 − X + X 2 − X 3 + X 4 ) × (1 + X 3 − X 5 + X10)
(il faut un logiciel de calcul formel si on veut factoriser sans efforts ; en revanche on liste facilement
les polynômes cyclotomiques du style Φpα = 1 + Xpα−1 + X2pα−1 + . . . et on teste s’ils sont diviseurs).
9.2.4 Pavage modulo 2
La plus petite solution pour paver modulo 2 avec A = (0, 1, 3) est B = (0, 2, 3). En effet A + B =
{0, 1, 2, 3, 3, 3, 4, 5, 6}.
FIG. 16 – pavage modulo 2 avec (0,1,3)
9.2.5 Éléments d’ordre 15 dans le corps à 16 éléments
Dans la discussion du problème de JOHNSON, on a vu que le polynôme (irréductible sur F2[X])
J(X) = 1 + X + X 4 a 4 racines d’ordre 15 dans F16. Leurs inverses sont aussi d’ordre 15, elles sont
racines du polynôme réciproque 1 + X 3 + X 4 . On a alors Φ(15) = 8 éléments d’ordre 15, on n’en
trouvera pas plus (Φ désignant la fonction d’EULER).
16

9.2.6 Transformation affine
Si α est premier avec n, alors l’application k ↦→ αk mod est une bijection. Donc A et αA sont en
correspondance bijective, et leurs polynômes associés sont bien 0-1.
9.2.7 Nombres algébriques de module 1 non racines de l’unité
Ma plus petite solution est A(x) = 1 + x + x 3 + x 5 + x 6 , dont les quatre racines non réelles peuvent
être exprimées par radicaux (poser y = x + 1/x) et sont de module 1, mais ce ne sont pas des racines
de l’unité (sinon A(x) aurait un facteur cyclotomique).
Références
[1] Al Fakir, S., Algèbre et théorie des nombres, T2, Ellipses 2004.
[2] 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.
[3] Amiot, E., Rhythmic canons and Galois theory, Grazer Math. Ber., 347 (2005), 1–25.
[4] Andreatta, M., On group-theoretical methods applied to music : some compositional and implementational
aspects, in : E. Lluis-Puebla, G. Mazzola et T. Noll (eds.), Perspectives of Mathematical
and Computer-Aided Music Theory, EpOs, 122–162, Universität Osnabrück, 2004.
[5] Coven, E., and Meyerowitz, A. Tiling the integers with one finite set, in : J. Alg., 212 :161-174,
1999.
[6] DeBruijn, N.G., On Number Systems, Nieuw. Arch. Wisk. (3) 4, 1956, 15–17.
[7] Fripertinger, H. Remarks on Rhythmical Canons, Grazer Math. Ber., 347 (2005), 55–68.
[8] Fripertinger, H. Tiling problems in music theory, in : E. Lluis-Puebla, G. Mazzola et T. Noll (eds.),
Perspectives of Mathematical and Computer-Aided Music Theory, EpOs, 149–164, Universität Osnabrück,
2004.
[9] Hajós, G., Sur les factorisations des groupes abéliens, in : Casopsis Pest. Mat. Fys.,74 :157-
162,1954.
[10] Johnson, T., Tiling The Line, proceedings of J.I.M., Royan, 2001.
[11] Jedrzejewski, F., A simple way to compute Vuza canons, MaMuX seminar, January 2004,
http ://www.ircam.fr/equipes/repmus/mamux/.
[12] Kolountzakis, M. & Matolcsi, M., Complex Hadamard Matrices and the spectral set conjecture ,
http ://arxiv.org/abs/math.CA/0411512.
[13] Laba, I., The spectral set conjecture and multiplicative properties of roots of polynomials, J. London
Math. Soc. 65 (2002), 661-671.
[14] Laba, I., and Konyagin, S., Spectra of certain types of polynomials and tiling of integers with
translates of finite sets, J. Num. Th. 103 (2003), no. 2, 267-280.
[15] Lagarias, J., and Wang, Y. Tiling the line with translates of one tile, in : Inv. Math., 124 :341-365,
1996.
[16] Tangian, A., The Sieve of Eratosthene for Diophantine Equations in Integer Polynomials and Johnson’s
problem, disc. paper N ◦ 309 Fern Universität Hagen.
[17] Tijdeman, R., Decomposition of the Integers as a direct sum of two subsets, in : Séminaire de
théorie des nombres de Paris, 3D, p.261-276, Cambridge U.P, 1995.
[18] Sands, A.D., The Factorization of abelian groups, Quart. J. Math. Oxford, 10(2) :45–54.
[19] Szabó, S., A type of factorization of finite abelian groups, Discrete Math. 54 (1985), 121–124.
[20] Vuza, D.T., Supplementary Sets and Regular Complementary Unending Canons, in four parts in :
Canons. Persp. of New Music, nos 29(2) pp.22-49 ; 30(1), pp. 184-207 ; 30(2), pp. 102-125 ; 31(1),
pp. 270-305, 1991-1992.
[21] Warusfel, Structures Algébriques finies, Classiques Hachette, 1971.
[22] Wild, J., Tessellating the chromatic, Perspectives of New Music, 2002.
[23] Midi files for the illustrations in this paper, as part of my website dedicated to rhythmic canons,
at http ://canonsrythmiques.free.fr/midiFiles/, 2004.
17

Journal of Mathematics and Music
Vol. 01, No. 03, December 2007, 1–21
David Lewin and Maximally Even Sets
Emmanuel AMIOT
1 rue du Centre, F 66570 St NAZAIRE, France
(v1.0.0 released june 2007)
David Lewin originated an impressive number of new ideas in musical formalized analysis. This paper formally proves and expands one
of the numerous innovative ideas issued by Ian Quinn in his dissertation [17], to the import that Lewin might have invented the much
later notion of Maximally Even Sets with but a small extension of his very first published idea, where he made use of Discrete Fourier
Transform (DFT) for investigating the intervallic differences between two pc-sets. Many aspects of Maximally Even Sets (ME sets) and,
more generally, of generated scales, appear obvious from this original starting point, which would deserve in our opinion to become
standard. In order to vindicate this opinion, we develop a complete classification of ME sets starting from this new definition. As a
pleasant by-product we mention a neat proof of the hexachord theorem, which might have been the motivation for Lewin’s use of DFT
in pc-sets in the first place. The nice inclusion property between a ME set and its complement (up to translation) is also developed, as
it occurs in actual music.
Keywords: Maximally Even Sets, Discrete Fourier Transform, David Lewin.
Notations : the cyclic group of order c is Zc. It models a chromatic universe with c pitch classes, and it
is as usual pictured as a regular polygon on the unit circle. In most actual examples c will be equal to 12.
x | y means the integer x divides y.
For the sake of readability we generally use the same notation for integers and their residue classes, the
context usually making clear whether a computation occurs in Z or in Zc.
The greatest common divisor of x, y is denoted by gcd(x, y).
We will use indiscriminately ‘Fourier transform’, ‘Discrete Fourier Transform’, or ‘DFT’.
The bracket notation is for the floor function.
The symbol X ⊕ Y means ‘all possible sums of an element of X and an element of Y ’, each result being
obtained in a unique way.
1 Fourier Transform of pc-sets
Part of our claim that Fourier Transforms provide the best way to define Maximally Even Sets relies on the
high musical significance of the DFT of pc-sets in general. This was salient in [17] for the special pc-sets
that Quinn collected as ‘prototypes’, among which the ME sets; and it was confirmed since by many other
cases. We thus feel it important to spend some time on the general DFT of pc-sets before turning to the
main topic, that is its application to ME sets proper.
1.1 History
In a short paper ( [13]), D. Lewin investigated intervallic relationships between two ‘note collections’ and
proved that, except in several listed exceptional cases, the interval function between the ‘note collections’
enables to reconstruct one from the other. He cursorily motivates the five exceptional cases by a final note,
wherein he puts forward that
(1) the interval function is a convolution product (of characteristic functions),
(2) the Fourier transform of such a product is the ordinary product of Fourier transforms.
Professor in Class Preps, Perpignan, France. Email: manu.amiot@free.fr
Journal of Mathematics and Music
ISSN 1745-9737 print / ISSN 1745-9745 online c○ 2007 Taylor & Francis Ltd.
http://www.tandf.co.uk/journals
DOI: 10.1080/17459730xxxxxxxxx

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},
the characteristic fuction of X as 1X(t) =
1
0
if t ∈ X
, IF unc appears immediately as the convolution
if t /∈ X
product of the characteristic functions of −A and B:
1−A ⋆ 1B : t ↦→
1−A(k)1B(t − k) =
1A(k)1B(t + k) = IF unc(A, B)(t)
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
transform of a convolution product,
k∈Zc
F(IF unc(A, B)) = F(1−A) × F(1B)
where F(f) stands for the discrete Fourier transform of a map f.
We will not quote the formula given by Lewin himself, as it is hardly understandable: his notations are
undefined and the computations extremely cursory. Of course this is not for lack of rigor: as the following
quotation suggests, Lewin did not really hope to be understood when making use of mathematics.
The mathematical reasoning by which I arrived at this result is not communicable to a reader who does not
have considerable mathematical training. For those who have such a training, I append a sketch of the proof :
consider the group algebra [. . . ] [13]
Reading Lewin’s paper gives one a strong feeling that he wrote as little as possible on the mathematical
tools that underlay his results. Indeed, what little he mentioned did rouse some readers to righteous ire in
the next issue of JMT.
Nowadays such a ‘considerable mathematical training’ will be considered basic by many readers of this
journal; for instance D.T. Vuza made an essential use of the equation above in the 80’s in the course of
his seminal work about rhythmic canons (see [21], lemma 1.9 sqq), wherein he stressed the importance of
Lewin’s use of DFT of characteristic functions.
And as we will endeavour to prove, this approach enables to define ME sets (in equal temperament) in
a way perhaps more suggestive and even intuitive, than historical/usual definitions.
1.2 A quick summary of Fourier transforms of subsets of Zc
1.2.1 First moves.
Definition 1.1 Following Lewin, we will define the Fourier transform of a pc-set A ∈ Zc as the Fourier
transform of its characteristic function 1A:
FA = F(1A) : t ↦→
e −2iπkt/c
The values FA(t), t ∈ Zc, are the Fourier coefficients.
1A is a map from Zc to C, whose DFT is well defined for t mod c, as FA(t + c) = FA(t).
The DFT of a single pc a is a single exponential function t ↦→ e−2iπat/c , the DFT of the whole chromatic
c−1
scale is FZc (t) = e−2iπkt/c = 0 for all t ∈ Zc except t = 0.
k=0
But FA + FZc\A = FZc , hence
k∈A

alternative title 3
Lemma 1.2 The Fourier transforms of a pc-set A and of its complement Zc \ A have opposite values,
except when t = 0:
∀t ∈ Zc, t = 0, F Zc\A(t) = −FA(t)
Furthermore, we get F Zc\A(0) = FA(0) if and only if Card A = c/2, as
Lemma 1.3 The Fourier transform of A in 0 is equal to the cardinality of A: FA(0) = Card A.
For other coefficients, taking into account lemma 1.2 and the triangular inequality one gets
Lemma 1.4 ∀ t ∈ Zc, t = 0 ⇒ |FA(t)| ≤ min(d, c − d).
The DFT FA characterizes the pc-set A, by the following identity (Inverse Fourier transform)
1A(t) = 1
c
k∈Zc
e +2ik tπ/c FA(k)
easily derived from the definition of FA. Thus the DFT yields the same information as the pc-set, but in
a form that stresses musically relevant concepts. More precisely, there is preservation of the absolute value
of DFT under all usual 1 musical transformations. For instance,
Theorem 1.5 The length of the Fourier transform, i.e. the map |FA| : t ↦→ |FA(t)|, is invariant by
(musical) transposition or inversion of the pc-set A. More precisely, for any p, t ∈ Zc
• FA+p(t) = e −2ipπt/c FA(t) (invariance under transposition)
• F−A(t) = FA(t) (invariance under inversion)
and also under complementation (except in 0 when Card A = c/2).
Let us say that A, B are Lewin-related when maps |FA| and |FB| are identical. It is the case whenever
A, B are exchanged by the T/I group of musical transformations, but the reverse is not true (see below).
All the same, the map |FA| appears to be a very good snapshot of the relevant musical information of a
given pc-set: by dropping the information of the phase of the Fourier coefficients and retaining only the
absolute value, we seem to keep the best part, in a way reminiscent of the Helmoltzian approach of sound,
which showed that the phase of a sine wave can (mostly) be neglected, as the frequency is the part that
generates the perception of pitch. This strongly vindicates and to some measure extends Quinn’s ( [17])
notion of ‘chord quality’, which appears in the last section of his dissertation with a value that is precisely
|FA(d)|, d = Card A, and is measured in ‘lewins’.
As as nice application of these invariance properties, we may characterize periodic subsets:
Proposition 1.6 A ⊂ Zc is periodic , meaning A + τ = A for some τ, if and only if FA(t) = 0 except
when t belongs to some subgroup of Zc.
The proof is left to the reader (see also Supplementary II online).
Remark 1
• Some may well claim this proposition is obvious: a subset A ∈ Zc is the set of residues of a periodic
set A ⊂ Z, with period c. This periodicity means precisely that 1A (or 1 b A , with the same formula) can
be expressed as a combination of c exponential functions, the t ↦→ e 2iπ k t/c : this is the inverse Fourier
transform formula and the very reason Fourier transform works. The existence of a smaller period m | c
means that m exponentials functions only are sufficient, e.g the t ↦→ e 2iπ k t/m .
• In Z12, the octatonic scale (0 1 3 4 6 7 9 10) is an interesting example of such a periodic subset. Its
group of periods is 3 Z12. Periodic subsets of Z12 are well known as Messiaen’s Modes à Transposition
Limitées.
1 Less usual transformations, like t ↦→ 7t mod 12, permute the Fourier coefficients.

4 title on some pages
1.2.2 DFT and intervallic content. The following theorem is based on the idea to interpret the multiplicities
of pc intervals within a pc-set A as complex numbers (such as we did with the values 0 and
1 of the characteristic functions 1A). The interval content is treated as function from Zc to the complex
numbers and is defined on the c (oriented) possible intervals 1 .
Theorem 1.7 (Lewin’s Lemma)
Define the interval content of a subset A ∈ Zc as
ICA(k) = IF unc(A, A)(k) = Card{(i, j) ∈ A 2 , i − j = k}
Then the DFT of the intervallic content is equal to the square of the length of the DFT of the set:
F(ICA) = |FA| 2
Proof Let A be a pc-set; as Lewin observed (for the more general interval function between two subsets),
the ‘intervallic function’ from pc-set A to itself is 2 the convolution product
ICA = 1−A ⋆ 1A
But as we recalled earlier, the Fourier transform of a convolution product is the ordinary product of Fourier
transforms, i.e. (using last part of theorem 1.5)
F(ICA) = FA × F−A = FA × FA = |FA| 2
Note that the Fourier transform of any IC is a real positive valued function, an uncommon occurence
among DFT of integer-valued functions 3 . Now we see that the Lewin relation is the equivalence closure
of the Z-relation:
Proposition 1.8 A, B ⊂ Zc are Lewin-related (|FA| = |FB|) if and only if they share the same interval
content.
The equivalence stands because |FA| holds all the information about ICA by inverse Fourier transform 4
– this case follows directly here from theorem 1.5.
From there we also get a very short proof of the hexachord theorem, one of the most striking mathematical
results in music theory.
At the time he issued his first paper, Lewin had come to work with Milton Babbitt, who was trying to
prove the hexachord theorem (see fig. in Supplementary II online):
Theorem 1.9 If two hexachords (i.e. 6 notes subsets of Z12) are complementary pc-sets in Z12, then they
have the same intervallic content (same numbers of same intervals).
A simple derivation of this theorem in Zc for any even c ensues from the elementary properties of DFT
already listed:
1 Usually, textbooks define interval content for T/I-classes of intervals.
2 This relation has been quoted, in musical context, by several authors: for [21], it might be the most important single contribution
by David Lewin: ”It is therefore my conviction that in the near future music theory will integrate convolution and Fourier transform as
effective investigation tools, music theorists being able to use them in the same way as presently they make use of groups, homomorphisms,
group actions, and so forth;” ; it also appears for instance in the recent [16].
3 The DFT of a real valued function is non real in general, it only verifies F(f)(−t) = F(f)(t).
4 Please note that we endeavour here to define a true equivalence relation, contrarily to the Fortean tradition which excludes the ‘easy
case’, when A, B are T/I related. This traditional position is weird; another argument against it is that some classes of ‘Z-related’ chords
are indeed exchanged through action of a larger group than T/I, like the two famous all-intervals (0 1 4 6) and (0 1 3 6) in Z12, which
are affine-related (see [20], pp.102 sqq)– and this is a general situation, as any affine transform of an all-interval set will be Z-related.
Jon Wild pointed out to me that the reverse is false.

alternative title 5
Proof If A ∈ Zc has c/2 elements, then as mentioned above, F Zc\A = −FA. So
F(ICA) = |FA| 2 = |F Zc\A| 2 = F(IC Zc\A) Hence (by inverse DFT) ICA = IC Zc\A.
As far as I know, this short proof was first published in [1] after I mentioned it during the J. Clough
memorial days in july 2005. But considering the coincidence in time of Lewin’s first paper and his meeting
with Babbitt, it is almost certain that he was aware of it. Perhaps the harsh reactions to the mathematics
in his first paper explain why he did not publish it. It is left to the reader, as a good and entertaining
exercise, to prove in the same way the Generalized Hexachord Theorem, as expounded in [18], [20], [16]
and many others.
2 Maximally Even Sets and their Fourier Transforms
The attribute ‘maximally even’ applies to pitch class sets, which — in comparison to all pitch class sets of
the same cardinality — are as evenly as possible distributed within Zc. This is obviously the case for totally
regular sets, which exist only for cardinalities d dividing the number c of pitch classes. The opposite special
case — where d and c are mutually coprime — was well studied in [8]. The point of departure for the
extensive study of the general case in [7] is an explicit construction of generalized diatonic sets in [8]. The
formula for this construction was later termed J-function. It departs from the arithmetic series of rational
numbers 0, c
c
, ..., (d − 1)
d d and ‘digitizes’ them within Zc in terms of the residue classes of the floor-values
of these ratios mod c: 0, c c
, ..., (d − 1) mod c. The J-function includes a translation parameter α:
d
d
J α c,d : k ↦→ kc + α
, k = 0 . . . d − 1.
d
In this section we accomplish the theory of maximally even sets with an alternative definition via Fourier-
Coefficients and derive the main known results directly from this definition. Our ‘Lewinesque’ definition
matches the semantics of the term ’maximally even’ better than the explicit J−function, which lacks the
aspect of comparison. See Supplementary I of the online edition for a compilation of facts and arguments
around maximally even sets, or the recent [10].
2.1 An illuminating remark by Ian Quinn
Discussing a general typology of chords (or pc-sets), Ian Quinn noticed ( [17], 3.2.1) that what he calls
‘generic prototypes’ are the ME sets, and that they share an extremal property in terms of Fourier ‘weight’ 1 .
This is what we will now adopt as a definition; Quinn’s impressive survey and classification of the landscape
of all chords was not focused exclusively on ME sets, and as his redaction voluntarily avoided, to quote him,
the ‘stultifying’ quality inherent to dry mathematical generalizations, he left room for a formal proof that
this definition is equivalent to the traditional ones (we will prove the following definition is equivalent to
the classicical description, up to and including the formula with J functions; see [7] and [10] for equivalence
between all previous definitions).
Moreover, and this is in itself justification enough for what follows, many properties of ME sets will now
appear obvious from this starting point. Finally, the only quantity involved is |FA|, the invariant of the
Lewin relation which is, as we have seen, in many ways the most natural musical invariant for pc-sets.
2.2 A Lewinesque definition of ME sets and derived properties
1 “ We note that generic prototypicality may be interpreted as maximal imbalance on the associated Fourier balance – at least to the
extent that a generic prototype tips its associated Fourier balance more than any other chord of the same cardinality possibly can”.

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 ′

8 title on some pages
2.5 Huddling Lemma
This subsection is dedicated to the analysis of the maximality condition for the absolute values of the 1-st
Fourier coefficients for multisets associated with pc-sets A.
Lemma 2.7 (Huddling Lemma) The absolute value of the 1-st Fourier coefficient |F(ζ)(1)| of a m|d
multiset A ′ with characteristic function ζ is maximal among the values |F(ξ)(1)| for all m|d-multisets ξ
in Zc ′ iff ξ is a contiguous cluster of d′ pitch classes of multiplicity m, i.e. iff there is a l0 ∈ Zc ′ such that
ζ is of the form
ζ(l) =
m for l − l0 ∈ {0, ..., d ′ − 1},
0 for l − l0 ∈ {d ′ , ..., c ′ − 1}.
Just for the sake of illustration, we point out the two simple subcases:
• When c, d are coprime, πd is bijective and A ′ = d A is an ordinary subset of Zc. The definition of ME
sets, corollary 2.6 and the huddling lemma above mean that A ′ is a chromatic cluster, i.e. some translate
of {1, 2, . . . , d}. Hence A = d −1 A ′ is an arithmetic sequence with ratio d −1 , as is well known since [8].
The seminal example is the major scale, generated by a cycle of fifths.
• When d is a divisor of c, then A ′ is a multi-singleton set { d a ′ } as then the value |F(ζ)(1)| = d is
clearly maximal – here the huddling lemma is obvious. This means that A is a saturated preimage, i.e.
A = π −1 (a ′ ) = a + c ′ Zc = a + ker πd with πd(a) = a ′ , i.e. A is a regular polygon: see figure 2 1 .
x 4
Figure 2. All exponentials superimposed
Now for the technical proof of the huddling lemma. It relies basically on the very old geometrical fact
that the sum of two vectors making an acute angle is grater than both.
Proof We consider a m|d multiset A ′ in Zc ′ such that ξ does not have the contiguous form given in the
lemma, and prove that |F(ξ)(1)| = |FA ′(1)| is not maximal; the heuristic idea is that ‘filling in the holes’
increases the length of the sum.
Let us enumerate the elements of A ′ as r real integers in some increasing order: k1 < k2 < . . . kr < k1 + c
(the span kr − k1 could be chosen minimal, but it is sufficient that it be < c). Assume that A ′ is not
a translate of { m 0, m 1, m 2, . . . m d ′ − 1}, then there must be some element k ∈ [k1, kr] with multiplicity
0 ≤ ξ(k) < m (and r > d ′ ).
• Say there is such a k with multiplicity < m, aka ‘hole’, with k1 < k < kr; I claim that |FA ′(1)| strictly
increases when (say) k1 is replaced by k, i.e. when ξ(k) is incremented while ξ(k1) is decremented:
1 This exemplifies that the Lewinesque definition aims at looking for the best approximation to a regular polygon — obviously it will
be only an approximation when d does not divide c, for instance there is no regular heptagon inside the 12 notes universe. Indeed the
solution (the major scale A =(0 2 4 5 7 9 11) or any translate thereof) achieves |FA(7)| = 2 + √ 3 ≈ 3.73, still far from the unattainable
value 7 (or rather 5, for the complement), but still the largest value possible.

in so doing, the sum S = FA ′(1) =
l∈Z ′ c
alternative title 9
ξ(l)e −2iπl/c′
is replaced with S ′ = S + e −2iπk/c′
− e−2iπk1/c′ .
If S = 0 then clearly |S ′ | > |S|.
If not, let S = r e−iθ . We can choose a determination of θ mod 2π (or rather choose the ki’s) such
that 2πk1/c ′ < θ < 2πkr/c ′ , and I will assume that θ is closer to 2πk1 (if not, the proof is the same but
with kr) i.e. 0 < θ − 2πk1/c ′ < π. As V = e −2iπk/c′
− e−2iπk1/c′ π(k − k1)
= 2 sin
c ′ e−iπ(k+k1)/c′ +iπ/2 and
0 < θ − π(k + k1)/c ′ + π/2 < π/2 by our assumption that θ is ‘close’ to 2πk1/c ′ , the vectors S, V with
directions respectively −θ and −π(k + k1)/c ′ + π/2 make an acute angle. Hence their sum S ′ is longer
than both, qed (see fig. 3).
This can be done until no ‘holes’ remain between k1 and kr, i.e. ξ(k) = m for all k1 < k < kr.
• Eventually we reach the last case: the vector of multiplicities must then be
ξ(k1) = µ, ξ(k2) = m = ξ(k3) = . . . ξ(kd), ξ(kd+1) = m − µ
Say for instance µ ≥ m − µ. Then the direction θ of
FA ′(1) = r e−iθ d+1
= m
k=1
e −2iπk/c + (m − µ)(e −2iπk1/c − e −2iπkd+1/c )
lies between 2πk1/c and the mean value π(k1 + kd+1)/c (convexity). Hence as above, moving one point
from position kd+1 to position k1, i.e. incremeting ξ(k1) while decrementing ξ(kd), i.e. adding e−2iπk1/c′ −
e−2iπkd+1/c′ to S, increases its length, as the two vectors makes an acute angle.
Iteration of this process increases S strictly until it is no longer possible, which happens when A ′ is made
of d ′ consecutive points with multiplicity m, qed.
k r
k1
k k new FA' (1)
FA' (1)
k r
k1
Figure 3. Maximizing the sum on a multiset
Notice that for m = 1, the maximal solution is simply a chromatic cluster: A ′ is an ordinary set with d
consecutive points.
2.6 Maximally Even Sets Revisited
It remains to be justified that our Lewinesque definition of maximally evenness is indeed equivalent to the
traditional definitions. In the following subsection we recover the definition via J-functions. In the present
subsection we explore the pre-images π −1
d (ζ) of contiguous clusters as described in the huddling lemma.
This leads to the well-know taxonomy of maximally even sets:

10 title on some pages
• The regular polygon type: When m = d and hence d ′ = 1, as mentioned above the associated
multiset νd A is a multi-singleton {ml0} of multiplicity m which corresponds to the complete pre-image
π −1
d (l0) = k0 + {0, c ′ , ..., (m − 1)c ′ } for some k0 ∈ π −1
d (l0) and hence is a regular polygon in Zc.
• The Clough/Myerson type: When m = 1 and hence c = c ′ the map πd = ϕd = ϕd ′ is an automorphism
of Zc and the associated multiset νd A is the characteristic function of an ordinary cluster of
cardinality d co-prime with c. We have found again the result of [8], e.g. that maximally even sets of cardinality
d which are co-prime with the chromatic cardinality c are generated by the inverse d−1 mod c1 .
• The general Clough/Douthett type: From our construction,
A = π −1
d (m {l0, . . . m ld ′ −1}) = π −1
d (m l0)⊔. . . π −1
d (m ld ′ −1) = (a0+m Zc)⊔. . . (ad ′ −1+m Zc) = {a0, . . . ad ′ −1}⊕m Zc
meaning, in accordance with the known facts from [7], that general maximally even sets are Cartesian
products of the two previous types, i.e. bundles of regular polygons which are anchored in a
Clough/Meyerson type maximally even set. For example with the octatonic scale, we have A ′ = { 4 0, 4 2},
with preimages 0, 3, 6, 9 for 4 and 1, 4, 7, 11, for 2: A = {0, 1} ⊕ {0, 3, 6, 9} = B ⊕ 3 Z12.
There is a nice Fourier interpretation of this last and most complicated case: as seen above, A is periodic
with period c ′ .
Let us introduce for clarity B = {0, 1 . . . c ′ − 1} ∩ A = {0, 1 . . . c ′ − 1} ∩ π −1
d (A′ ) = πc ′(A) ⊂ Zc ′. We have
shown that
Theorem 2.8 A is a ME set in Zc if and only if A = B ⊕ m Zc and B is a ME set in Zc ′.
This is pleasantly related to the the following simple equation between Fourier transforms:
Remark 1 If A = B ⊕ m Zc then FA(d) = m FB(d ′ ) (B being considered as a subset of Zc ′).
This number is of maximal length if and only if B is a ME set in Zc ′, which is precisely the above
theorem.
Indeed the Fourier coefficients of B are (up to the m factor) the meaningful values of FA(d) as when
A is c ′ −periodic, all coefficients FA(k) vanish for k not a multiple of c ′ (Prop. 1.6). This is clearly visible
on figure 4, with Fourier transforms of the ME set (0 2 4) in Z7 and its counterpart (0 2 4)⊕ (0 7 14 21)
in Z28. This argument seems to us more illuminating than purely algebraic computations, as it enhances
the fact that the “characteristic domain” B concentrates its energy in the sense of the huddling lemma,
in order for A to do the same.
We get from there the complete enumeration of ME sets, which is developed in the end of Supplementary
I in the online version of this paper.
2.7 Expression by way of J functions
For the sake of completeness we add this technical but quick derivation of all ME sets:
Theorem 2.9 Let A ⊂ Zc be the pc set, whose elements are given by the J function, i.e.
A = {J α c,d (k) | k = 0 . . . d − 1} = {kc + α
, k = 0 . . . d − 1.}
d
Then πd(A) is a contiguous cluster of d ′ pitch classes of multiplicity m, i.e. A is maximally even.
Proof We compute values of the floor-function in Z, but interpret the results in Zc and Zc ′. Further we
suppose α = 0 w.l.o.g.
1 Contiguous order of cluster A ′ = l0 + {0, ..., d − 1} represents generation order of ME set A = l0d −1 + {0, d −1 , ..., (d − 1)d −1 }.

12
10
8
6
4
2
4
0
ME7,3
4
alternative title 11
2
12
10
Figure 4. Maximizing for B is maximizing for A
8
6
4
2
ME28,12
(0 2 4)
4 8 12 16 20 24
From the equations (k + d ′ )c
kc
=
d d + d′ c
kc
= + c ′ , we conclude first that A is a disjoint
d d
union of m translates of the set B = { kc
, k = 0 . . . d ′ − 1}, with multiples of c ′ as displacements, i.e.
d
A = B ⊔ c ′ + B ⊔ ... ⊔ (m − 1)c ′ + B. Thus, each element in the multiset πd(A) has multiplicity m. It
remains to be shown that πd(B) is a contiguous cluster.
We will use the fact that the fractional parts of the rational numbers kc
d
= kc′
d ′ take d′ different values
when k runs from 0 to d ′ − 1. This is true because c ′ and d ′ are coprime. To see this choose 0 ≤ k, k ′ < d ′ :
k ′ c ′ kc′
−
d ′ d ′ = n ∈ Z ⇒ (k′ − k)c ′ = d ′ n ⇒ d ′ | (k ′ − k) ⇒ k ′ = k as |k ′ − k| < d ′
From the d ′ different fractional parts 0 ≤ kc′ ′
−kc
d ′ d ′
kc ′
< 1 we obtain d ′ different integers 0 ≤ kc ′ −d ′
d ′
≤
d ′ − 1, which are in fact all the integers 0, ..., d ′ . Reduction of the elements kc ′ ′
− d ′kc
d ′
modulo c ′ yields
the set −πd(B) = −d ′ B mod c ′ . Thus πd(B) is a cluster, namely πd(B) = {c ′ − d ′ + 1, c ′ − d ′ + 2, ..., c ′ }.
3 Generated Sets and Groups Generated by a Set
The Fourier approach offers further directions of investigation. Here we restrict ourselves to maximality
conditions for the absolute values for Fourier coefficients. As we have seen in Section 2 it is the index
d ∈ Zc, i.e. the residue class of the chords cardinality to which the maximality condition for maximal
evenness is attached.

12 title on some pages
What about the other coefficients? It is illuminating to investigate the maximal Fourier coefficients
among invertible indices t ∈ Z ∗ c as well as among all non-zero indices t ∈ Zc\{0}. In the definition below
we exclude the index t = 0, because the maximum |FA(0)| = d is shared by all sets A with d elements.
Definition 3.1 For any pitch class set A ⊂ Zc let FA = max
t∈Zc,t=0 |FA(t)| and FA∗ = max
t∈Z∗ |FA(t)| be
c
respectively the maximal absolute value among all Fourier coefficients at non-zero indices, and the maximal
value of Fourier coefficients at invertible indices.
First notice that if f : x ↦→ a x + b, a ∈ Zc ′, is a bijective affine map then for any subset A
F f(A) ∗ = FA ∗
as ∀t ∈ Zc |F f(A)(t)| = |FA(a t)|
(the Fourier coefficients are permuted by affine maps). Same for FA: these quantities are invariant on
affine orbits of subsets.
There are three plausible values for the maximum FA or F ∗ A .
The first is the value characterizing ME sets:
Proposition 3.2 Fix a cardinality d coprime with c. Let µ(c, d) = |FB(d)| for some (c, d) ME set B. For
all d-element subsets of A ⊂ Zc, we find that FA ∗ ≤ µ(c, d) The equality occurs iff A = r · B + t for
suitable r ∈ Z ∗ c and t ∈ Zc, or equivalently A = a0 + {0, f, ..., (d − 1)f} is generated by a residue f ∈ Z ∗ c
(coprime with c).
The second plausible value is sin(π d/c)/ sin(π/c) which is equal to |FC(1)| for C a cluster, eg C =
{1, 2 . . . d}. The affine images of C are the generated scales with cardinality d, and we have a similar
proposition:
Proposition 3.3 Let ρ(c, d) = |F {1,2...d}(1)|. For all d-element subsets of A ⊂ Zc, we find that FA ∗ ≤
ρ(c, d). The equality FA ∗ = ρ(c, d) occurs if and only if A = a0 + {0, f, ..., (d − 1)f}, i.e. A is generated
by a residue f ∈ Z ∗ c coprime with c.
The last interesting value is d itself, as we have seen that FA(t)| ≤ d ∀t. First of all, remember that
from prop. 1.2, F Zc\A = FA is at most the lowest of d, c − d, so it is enough to work out the case
d ≤ c/2: dealing with a ‘large’ ME set (d > c/2) is equivalent to dealing with a ‘small’ one (d ≤ c/2), its
complement. Henceforth we will assume the latter case.
Proposition 3.4 FA = d iff A is contained in a regular polygon, i.e.
∃r ∈ N, a0 ∈ Zc, 1 < r < c, A ⊂ a0 + rZc
Notice that, although this includes the generated scales that we missed in the last proposition, other
cases are possible: C = {0, 2, 6} ∈ Z12 also checks FC(6) = 3.
The proofs of these propositions and a discussion of the remaining chords with maximal FA which are
not of the previous types are to be found in Supplementaryary III of the online version.
4 Chopin’s theorem
As the inverse of a ME set (in the musical sense) is also maximally even, either f ′ = d ′−1 or its opposite −f ′
will generate a 〈c ′ , d ′ 〉 ME set1 . This has a consequence on complementary ME sets classes: as gcd(c, c−d) =
gcd(c, d) = m, when one replaces d by c − d, one gets the same c ′ , and replaces d ′ c − d
by
m = c′ − d ′ ≡ −d ′
mod c ′ ; hence
Lemma 4.1 A same generator f ′ can be used for the construction of both 〈c, d〉 and 〈c, c − d〉 ME sets.
1 The interesting question of all generators of a scale (not only for ME sets) is to be elucidated in [2].

alternative title 13
For instance, the fifth f ′ = f = 7 generates both the pentatonic and the major scales, when c = 12.
For, say, c = 20 and d = 8, one gets m = 4, d ′ = 2, c ′ = 5, f ′ = 3 and the generated ME sets with 8
and 12 elements are {0, 3} ⊕ {0, 5, 10, 15} and {0, 3, 6, 9} ⊕ {0, 5, 10, 15} = {0, 3, 1, 4} ⊕ {0, 5, 10, 15}. More
generally,
Theorem 4.2 Let 1 < d ≤ c/2; then any given 〈c, c − d〉 ME set contains several (exactly c ′ − 2d ′ + 1)
〈c, d〉 ME sets. In other words, any ‘small’ ME set is contained in several translates of its complement
Proof A 〈c, d〉 ME set is constructed by truncating to just d ′ consecutive values the sequence
{f ′ , 2f ′ , . . . (c ′ − d ′ )f ′ } mod c ′ , which generates (adding up c ′ Zc) the given 〈c, c − d〉 ME set A. This
can be done in precisely c ′ − 2d ′ + 1 ways.
From there, as seen in thm. 2.8, it suffices to add c ′ Zc to get both whole ME sets, since c ′ is the same
for d and c − d, preserving the inclusion relation all the time.
We would like to baptize this result Chopin’s theorem in reference to the Etude op 10 N ◦ 5 (see fig.
in Supplementary II of the online version) where the right hand plays the pentatonic (black keys only)
while the left hand wanders through several keys, G flat and D flat major for instance. This result has
been observed (especially in this pentatonic ⊂ major scale case) and commented 1 although perhaps it has
not been stated and proved as a quality of all ME sets (or, alternatively, generated scales).
So David Lewin, who almost invented ME sets as we have seen, might also have originated set-complex
Kh−theory too in one fell swoop.
5 Coda
We have examined the definition of the DFT of a pc-set, according to David Lewin. Several interesting
features of the pc-set are encapsulated in the absolute value of this function.
Following then Ian Quinn, we were led to advance an original definition of Maximally Even sets, which
appears to be geometrical, concise, elegant, and illuminating 2 . We hope that this definition will become a
productive one.
Acknowledgements
First of all to Ian Quinn who not only spelled out the property which makes the gist of this paper, but also
drew our attention, through his comprehensive study of chords landscape, to the impressive advantages
of the DFT of chords, and not only ME sets and other ‘prototypes’. David Clampitt kindly explained the
subtleties of WF scales vs ME sets and most of the history of these fascinating notions. Equally important
to the field of ‘mathemusical’ knowledge is the continued contribution of Jack Douthett (with the late
John Clough and other partners).
Several reviewers have been instrumental in bringing this paper up to the quality level of the Journal,
an undomitable task for a lone writer. I would like to thank especially Dmitri Tymocsko, Robert Peck and
particularly Thomas Noll, in that respect.
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14 title on some pages
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Models and Human Reasoning - Bernd Mahr zum 60. Geburtstag. W&T Verlag, Berlin.
[20] Rahn, J., Basic Atonal Theory, Longman, New York, 1980.
[21] Vuza, D.T., 1991-1992, Supplementaryary Sets and Regular Complementary Unending Canons, in four parts in: Canons. Persp. of
New Music, 29:2, 22-49; 30:1, 184-207; 30:2, 102-125; 31:1, 270-305.
Supplementary I: about Maximally Even Sets
5.1 A short history of ME sets
Maximally Even Sets, or ME sets in short, were defined in [8], generalized in [7] and later extended to
Well Formed Scales, which exist also in non equal temperaments (see [6]). The name refers to the intuitive
feature of being ‘as evenly distributed in the chromatic circle as possible’. As we will see, it is not so
easy to make this idea rigorous: many different though equivalent definitions exist, and our main objective
in this paper is to ground firmly the notion of ME sets on a DFT-based definition. We include a short
paragraph for readers who might still be unfamiliar with the notion, followed by a discussion of several
existing definitions. A very thorough paper on state-of-the-art applications of ME sets is [10].
Originally, Clough, Myerson and soon after Douthett observed this yet informal notion of ‘maximal
evenness’ in a collection of famous scales: whole tone scale, major scale, pentatonic, octatonic. . . For musicological
reasons, and perhaps also because of mathematical difficulties we will mention below, their
definition was rather indirect.
In the minor scale there are three different values of intervals between consecutive notes. Not so for the
major scale, or the melodic (ascending) minor; but the latter features three different fifths.
From these examples, and others, ME sets were defined in regard with the different (some say ‘diatonic’)
possible values of intervals inside the scale: for instance, the major scale and the pentatonic alike have
only two different interval sizes between consecutive notes – tones and semi-tones for the one, tones and
minor thirds for the other. Also notice that the two semi-tones in the major scale, for instance, are as far
from one another as possible. This has some relevance to the organisation of black and white keys on a
keyboard, and hence to traditional musical notation in staves.
The common original definition (here reworded) states that
Definition 5.1 Let A be a subset of Zc. Let us for convenience’s sake call a ‘second’ any interval between
two adjacent elements of A, a ‘third’ an interval between every odd note, and so on.
Then A is maximally even if, and only if, there are at most two different kinds of ‘seconds’, ‘thirds’,
‘fourths’ aso.
This definition suffers from the common blemish of many formalized musicological definitions, that take
for granted many notions with intuitive, musical support (like diatonic intervals, adjacency of notes, etc.)
which are not so obvious to define mathematically 1 .
1 To be fair, pre-Hilbert mathematics (and some post-Hilbert, too) often relied too heavily on intuitions of the physical world, as the

alternative title 15
To state it with numbers: if an ordered scale 2 is A = {a1, a2 . . . ad} with indexes taken modulo d and
values taken modulo c, for each value of k there should be at most two different values of ai+k − ai when
i varies.
This was named the ‘Myhill property’ in [8] 1 ) and it is not at all straightforward.
Figure 5. All intervals come in two sizes
Worse still, in our opinion, this definition necessitates an ordering, or reordering, of the notes : (C E D
G A) is not a ME set, though (C D E G A) is ! This verges on the unsatisfactory, if one is interested in
pc-sets and not (ordered) scales.
Many geometrical criteria have been given, and proved equivalent (see [10]); we especially like the ‘black
and white’ definition in [7], very intuitive though hardly practical (see figure 6): plot two regular polygons,
one white with d vertexes and one black with c − d vertexes. Then rearrange all the vertexes, preserving
order, with identical distance between consecutive points. Both black and white subsets are ME sets.
mixing two regular polygons the same rearranged
Figure 6. Rearranging the points of two intertwined regular polygons
The most effective way to actually compute ME sets is as follows: taking c as the cardinality of the
ambient chromatic space, d the number of notes of the looked-for set, and α some arbitrary number, the
J functions
J α c,d : k ↦→ kc + α
, k = 0 . . . d − 1
d
quarrel on non-euclidean geometries made clear.
2 We skip a formal definition of ‘ordered’ in Zc, which will be useless in our approach.
1 Note that in general, it is not enough that Myhill property holds for adjacent notes, e.g. having only two kinds of ‘seconds’ does not
ensure we have a ME set, as shown by the example of the melodic minor scale.

16 title on some pages
already introduced in [8], give all ME sets with cardinality d by their sets of values
J α c,d (0), J α c,d (1), . . . J α c,d (d − 1)
(taken modulo c): for instance with c = 12, d = 5, α = 12 one gets the pentatonic (0 2 4 7 9); but relevance
to the intuitive idea of maximum evenness, or even to sizes of intervals, is less than obvious.
The most natural definition might be to try and maximize the mutual distances between all the notes,
eg
a,a ′ ∈A δ(a, a′ ), but the result depends on the chosen distance function δ, and is not satisfactory for
the (arguably) most natural one, the interval metric :
δ(u, v) = min |u − v + kc|
k∈Z
as several unexpected 1 extraneous solutions crop up, as in figure 7 . A ‘good’ definition would be expected
to give one characteristic shape for a given pair (c, d), not so many. This exemplifies why there is no
universal, or obvious, definition for the naïve concept of ‘Evenness’.
Figure 7. Some sets maximizing the sums of distances for the interval metric – c = 15, d = 6
It is because none of these definitions (or others) appears completely satisfactory in our opinion, that
we ventured to propose another one.
5.2 Symmetries of ME sets
This is the sequel of subsection 2.6.
Corollary 5.2 The number of different ME sets of cardinality d in Zc is c ′ = c/ gcd(c, d) (the number
of different possible B’s). All are translates of one another (the group of translations acts transitively on
ME sets) 2 .
For each couple (c, d) there is but one translation class of ME sets with d points in Zc. Henceforth we
will denote such a ME set class as 〈c, d〉. An actual ME set will be ‘a 〈c, d〉 ME set’. For example there
are exactly three different 〈12, 8〉 ME sets, i.e.the octatonic scales.
Remark 1 Each individual 〈c, d〉 ME set is invariant under the m translations of step c ′ and multiples. We
have seen (1.5) that the inversion operation preserves the class of ME sets: this means that the inverse of
a ME set is one of its translates. Indeed a ME set is its own image under exactly 3 2 × m operations, m
translations and m inversions in the dihedral group T/I of transformations of type x ↦→ x+τ and x ↦→ ℓ−x
in Zc. For instance, inversions x ↦→ −x, 3 − x, 6 − x, 9 − x preserve the above octatonic.
1 But all strictly convex distance functions on the unit circle will give maximums on the same pc-sets, which are the ME sets, as
shown in [12]. Nonetheless, such a distance (like the chordal distance, length of the line segment between two points of the circle) has
little musical meaning.
2 Only when m = 1 do we have simple transitivity, i.e. an interval group in the sense of [15].
3 The stabilizer of any pc-set in T/I, isomorphic to the dihedral group Dc, is either a cyclic or a dihedral group. For 〈c, d〉, it is always
a Dm.

Supplementary II: pictures and proofs
alternative title 17
Figure 8. These two hexachords share intervallic content
On figure 8 with the two complementary hexachords, the fifths have been signaled with arrows. Each
hexachord has the same number of fifths, three in this example.
Proof of prop. 1.6:
Proof From Thm. 1.5 we have
Figure 9. Etude N ◦ 5 opus 10, Frédéric Chopin
A is τ−periodic ⇐⇒ ∀t ∈ Zc FA(t) = e −2iπτ t/c FA(t) ⇐⇒ ∀t ∈ Zc (1 − e −2iπτ t/c )FA(t) = 0
Unless e −2iπτ t/c = 1, this compels FA(t) to be 0. Now the condition e −2iπτ t/c = 1 is equivalent to c | τt,
i.e. t multiple of m = c/ gcd(c, τ) – this makes sense for any representative of the residue classes τ and t.
This is compatible with reduction modulo c, and means t ∈ m Zc ⊂ Zc. Conversely, if FA is nil except on
a subgroup, say m Zc with 0 < m | c in Z (we recall all subgroups of Zc are cyclic) then, by inverse Fourier
Transform
∀k ∈ Zc
1A(k) = 1
FA(t)e
c
t∈Zc
2iπ k t/c = 1
c
t ′ ∈m Zc
FA(t ′ )e 2iπ k t′ /c = 1
c
t ′′ =1... c
m
FA(m t ′′ )e 2iπ k t′′ m/c
and this is obviously periodic with (the residue class of ) c
c
as a period, as each term in the sum is
m m
periodic.
Proof of prop 2.2:

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.

3
chunk of whole tone
1 2 3 4 5 6 7 8 9 10 11
alternative title 19
another pc set with maximal DFT
3
1 2 3 4 5 6 7 8 9 10 11
Figure 10. DFT of (0 2 4) and (0 2 6) modulo 12 share maximal value in 6
Definition 5.4 For any pitch class set A ⊂ Zc let G[A] ⊂ Zc denote the interval group 1 of A. It is
generated by the differences in A: G[A] := 〈A − A〉 = {r · (k1 − k2) | k1, k2 ∈ A, r ∈ Zc}. One can see that
G[A] = 〈{a0 − k | k ∈ A}〉 independently of the choice of a0 ∈ A (c.f. [18], p. 125).
It will be impossible to reach FA = d for a ‘large’ ME set, i.e. when d > c/2, as in general FA ≤
min(d, c − d). So we work in the case d ≤ c/2.
Theorem 5.5 FA = d ⇐⇒ G[A] = Zc. Any subgroup of Zc being cyclic, say G[A] = r Zc (taking r
minimal); this means A ⊂ a0 + rZc.
This can happen if and only if d is lower than some strict divisor c ′ = c/r of c (for instance whenever c
is even).
Proof Assume FA = d.
Then |FA(t0)| =
k∈A e−2iπ k t0/c = d for some t0 = 0; but from Cauchy-Schwarz inequality’s case
of equality, this means that all exponentials, each with length 1, are equal. In other words, multiset
t0 A = { d a} is a singleton with multiplicity d (and t0 cannot be invertible). Hence A is a subset of the
preimages of a, i.e. A = a0 + ker ϕt0 i.e. G[A] = ker ϕt0. As we have seen when studying maps ϕd, this
kernel is a regular polygon with c ′ = c/ gcd(c, t0) elements.
So Card(A) ≤ c ′ , a strict divisor of c.
Conversely, assume d ≤ c ′ = c/m, a strict divisor of c. Then there are subsets A of c ′ Zc with cardinality
d, any of which will check |FA(m)| = d.
It is notable that in that case, the maximum is reached for members of a subgroup:
|FA(t)| = d ⇐⇒ t ∈ m Zc
1 In a more general context Mazzola [18], p. 125 - 127 calls this the module of a local composition.

20 title on some pages
Notice that, although this includes generated scales, other cases are possible: C = {0, 2, 6} ∈ Z12 also
checks FC(6) = 3. This includes the ‘secondary’ and many ternary ‘prototypes’ 1 in [17], 2.4, as it seems
that Quinn had noticed. This class of maximal pc-sets includes generated scales, but is somewhat wider.
Now the general question arises: for a given pair (c, d), what are the subsets A ⊂ Zc with cardinality d
that yield the maximal value M of all FA ? There are three cases:
(i) The maximum value is d: this is well understood from the last theorem. It happens whenever d is lower
than some strict divisor of c.
(ii) The maximum value is the one for ME sets, i.e. m sin πd′
c ′
π sin c ′ . This is very often the case. For instance
for c = 12, d = 4 the cluster (0 1 2 3) is not maximal for FA: the winner is the 〈12, 4〉 ME set (0 3 6
9).
(iii) Sometimes the maximum is not of one of the previous types. For instance for c = 75, d = 27 when d is
larger than all divisors of c, one gets µ(c, d) = 21.6581 for ME sets or their affine image; the value for
clusters or generated scales is lower, ρ(c, d) = 21.6075 (this is general), but for
A = {0, 1, 3, 4, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72}
i.e. 3Zc ∪ {1, 4}, one gets FA = √ 579 = 24.062188. The principle involved is, just as in the huddling
lemma but in greater generality, to have for some k the multiset A ′ = k A ⊂ Zc as ‘clustered’ as possible
[we depart here from the definition of A ′ in fig. 1]. Ideally, from the huddling lemma philosophy, one
should aim at a few multipoints as close as possible in k Zc, with maximum multiplicities. Let us clarify
this, without working out the complete theory.
We are working with a subset A with cardinality d, greater than any strict divisor of c, which is an
odd number. The maximum value of |FA| is some |FA(k) = FA ′(1)| where A′ is the multiset k A; k
is not coprime with c or else we get a previous case, and as permutations of Fourier coefficients are
irrelevant, we can assume (up to an affine transform of A) that k is a divisor of c.
The kernel of πk : t ↦→ k t mod c is c ′ Zc with c ′ = c/k, it has k elements ; this is the maximum
possible multiplicity for a point of A ′ . The distance between consecutive points of A ′ [in Zc] will be k,
hence the number of different points in A ′ is d ′ = ⌈ d
⌉. All these points will have multiplicity k, except
k
one on the border with multiplicity d − k d ′ . The maximum shall be found among the different subsets
obtained in this way, checking on all values of k | c.
In the example, A ′ has one point (0) with multiplicity k = 25 and another one (25) with multiplicity
2. B ′ has two points, with multiplicity 15 and 12 respectively, and C ′ has 5 points with multiplicity 5
and one with multiplicity 2 [other possible values of k have been left aside].
As it happens, the respective values are
FA = √ 579 = 24.0624 FB =
909 − 180 √ 5 = 22.5057 FC = 21.529
In that case, the winner is A, corresponding with the greatest divisor (k = 25). But in other cases, it
may pay off to restrict the angular span of the associated multiset – this means that (d ′ −1) 2π
c ′ is made
as low as possible, by choice of the divisor k of c. For instance, with c = 75, d = 29, the set analogous
to B (i.e. one point with maximum multiplicity k = 15 and another one with multiplicity 14) is the
winner, as FA = 23.2594 < FB = 23.4689 and ||FC|| = 22.3884.
So there is an algorithm, but no hard and fast rule for constructing the subsets with the biggest
FA. The solutions approximate (affine transforms of) ME sets, add or drop a few points. Maybe
these chords, or scales, which generalize ME sets in a way (look at the multisets on fig. 11 ), and are
defined modulo the action of the affine group, should be catalogued as for instance the tiling subsets of
1 (0 2 6) mod 10 falls under the last theorem, but not (0 1 2 3 4 5 7 8 9).

alternative title 21
A B C
A' B' C'
Figure 11. Three candidates for a maximum FA.
Zn have been. This is one of many interesting directions for future research on the subject of musically
relevant features of the DFT of discrete structures..

Journal of Mathematics and Music
Vol. 03, No. 01, February 2009, 1–26
Autosimilar Melodies
Emmanuel AMIOT
1 rue du Centre, F 66570 St NAZAIRE, France
(v1.0.2 released September 2009)
The present work originated from purely musical topics, namely the notion of ‘selfRep melody’ as defined by composer Tom Johnson
in [9], and reaped interesting mathematical as well as musical rewards. It is about melodies that contain themselves in augmentation,
and some generalizations thereof. Some of the mathematical by-results are new to the best of the author’s knowledge. Applications range
from melodies to rhythms, and include some new results on mosaics. Finally, extensions to approximate and ‘non invertible’ autosimilar
melodies suggest that this notion is both widespread and universal.
Notations: Zn stands for the cyclic set with n elements, with its group or ring structure if needed.
Quite often, calculations make sense both in Zn and Z. When in doubt, consider that a ∈ Zn can be
identified with the smallest non negative integer in Z with residue a.
Divisibility (usually in Z) is denoted by |: for instance 8 | 4 in Z12, as 4 = 5 × 8 mod 12.
The invertible elements of (Zn, ×) are the generators of the additive group (Zn, +); they form a multiplicative
group, Z ∗ n.
Any set might be given by the list of its elements: (0, 3, 5); or by some defining property, e.g. Z ∗ n = {a ∈
Zn, gcd(a, n) = 1}.
The subgroup generated by some element g of a group G is denoted by gr(g). For instance gr(a) =
(Zn, +) ⇐⇒ a ∈ Z ∗ n.
A periodic melody M is a map from Zn into some musical space, usually pitches or notes, or equivalently
a periodic sequence: ∀k ∈ Z, Mk+n = Mk. So Mk is well defined for k ∈ Zn.
The affine group modulo n is the set Affn of affine bijections in Zn, i.e. x ↦→ a x + b for (a, b) ∈ Z ∗ n × Zn.
The order of an element g of a group G is the cardinality of gr(g), i.e. the smallest integer r > 0 with
g r = e, the unit element of group G. It is classically characterized by the following equivalence:
g k = e ⇐⇒ o(g) is a divisor of k
The cardinality of any set G is denoted |G|: e.g. o(g) = | gr(g)|.
Introduction
Symmetries of Zn have been thoroughly explored under the group of translations and inversion (T/I), for
instance in American Set Theory. Such transformations are expressed by maps x ↦→ ±x + b. But despite
the obvious interest of more general affine transforms in Zn, e.g. x ↦→ a x + b, much less research has been
made on orbits of affine maps, or subgroups of the affine group. 1 . This is surprising, as many interesting
notions are invariant under affine transformations: interval content (up to permutation), all-interval sets,
limited transposition modes, tiling (mosaic) property, series and all-interval series, to name but a few.
This paper both presents a mathematical study of orbits of affine maps operating on a cyclic group, and
develops a musically interesting notion of autosimilarity, just like famous fractals (the Cantor Set, Koch
Professor in Class Preps, Perpignan, France. Email: manu.amiot@free.fr
1Except of course for c = 12, well studied for instance in [12] or [11], but with great loss of generality, as stigmatised in [10][section
11.5.4.2]: for instance there is no element of order greater than 2 in (Z∗ 12 , ×). I am indebted to a reviewer who mentioned the work of
Batstone ( [6]) in the case c = 2n − 1, also well covered by Johnson with the natural multiplier 2 and orbits of length n, see below.
Journal of Mathematics and Music
ISSN 1745-9737 print / ISSN 1745-9745 online c○ 2009 Taylor & Francis Ltd.
http://www.tandf.co.uk/journals
DOI: 10.1080/17459730xxxxxxxxx

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

GGGE♭ 3
Figure 2. Thema of ’in the Mood’
But it will surprise many readers to realise that much more ancient Western music features autosimilarity:
it can be found in Beethoven’s Fifth Symphony, though the ratio 3 autosimilarity is not blatant at all (fig.
4). The ubiquitous Alberti Bass (fig. 3), well known from (for instance) the beginning of Mozart’s Sonata
in C major K. 545, is an excellent example, with autosimilarities at ratios 3, 5 and generally every odd
number. In Mozart’s first bar, left hand is exactly autosimilar while right hand significantly plays the same
three notes. Ratio 4 autosimilarity is even explicit in the first two bars of Scarlatti’s Sonata Kirk. 9 in D
minor, as pointed by an anonymous reviewer.
1.3 Mathematical generation.
Figure 3. Alberti Bass with augmentation by 3
Theorem 1.2 Any autosimilar melody with ratio a and period n is built from orbits of the affine map
x ↦→ a x mod n: denoting the orbit of x as
Ox = {a k x mod n, k ∈ Z} = a Z .x,
for each note p of the melody, the subset of indexes M −1 (p) = {i ∈ Zn, Mi = p} is one such orbit, or a
union of several ones.
Proof It is sufficient to prove that if Mx = p then Mk = p for all k ∈ Ox, hence every orbit will come in
toto for a given note. But this is obvious from the definition, as
Mk = Ma m x = Ma m−1 x = . . . Mx = p.
A basic fact about orbits is worth recalling here, namely that x ∈ Oy ⇐⇒ y ∈ Ox.
In group theory, these orbits are the classes of the action of the cyclic subgroup generated by the map
f : x ↦→ a x mod n.
At this point it seems a good idea to demand that a (or f) should generate a subgroup, which means
that a is coprime with n, or equivalently a ∈ Z ∗ n. As will be seen below, interesting situations arise when
this condition is dropped. But until section V,
The ratio a of an autosimilar melody is assumed to be coprime with the period n.
before WW II. Quoting Henry Cowell in his preface to [13], “The idea behind the Schillinger System is simple and inevitable: it undertakes
the application of mathematical logic to all the materials of music and to their functions, so that the student may know the unifying
principles behind these functions, may grasp the method of analyzing and synthesizing any musical materials that he may find anywhere
or may discover for himself, and may perceive how to develop new materials as he feels the need for them.”

4 autosimilar melodies
Example 1.3 The abstract melody generated by ratio 3 modulo 8 has five orbits: 0 is sent to 0 so (0)
is one orbit. 1 is sent to 3 and 3 sent to 1, so (1 3) is another one. (4), (5 7), (2 6) are the remaining
orbits, hence the general structure, x, y . . . denoting arbitrary pitches, is: xyzytuzuxyzytuzu . . . .
The Alberti Bass (cf. fig. 3) has less than five notes, because identical notes are used for different orbits:
putting arbitrarily C on 0 and 4, G on odd beats i.e. (1 3) and (5 7), and E on (2 6) we get: C G E G C
G E G C G . . . , still autosimilar with ratio 3.
Here several orbits have been collapsed on identical notes: z = E, x = t = C, u = y = G. Hence it is
necessary to define:
Definition 1.4 A primitive autosimilar melody is a melody generated by a ratio a and a modulo n with
different notes for different orbits. In other words, there are as many different notes as possible for the
given n, a.
As will be seen below, several mathematical results will only stand for primitive melodies: obviously the
possibility of collapsing all the orbits into as few as one (a one note melody, or even a melody of silences. . .
like Cage’s 4’33” !) impairs any attempt at a classification of symmetries.
1.4 The simple case of n prime.
In this paragraph we assume that the period n is a prime number. This is not really necessary for further
study, but it helps come to grips with the notion of autosimilar melodies.
Proposition 1.5 One orbit has only one note : O0 = (0). All others share the same cardinality, which is
the multiplicative order of a:
o(a) = |O1| = |{a k mod n, k ∈ Z}| = min{k > 0, a k = 1 mod n}
Proof The orbit of 1 is exactly the subgroup of Zn ∗ generated by a, e.g. all different powers of a mod n:
hence |O1| = o(a).
Let x = 0, now the map y ↦→ y x is one-to-one (as x must be invertible, Zn being a field when n is prime)
and maps precisely O1 onto Ox: hence |Ox| = |O1| = o(a).
n − 1
This is the one case where the number of different notes in the melody is easily computed, i.e. 1 +
o(a) .
Musically it is a natural idea to put a rest (or silence) on the singleton 0, as Tom Johnson does in many
instances.
There is another simple result, given in [7] for n prime, but both true and simpler in the general case:
Proposition 1.6 For any period n > 1, the ‘melody’ xyyyyyyy . . . xyyyyyy . . . is autosimilar 1 for any
ratio a (coprime with n).
This is obvious: x stands on the orbit of 0, and the y’s are just on the union of all other orbits, whatever
the value of a.
Of course, such a melody (with one single note repeated on n−1 beats out of n) seems a little simplistic.
Still the following example (fig. 4) will sound familiar to most readers (from mes. 256 sqq) and it is
autosimilar with n = 4, a = 3.
1.5 Some figures.
In the case of x ↦→ a x mod n where a is no longer prime, we can give a few explicit formulas. As will be
proved in the next section, in the most general case these formulas remain valid about half of the time.
1 But it is not primitive in general.

GGGE♭ 5
Figure 4. Beethovenian autosimilarity
1.5.1 Number of occurences of one note. Some special cases are easy: O0 always has a single element,
and O1 is the multiplicative subgroup generated by a, i.e. the set of powers of a, and its cardinality is
equal to the order o(a) of a. This is exactly what happened for n prime.
The group Z ∗ n however, though abelian, is fairly complicated. In particular, the maximal order of any
element a ∈ Z ∗ n, that is to say the largest possible number of occurences of one note in a n−periodic
primitive autosimilar melody, is given by Carmichael’s function Λ. 1 Also, even for simple a’s, for instance
for a = p prime, the lengths of orbits are by no mean easy to compute. 2 This comes from a new behaviour:
contrariwise to the n prime case, the map
is no longer one to one.
O1 ∋ a k t↦→x t
↦−→ x a k ∈ Ox
Proposition 1.7 The length of any orbit, i.e. the number of occurences of a given note in a primitive
autosimilar melody of ratio a and period n, is a divisor of o(a).
Proof In short, f acts on every Ox as a circular permutation.
It is helpful to visualise (fig. 5) all these orbits as little clocks of different sizes, ticking at different speeds,
with at least one great clock whose size is a multiple of all others. Each multiplication by a (mod n) ticks
every clock, and after a whole ‘day’, e.g. after o(a) ticks, all clocks must have resumed their initial positions.
This is the substance of the above proposition. The perception of an autosimilar melody is then explained
as an aural illusion: multiplying by a, i.e. picking one note every a beat, completely rearranges the order
of all the notes inside, but as the different onsets of a given note belong to the same orbit, we assume that
we hear the same note at that moment.
0
7
14
12
3
6
11
16
Figure 5. Several clocks ticking together: n = 21, a = 2
The length of the orbit of a given x is given exactly by:
1 The Carmichael function Λ verifies Λ(p α q β . . . ) = lcm[Λ(p α ), Λ(q β ), . . . ] with Λ(p α ) = Φ(p α ) = (p−1)p α−1 except when p = 2 < α.
See http://mathworld.wolfram.com/CarmichaelFunction.html for details.
2 The orbits (x, p x, p 2 x . . . ) are called cyclotomic orbits, their lengths are the degrees of the irreducible factors of X n − 1 in the ring
of polynomials on the field with p elements, see [1] for a musical application to a species of rhythmic canons.
1
8
2
4
13
17
5
19
10
20

6 autosimilar melodies
Proposition 1.8 Let d = gcd(x, n); the length of Ox = (x, a x, a 2 x, . . . ) is the order of a modulo n/d,
i.e. the smallest integer k > 0 with a k = 1 mod n/d. In particular, Ox is of maximal length whenever x
and n are coprime.
Proof Let us begin with a particular case:
Lemma 1.9 If gcd(y, m) = 1 then the length of the orbit of y in Zm is exactly the order of a (mod m).
Proof The map Ψy is one-to-one and maps O1 to Oy.
The general case follows by considering the orbit of x as a part of the cyclic subgroup x Zn = d Zn, where
d = gcd(x, n), which is isomorphic to Zm, m = n/d where the orbit Ox ⊂ Zn is mapped onto O x/d ⊂ Zm
for the map y ↦→ (a/d)y ∈ Zm: therein the lemma applies.
We draw the reader’s attention to the fact that the automorphisms of the additive group (Zn, +), which
are the x ↦→ b x with b ∈ Zn ∗ , permute these maximal orbits. Indeed they permute all orbits, preserving
their sizes.
1.5.2 Number of single notes. A last particular case is that of one note orbits, i.e. single notes, i.e.
fixed points of the map x ↦→ a x:
Ox = {x} ⇐⇒ a x = x ⇐⇒ (a − 1)x = 0 (mod n).
This means exactly that x contains the prime factors of n that are missing in a−1. For instance, say a = 4
and n = 15: such x’s are simply the multiples of 5. Hence
Proposition 1.10 The number of single notes in a primitive autosimilar melody of ratio a and period n
is gcd(a − 1, n). They are the multiples of n/ gcd(a − 1, n).
This will come as a special case of Prop. 2.14.
Looking for an autosimilar melody with ratio 3 and period 8, we can thus predict gcd(3 − 1, 8) = 2
singletons, and indeed 0 and 4 are the only fixed points of the map x ↦→ 3x (mod 8) (both are note C in
the Mozart example). This result will be generalized in the following section.
Tom Johnson conjectured that the total number of different notes in a melody with period n is at most
3n/4, as happens for the melody in fig. 6 (n = 8, a = 5, 6 different notes):
Figure 6. Melody with many different notes
This follows from the last proposition, even in the most general case (see Thm. 2.16).
2 Extension to general affine automorphisms
A more general case, especially from an aural point of view, is the action of any affine automorphism.
Practically, the following definition means that by extracting one note every a notes in the melody, one
hears the same melody, though perhaps with a different starting point (namely b) – but this is surely
irrelevant mathematically, as periodic melodies do not have a starting point.

GGGE♭ 7
Definition 2.1 Let M be a periodic melody with period n. M is autosimilar with ratio a and offset b iff
∀k ∈ Zn Ma k+b = Mk.
An example is the following common rhythmic beat, which is autosimilar with ratio 3 and offset 1:
Figure 7. Autosimilarity with offset
Theorem 2.2 Any autosimilar melody of ratio a, period n, and offset b is built up from orbits of the affine
map x ↦→ a x + b (mod n).
The proof is identical to that of Thm. 1.2. 1
Remark 1 This new, more general setting, includes the case a = 1 with melodies invariant under x ↦→ x+τ,
i.e. maps with a period smaller than n. Each orbit, and hence each preimage
M −1 (p) = {k, Mk = p}
is then a Limited Transposition Subset of Zn. We will henceforth exclude this case and assume a = 1. 2
For instance, the Kientsy Loops 3 melody G F E D E F G D G F E D E F G D G F E. . . can be viewed
as generated by x ↦→ 3x + 6 (mod 8) if we set origin at G=0.
It can be viewed more simply as generated by x ↦→ 3 x if we decide that the consecutive F, E are labelled
0, 1 instead of 1, 2. This ambivalence will be elucidated below.
2.1 Orbit lengths
Lemma 2.3 The order of map f : x ↦→ a x + b (mod n) (e.g. the size of the subgroup gr(f) generated by
f in Affn) is:
Proof Easily from formula
o(f) = min k > 0 such that gcd(a − 1, b) (1 + a + . . . a k−1 ) = 0 (mod n) (2)
f k (x) = f ◦ f ◦ . . . f(x) = a k x + b (1 + a + . . . a k−1 ) (3)
In practice, compute the ‘missing factor’ mf = n/ gcd(a − 1, b, n), and look up the first number 1 + a +
. . . a k−1 that is a multiple of mf.
1 The mathematical standpoint would be here to define an action of Affn on the set of maps M : Zn → (pcs) by f • M = M ◦ f.
2 The setting of affine maps modulo n might be unfamiliar to many readers, and a few reminders may be useful. The main point is to
distinguish between the monoid of general affine maps, and the group of affine transformations, which are one-to-one maps; these
last are exactly the x ↦→ a x + b (mod n) with gcd(a, n) = 1. Their group Affn is a semi-direct product of its translation subgroup (all
x ↦→ x + b), isomorphic to the group Zn, and its homotheties subgroup (all x ↦→ a x, gcd(a, n) = 1) which is isomorphic to Z ∗ n; Affn is
not abelian and several open problems remain about its structure. [10] is justified in demanding that the exact sequence
0 → (Zn, +) → (Affn, ◦) → (Z ∗ n , ×) → 1 (ES) (1)
(which is another way of expressing that Affn = Z ∗ n ⋉ Zn) be taken into account; but it does not explain all that follows.
3 CD Pogus productions, P21033-2, 2004.

8 autosimilar melodies
Proposition 2.4 o(f) is a multiple of o(a), and a divisor of the smallest integer s satisfying
1 + a + a 2 + . . . a s−1 = 0
We leave the proof to the reader.
All these numbers are identical for instance when a − 1 is coprime with n, as then
1 + a + a 2 + . . . a s−1 = 0 ⇐⇒ (1 − a)(1 + a + a 2 + . . . a s−1 ) = 0 ⇐⇒ 1 − a s = 0 ⇐⇒ a s = 1
A technical result brings in more precision:
Proposition 2.5 Let τ be the number of translations in gr(f) where f : x ↦→ a x+b: then τ = n/ gcd(k, n)
and o(f)/o(a) = τ.
Proof Straightforward from the exact short sequence
0 −→ k Zn ≈ Zτ ≈ gr(x ↦→ x + k) −→ gr(f)
(x↦→a x+b)→a
−→ gr(a) ⊂ Z ∗ n −→ 1,
which mimicks the exact sequence 0 → (Zn, +) → (Affn, ◦) → (Z ∗ n, ×) → 1.
This means heuristically that if a has many (o(a)) different powers, then the melody will have few (τ)
periods.
Example 2.6 : consider map f : x ↦→ 3 x + 1 (mod 8). Then f ◦ f(x) = 3(3 x + 1) + 1 = 9x + 4 = x + 4
(mod 8) hence f 4 (x) = x (mod 8), τ = 2 = o(a) and o(f) = 4, as predicted.
As in section I, we get
Proposition 2.7 The length of any orbit is a divisor of o(f).
The metaphor of the clocks (fig. 5) is the same as in the case b = 0: all smaller clocks must have looped
to initial position when a ‘day’ is spent, that is to say when f has been iterated a number of times that
equals it to identity. 1
This means that the number of occurences of a given note divides o(f) – but this stands only if the
melody is primitive: random reunions of orbits would of course play havoc with this result. Surprisingly,
one result from the easy case b = 0 remains valid:
2.2 Orbits with maximal length.
Theorem 2.8 There exists at least one orbit with length exactly o(f).
Consequently, o(f) ≤ n. This 2 is NOT obvious, as
• the group of all affine automorphisms has cardinality n Φ(n) and is non commutative; 3 a subgroup (like
the one generated by f) of such a finite group might well have more than n elements;
• in general, the order of a general permutation of n objects may be much greater than n: French composer
J. Barraqué made use of this when he translated twelve tone rows into permutations of 12 objects,
and generated up to 60 series from a single one in this way. 4 The general formula is that the order of a
map that permutes n objects is the lowest common multiple of the cardinalities of its orbits, which is
usually more than the greatest among these cardinalities.
1 This is a special case of a general result in group action theory: cardinals of orbits of a finite group divide the cardinality of the
group, and more precisely the quotient is the cardinality of the subgroup fixing some given element of the orbit. Here this subgroup is
gr(f).
2 o(f) = n for instance when f(x) = x + b, b ∈ Z ∗ n . These maps are conjugates of the basic translation x ↦→ x + 1 in Affn. It happens
also, surprisingly, for non translations, like x ↦→ 5 x + 1 (mod 8) or x ↦→ 16 x + b (mod 45), gcd(b, 45) = 1. These maps are related to
circulant graphs, cf. [15].
3 Except for n = 2; here Φ denotes Euler’s totient function.
4 A permutation with cycles (= orbits) of lengths 3, 4 and 5 has order 60.

Proof We need Dirichlet’s famous theorem on arithmetical sequences:
GGGE♭ 9
In any sequence {v, u + v, . . . k u + v, . . . } with gcd(u, v) = 1, there exists an infinity of prime numbers.
a − 1 b
Let α = gcd(a − 1, b) and u = , v = : u, v are coprime integers, hence we can produce some large
α α
prime p = u x0 + v > n in the sequence above, and:
Lemma 2.9 There exists x0 with p = u x0 + v =
We then compute the length of Ox0:
a − 1
α x0 + b
α
invertible modulo n.
f k (x0) = x0 ⇐⇒ 0Zn = (a−1)x0+b (1+a+. . . a k−1 ) = (u x0+v)α(1+a+. . . a k−1 ) ⇐⇒ α(1+a+. . . a k−1 ) = 0
As seen in Lemma 2.3, this implies that o(f) is a divisor of k, which completes the proof.
Example 2.10 n = 10, a = 3, b = 4.
We compute α = gcd(a − 1, b) = 2. Also r = o(f) = 4 as α (1 + 3 + 9 + 27) = 0 (mod 10). And indeed
there is one orbit of length 4, O0 = (0, 4, 6, 2). Other orbits like O3 or O1 have lengths 1, 2 or 4.
2.3 Unexpected lengths.
Example 2.11 : Sometimes, orbit lengths appear unrelated to the order of a. For instance, though 7 2 = 1
(mod 12), the orbits of x ↦→ 7 x + 2 mod 12 have length 3 or 6 (this last being the order of f). 1
On the other hand, not all divisors of o(f) are lengths of orbits, just as not any divisor of the order of
a group corresponds to a subgroup. Still a composer might wish to repeat some note a given number of
times, and so try and find a map f ∈ Affn with appropriate order. For instance, by Cauchy’s Lemma,
there exists an element of order p in any group whose cardinality is divisible by p, hence if p is a prime
factor of n or Φ(n), there is an element of f ∈ Affn with order o(f) = p.
More generally, any result on the order of elements of group Affn, e.g. Sylow theorems and their like, 2
can be interpreted as a result on lengths of orbits. 3
2.4 Orbits with one element.
The shortest orbits are given by fixed points of the affine map. There is a nice geometric characterization:
2.4.1 Fixed points are for homotheties.
Theorem 2.12 Any autosimilar melody with ratio a and offset b admitting at least one lone note is
generated by a homothety x ↦→ a x (mod n) for some a – if this lone note is chosen as the origin 0 on Zn.
In algebraic terms this means that the map is a conjugate of a homothety.
This means that, at least musically, more than half of all autosimilar melodies belong to the simple case
we studied in the first place. We have already observed this behaviour with the melody of Kientsy Loops
above.
Geometrically this is obvious, if one is willing to convey his or her intuition of affine maps into Zn: if f
fixes z ∈ Zn, then as f is affine it is completely determined by the value of f(z + 1) = f(z) + a. But the
1 The classification of these lengths for c = 12 apparently goes back to [14].
2 Tom Johnson advocated that these textbook theorems on finite groups be applied to the context of autosimilar melodies./
3 For instance, if n = 2 v(2) p v(p) q v(q) . . . , it could be shown that there exists an element of Z ∗ n with order p v(p) q v(q) . . . 2 v(2)−2 , or any
divisor of this. This maximal value is called Λ(n), see note on Carmichael’s function above. See also Sloane’s integer sequence A046790.

10 autosimilar melodies
homothety hz,a with center z and ratio a gives the same values in z, z + 1, so hz,a = f. There is also a
numerical test, that the reader may check with a direct computation:
Theorem 2.13 f is a homothety up to a change of origin iff f(x) = a x + b with b a multiple of a − 1 in
Zn (if b is to be read as a true integer in N, this reads as “ b must be a multiple of gcd(a − 1, n)”).
It is worthy of note that
• if f is a homothety (i.e. admits some fixed point) then o(f) = o(a), but
• the converse is not true: map x ↦→ 3 x + 1 (mod 10) has no fixed point at all but is of order 4, just the
same as its ratio: 3 4 = 81 = 1 (mod 10) and 3(3 x + 1) + 1 = 9 x + 4 = 4 − x (mod 10), 4 − (4 − x) = x.
Two orbits have length 4, and the other has length 2.
2.4.2 Number of fixed points. Contrarily to our intuition in planar geometry, an affine map mod n
may well have several fixed points, e.g. ‘centers’.
Proposition 2.14 Let d = gcd(a − 1, n): if d | b (in N) then we get d fixed points. Else there is none.
Proof x0 is a fixed point ⇐⇒ (a − 1)x0 = −b (mod n).
If d = 1, then a − 1 is invertible, and x0 = (a − 1) −1 b is the only possible fixed point.
If d divides a − 1 and n but not b, we get an impossibility.
Assume d does divide b : b = k d: the equation now reads
(a − 1)x0 = b (mod n) ⇐⇒
a − 1
d x0 = b
d
(mod n
d ),
a − 1
and as and
d
n
are coprime, we get one solution modulo n/d, that is to say d solutions modulo n.
d
So a homothety might have different centers, that is to say an autosimilar melody can be related to
the simplest case in more ways than one. Musically this enables to render the autosimilarity on different
circular permutations of the initial melody.
2.4.3 Number of homotheties. From theorem 2.13 above we can easily compute the number of homotheties
in Affn, as it is a simple matter to enumerate all acceptable b’s for a given a, which are all multiples
of gcd(n, a − 1):
Proposition 2.15 The number of homotheties in Affn, identity map excluded, is given by formula
Nhom(n) =
2≤a≤n−1
gcd(a,n)=1
n
gcd(n, a − 1)
Its maximum value is achieved when n is prime, as for all values of a, gcd(n, a − 1) = 1 and hence
Nhom(n) = n(n − 2) = (n − 1) 2 − 1, among n 2 − n affine maps. By contrast, Nhom(30) = 63 only; and
almost one affine map out of 3 is a homothety when n is a power of 2. It seems that 4, 6 and 12 are the
only values of n with Nhom(n) < n. The proportion of homotheties among general affine maps, depending
mostly on the factorisation of n, is erratic, cf. fig. 8:
On average and for practical purposes, the proportion of homotheties in Affn is around 57% for n ≤ 200.
2.5 Number of different notes
The question of the maximal possible number of different notes (that is to say the number of orbits) for a
map f not equal to identity was answered in the simpler case b = 0. The following result states that the

1.0
0.8
0.6
0.4
GGGE♭ 11
10 20 30 40 50
Figure 8. Proportion of homotheties in Affn.
simpler case is also the general one. We leave the proof to the reader (or Online Supplementary I).
Theorem 2.16 The maximum number of different notes for an autosimilar melody with period n is 3n/4,
which is reached exactly when n = 4k, a = 2k + 1 and b is 0 or n/2.
In general, the total number of notes (or orbites) varies wildly with the modulo and ratio. There is a
formula, making use of stabilizer groups and the celebrated ‘Lemma that is not Burnside’s ’ of group theory
and combinatorics, but it is computationally more efficient just to compute all orbits and enumerate them.
Here it is, wherein X(g) is the number of fixed points of g ∈ Affn (computed from Prop. 2.14):
Proposition 2.17 Let r = o(f), dk = gcd(ak − 1, n) and X(f k
dk if dk | b(1 + a + . . . a
) =
k−1 )
.
0 if not
The total number of notes, i.e. of orbits of f, i.e. of the group G = gr(f) ⊂ Affn, is
g∈G
X(g) |G| = 1
r
r
X(f k )
Here is a plot (fig. 9) with the mean value of the number of different notes for any given n, mean taken
on all ratios a > 1 coprime with n and all possible offsets b.
15
10
5
k=1
20 40 60 80
Figure 9. Average number of notes

12 autosimilar melodies
These computations enable to compute a fairly reasonable 1 value of the probability for a melody to be
[primitive] autosimilar, namely the number of partitions of Zn into affine orbits, over the number of all
partitions (e.g. 2 n ). This probability decreases quickly, for n = 20 it is p = 0.000084877 and for n = 72, with
only 480 partitions into affine orbits, the probability is negligible (≈ 10 −19 ). This shows that autosimilar
melodies are highly organised material, and that autosimilarity is a significant feature.
3 Other symmetries
We will remain in the more general context of affine automorphisms x ↦→ a x + b, not only homotheties.
3.1 Symmetry group
Definition 3.1 The symmetry group of a (periodic) melody M is the subgroup of Affn containing all
maps g satisfying M ◦ g = M, that is to say ∀k M g(k) = Mk. One says that g stabilizes M.
This generalizes Johnson’s remark that a melody invariant under two ratios is also invariant under their
product.
Two extreme examples are a melody that is not autosimilar, meaning its symmetry group contains only
the identity map; and the melody with only one note, which has the whole group Affn for symmetries.
The Alberti bass C G E G C G E G . . . admits all odd ratios for autosimilarity, and more precisely its
symmetry group is made of eight distinct maps mod 8 (this is an abelian group):
x ↦→ x, 3x, 5x, 7x, x + 4, 3x + 4, 5x + 4, 7x + 4
As any autosimilar melody is built up from some map f ∈ Affn, it is obvious that any f k stabilises the
melody. Indeed this means exactly that the melody is autosimilar under map f. The reverse is partially
true:
Theorem 3.2 Let M be a primitive autosimilar melody generated by map f : x ↦→ a x. Then any homothety
g ∈ Affn, e.g. g(x) = c x that stabilises M, is a power of f, e.g. ∃k g = f k , i.e. c = a k .
Maps g(x) = c x where c is not a power of a permute the orbits, that is to say stabilises the rhythmic
structure of the melody, while exchanging its notes.
Proof Assume g(x) = c x stabilises M. In particular, the orbit O1 which contains powers of a is globally
invariant under g, meaning g(1) = c ∈ O1 is some power of a.
If c is not a power of a, as we have seen already, maps of the kind x ↦→ c x turn Ox into Oc x.
This means, quite significantly, that in considering only the simpler affine maps (homotheties), only the
obvious symmetries will occur. The picture is of course different in the whole affine group, and we do not
have a general result. Of course, nothing can be said when the melody is not primitive, since collapsing
some orbits together will increase the symmetry group. Apart from the Alberti Bass, we quote below (fig.
13) one page of the score of Loops for Orchestra by Tom Johnson, wherein the melody admits several
different ratios. The result stands of course for an affine map that is a homothety, up to a change of origin
– the most frequent occurence as we have seen.
Remark 1 One could also look for the larger subgroup of Affn preserving the set structure of orbits –
meaning that exchanges of notes are allowed. In the case of the theorem above we fall back on the whole
group of homotheties, isomorphic to Z ∗ n.
In the more general case, the situation can be less simple: for instance the map x ↦→ 3x + 1 (mod 8)
(cf. Fig. 7) yields the melody CCGGCCGGCCGGCCGG. . . which admits 8 symmetries, like the Alberti
1 There are many different ways to define a probability space on melodies, and about as many different probability values. The order
of magnitude of the result stands, however, regardless of the chosen probabilized space.

ass:
GGGE♭ 13
x ↦→ x, x + 4, 3x + 1, 3x + 5, 5x, 5x + 4, 7x + 1, 7x + 5
Complicated symmetry groups are possible (the last one, often called H8, is not abelian). As all powers
of f are in the symmetry group, we can at least predict from Lagrange’s theorem that
Lemma 3.3 The order of the group of symmetries of M is a multiple of o(f).
It is interesting for composers, and maybe analysts alike, to find a melody with a given symmetry group.
For instance one may wish to find an 8-periodic melody, palindromic, and autosimilar with ratio 3. Hence
the orbit of 1 must contain −1 = 7 (for palindromicity); as it contains 3, 5 = −3 is there too. Hence O1
contains all odd numbers.
Acting similarly with the remaining residue classes, one finds the primitive solution O1 = (1, 3, 5, 7), O2 =
(2, 6) with 4 and 8 standing alone as fixed points. A rendering of this unique solution is the Alberti Bass.
This leads to the most general definition so far, which indeed should be the starting point for the study of
melodies invariant under some affine maps:
Definition 3.4 An autosimilar melody M with period n and symmetry group G ⊂ Affn is any map
M : Zn → (some pitch set) that satisfies
∀f ∈ G ∀k M f(k) = Mk
Theorem 3.5 Such a melody is built from the orbits of G, i.e. each pitch appears on indexes that are a
union of orbits Ox = {f(x), f ∈ G}.
Algorithmically speaking, one usually wishes to consider only some symmetries in G. An orbit will be
produced by repeatedly applying the given affine maps, starting with a given seed, until no new number
is produced. 1 This last definition reduces to the previous one when the group of symmetries is cyclic,
generated by just one map.
It must be pointed out that the group of symmetries eventually achieved is usually larger than the group
one starts from; not any odd group is the symmetry group of some object: for instance modulo 8, this
Klein group
K = {x ↦→ x, x ↦→ 3x + 1, x ↦→ x + 4, x ↦→ 3x + 5}
is not the symmetry group of a melody: its orbits are (0, 1, 4, 5), (2, 3, 6, 7), and the symmetry group of any
melody built up from these [say CCGGCCGG. . . ] is strictly larger (for instance it contains x ↦→ 5x + 4),
with 8 elements.
Remark 2 Musically speaking, for such an autosimilar melody with a non cyclic group of affine symmetries,
it is possible to extract the melody from itself at different ratios. A similar situation will arise in the last
section.
3.2 Palindroms
Now we can clarify when a given autosimilar melody will be a palindrome, as this means exactly that
x ↦→ −x is an element of the group G of symmetries. The answer is obvious when G is generated by
ha : x ↦→ a x, as we have seen that the symmetries must be powers of ha:
1 The underlying idea is that: any orbit is a fixed point of the action of any set of generators of G on the set of all subsets of Zn. This
was implemented in OpenMusic, cf. [3].

14 autosimilar melodies
Theorem 3.6 A primitive autosimilar melody M with ratio a, period n and offset 0, will be palindromic
⇐⇒ some power of a is equal to −1 (mod n).
The question of whether -1 is a square modulo n (a quadratic residue) is familiar in number theory,
but the concept of -1 being a power residue appears to be novel. The sequence of moduli n admitting
such a possibility: 5, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33. . . has been
added to Sloane’s online encyclopedy of integer sequences under number A126949. 1 It is a rather common
occurence.
For instance 19 belongs to the sequence because 10 9 = −1 (mod 19).
Again, this stands only for primitive melodies: it is always possible to build a palindromic (autosimilar)
melody from any autosimilar melody, by setting identical notes on pairs of orbits Ox and O−x, e.g. turning
C G E G B A E A. . . into the Alberti bass.
What of a melody autosimilar with offset ? The infinite melody A A B B A A B B . . . is only a palindrom
up to an offset of the origin. Here is a sufficient condition:
Theorem 3.7 If M is a primitive autosimilar melody generated by f : x ↦→ ax+b (mod n); if some power
of a is equal to −1 (mod n), then M will be palindromic up to some offset.
Proof Assume that a r = −1. Then f r (x) = −x + c for some c and hence Mk = M f r (k) = Mc−k, i.e. M is
palindromic for some starting point.
The reverse, unfortunately, is false when the map is not a homothety: no power of 3 equals -1 modulo
8, though x ↦→ 3x + 1 generates a palindromic melody with offset. And for some pairs (a, n), no value of b
will allow x ↦→ a x + b to generate a palindrom, for instance x ↦→ 8x + b (mod 15).
4 Autosimilarity and tilings
This is about some mosaics, or tilings, which are deduced from some autosimilar melodies. Most of this
section had to be moved to the Online Supplementary in order to save space.
4.1 Autosimilar tesselations
The problem of tesselating Zn is an ancient and difficult one (see [1, 4, 8]). It can be asked whether an
autosimilar melody tesselates Zn, meaning that all orbits are translates of one another.
Example 4.1 All maps of form x ↦→ x + b will give some tesselation (with only one orbit when b ∈ Z ∗ n).
A less trivial case: 7x + 7 (mod 12) has two orbits in Z12, (0 3 4 7 8 11) and (1 2 5 6 9 10), which make
up a tiling.
It is rather difficult to meet the tight requirements for an autosimilar melody to be a tiling: the orbits
must be not only the same size, but also the same shape. Also the last example shows why tiles will often
be periodic themselves (i.e. all invariant under some same translation): if some power of the map f is a
translation x ↦→ x + τ, then all orbits must be τ−periodic. We have only a very partial result:
Theorem 4.2 A (primitive) melody autosimilar with ratio a = 1 and period n > 4 gives a tiling of Zn by
translations of a 2-tile iff n = 4k, a = 1 + 2k, b = ±k with k odd.
Remark 1 It is easier to build tiles with unions of orbits of an affine map but we have no general result.
Indeed the question of tiling vs autosimilarity arose when Tom Johnson found (0, 1, 3, 7, 9), which tiles Z20
by translation and also with ratio 3.
1 http://www.research.att.com/ njas/sequences/A126949

4.2 Tilings with augmentations
GGGE♭ 15
In the simplest case n prime, b = 0, we get tilings of Z ∗ n with augmentations, as any orbit except O0 = (0)
is an augmentation of O1: Ox = xO1. For instance, (1 2 4) and its augmentation (3 6 12) tile together Z ∗ 7
as (3 6 12)=(3 5 6) mod 7.
But there is a better way to obtain a whole family of tilings with particular affine maps. But the tiles
will no longer be the orbits: on the contrary they will be sets transverse to them.
Lemma 4.3 Any autosimilar melody whose orbits share the same length enables to build tilings with
augmentation.
Proof Consider any set transverse with the orbits, i.e. X containing one point and only one from each
orbit. Then X, f(X), . . . f r−1 (X) partition Zn, i.e. X tiles with augmentations a X + b a.s.o.
Example 4.4 All the orbits of f : x ↦→ 13 x + 3 (mod 20) have length 4: (0 2 3 9), (1 6 11 16), (4 15
17 18), (5 7 8 14) and (10 12 13 19). Take for instance the first elements: X =(0 1 4 5 10). Applying f
yields all following members of each orbit: f(X) = (3 16 15 8 13). Iteration of the process gives a mosaic,
where all motives are images of the preceding one by the map f. Notice that one can choose any starting
element in each orbit.
For this to happen, all orbits must share the same length. A discussion of this condition is enclosed in
Online Supplementary I.
5 Approximate autosimilarity
In a way, autosimilarity in a (periodic) melody is a special form of redundancy: as we have seen by now,
autosimilarity is an aural illusion, where identical notes are identified though lying in fact in different
positions in the original melody and in its augmentation. It is a legitimate question to ask for approximate
autosimilarity: what if some melody is autosimilar except for a few notes ? With which ratio ? It turns out
that this can be investigated with a simple and fast algorithm, and also that such relaxed autosimilarity
appears surprisingly often in the corpus of classical music.
5.1 Algorithm
Let M be some melody with period n. We define the periodic augmentations of M as the a M + b in
symbolic notation, meaning here the sequences (M a k+b (mod n))k∈N. As seen above, M is autosimilar iff
a M + b = M for some a, b. We can compute a correlation coefficient between all a M + b and M itself
by checking the proportion of notes that are the same – technically it is a comparison of circular lists, or
circlists, between M and a M. Checking for maximum on b, then a, allows to find the best candidate for
autosimilarity. Here is an implementation in Mathematica R○ . Comparison of circlists uses a trick: Union is
applied to pairs {Mk, Ma k+b}; if it gives a singleton, there is coincidence.
correl[melo_, a_]:= Module[{n=Length[melo], meloBis},
(* local variables *)
meloBis = Table[melo[[Mod[a*(k+decalage)-1 , n]+1]], {k,n},{decalage, n}];
(* these are the alternate melodies a*M + decalage *)
Max[(Count[Length /@ (Union /@ Transpose[{melo, #}]), 1])& /@ meloBis]/n]
For instance, trying this function on a modified Alberti bass:
Table[correl[{C, G, E, A, C, G, E, G},k], {k, {3,5,7}}]
yields the correlation coefficients
3 3
4 , 1, 4 . So though 3 is no longer a ratio for autosimilarity, 5 still is.

16 autosimilar melodies
5.2 Example
It is fairly obvious that no autosimilarity will be found when all notes are different – the melody cannot
be broken down into orbits in that case. But with repeated rythmic motives involving repeated notes,
approximate autosimilarity may well be found. The first melodic sentence of Beethoven’s Fifth exhibits
very good autosimilarity when cut down to 12 notes: G, G, G, E♭, A♭, A♭, A♭, G, E♭, E♭, E♭, C has a
correlation coefficient of 5/6 for x ↦→ 7 x + 6. Musically this means that only two pcs are not identical in
the augmented version. These alien pcs have been signaled as chords together with the ‘expected’ note on
the score fig. 10 (notice also the octave identification for E♭’s).
Figure 10. A famous almost autosimilar melody
With a little bad faith, strong approximate autosimilarity could be found widely in classical or Jazz
music, as shown by this very first try. It is also part of the routines looking for ‘interesting melodies’ in
the OMax software for improvisation developed in Ircam.
6 About general affine maps (not one to one)
Most of this section also had to be transferred to Online Supplementary II.
6.1 Definition
The condition that the ratio a be invertible modulo n may seem a little artificial, a contrivance in order
to allow the mathematical tools to come in. Actually the initial definition can work well with any ratio
(not zero), as for instance melody
D, G, F, G, D, G, F, G, D, G, F, G, D, G, F, G, D, G, F, G, D, G, F, G . . .
with period 24 is certainly autosimilar with ratio 3, though 3 is not coprime with 24. So we have to consider
what happens when iterating an affine map that is not bijective.
There is a very pleasant theorem, establishing autosimilar melodies as universal objects.

GGGE♭ 17
Theorem 6.1 The iteration of any affine map f modulo n (not one to one) eventually reduces to iterating
an affine transformation on some subset of Zn. Musically, this means that one hears an autosimilar
melody after several augmentations of any periodic melody. Mathematically, it means that the submelody
M = M ◦ (f p ) =
Mf p (k) k∈Z is autosimilar under some power of f: M ◦ (f q ) = M for some p, q.
Example 6.2 Consider this seemingly random sequence of 36 notes as a periodic melody:
D, C, G, G, B, F, A, B, F, F ♯, B, E, B, B, G, F ♯, F ♯, C, E, C, C, E, E, E, F ♯, F ♯, E, A, C, E, E, E, D, F, F ♯, E, (D, C, G .
The two first iterations of map x ↦→ 3x − 1 (mod 36), that is to say picking one note out of three starting
with the second, yield successively
C, B, B, B, B, F ♯, C, E, F ♯, C, E, F ♯, C, B, B, B, B, F ♯, C, E, F ♯, C, E, F ♯, C, B, B, B, B . . .
B, B, E, E, B, B, E, E, B, B, E, E, B, B, E, E, B, B, E, E, B, B, E, E, B, B, E, E, . . .
the last of which is periodic and autosimilar: further iterations of the same transform will yield the same
melody. We notice that several notes have disappeared, and that the ultimate period is smaller than 36.
The algorithm that enables to construct such an ultimately autosimilar melody is straightforward:
Definition 6.3 We generalise the definition of orbits to the attractor of x: Ax = {f k (x) | k ≥ p) (where
p is defined in thm. 6.1). It is the part of the sequence f k (x) that loops (beware ! usually x /∈ Ax. . . ).
Now it only remains to make sure that, as in the preceding theorems about building autosimilar melodies,
all notes with indexes in the same Ax are identical. In the above example, we have two attractors, A =
(5, 14) and B = (23, 32). The initially completely random melody M was modified by setting M14 = M5 =
B, M23 = M32 = E. All other notes are irrelevant. Musical applications could involve extracting a simple,
autosimilar beat, from a complex melody.
Another nice application is to arrange the initial melody in order to support several extractions of
ultimately autosimilar melodies. For instance,
E, F ♯, B, G, C, G, F ♯, G, E, A, F ♯, F ♯, C, C, B, G, F, G, C, G, F ♯, G, G, F ♯, A, F, G, G, E, F ♯, F ♯, G, F, G, F, B . . .
gives two autosimilar melodies when augmented by 2 or by 3 by applying the same trick as above, both
to the attractors of x ↦→ 3 x + 1 and those of x ↦→ 2 x. It is plainly visible on the ‘score’ below that two
(simple) autosimilar melodies emerge (the initial melody is the middle voice).
Figure 11. Two attractors for one melody
Remark 1 What remains after m iterations of f on the original melody is not necessarily invariant under
f itself, but as proved in the supplementary, it is always invariant under some f q . In other words, it has

18 autosimilar melodies
at most q ‘alternate melodies’ (under iteration of f). Though in theory one might stumble on the case
f q = id (for instance if the original melody has n distinct notes!), in practice one frequently finds some non
trivial attractor. In this sense, autosimilar melodies are exactly the attractors of any affine map operating
on any initial (periodic) melody, thus reaching the elated status of universal object.
7 Perspectives
7.1 More symmetries
Other generalizations are of course possible. Uncharacteristically, Tom Johnson put forward in [9] a false
conjecture, refuted by Feldman ( [7]), that hints at transformations of the pitch- or time-space more general
than inversion or retrogradation. The conjecture was:
A related melody [= an augmentation of the original melody] produced by playing a melodic loop [= a periodic
melody] at some ratio other than 1:1, can never be the inversion of the original loop, unless it is also a retrograde
of the original loop.
The melodies satisfying this conjecture can be formalized and generalized in the following way:
Definition 7.1 Let G be some finite order transformation of the pitch (classes) domain, and f ∈ Affn.
We define a melody M0, M1, Mn = M0 with period n autosimilar under f, with respect to G, by the
condition ∀k ∈ Zn M f(k) = G(Mk).
We state without proof that this occurs whenever (Mk) is built from the iteration of g on the orbits of
f:
∀x ∈ Zn, ∀k ∈ Z, M f k (x) = G k (Mx).
Also, for this to happen, the order of G must divide all the orbit lengths.
David Feldman, who cast the first shrewd mathematical look on these melodies (he used some himself as a
composer) yet unfortunately only published one page in [7] about them, explains why Johnson’s conjecture
will be true in most cases, and provides a counterexample with period 15. Inversion is G(m) = −m, but
other operators are possible: a reviewer suggested G(m) = m + 3 (mod 12) [on length 4 orbits]. Feldman’s
example is similar to the following: take a = 2, n = 15 and fill in the ordered orbits (1 2 4 8), (3 6 12 9),
(5 10), (7 14 13 11) with alternate opposite values of one note (0 is C, 1 is C♯, 2 is B, a.s.o.), we get by
construction M f(k) = −Mk ∀k, see fig. [?] with the inverted elements of orbits in blue:
Figure 12. One note out of two gives inversion, not retrograde
7.1.1 The ratio that retrogrades. The above suggests looking for the retrograde among the augmentations
of a melody. While composing an electronic piece, Orion, we found (C D E C D F C D G C D
E . . . ), autosimilar with period 9, ratio 4 and offset 6: picking every odd note turned the melody into its
retrograde (up to offsetting): (C E D C G D C F D C E D C . . . ), i.e. M2k+1 = M8−k. As it happens, this
is a general phenomenon:

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.

20 autosimilar melodies
[8] Fripertinger, H., Tiling Problems in Music Theory in in: E. Lluis-Puebla, G. Mazzola et T. Noll (eds.), Perspectives of Mathematical
and Computer-Aided Music Theory, EpOs, 153–168, Universität Osnabrück, 2004.
[9] Johnson, T., Self-Similar melodies, (1996), Two-Eighteen Press, NY.
[10] Mazzola, G., et alli, Topos of Music (2002), Birkhaüser.
[11] Morris, R., Composition With Pitch Classes: A Theory of Compositional Design, (1987), New Haven, Yale University Press.
[12] Rahn, J., Basic Atonal Theory, (1980), Longman, New York.
[13] Schillinger, J.,The Schillinger System of Composition, (1946), Carl Fischer Inc, New York
[14] Starr, D., Morris, R., A General Theory of Combinatoriality and the Aggregate, (1977-78) Perspectives of New Music 16(1)
[15] Weisstein, Eric W. Circulant Graph., from MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CirculantGraph.html
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Online Supplementary I
7.3 Proof of Thm. 2.16
GGGE♭ 21
The theorem states that the maximum number of orbits (and hence of different notes in the melody) is
3n/4.
Proof Consider some autosimilar melody with period n generated by map f : x ↦→ a x + b. Obviously, for
the number of orbits to be maximal, their sizes must be minimal. Rejecting the case f = id, this means
that there are as many one note orbits (i.e. fixed points) as possible, the rest being arranged in two note
orbits. But from Prop. 2.14, the number of fixed points is the gcd of b, a − 1, n, and thus we can do no
better than a − 1 = b = n/2 (with n even). n/2 points remain; in order to arrange them by two note orbits
n/2 must be even also. So necessarily n = 4k, b = 2k, a = 2k + 1.
Now we just have to check the converse: for such n, a, b there are n/2 fixed points:
f(x) = x ⇐⇒ (2k + 1)x + 2k = x ⇐⇒ 2k(x + 1) = 0 (mod 4k) ⇐⇒ x is odd
and the other points (even values of x) come in pair orbits : x, f(x) = (2k + 1)x + 2k as then x = f(x) and
f(f(x)) = (2k + 1)[(2k + 1)x + 2k] + 2k = 4k((k + 1)x + 1) + x = x (mod n)
We have n/2 orbits with 1 element and n/4 orbits with two elements, which totals 3n/4 different notes.
7.4 Tesselations with autosimilarity
7.4.1 Proof of Thm. 4.2. The theorem characterized autosimilar melodies that tile by translation with
a 2-tile.
Proof It is easy to check that the given condition provides 2k orbits with two elements, as f : x ↦→
(2k + 1)x ± k has order 2:
f(f(x)) = (4k 2 + 4k + 1)x ± k(1 + 2k + 1) = x ± 2k(k + 1) = x as k s odd and 4k = 0
and f(x) − x = 2k x ± k is equal to ±k when x is even, and to 2k ± k = ∓k (mod 4k) when x is odd: so
all orbits, being pairs with equal diameter, are translates of one another.
Conversely, let us assume that f has order two: f 2 = id i.e. a2 = 1 and b(1 + a) = b + a b = 0.
If (as assumed) O0 = (0, b) is a translate of O1 = (1, a + b), then both pairs have same diameter:
either b − 0 = a + b − 1 or b − 0 = 1 − (a + b). The first case is forbidden as a = 1. So a + 2b = 1. We
must now consider O2 = {2, 2 a + b} (unless 2 is a member of O1 or O2, which case is easily excluded) :
either 2 a + b − 2 = b − 0 or 2 a + b − 2 = 0 − b, i.e. 2a + 2b = 2 which would imply again (substracting
a + 2b = 1) a = 1. Only the former case is possible: hence 2a = 2 which is allowed only if a = n
2 + 1. Then
2b = 1 − a = n
2 , implying that n must be a multiple of 4. Finally n = 4k, a = 2k + 1 and 2(b + k) = 0
(mod n) i.e. b = ±k. Moreover k must be odd, or else b(1 + a) = ±2k(k + 1) = 0 (mod n = 4k).
Going from one orbit to the next is like braiding a girdle, as the first note of one orbit is translated to
the last note of the next, and vice versa.
7.4.2 Tilings with augmentations. The main text has stated that
Any autosimilar melody whose orbits share the same length enables to build tilings with augmentation.
Under this condition, a tile is just any set with one note exactly in each orbit.

22 autosimilar melodies
Now we would like to know when this can happen. We have no simple arithmetic characterization, but
a few interesting conditions. Notice that a − 1 is not allowed to divide b (in Zn) – which excludes a − 1
being coprime with n in Z, or b = 0 – or else we have fixed points, by Prop. 2.14.
Lemma 7.4 All orbits have the same length whenever the smallest orbit has length (some multiple of)
o(a).
Proof Let x0 belong to the smallest orbit with length m: ((a − 1)x0 + b)(1 + a + . . . a m−1 ) = 0 and
∀x, ((a − 1)x + b)(1 + a + . . . a r−1 ) = 0 ⇒ r ≥ m
All orbits will share the same length m iff r = m works, i.e.
∀x 0 = 0−0 = ((a−1)x+b)(1+a+. . . a m−1 )−((a−1)x0+b)(1+a+. . . a m−1 ) = (a−1)(x−x0)(1+a+. . . a m−1 )
As (for x = x0 + 1) (a − 1)(1 + a + . . . a m−1 ) = a m − 1 = 0 iff m is a multiple of o(a), and ((a − 1)x0 +
b)(1 + a + . . . a m−1 ) = 0 by assumption, this works.
Reverse implication: if all orbits share the same length m, we know that this length is the order of f,
which is always a multiple of o(a).
Computationally,
Theorem 7.5 All orbits will have the same length m ⇐⇒ ∀x
((a − 1)x + b)(1 + a + . . . a m−1
) = 0 ⇒ o(a) | m
that is to say one cannot have ((a − 1)x + b)(1 + a + . . . a m−1 ) = 0 without o(a) dividing m.
Proof Assume all orbits have the same length m; from the Lemma above, that length is a multiple of o(a).
The length of Ox is the smallest r such that
a r x + b (1 + a + . . . a r−1 ) = x ⇐⇒ ((a − 1)x + b) (1 + a + . . . a r−1 ) = 0
By assumption, this r is a multiple of o(a). Hence a r = 1. Hence also b (1 + a + . . . a r−1 ) = 0.
Now for any m > 0 with
((a − 1)x + b) (1 + a + . . . a m−1 ) = 0
this means f m (x) = x and m is a period of the restriction of f to Ox. As seen previously, this means that
m is a multiple of r, length of Ox: again a m = a kr = (a r ) k = 1 i.e. o(a) | m.
Reverse implication: assume that condition ((a − 1)x + b) (1 + a + . . . a m−1 ) = 0 always implies that
a m = 1, i.e. o(a) | m. This means literally that all orbit lengths m are multiples of o(a); by the lemma
above, it proves that all orbits have the same length.
There is little hope of simplifying this condition: one has to look into the sequence 1 + a + . . . a m−1 (m
varies from 1 to some divisor of o(f)) for factors c common with n, and look for arithmetic sequences
(b + x(a − 1)) (x from 0 to n/ gcd(n, a − 1)) that do NOT provide the missing factors n/c. In the last
example, the only possible case was with m = 2, 1 + a = 14, common factor 2: one had to find arithmetic
sequences with ratio 12, where no term is a multiple of 20/2 = 10.
Example 7.6 For instance, for f : x ↦→ 13 x+3 (mod 20), the sequence of the 1+a+. . . a m−1 takes values
1, 14, 3, 0 cyclically. The order of a = 13 is 4 mod 20. For b = 3, one computes b+x(a−1) = 3, 15, 7, 19, 11,
neither of which gives 0 when multiplied by 1, 14 or 3. So the condition is verified, as it is for any odd b.
But for (say) b = 2, it does not work (|O4| = 2). The sequence (12 x + 2)x=0...4 contains 10 (for x = 4),
and this enables ((a − 1)x + b) (1 + a + . . . a m−1 ) = 0 for m = 2 < 4, i.e. an orbit of length 2 instead of 4.

GGGE♭ 23
The smallest odd value of n satisfying this condition is: n = 21, with a = 4.
Remark 1 When equal length orbits are longer than o(a), it means that o(f) > o(a). In that case, all
orbits will be periodic, as they will be invariant under f o(a) , which must be a translation as f o(a) (x) =
a o(a) x + · · · = x + . . . (not x + 0 by assumption). For instance with
f : x ↦→ 3 x + 1 (mod 8) O0 = (0, 1, 4, 5) O2 = (2, 3, 6, 7) f 2 (x) = x + 4
But such periodicities of orbits cannot happen when o(a) = o(f), as we have seen with x ↦→ 13 x + 3
(mod 20).
Remark 2 Say a = −1 for some even n = 2k and b = 2k + 1: f : x ↦→ 2k + 1 − x (mod n) is a map with
all orbits of length 2. Barring those ‘trivial’ solutions, the first values of (n, a) giving such tilings are (with
adequate values for b)
(8, 3) (8, 5) (12, 5) (12, 7) (16, 2k + 1) (18, 7) (18, 13) (20, 9) (20, 11) (20, 13) . . .
Some other families of solutions could be devised likewise, e.g. when n = 4k, a = 2k±1, f : x ↦→ (2k±1)x+1
has orbits of length four, and provides a tiling. But musically (following Tom Johnson’s advice) it is better
to keep to small values of a, so we won’t pursue this line.
7.5 Multiple symmetries
7.5.1 A false conjecture. This was the conjecture stated in [9]:
A related melody produced by playing a melodic loop [= a periodic melody] at some ratio other than 1:1, can
never be the inversion of the original loop, unless it is also a retrograde of the original loop.
What Tom Johnson means by ‘related melody’ is just some f[M] = (Mf(k))k∈Zn ; an autosimilar melody
is precisely a melody M that is equal to one of its related melodies.
Here we are looking for periodic melodies satisfying a condition
M f(k) = p − Mk ∀k ∈ Zn (where p is some constant)
For this to happen, we need the musical space wherein M takes its values to possess some minimal algebraic
structure, which is usually true in most models (eg pc-space).
The conjecture states that the above condition implies
∀k M−k = M f(k). (4)
Feldman shrewdly points out that Tom’s conjecture will be true when n is prime [and f is homothetic]
and provides a counterexample with period 15. Let us have a closer look.
The inversion condition implies that M itself is autosimilar under f 2 , as
∀k Mf 2 (k) = p − Mf(k) = p −
p − Mk = Mk
As the inversion acts on the orbits of f, they must all have even cardinality: let us consider the simpler
case f(x) = a x, say note 1 is pitch x, then note a is pitch p − x, note a 2 is pitch x again, a.s.o. What we
want to avoid in order to disprove the conjecture is −1 ∈ O1, as then the melody would be invariant under
x ↦→ −a x. As seen above, -1 is often a power residue (this explains Johnson’s error) and, for instance when
n is prime, if a is of even order 2k, then −1 = a k as a k = 1 is solution of X 2 = 1 in the field Zn, where
this equation has only two solutions ±1 (this is Feldman’s argument).

24 autosimilar melodies
But in Z15 for instance, X 2 = 1 has other solutions (eg 4); taking a = 2, n = 15 and filling in the ordered
orbits (1 2 4 8), (3 6 12 9), (5 10), (7 14 13 11) with alternate opposite values of one note (0 is C, 1 is C♯,
2 is B, a.s.o.), we get by construction M f(k) = −Mk ∀k, close to Feldman’s example.
7.5.2 Proof of Thm. 7.2. Proof Consider an autosimilar melody generated by f : x ↦→ a x + b and let
r = −a −1 , which is coprime with n; hence the sequence
M0, Mr, M2r, . . .
contains all the notes of the sequence M0, M1, . . . though in a different order. We know by construction
that
Hence (putting y = a x + b)
∀x, Ma x+b = Mx.
∀y, My = M−r y+c ⇐⇒ Mr y = Md−y
for some offsets c, d. This is what we endeavoured to prove: that some augmentation of the melody is one
of its retrogrades.
7.5.3 Inverse-retrograde symmetry. The last situation is about melodies whose inverse IS the retrograde
(like in Tom Johnson’s conjecture).
For instance, with f : x ↦→ 3x + 1 mod 26: see fig. 14.
Figure 14. Autosimilar melody (ratio 3) with inverse-retrograde symmetry
It can be seen, and even better, heard, that O0 (the B flat’s) and O8 (the C’s) (resp. O2 and O3)
are retrogrades one of another. This allows a pretty rendition of the melody, setting opposite notes for
symmetric orbits: then the retrograde of the melody will be its inversion, as seen on figure 14 (the symmetry
axis for pitches is around F).
It is somewhat difficult in fact, to construct an example of an autosimilar (primitive) structure without
such a symmetry (this is akin to Johnson’s conjecture). For one thing, if f is a homothety (recall this
happens whenever a − 1 | b in Zn, not only when b = 0), then x ↦→ −x permutes the orbits, as any other
homothety does. Also if some power of a is equal to c − 1, we get directly a palindrom.
Still, an autosimilar melody built from 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 given in the main text a condition ensuring that
such retrogradation symmetries between orbits exist: Thm. 7.3. Here is its proof.
Proof We assume that 2b = c(a − 1) and consider symmetry S : x ↦→ −c − x. The theorem states that S
acts on the set of all orbits, i.e. the symmetric of any orbit is some orbit, i.e. for any x, S(Ox) = O S(x).
Recall Ox = {x, f(x), f 2 (x), . . . } where f(x) = a x + b; it is sufficient to notice that f ◦ S = S ◦ f, i.e.
f(S(x)) = a (−x − c) + b = −a x + b − a c = −a x − b − c = −c − (a x + b) = S(f(x))

GGGE♭ 25
as a c − b = b + c by hypothesis.
From there we get immediately S(f k (x)) = f k (S(x)) and hence S(Ox) ⊂ O S(x). By symmetry (reasoning
on S −1 which is none other than S itself !) the inclusion is an equality.
These examples open a new alley for future research, combining inner symmetries (the autosimilarity)
of a melody with outer symmetries (e.g. inversion), using some structural features of the space of musical
events.
8 Online Supplementary II: about general affine maps
Here we consider what happens when iterating an affine map that is not bijective.
8.1 Universal property
Theorem 6.1, establishes autosimilar melodies as universal objects. Musically this means that one hears
an autosimilar melody after several augmentations of any periodic melody.
The iteration of any affine map f modulo n (not one to one) eventually reduces to iterating an affine trans-
formation on some subset of Zn.
Mathematically, it means that the submelody M = f p (M) =
Mf p (k) is autosimilar by some power of
k∈Z
f: f q [ M] = M for some p, q > 0.
Proof The set (algebraically, a monoid) of all affine maps modulo n is finite. Thus the sequence of powers
of f will only take a limited number of different values. So there must exist two different exponents p, p + q
with f p = f p+q . Now for any r > p,
f r+q = f (p+q)+(r−p) = f p+q ◦ f r−p = f p ◦ f r−p = f p+(r−p) = f r
We have just shown that the sequence of powers of f is ultimately periodic. So is for any x ∈ Zn, the
sequence f k (x). This means that after p iterations of f, any further iteration of f q will preserve the
sequence.
8.2 A Fitting ending
We will round up this last theorem with a more detailed explanation in the simpler case of homotheties,
which links this result with the abstract Fitting Lemma already connected with several musicological
situations (Anatol Vieru’s iteration of the difference operator, [5]). 1 A connection to the general case is
that the ultimate period of f ∈ Affn : x ↦→ a x + b is a multiple of the ultimate period of its linear part
f : x ↦→ a x.
Let us consider this map x ↦→ a x mod n. First we will assume for simplicity’s sake that gcd(a, n) = p
is a prime factor. This means n = p m q where q is coprime with p. Now x ↦→ a x maps Zn into the cyclic
subgroup of index p, namely p Zn, isomorphic with Zp m−1 q. After m iterations we are working in p m Zn,
cyclic subgroup of Zn isomorphic with Zq. There x ↦→ a x is one-to-one, at long last.
Proposition 8.1 The attractors Ax = {a k x, k ≥ m} are the orbits of f : x ↦→ a x operating on Z/qZ,
identified to subgroup p m (Z/nZ).
1 It is worth noticing that orbits, for homotheties, or their difference sets, for general affine maps, are exactly the eigenvectors of
Vieru’s difference operator acting on subsets of Zn:
∆(x, a x, a 2 x . . . ) = (a x − x, a 2 x − a x, . . . ) = (a − 1)(x, a x, a 2 x . . . )

26 autosimilar melodies
At this juncture, everything is like in section I: f cycles around the Ax, generating an autosimilar melody.
In a more general setting, this is a case of the Fitting Lemma, a very abstract result on decomposition of
modules in commutative algebra:
Theorem 8.2 Let p1, . . . pr be the prime factors belonging to both a and n:
a = p m1
1 . . . p mr
r . . . p mℓ
ℓ (ℓ ≥ r) n = p n1
1 . . . pnr r × Q = P × Q, with gcd(P, Q) = 1
then the sequence (a k x)k∈N is ultimately periodic, from at least the rank m verifying (n/Q) | a m , i.e. the
smallest integer exceeding all ratios ni/mi.
The periodic parts of this sequence, i.e. the attractors Ax = {a k x, k > m}, partition the sub-group n
Q Zn
of Zn, isomorphic with ZQ.
Proof We use the Chinese Remainder Theorem: the ring Zn is isomorphic with the ring product ZP × ZQ
(meaning essentially that any residue class modulo n is well and truly determined by its residues modulo P
and modulo Q). Thus any (affine) map in Zn can be decomposed into two (affine) components on ZP and
ZQ: if x ∈ Zn corresponds to (y, z) ∈ ZP × ZQ then f(x) corresponds to ( f(y), f(z)) where f(y) = f(x)
mod P, f(z) = f(x) mod Q.
Consider map f : x ↦→ a x ∈ Zn: as a m = 0 mod P , we have f m = 0 (the null map), i.e. f is nilpotent;
conversely, as a is coprime with Q, the other component f is one to one. After k ≥ m iterations, f k reduces
to ( f k , f k ) = (0, f k ) and we are back to section I.
Remark 1 The ultimate period, q, is the order of a in Z ∗ Q .
Example 8.3 Take f : x ↦→ 10 x in Z84. Here P = 4, Q = 21. The projections of f are f(y) = 10y
(mod 4) = 2y (mod 4), f(z) = 10y (mod 21)). We get f 2 (y) = 0, but f cycles on length 7 orbits. Hence
after two rounds, we get an autosimilar melody, with each note repeated 7 times. For instance, iterating
f on 1 yields 1, 10, 16, 76, 4, 40, 64, 52, 16, 76, 4, 40, . . . corresponding to the iterates of f : 1, 2, 0, 0, 0, 0 . . .
and of f : 1, 10, 16, 13, 4, 19, 1, 10, 16, 13, 4, 19, 1, 10, . . .

AUTHOR: AMIOT Emmanuel
TITLE: Discrete Fourier Transform and Bach’s Good Temperament
KEYWORDS: Temperament, tuning, Fourier Transform, Lehman, J.S. Bach, Wohl Temperierte Klavier.
Emmanuel Amiot
Prof. in CPGE, Perpignan.
1, rue du Centre
F 66570 St Nazaire
manu.amiot@free.fr
ABSTRACT:
The statement by Bradley Lehman that the scribble at the beginning of the first page of WTC shows how J.S.
Bach tuned his harpsichord is much disputed. The use of a mathematical and rather abstract characterization
of major scales puts forward a quality of the Lehman temperament that singles it out among all competing
temperaments, thereby sustaining his claim.
Accompanying files: 2gammes.pdf, program.pdf, huddling.pdf, minors.pdf, tableauMSS.pdf, wtc.gif, algo.pdf,
heptagon.pdf, justFifth.wav, shortFifth.wav, mediumFifth.wav, NearestHeptagons.mov.
Introduction
0.1. Lehman’s hypothesis. In two papers ([6, 7]) published in 2005, Bradley Lehman introduced the view
that the recipe for the long lost temperament of Johann Sebastian Bach, had in fact been lying for all to
see – not unlike Poe’s Purloined Letter – as a scribbling on the front page of the autograph edition of Das
Wohltemperirte Clavier, the ‘Well Tempered Clavier’ (henceforth abbreviated as WTC). The two papers are
downloadable here. As Lehman’s own learned comparisons with many previously known tunings make clear,
this is a vexed question, and his ‘discovery’ aroused a number of refutations, on various grounds.
Lehman’s homepage is accessible by clicking here, with abundant discussions, and links. Here is a short summary
of his interpretation of picture 1.
Figure 1. Bach’s diagram on the first page of WTC.
Let us recall the gist of the problem of tuning. Twelve just fifths (frequency ratio 3/2) amount to seven octaves
plus a Pythagorean comma. Using eleven just fifths leaves a ‘wolf’, e.g. one ugly fifth, and all tonalities featuring
this fifth will sound wrong. The abstract solution is equal temperament, wherein all fifths are reduced by one
twelfth of a comma, but then all intervals are out of tune, and all scales sound (equally) disharmonious. The
problem of tuning thus consists of adjusting with different fifth sizes, aiming at a temperament wherein all
tonalities sound well. This is exactly the meaning of ‘Wohl Temperirte’ as Bach explains it on the first page of
WTC. According to Lehman, the picture before the text (fig. 1) gives the directions for such a temperament.
First the picture must be turned over; now consider all those loops as directives on how to tune successive fifths,
beginning with F-C. The small inner loops represent nudges, slight moves of the tuning-pin. 1 There are either
2, 0 or 1 nudges. The total is 13 nudges (=2+2+2+2+2+0+0+0+1+1+1), suggesting that each nudge should
be one thirteenth of the Pythagorean comma 2 in order to get round the circle of fifths smoothly. Eventually, the
1 Counterclockwise, i.e. decreasing the size of the fifth.
2 Lehman uses one twelfth of a comma, which is indistinguishable in practice.
1

2
circle of fifths begins with five short fifths (like numerous generalized meantone temperaments), carries on with
three pure fifths and ends with intermediate-size fifths (click for the sound files). It is fairly simple to realize
in practice, in accordance with the well-known fact (or legend) that Bach tuned a harpsichord in a quarter of
an hour. Lehman made a movie showing the tuning of an octave on YouTube, and accessible here. The three
different sizes of fifths are hard to distinguish when played alone, Lehman’s videos are much more explicative.
0.2. Why it is unprovable. . . The unsolvable problem of tuning is well understood since Ptolemaios at least:
twelve just fifths are slightly in excess of seven octaves, (3/2) 12 ≈ 129.75 > 2 7 = 128, and more generally, there
is no way any number of fifths can equal some number of octaves, as the equation (3/2) n = 2 p admits no
solutions in integers n, p > 0. Fiddling with the sizes of fifths introduces problems with other ‘pure’ intervals,
like the major third, that should ideally be close to 5/4.
There can be no ideal solution, since a number of different properties are desirable, competing and vying with
one another. Hence the musicological debate on the quality of Lehman’s temperament (henceforth called LT)
might go on for ever, as proponents of any other temperament will put forward (usually in good faith) diverse
qualities, 3 often sporting personal preferences instead of facts (see for instance the acerbic [9] and its refutation
by Lehman).
0.3. . . . or is it ? We need some objective quality to assess a temperament, preferably addressing the whole
collection of major (and minor) scales – as Bach did, allowing arbitrary modulation in the composition and
play, e.g. tuning the harpsichord once and for all before playing through the 24 tonalities in WTC.
The present paper puts forward a geometric quality of temperaments, measured with a single number (the
Musical Sameness of Scales, or MSS); as it happens, the comparison of values of MSS for the different tunings
in competition so far, singles out LT as a clear winner, primus inter pares.
This MSS makes use of the Discrete Fourier Transform (DFT), a mathematical tool whose relevance to major
scales was discovered very recently (see [11, 1]). Despite its technicality, the MSS puts forward a musical quality
that Bach would have found desirable.
Its discovery was serendipitous, and has nothing to do with Bach and WTC. I was investigating DFT of major
scales for purely theoretical reasons, but endeavoured to try it on unequal temperaments as an illustration for
the Helmholtz ‘Klang und Ton’ workshop in Berlin, May 2007. I came across Lehman’s story while browsing the
internet, and included it in my list of various temperaments out of curiosity. Computing values of DFT for all
major scales in all these temperaments, I noticed an exceptional quality of LT, only equaled by Werckmeister’s
IV, or septenarius. 4 The notion of MSS was formulated in order to sum up this quality in a single number, for
the sake of simplicity.
Of course, I do not claim that the best temperaments (musically speaking) are just the ones with the greatest
MSS. For instance, equal temperament gives an infinite value for MSS, but nowadays specialists agree on the fact
that the equal temperament was abhorrent to Bach, as it still is to baroque musicians. The computation of MSS
should come after all musicological arguments, enabling to choose among temperaments already acknowledged
as musically interesting.
Readers are invited at this point to listen to the pleasant sound of the recordings in LT, which can be found on
this page. Together with the record value of MSS, this makes a convincing case for Lehman’s hypothesis. But
arguments, like a razor, 5 cut both ways: some lesser known temperament might achieve a better MSS than LT.
Readers are strongly invited to test the temperaments they like best against LT with the formula for MSS given in
section I. The computation is possible even with a pocket calculator (especially if it manages complex numbers)
and only takes a few lines of programming with a computer. For instance, some alternative interpretations of
Bach’s scribble have been offered, and the values of MSS for these temperaments are of interest.
On the other hand, should some other temperament with a greater MSS supersede LT one day, the fact will
remain that most or all temperaments reportedly in use around Bach’s time have a much smaller MSS, and this
will remain a significant hallmark of LT.
3 For instance, many pages have been written about the quality of the single third E-G# in LT versus other ways of tuning; but
the overall number of nice ‘tertiam minorem’, as Bach calls them, might also be considered of equal or greater importance than that
of major thirds, or the size of whole tones, or any other plausible feature of a given temperament.
4 In the rectified version, assuming the value 176/196 in the division of the monochord should be read as 175/196, as many have
corrected.
5 Occam’s razor? See [10].

1. Sameness of Scales in a Temperament
I fear it is impossible to appreciate the import of the MSS value without some attempt at understanding the
math behind it: most of the strength of the argument derives from Theorem 1. I have made use of footnotes
and annexes in order to lighten the reading as much as possible.
Roughly speaking, the DFT defined below enables one to appraise how close a scale 6 is to the mathematical
paradigm of a regular polygon. This makes musical sense, because the largest DFT values characterize the
major scales. As it happens, the distribution of these values calculated for the 12 major scales exhibits a very
special feature in the case of LT.
The next subsection attempts a non-technical description. Readers averse to mathematics are invited to read it
and then skip to section II, with tables of values of MSS for a number of temperaments. The precise definitions
leading to the MSS are given in the sequel of this section, but the truly technical part (the proofs of the Lemma
and Theorem) were confined to the last section.
We hope that most readers will go through the whole section, thus getting the gist of the argument, even if
understanding the actuals proofs requires some training in that prolific branch of mathematics called harmonic
analysis. The complexity of the diatonic, which underlies the major scale and is hence responsible for a large
part of the beauty of western classic music and the fulfilling emotions that we experience with it,
1.1. Major scales are regular heptagons (almost). Consider the C major scale in fifth order: F C G D A
E B. A tone-deaf person might not perceive that the interval between the extreme notes, B-F, is smaller than
the other fifths. After all, the difference is the smallest possible inside the twelve tone universe, e.g. a semitone
(or augmented prime). This means that the sequence of seven notes in the major scale is as close as can be, in
the twelve note ambient universe, to a completely regular sequence (a perfectly regular sequence is the whole
tone scale, for instance). This can be seen graphically on this movie.
The mathematical tool that measures such regularities is the Fourier Transform (rigorously defined in the next
subsection). It enables to simplify considerably the data for periodic phenomena: in the domain of sound for
instance, a sound file recording one note played by a musical instrument can be summarized by a few figures
giving the Fourier coefficients, which are in this case the weights of the different overtones. In the study of
scales, we use a slightly simplified version, called the Discrete Fourier Transform.
For a perfectly regular scale, like the whole-tone, the value FA(1) of this Fourier Transform would be equal to
1, or 100%. The major scales are the twelve seven-note scales with the greatest value of this FA(1) – not 100%,
but about 98%. This is the most important point about FA(1): it characterizes the major scales among all
possible seven-note scales.
Now in unequal temperaments, all scales are different, if only slightly. Hence for each tonality A, we compute a
different value of FA(1). The MSS is then defined as a measure of the discrepancy of the twelve values of FA(1)
for the twelve major scales: the higher the value of MSS, the closer the values of FA(1) are to one another.
For high values of MSS, we expect all (major) scales to be equally close to their abstract model. Indeed an
infinite value of MSS means an equal temperament, wherein all scales sound identically. Hence MSS can be
taken as a measure of how much a temperament achieves Bach’s goal, having all tonalities coexist peacefully
inside a same temperament.
1.2. Discrete Fourier Transform of a Scale. Research on DFT of scales originated in the preparation of
the John Clough Memorial Days in Chicago (July 2005), organized by Richard Cohn and David Clampitt,
when several researchers had their interest about DFT aroused by Ian Quinn’s ground-breaking dissertation
[11], wherein he rekindled an original idea of the late David Lewin [8]. Fresh developments may be found in
[1]. But a slightly different track, initiated a few months later by Thomas Noll, led to the present indicator of
sameness of scales.
Definition 1. Let all notes inside an octave be given by a real number between 0 and 1; this means chosing a
reference note (say C) and measure all intervals from there in cents/1200. Then the Discrete Fourier Transform
of a scale A = (a1, a2, . . . an) ⊂ [0, 1] is the map
FA : t ↦→ 1
n
n
k=1
6 Meaning a sequence of notes in a given temperament
e 2iπak e −2iπk t/n = 1
n
n
k=1
e 2iπ
ak−k t/n
3

4
(it is the Fourier Transform of the map k ↦→ e 2iπak ).
The values FA(0), FA(1) . . . FA(n − 1) are the Fourier coefficients of scale A.
For instance, the (equal) tempered C major scale would be CM = (0, 1/6, 1/3, 5/12, 7/12, 3/4, 11/12) and its
DFT is
1
2iπt 4iπt 6iπt 8iπt 10iπt 12iπt
− − − − − −
FCM : t ↦→ 1 + e 7 + e 7 + e 7 + e 7 + e 7 + e 7
7
It may be useful to recall that the Fourier Transform is in general a tool for checking the periodicities of a given
phenomenon. Here, it is easily seen that the map FA is defined from the cyclic group with n elements Zn to
the field C of complex numbers. 7 The notes can be visualised as the points e2iπak on the unit circle, which is
the quotient structure of frequencies modulo octave: see fig. 2.
1.3. DFT and Maximal Evenness.
Lemma 1. |FA(1)| = 1 ⇐⇒ A is a ‘regular polygon’, i.e.
a2 − a1 = a3 − a2 = · · · = a1 − an = 1
n
For any other scale, for any t, |FA(t)| ≤ 1.
mod 1
The first case |FA(1)| = 1 occurs for instance when A is a whole tone scale (in equal temperament), or an
augmented triad, or diminished seventh. But of course a seven note scale in a (decent) twelve note temperament
cannot be a regular polygon, as 12 cannot be divided into 7 equal integral parts (see fig. 2). Still, best
approximations of regular polygons are musically interesting scales:
Theorem 1. Let S be the set of scales of n notes chosen in some equal temperament with m notes (m > n),
meaning
∀A ∈ S, ∀a ∈ A, m a is an integer.
Then the scales in S with biggest value of |FA(1)| are the Maximally Even Sets.
What are Maximally Even Sets ? The ME Sets were defined in [4], extended in [3]. A recent and thorough paper
on their features is [5] and the connection with DFT, discovered in [11], is investigated in [1]. The proofs of the
above Lemma and Theorem are relegated to the annex, together with technical definitions. For the moment let
it suffice to mention that
(1) Informally, a ME set is the ‘most regular’ repartition of n notes in a given temperament.
(2) ME sets (in equal temperament) are musically prominent scales: for instance the pentatonic, whole
tone, major and octatonic scales all are ME sets in twelve tone temperament.
(3) In several cases, including the major (and pentatonic) scales, this Maximal Evenness implies the ‘generated’
quality, e.g. the major scale is generated by consecutive fifths.
This means that in a given context (here an equal tempered chromatic ambient universe) the size of |FA(1)|
measures the regularity of the scale, i.e. its closeness to a regular polygon. The above theorem, as the next
proposition, can bear some degree of approximation, 8 and hence both still stand in all common tunings, which
are close to equal temperament.
However, we only need the case of 7 note ME sets in 12 tone temperaments:
Proposition 1. In 12 tone equal temperament, the Maximally Even Scales with 7 notes (e.g. the seven-note
scales A with greatest value of |FA(1)|) are precisely the 12 major scales.
Now we can see that |FA(1)| really measures the closeness of 7 note-scale A to the theoretical regular heptagon,
which appears as a common goal, a Platonic model, that major scales strive to approximate as best they can:
this is the meaning of the last proposition. 9
7 Adding n to t does not change the value of FA(t).
8 The DFT is a continuous map: if |FA(1)| > |FB(1)| in equal temperament, the small modifications of arguments A and B
resulting from nudging some notes away from their equal tempered position, will only slightly disturb the values |FA(1)|, |FB(1)|
and the inequality will still stand.
9 Obviously this appears as a mathematical feature, not a musical one. The fact that the major scale is Maximally Even is highly
relevant to tonal functions. This is discussed in the literature on ME sets and the reader is invited to consult the references given
in the bibliography.

A
A
G
B
G
C
C major
Figure 2. Major scales are best approximations of regular heptagons
F
From now on, all we need to remember is this meaning of |FA(1)|. For instance, if scale A is a regular heptagon
(say a ‘whole tone scale’ in 14 note equal temperament) then |FA(1)| = 1; for any major scale A in 12 tone
equal temperament, |FA(1)| ≈ 0.988846. This is a fundamental feature of major scales and not a mere curio:
a basic fact about DFT, Parseval’s formula, states that
t=0...n−1 |FA(t)| 2 = 1 = 100%. Hence, when |FA(1)|
evaluates around 0.98 for a major scale, it means that almost all the energy of the scale is concentrated in this
coefficient, as all other Fourier coefficients are negligible. In the case of other, random, seven note scales, the
energy is spread between the seven coefficients: for a chromatic succession of 7 notes, |FA(1)| is roughly 0.74,
see fig. 3. It is also worth considering the scales with second best value of |FA(1)|, which are quite different
from major scales ([0234568], e.g. G♯, A♯, B, C, C♯, D, E).
1
0.5
1 2 3 4 5 6
Figure 3. DFT of a major scale and a chromatic 7 note scale
We are now ready to define the sameness coefficient of scales (MSS) inside a given temperament.
1.4. Sameness Coefficient. A temperament is simply a collection of 12 tones, i.e. a subset T of [0, 1] with
12 elements. Henceforth we will only consider scales with seven notes, that is to say ordered sequences of seven
elements of T . Let us assume that the elements of T are given in order:
C
F
D
E
D
5

6
Definition 2. A temperament, or tuning, is an ordered sequence of twelve different notes:
0 ≤ t0 < t1 < t2 < . . . t11 < 1
Definition 3. A major scale in temperament T is a sequence of the form
Aα = (a0, . . . a6) with ai = t (ki+α mod 12)
where α is a constant and the ki’s are the indexes of the standard C major scale:
(k0, k1 . . . k6) = (0, 2, 4, 5, 7, 9, 11)
Example: say α = 5, we get the notes ai with i = 5, 7, 9, 10, 12 = 0, 14 = 2, 16 = 4, i.e. F major.
This enables to compute |FAα(1)| for all α = 0 . . . 11, i.e. for the 12 major scales in T . For instance, taking for
T the so-called pythagorean tuning with the ‘wolf fifth’ between A# and F, we get the following values for all
major scales (in semi-tone order):
0.9891, 0.9891, 0.9856, 0.9927, 0.9856, 0.9915, 0.9856, 0.9856, 0.9915, 0.9856, 0.9927, 0.9856
Notice that all scales that do not contain the wolf fifth are isometrical, and hence share the same absolute value
of the Fourier coefficient (for equal temperament). The most striking fact about these values is their closeness
to 1, which but reflects the characterization of ME sets in theorem 1. But the most important feature in a given
temperament is the distribution of these values, which are exceptionally close to one another in the case of LT.
In order to visualize this phenomenon more easily, we define
Definition 4. The Major Scale Sameness (MSS) of temperament T is the inverse of the biggest discrepancy
between values of coefficients |FAα(1)| for all twelve major scales in T :
MSS(T ) =
1
max
α (|FAα(1)|) − min
α (|FAα(1)|)
This quantity is highest when all values of |FAα(1)| (for all 12 major scales) are the closest, i.e. when all major
scales are almost equally similar to the ideal (theoretical) model of the regular heptagon. For instance for
Pythagorean tuning, we get a maximum (resp. minimum) value of 0.9927 (resp. 0.9856) and hence
1
MSS(Pyth) =
0.9927 − 0.9856 =
1
≈ 140.
0.0071
For LT, no major scale comes higher than |FAα(1)| = 0.991, but none comes lower than 0.987, so the MSS is
particularly high: MSS(LT)=
1
≈ 250 (actually a little more, see fig. 5).
0.991 − 0.987
1.5. Musical relevance: playing in all tonalities. Most analyses of different temperaments, including
Lehman’s, put forward some particular intervals. The MSS coefficient on the other hand is comprehensive: it
gives a measure of the sameness of all major scales in T . Though it tells nothing about the closeness of E-G♯
to the pure third, it is well in tune with Bach’s own project in WTC. 10 A tuning with high MSS means that
ALL major scales are similarly close to the theoretical heptagon, i.e. similarly ‘good’. Or should one say, ‘wohl’
? This movie online shows all diatonic scales in LT, together with their closest heptagons. It can be observed
that though the 12 scales differ among themselves, they are quite similar in the way they resemble the regular
heptagon: this is what MSS is all about.
2. Comparison of MSS for different tunings
This section is devoted to numerical results, i.e. tables of values of MSS.
10 Admittedly Bach does mention major and minor thirds on the front page, plus tones and semi tones (see 1), but he insists on
the possibility of playing all these intervals in all 24 tonalities.

2.1. Tables. Let us begin with a computer program for the computation of MSS on a given T . It is convenient
to put the values of the 12 elements of T in some table:
T = {t0, t1, . . . t11}
To do this, take any table of a temperament in cents and divide by 1200 to get the ai’s. This will be the input
of the function that computes the MSS. Now we loop through the twelve major scales:
For α = 0 to 11, compute the scale Aα and its first Fourier coefficient cα:
Aα = (ai = t (ki+α) mod 12)i=0...6 cα = 1
6
e
7
2iπ(ak−k/7)
Lastly, find the minimum and the maximum in those 12 values; subtract the one from the other, and take the
inverse11 of the result:
1
MSS(T ) =
maxα(cα) − minα(cα)
Fig. 4 gives this algorithm in Mathematica R○ , for scales with any number n of notes.
fou1gamme : Modulen Lengthgamme,
N
2 Π gamme 2 Π k
.Table n , k, 0, n 1
n
consistencyscale : ModulemajScales, top, bottom,
majScales Tablefou1
scaleSortr 0, 2, 4, 5, 7, 9, 11 1 mod 12 . 0 12,
r, 0, 11;
top maxmajScales; bottom minmajScales;
1
, top, bottom
top bottom
Figure 4. A program for computing MSS
In Table 5, consecutive fifths are given in cents from the origin; so in program 4, the variable gamme had to be
divided by 1200. All origins (A) have been set to the same value, 0. Of course, different origins can be given
for each tuning, but this is equivalent to some rotation on the circle of notes; and the quantity MSS, being
geometric in essence, is invariant under such transformations. 12
I selected only a few elements in the extensive family of meantone tunings. The Neidhardt tuning used here is
the 1732 one, as quoted by Lehmann in his answer to Lindley and Ortgies’s acerbic refutation (see [9]). Lindley’s
tunings are tabulated from the same source: the first two are built respectively with sevenths of Pythagorean
and syntonic comma, and the last is the tuning used in the Michaelstein conference (this one found favor with
Lehman). Similarly I added the Sparschuh tuning and the previous Lehman proposition (dated 1994) as possible
competitors, other interpretations of the scribble being of course of particular interest.
I hope that through the process of chain-quotation, the exact values of these tunings have been preserved. As
the computations have been carried through the professional software Mathematica R○ , with 64 bit arithmetic,
we can hope to be rid of the rounding errors that blemished several controversies, and that resulted from
incompatible unit conventions and inadequate software (Excel R○ ).
It is quite clear on these values that LT achieves by far the best value of MSS, except for Werckmeister IV 13
which reaches almost equal value. It is interesting to see which classical tunings figure well in this table:
Kirnberger’s are in good stand, so is Valloti, but it is a little surprising that the Pythagorean tuning supersedes
11In order to facilitate comparison.
12 Incidentally, it is also invariant under inversion (which amounts to reversing the lists), as the inverse of any major scale is also
a major scale.
13 I wonder whether this temperament ever had practical importance – who ever managed to tune a 196/139 interval by ear ?
k=0
7

8
MSS A Bb B C C D Eb E F F G G
Pythagore 141.572 0. 113.69 203.91 317.6 407.82 521.51 611.73 701.96 815.64 905.87 1019.6 1109.8
Zarlino 47.8701 0. 111.73 203.91 315.64 386.31 498.04 568.72 701.96 813.69 884.36 1017.6 1088.3
Kirnberger2 146.869 0. 90.225 203.91 294.13 386.31 498.04 590.22 701.96 792.18 895.15 996.09 1088.3
Kirnberger3 164.332 0. 90.379 195.19 294.01 386.45 498.05 590.3 698.22 792.33 890.27 995.97 1088.2
Neidhardt 31.6585 0. 113.69 203.91 317.6 384.36 474.58 611.73 748.88 815.64 882.4 1062.9 1066.5
Werkmeister1 180.602 0. 90.225 192.18 294.13 390.22 498.04 588.27 696.09 792.18 888.27 996.09 1092.2
Werkmeister2 119.663 0. 82.405 196.09 294.13 392.18 498.04 588.27 694.13 784.36 890.22 1003.9 1086.3
Werkmeister3 234.585 0. 96.09 203.91 300. 396.09 503.91 600. 701.96 792.18 900. 1002. 1098.
Werkmeister4 268.489 0. 90.661 196.2 298.07 395.17 498.04 594.92 697.54 792.62 893.21 1000. 1097.1
Werkmeister5 27.5673 0. 43.305 133.53 176.83 337.44 451.12 470.97 561.19 745.26 835.48 878.79 969.01
meanTone15 80.1584 0 113.69 194.53 308.21 389 502.74 616.4 697.26 810.95 891.79 1005.5 1086.3
meanTone16 117.405 0 109.78 196.09 305.87 392.18 501.96 611.73 698.04 807.2 894.13 1003.9 1090.2
Vallotti 164.255 0 94.13 196.09 298.04 392.18 501.96 592.18 698.04 796.09 894.13 1000 1090.2
BachLehman 266.823 0 103.91 200 305.87 403.91 501.96 603.91 698.04 807.2 901.96 1003.9 1103.9
Lindley 45.6815 0. 125.81 187.88 289.44 379.67 493.35 583.58 685.92 827.76 893.74 1003.5 1121.9
LindleyBis 76.1371 0. 113.69 193.74 296.48 386.71 500.39 590.62 692.18 813.29 903.52 1027.4 1117.6
Lindley94 61.4739 0. 113.69 173.41 273.02 363.25 476.93 567.16 671.46 801.56 891.79 1005.5 1109.8
Sparschuh 47.5277 0. 121.51 189.83 285.53 394.13 495.31 635.19 725.42 839.1 909.78 1016.8 1116.
Lehman94 50.7568 0. 135.19 182.4 296.09 386.31 500. 590.22 723.46 837.15 927.37 1019.6 1107.8
Figure 5. Values of MSS for different tunings
Zarlino and the meantone temperaments. Remember that MSS measures a closeness between major scales, not
the quality of, for instance, the twelve fifths.
2.2. Minor scales. As watchful readers will have noticed, the MSS has so far left aside half of WTC: J.S. Bach
also gave his preludes and fugas in all minor tonalities. MSS is originally a measure of ‘diatonicity’: could it
also be used for testing minor scales ? If by ‘minor’, one should understand the harmonic minor scale, then
the theoretical relevance is poor. 14 But this natural assumption, taught to every kid in music school, has long
been challenged by specialists. 15 Perhaps the most ‘natural’ form of the minor is also diatonic, meaning that
MSS is relevant both for minor and major tonalities.
It might be interesting though to try this comparison on other familiar scales, like the ascending melodic minor
[023579a].
2.3. First five fifths equal. Permutations of the values of the different fifths of LT can be tried. The result is
illuminating. As we have already discussed, an optimal value of MSS (= ∞) is attainable, in equal temperament.
This is not realistic, as most musicologists now agree that equal temperament was as abhorrent to Bach as it
was to many, before and afterwards.
So it should come as no surprise that some possible (non equal) temperaments improve the value of MSS. It is
the case for Werckmeister IV, which, though rational-valued, is a close approximation to equal temperament. 16
I found several such tunings in trying up all permutations of the values of fifths that Lehman attributed to
the three different kinds of loops in Bach’s scribble. For instance the one which corresponds to the sequence of
adjustments 0, −2, −2, −2, 1, −2, 1, −1, 0, −1, 0, −2 17 , has a much higher value of MSS.
Such artefacts are generally very close to equal temperament. I did not recognize any known tuning in them,
though learned readers might.
But strikingly again, LT is still a winner when these permutations are confined to the six last fifths, i.e. when
we keep the first five fifths equal: here I am discussing adjustments of the form −2, −2, −2, −2, −2 followed by
any permutation of 0, 0, 0, −1, −1, −1. This must be relevant, as it was and still is common practice to start
the tuning of a keyboard instrument by five equal fifths (often slightly shorter than a pure fifth), thus tuning
equally the Guidonian hexachord FCGDAE. I certainly did, long before I ever heard of Bradley Lehman. This
is true of all meantone tunings for instance, and also very obvious for a musician accustomed to tonal music,
14 Still I felt compelled to compute the mSS, the equivalent of MSS for harmonic minor scales, and rather strangely, it concurs
with Table 5.
15 ‘It may be of some interest that Johann Sebastian Bach in his manual on thorough bass (reprinted in Phillip Spitta’s biography
of Bach as Appendix XII to Vol. VII) expressly states the identity of the minor mode with the Aeolian system...” (p. 46, Heinrich
Schenker [Harmony], Oswald Jonas, ed., Elisabeth Mann Borghese, trans. 1954, reprinted MIT Press 1973).
16 Hearing WTC in this tuning should be an interesting experience, considering it is so far the only competitor to LT for MSS.
17 Whereas LT gives −2, −2, −2, −2, −2, 0, 0, 0, −1, −1, −1[, 0]. The unit chosen by Lehman is a twelfth of a pythagorean comma.

since it is still desirable to have many equal fifths inside the scale(s) he or she plays in. Many harpsichord,
organ, or piano-forte playing readers will acknowledge that they begin their own tunings by equal fifths between
F-C-G-D-A-E. See below a discussion of O’Donnell’s objection, though [10].
2.4. Other values of the wiggles. One weak point of Lehman’s proposition is his arbitrary calibration of
the wiggles as multiples of PC/12 (PC = a pythagorean comma). On the other hand, changing this value
ever so slightly falls within the approximation involved in any practical tuning – this is not exact science ! On
investigation, I tried variants of LT, replacing the unit PC/12, or 2.346 cents, by values close to it. The value
of MSS can thus be increased slightly: if the fundamental wiggle is reduced to 2.16 cents, MSS reaches 278.7
instead of 266. This is the best possible improvement in that direction, with
T = {0, 95.685, 196.71, 297.795, 393.42, 498.105, 593.73, 698.355, 797.64, 895.065, 997.95, 1091.78}
Of course, a difference of 0.186 cents is hardly perceptible by ear – it is roughly a 300th of a comma ! It is only
the different order of fifths, the geometry of the scale, and not the change of comma, that explains the record
value of MSS; the different value of the comma in Lehman’s first proposition in 1994 (labeled Lehman94 in fig.
5) has nothing to do with its poor value of MSS.
2.5. Other interpretations of the wiggles. Some have questioned the original note of the tuning: is it F –
a common practice – or do some further curlicues along the scribble (look again at fig 1) indicate the position of
C, or D, or some other note ? As far as the MSS is concerned, this is irrelevant, as a ‘well tempered’ instrument
will remain so even if the tuning is transposed, in the sense that all tonalities still sound well. Of course it
would be nice to know exactly Bach’s tuning (giving specific different colours, i.e. Affekt, to different tonalities)
but MSS cannot discriminate between tunings there, as it is invariant under a change of origin of the tuning. 18
Similarly, MSS cannot help us decide whether turning around the page (as Lehman did) is permitted, or
mandatory: the inversion of a tuning shares the same geometric values 19 , and the sequence of wiggles (-1,-
1,-1,0,0,0,-2,-2,-2,-2,-2) has the same MSS as LT – a nice feature for counterpointists. In other words, using
Lehman’s recipe with fourths instead of fifths yields exactly the same value of MSS.
MSS can help, though, to discuss whether the wiggles mean consecutive fifths or consecutive semitones. This
is a strong argument, as O’Donnell points out ([10]) that the order of tonalities in WTC is given in semitones
not in fifths – but Chopin, for instance, did contrariwise in his Preludes, which are clearly referring to WTC.
I am not sure that I understood O’Donnell rightly, as this sequence of five short successive semitones yields
one very poor fourth interval (or a very large fifth); but I computed MSS for those tunings given in Lehman’s
comprehensive answer to O’Donnell, in Table 6:
MSS A Bb B C C D Eb E F F G G
BachLehman 266.823 0 103.91 200 305.87 403.91 501.96 603.91 698.04 807.2 901.96 1003.9 1103.9
O'Donnell 40.0079 0. 137.15 227.37 294.13 384.36 498.04 635.19 678.49 839.1 905.87 1019.6 1109.8
Neidhardt4 110.593 0. 43.305 180.45 247.21 337.44 451.12 588.27 631.57 792.18 858.94 972.63 1062.9
Neidhardt 1732 53.9361 0. 137.15 227.37 294.13 384.36 498.04 588.27 701.96 815.64 929.33 1043. 1109.8
Sorge 54.7281 0. 113.69 227.37 294.13 384.36 498.04 635.19 678.49 815.64 905.87 1043. 1133.2
Figure 6. MSS for O’Donnell-like tunings
Again MSS is much greater for LT than for all other tunings.
Several falsifications, in the sense of Popper, have been tried. In each case, LT emerges a winner. It is clearly
a very special temperament, not only in its closeness to Equal Temperament, but in the homogeneity of all 24
scales. This constitutes a strong vindication of Lehman’s claim.
3. Conclusion and perspectives
The MSS criterion arguably puts emphasis on the sameness of fifths inside the temperament. This goes some
way towards explaining why permutations of the wiggles fail to improve MSS, when the first five are kept
identical (see subsection 2.3); it also explains the fair value of MSS for Pythagorean (Ptolemaic) tuning, for
instance. But this argument has limits, as the discussion on all permutations of the wiggles proves. Still is
is quite important musically, as keyboard intrument players will readily agree, to find in any major scale a
18 Transposition, in the musical sense, does not change the absolute value of the DFT.
19 The DFT is turned into its conjugate. See [1].
9

10
sequence of 3 (and preferably more) identical fifths in a row. This makes it easier to tune together with string
instruments, for instance.
Apart from aesthetic and other indefinitely debatable arguments, the computation of MSS gives a steadfast
comparison point for tunings aimed at tonal music. Of course it should not be considered as the sole criterion
for the quality of a tuning – for instance, equal temperament would then be unbeatable, and Pythagorean
tuning would be better than Kirnberger’s – but it does make sense to use it, among other criteria, to compare
tunings already short-listed for historical and musical reasons.
A limitation to the discriminating power of MSS is its indifference towards isometric transformations. Starting
the tuning with C or F# or any note does not change the value of MSS, neither does tuning in fourths instead
of fifths, i.e. reversing the cycle of fifths. This last point is one that raised much discussion: is Lehman
right in turning around the first page of WTC ? As we discussed previously, MSS brings no answer there. But
consideration of the phase of the Fourier coefficients (and not only of module), or maybe scrutiny of other Fourier
coefficients (e.g. FA(0)), might help refine the analysis (I am indebted to Thomas Noll for these suggestions).
Other measures of the sameness of scales inside a given temperament might, and should, be proposed, as this
criterion makes sense when considering WTC. 20 For one thing, the MMS is designed primarily for comparison
of the major scales, and perhaps some another measurement should give equal standing to the minor scale in
some form.
The computations made for this paper are by no means exhaustive, and I hope that some other researchers
will complete them. I intentionally refrained, though, from computing MSS on many tunings that either fall
far from Bach’s universe, or appear too arbitrary. Should some major tuning be lacking in this analysis, it is
because I was not aware of its existence.
A puzzling case is Werckmeister IV, the only serious challenger of LT for the value of MSS. It stems from most
modern authorities that the relationship Bach/Werckmeister was more about counterpoint than tuning, but
maybe this fresh evidence will rekindle interest in the notion that Bach shared at least some of Werckmeister’s
original ideas about tuning.
My modest intention was to add some non subjective evidence to the discussion of Lehman’s hypothesis. In
the present state of the calculations in Table 5, the MSS criterion unambiguously supports Lehman’s theory.
Maybe further efforts in the same line will help refine his claim – or crown some other tuning that I cannot
foresee.
3.1. Proof of lemma.
Appendix: the mathematics of MSS.
Proof. From Parseval’s identity (see below), if |FA(1)| = 1 then for all t = 1, FA(t) = 0. Hence by inverse
Fourier transform (Plancherel’s theorem) the elements of A are given by
k ↦→
t∈Zn
FA(t) × e +2iπ kt/n +2iπ k/n
= FA(1) × e
which puts them on a regular n−gon.
The other assertion |FA(t)| ≤ 1 is a consequence of Parseval’s identity:
|FA(1)| 2 +
|FA(t)| 2 =
|FA(t)| 2 = 1.
t=2...n
3.2. Proof of theorem. I give the proof of the theorem in the simpler case 21 when the number of notes n is
coprime with the cardinality of the temperament, m. In this paper we have n = 7, m = 12. In such cases, a
ME set is generated (cf. [4]), i.e. the indexes of its notes are in arithmetic sequence modulo m, with a ratio f
of the sequence that is the multiplicative inverse of n mod m. In the case n = 7, m = 12, this expresses the
fact that the major scale is a sequence of fifths, i.e. (0, 7, 7 + 7 ≡ 2, 9, 4, 11, 6) or any transposition thereof. The
ratio f = 7 is the inverse of n = 7: it checks f × n = 7 × 7 = 1 + 4 × 12 ≡ 1 mod 12.
t=1...n
20 For instance, some Haussdorf distance between polygons up to isometric transformations.
21 It still stands in the most general case, following the framework of [1].

Proof. I begin with pointing out that the map (k, j) ↦→ n × k − m × j is one-to-one (and onto) from Zm × Zn to
Zn×m, where Zp stands for the cyclic group with p elements. This morphism (it is well defined, and obviously
linear) of Z−modules is injective:
n × k − m × j ≡ 0 (mod n m) ⇐⇒ ∃ℓ, n k = m j + ℓ × m n ⇐⇒ m | k and n | j
⇐⇒ k ≡ 0 (mod m) and j ≡ 0 (mod n)
using Gauss’s lemma (m divides n k but is coprime with n, hence divides k, similarly for n); and hence bijective
because the cardinalities of both domain and codomain are finite and equal.
This enables to choose n couples (k0, 0), (k1, 1) . . . (kn−1, n − 1) in Zm × Zn with n kj − m j ∈ {0, 1, . . . n − 1}
(mod m) × n, (choosing j first then kj) hence kj j
n − 1 1
− stays between 0 and <
m n n m m .
Hence the vectors occuring in the computation of FA(1), i.e. the e2iπ( kj j
− m n ) , are as close together as possible.
This maximizes their sum, as the cosines of their projections on the direction of their sum get as close to 1 as
possible. This is proved by the following ‘huddling lemma’, whose general form may be found in [1].
Lemma 2. Let e2ikjπ/m
, i = 1 . . . n be n different mth
roots of unity. Their sum has length at most e
1≤k≤n
2ikπ/m
i.e. the maximum value is obtained when all these points are huddled together.
Figure 7. The sum increases when the points are closer
Proof. Put S = e 2ikjπ/m . Then |S| ≥ Re S = cos(2kjπ/m). Say for instance that n is odd: then there
are exactly n integers between
1 − n
2
and
n − 1
.
2
(n − 1)
Unless the kj are precisely these integers, there exists some j0 with |kj0 | >
that is not one of the kj. As
2
and a value k ′ ∈ [
1 − n
,
2
n − 1
11
]
2
cos(2πkj0π/m) < cos(2πk′ π/m),
the sum of cosines is increased when k ′ is substituted to kj0 . This can be done till all the kj’s
1 − n
are in [ ,
2
n − 1
],
2
and as there are n such kj’s, they must be the n consecutive integers lying in this range. Then Re S is at
last maximal, and so is |S| = Re S (this last equality is now true by symmetry). The case ‘n even’ is a little
trickier but similar and will be left to the reader, as this paper only needs a proof when n = 7, m = 12.
We return to the proof of the theorem. The lemma proves that the maximum configuration occurs when
Multiplying by f = n −1 mod m yields
A = {k0, . . . kn−1} with {n kj − m j} = {0, 1, . . . n − 1}
{kj} =
mod m {kj − f × m × j} = {0, f, . . . (n − 1)f}

12
i.e. an arithmetic progression with ratio f. The most general case is obtained by translation (i.e. a transposition,
musically speaking) of this one.
Acknowledgements
I am indebted to the members of the American new wave of musicologists who renewed the interest in the study
of scales, through the novel notions of ME sets, and later Well Formed Scales: the late J. Clough, G. Myerson,
J. Douthett, N. Carey and particularly David Clampitt who, together with Richard Cohn, kindled my interest
in the topic by inviting me to the John Clough Memorial Days; to Ian Quinn whose fascinating work on ‘chord
quality’ revived David Lewin’s idea on Discrete Fourier Transform and gave me a starting point for fruitful
research; and to Thomas Noll who shrewdly perceived that DFT could be extended to non equal temperaments
with a slightly different formula, and also gave me the incentive to explore this alley when he invited me to
the Helmholtz ‘Klang und Ton’ workshop in Berlin. I also thank Bill Sethares, a great specialist of Fourier
Transform for consonance issues, for his useful advice and encouragements. Last but not least, my wife helped
me tremendously with the wording of this paper, despite its technicality.
References
[1] Amiot, E., David Lewin and Maximally Even Sets, 2007, Journal of Mathematics and Music, Taylor & Francis, I(3).
[2] Carey, N., Clampitt, D., 1989, Aspects of Well Formed Scales, Music Theory Spectrum, 11(2),187-206.
[3] Clough, J., Douthett, J., 1991, Maximally Even Sets, Journal of Music Theory, 35:93-173.
[4] Clough, J., Myerson, G., 1985, Variety and Multiplicity in Diatonic Systems, Journal of Music Theory, 29:249-70.
[5] Douthett, J., Krantz, R., 2007, Maximally even sets and configurations: common threads in mathematics, physics, and music,
Journal of Combinatorial Optimization, Springer. Online: http://www.springerlink.com/content/g1228n7t44570442
[6] Lehman, B., Bach’s extraordinary temperament: our Rosetta Stone-1, Early Music, February 2005; 33: 3 - 24.
[7] Lehman, B., Bach’s extraordinary temperament: our Rosetta Stone-2, Early Music, May 2005; 33: 211 - 232.
[8] Lewin, D., 1959, Re: Intervallic Relations between two collections of notes, Journal of Music Theory, 3: 298-301.
[9] Lindley, M., Ortgies, I., Bach-style keyboard tuning, Early Music, November 2006; 34: 613 - 624.
[10] O’Donnell, J., Bach’s temperament, Occam’s razor, and the Neidhardt factor, Early Music, November 2006; 34: 625 - 634.
[11] Quinn, I., 2004, A Unified Theory of Chord Quality in Equal Temperaments, Ph.D. dissertation, Eastman School of Music.

NEW PERSPECTIVES ON RHYTHMIC CANONS AND THE SPECTRAL CONJECTURE
EMMANUEL AMIOT
Abstract. The musical notion of rhythmic canons has proved to be relevant to some non trivial mathematical problems.
After a survey of the main concepts of tiling rhythmic canons, we discuss recent developments that enable to make, or
expect, definite progress on several open mathematical conjectures.
1. Introduction
1.1. Purpose. The story of rhythmic canons is long and fullsome, as this notion is in fact equivalent to the problem of
tiling the integers. Musical ideas arrived with Dan Tudor Vuza [26] in 1991, and gave rise to interesting developments,
especially after this theoretical model was implemented in the it OpenMusic visual programming language [10]. One
hard remaining problem is the computation of Vuza canons, which are fundamental bricks out of which all canons can
be constructed, and all their features derived. But as there is no known polynomial algorithm for this task, it had been
impossible until recently to enhance Fripertinger’s results, verifying that all Vuza canons up to a period of 108 were
classified.
Discussion of our results and ideas with specialists of more general tiling problems, also featured in this issue of the journal,
opened new alleys for exploration, including a complete list of all Vuza canons for the two next possible periods, n = 120
and n = 144. Fresh results also suggest future directions for exploration, and renew hope for solution of long standing
conjectures.
1.2. Some definitions. This might be redundant with other papers in this issue, but it seemed desirable for clarity’s
sake to state the basic definitions.
Definition 1. For the purposes of the present paper, a rhythmic canon (RC) is a tiling of the cyclic group Zn by
translations, i.e. Zn is the disjoint union of translates of some subset A ⊂ Zn:
A is called the inner rhythm and B the outer rhythm.
Zn = A ∪ (A + b1) + (A + b2) + · · · = A ⊕ {0, b1, . . . } = A ⊕ B
Musically this can be rendered as a canon (say with percussion instruments) repeating with period n, playing any motif
(modelized as a collection of integers corresponding to the beats played) that reduces to A modulo n, beginning each
instance of this motif on beats congruents modulo n to the bi’s. Already this model loses something of the actual musical
object, as motif (A) and offset list (B) are reduced modulo n. This makes sense, perceptually, for periodic rhythms with
enough repetitions for the periodicity to be perceived. Also, though the reference to rhythm is traditional, originating
in [26], the concept is really adequate to any discrete, periodic structure, such as pitch classes for instance. Perhaps one
might talk about ‘musical canons’ instead, but this is evocative of a fuzzier, contrapunctual, concept.
Figure 1. A tile and one of its rotated forms
The algebraic notion closest to the idea of a tile in Zn would be the orbit of A under the action of Zn by translation, i.e.
the set {A + k, k ∈ Zn} – think of the bass instrument in a classic tango, with A = {0, 3, 4, 6} and n = 8 for instance.
1

2 EMMANUEL AMIOT
If we visualize Zn as a circle, it means A is viewed up to rotation (see fig. 1) – for a periodic event, the question of a
‘starting beat’ is irrelevant, though musically of course this is different. In practice, we will assume that 0 ∈ A, and the
results will usually be stated up to such a rotation. Let it be said once and for all that it may be convenient to interpret
any tile A ⊂ Zn as a subset of the non negative integers, beginning with 0 and with minimal largest element:
A = {ai, i = 0 . . . k − 1}, 0 = a0 < a1 < . . . ak−1
Of course inner and outer rhythms can be exchanged: it is the duality of RC, see fig. 2.
Figure 2. A canon and its dual: Z24 = {0, 2, 7} ⊕ {0, 3, 6, 9, 12} = {0, 3, 6, 9, 12} ⊕ {0, 2, 7}.
Definition 2. A Vuza Canon (henceforth VC) is a tiling A⊕B = Zn wherein neither A nor B are periodic, i.e. A+p = A
and B + p = B for all 1 < p < n.
Nearly all researchers have made use of the polynomial representation of a tiling:
Definition 3. The associated polynomial of a subset A ⊂ Zn is A(X) =
Xk .
It is well defined in the quotient ring Z[X]/(X n − 1) of polynomials modulo X n − 1. Moreover,
A ⊕ B = Zn ⇐⇒ A(X) × B(X) = 1 + X + X 2 + . . . X n−1 = Xn − 1
X − 1 mod (Xn − 1)
1.3. Combinatorial explosion. One nice feature about VC is their (comparatively) limited number: for one thing, they
only occur in ‘bad groups’, which are scarce among cyclic groups (n may be equal to 72, 108, 120, 144, 168, 180. . . which
constitutes Sloane’s integer sequence A102562). 1 Also, among the hundreds of millions (or more) of RC, 2 in a bad group
only one canon out of several millions is a VC. Still, owing to the huge number of possible RC for a given n, it is
very important to develop effective and performant algorithms. Some recent, original ideas by Mate Matolcsi and Mihalis
Kolountzakis enlarge the field of possible computations and open new horizons, both for practical and theoretical purposes.
This enlargement, together with its practical and theoretical consequences, is the subject of the present paper.
2. State of the art
2.1. The beginnings. A more detailed narration can be found in [1]. In the 50’s, Hajós, Redei, DeBruijn, Sands and
others were working on the Hajós conjecture: in any tiling of a cyclic group is there necessarily one periodic factor? They
discovered counterexamples (first for n = 108), then classified which were the ‘good groups’, or Hajós groups, wherein the
conjecture is true, and which were the ‘bad groups’. The classification was accomplished several years later by Sands, see
[22] also for a bibliography of all prior efforts:
Theorem 1. Hajós groups, or good groups, are the cyclic groups with cardinality n of the following form (p, q, r, s denote
distinct prime numbers, and α a positive integer):
n = p α
n = p α q n = p 2 q 2
k∈A
n = p 2 qr n = pqrs
The smallest ‘bad group’ is Z72.
In his monumental 10-year work [26], Dan Tudor Vuza actually rediscovered and proved the collection of results by the
aforementioned mathematicians, with original methods. In the process, he also established several fundamental results
well before they were spotted by the mathematical community when tiling problems gathered renewed interest in the late
90’s – for instance Lemma 2.2 of [11], ‘fundamental’ in their own words, which states the invariance of the notion of tiling
under the affine group on Zn (see Prop. 2 below). Musically, an inner (or outer) rhythm that repeats itself within the
overall period of the canon will tend to be perceived as the smaller submotive, repeated. Hence Vuza naturally introduced
‘Rhythmic Canons of Maximal Category’ that we will call ‘Vuza Canons’ or VC for short in this paper. He stated and
proved the above theorem, providing on the way a construction (the Vuza algorithm) of several VC in any bad group. It
will be seen that the concept is mathematically important, as VC are the atoms in the construction of all RC and their
features are mirrored in one and every possible RC.
1 http://www.research.att.com/˜njas/sequences/
2 Harald Fripertinger established formulas for the enumeration of RC, see [14].

NEW PERSPECTIVES ON RHYTHMIC CANONS AND THE SPECTRAL CONJECTURE 3
2.2. The revival. The connection between Vuza’s work on RC and Hajós’ conjecture was drawn by Moreno Andreatta
while working on his tesi di laurea [8] and then in his PhD [9] and in [7]. He focused some interest of the musical
community on the ‘Rhythmic Canons of Maximal Categories’ (VC), and several musicians (especially composers) began
to experiment with them, particularly with transformation between rhythmic canons (RC). This allowed to understand
that Vuza’s algorithm, while it produced VC for all ‘bad groups’ (non Hajós groups), i.e. all possible values of n, did
not reach all possible VC. The point was discussed at least as early as 2003 at the MaMuTh meeting in Zürich, or even
previously: if, say A ⊕ B = Zn then by halving the tempo, and repeating motif A one beat after itself, one gets a tiling
of Z2n (read fig. 3 like a percussion score, each line standing for a different instrument playing at each black square):
or, more generally (see Prop. 4),
A = 2A ∪ (2A + 1) and A ⊕ B = Zn ⇐⇒ A ⊕ 2B = Z2n
A = kA ∪ (kA + 1) ∪ · · · ∪ (kA + k − 1) and A ⊕ B = Zn ⇐⇒ A ⊕ kB = Zk n
Figure 3. Example of ‘stuttering’ starting with {0, 5, 10} ⊕ {0, 3, 6, 9} = Z10
In word theory, this would mean applying the morphism 0 → 00, 1 → 11. More to the point, the new canon is also a VC
whenever the old one was. So in that way we constructed VC that were not available with the algorithms provided by
Vuza (or Hajós for that matter). There was a flurry of activity in 2003-2004 when we tried all kinds of transformations
in order to produce previously unchartered Vuza Canons (cf. [2]). One productive way was to look for all complements
of a tile already known to be a factor of a VC, and select the aperiodic ones. This enlarged the results provided by the
Vuza algorithm. Moreover, by exhaustive production of RC in a given cyclic group, Harald Fripertinger ([13]) managed
to find all VC for n = 72 and n = 108: as it happens, in the first case there are no others than the VC provided by Vuza’s
algorithm; in the latter, there are no others than the ones that we had found with our musical transformational techniques
of previously known VC (more on this in section 2.4).
Still the greatest breakthrough was arguably the seminal paper [11] in 1998, breaking new ground and introducing properly
the sets of cyclotomic indexes of a tile (like [19], I adapt slightly the original definition, aiming at tilings of Zn not Z: it
is well known since the 50’s that any tiling of Z by translations of a finite tile is periodic, hence it defines a tiling of some
cyclic group. See for instance [4, 1, 12] for details).
Recall that the dth cyclotomic polynomial Φd ∈ Z[X] is an irreducible polynomial whose roots are the generators of the
group of dth roots of unity. They can be computed recursively with the formula
Φd = X
d|n
n − 1. If we consider a tiling
A ⊕ B = Zn, all cyclotomic polynomials with index d, 1 < d | n, must divide A(X) or B(X) (sometimes both).
Definition 4. If A ⊂ Zn then the sets RA = {d | n, Φd is a divisor of A(X)} and SA = {d ∈ RA, d is a prime power} are
well-defined.
In particular, changing A(x) by a multiple of x n − 1 does not change RA. For instance, motif A = {0, 1, 8, 9, 17, 28} gives
A(x) = (1 + x) 1 − x + x 2 1 + x + x 2 1 − x 2 + x 4 1 − x 3 + x 6 1 + x 3 − x 4 − x 7 + x 8 − x 9 + x 11 − x 12 + x 13
The first factors are Φ2, Φ6, Φ3, Φ12, Φ18; the last one is not cyclotomic.
The pertinence of RA can be understood from the following theorem, proved in [11]:
Theorem 2. Consider two motifs A, A ′ with same cardinality.
If A ⊕ B = Zn and RA = RA ′, then A′ tiles with the same outer rhythm: A ′ ⊕ B = Zn.
This is very important in practice, as if we have a RC A ⊕ B = Zn then we can look for all others A ′ tiling with the same
B. Note that in fact condition RA ⊂ RA ′ is sufficient for A′ to tile with B.
Coven and Meyerowitz in [11] establish for the first time some conditions for a motif to tile:
Definition 5.
A ⊂ Zn satisfies (T1) if A(1) =
p k ∈SA
p, the product of the prime numbers p for each element p α of SA.
A ⊂ Zn satisfies (T2) if for any powers of different primes p α , q β , r γ . . . in SA, their product p α × q β . . . lies in RA.
They proved the following implications, the last of which is difficult:

4 EMMANUEL AMIOT
Theorem 3.
(1) Tiling ⇒ (T1);
(2) (T1) and (T2) ⇒ tiling;
(3) Tiling Zn where n has at most two different prime factors (n = p α q β ) implies (T2) [and (T1) as above].
It is not known whether there exist tilings where the tile does not satisfy condition (T2). In the example above, (T1) reads
“A(1) = 6 = 2 × 3” and (T2) reduces to the same algebraic identity 2 × 3 = 6, this time stating that 6 ∈ RA.
The obvious (obvious to state) “(T2) conjecture” is one that Coven and Meyerowitz refused to make explicitly:
(T2) conjecture. If A tiles some cyclic group, then (T2) is true.
Remember that the second assertion is constructive: if some A ⊂ Zn satisfies (T1) and (T2), then Coven and Meyerowitz
gave a formula, founded on SA, that exhibits a complement B such that A ⊕ B = Zn. This is featured in the ‘cyclotomic
rhythmic canons’ module of the software OpenMusic. It is also the foundation of the original method proposed by Matolcsi
in the present issue[19].
2.3. The spectral conjecture. The conjecture stated by Fuglede in 1974, also called Spectral Conjecture, is one of those
fundamental ideas 3 that link apparently unconnected domains of mathematics, here harmonic analysis and geometry. It
is a simple statement:
Fuglede’s Conjecture. Tiling ⇐⇒ Spectral.
Of course some precise definitions of those termes are required. But in the context of tilings of Zn it will be very simple
to characterize. For clarity’s sake I will give a definition that is not the most general.
Definition 6. A subset A of some vector space (say Rn ) is spectral iff it admits a Hilbert base of exponentials, i.e. if any
map f ∈ L2 (A) can be written
f(x) = fk exp(2iπλk.x)
for some fixed family of vectors (λk)k∈Z where the maps ek : x ↦→ exp(2iπλk.x) are mutually orthogonal (i.e.
A ekej = 0
whenever k = j) .
Without the technicalities, it means that maps on A admit a Fourier decomposition. A standard example: the tile
A = [0, 1). The maps from A to C are naturally seen as restrictions of 1-periodic maps on R, hence are sums of the
e 2iπnx , n ∈ Z: this is the standard Fourier decomposition.
Definition 7. The set A tiles E by translations iff there exists a set T such that A ⊕ T = E, the symbol ⊕ meaning that
the translates A + ti, ti ∈ T do not intersect each other except on sets of measure 0.
For instance A = [0, 1] tiles R with T = Z.
Fuglede himself proved his conjecture when A or T is a group (lattice); several other special cases have been proved.
But Field medalist Terence Tao made news in 2003 with a very simple idea nobody else had had before: he looked for a
counterexample, and found one – in high dimension ([25]). So far, the conjecture is known to be false for both implications,
in dimension greater than 3 [18]. Dimensions 1 and 2 are essentially open problems, 4 the one dimensional case of tiling is
precisely the study of RC (cf. [21]). Here is the (discrete ⇒ continuous) relationship:
A tiling A ⊕ B = Zn of Zn is easily uplifted to a tiling of the integers:
A + (B + nZ) = A + C = Z
Now we can turn it into a tiling of the line R : R = A ′ ⊕ C, where A ′ = [0, 1) + A. In this setting, the condition for being
spectral can be reformulated very simply (see [20]):
Proposition 1. A finite subset A ⊂ Z is spectral if there exists Λ ⊂ [0, 1) with ♯Λ = ♯A and for all λk = λj ∈ Λ, e 2iπ(λj−λk)
is a root of the associated polynomial A(X).
For instance, say A = {0, 1, 6, 7}. Then Λ = {0, 1 1 7
, , } is a spectrum for A, enabling to retrieve the roots of
12 2 12
A(X) = Φ2 × Φ12, e.g. e iπ , e ±iπ/6 , e ±5iπ/6 , There are some delicate questions about the ‘rationality of the spectrum’:
whether these roots (lying on the unit circle) are or are not roots of unity, i.e. of cyclotomic factors of A(X). 5 However,
in all cases known so far, a stronger fact than the spectral conjecture is true:
Theorem 4. (Isabella ̷Laba) If A is a finite subset of Z and both (T1) and (T2) are true, then A is spectral.
3Like the more famous Langlands program, or Taniyama-Weil’s conjecture.
4Izabella ̷Laba stated some results when the size of the group is not much larger than the width of the tile; but as Kolountzakis has shown,
this cannot be assumed in general.
5This must be the case whenever all roots of A(X) are on, or inside, the unit circle, from a well known result on polynomials.

6 EMMANUEL AMIOT
The inverse operation is possible whenever there is equirepartition of one tile modulo some divisor of n; that is to say
when (without loss of generality) the outer rhythm is divisible by some p. This is always the case, for instance, for RC
in Z72. But Szabo has exhibited tilings of large cyclic groups which do not have this feature ([24]), thus killing a few
conjectures. The smallest known RC (VC actually) without equirepartition happen for n = 900. Not unlike Kolountzakis’
3D construction (cf. [19]), Szabo starts from some very regular tiling and nudges it slightly, breaking the regularity while
preserving the tiling property. This appears to be a fruitful philosophy.
Figure 5. Multiplexing two periodic canons into a Vuza Canon
2.5. Relevance to the spectral conjecture. To sum up in one sentence several partial results of mine ([3]) that were
skillfully and recently completed by Edouard Gilbert [15],
Proposition 5. All the transformations in the above paragraph, namely concatenation, multiplexing (including the zooming
of the outer rhythm), affine multiplication, preserve both the spectral and the (T2) condition.
The consequence is straightforward: if there is a tiling with one factor that does not satisfy condition (T2), then either
it is a Vuza or not. If not, then it is concatenated from a smaller canon, where one factor at least does not satisfy (T2)
either. Iterating the process must end with a Vuza canon: if not, one would reduce eventually to the trivial canon 0 ⊕ 0,
but it satisfies (T2). So
Corollary 1. The spectral and (T2) conjectures are true for RC in general if and only if they are true for VC.
Hence the crucial importance of Vuza canons: if we know all about VC, then we know all about all RC – notably, whether
the Spectral Conjecture is true or not. Another consequence is one I stated a few years ago ([3]):
Corollary 2. The (T2) conjecture is true in any ‘good group’.
This adds a few cases to the one proved by [11], like n = pqrs or n = pqr 2 .
3. New perspectives
It is better to present the new techniques, which are detailed in [19], by way of an example, and show how they provide
insight on VC.
3.1. Example. For n = 120, the possibilities for SA, SB are the following:
SA = {2}, {4}, {8}, {3}, {5}, {2, 3}, {2, 4}, {2, 5}, {2, 8}, {3, 4}, {3, 5}, {3, 8}, {5, 8}
and SB is the complement set of SA in {2, 3, 4, 5, 8}.
Let us consider one of the most interesting cases, SA = {3, 4}.
We are looking for tilings of Z120 with motif A, where SA is supposed to be equal to {3, 4}. To begin with, by condition
(T1) A has 6 = Φ3(1) × Φ4(1) elements; if B is some outer rhythm such that A ⊕ B = Z120, then SB = {2, 8, 5}.
Also, A must provide a tiling of Z12, or, to be precise, the projection A ′ of A to Z12 must tile. This is easily seen
from an example (the general theorem will be addressed infra, see Thm. 10): say A = {0, 30, 32, 62, 88, 118}, then A
mod 12 = A ′ = {0, 6, 8, 2, 4, 10} is easily seen to tile Z12.
So one only has to start from the possible tilings of Z12 by A ′ ’s with SA ′ = SA. The order of the group being small, this
is quickly computed. For such compliant cyclic groups, one can use the General Coven-Meyerowitz Complement formula
given in [11]: it is known that for any tiling of Zn, n < 120, condition (T2) must be true; so the construction applies. Here
for each element of SB = {2}, we look for the largest factor in n = lcm(SA) = 12 that is coprime with it, e.g. 3. We form
Φ2(X 3 ) = 1 + X 3 . In general, the Universal CM Complement is the product of factors obtained in this manner. Here we
get B = {0, 3}. 6 The only possibilities for A tiling with B, up to rotation, are
A ′ 1 = {0, 1, 5, 6, 7, 11}, A ′ 2 = {0, 2, 4, 6, 8, 10}.
As we will see, they open two very different alleys, the latter giving rise to Vuza canons while the former does not. Notice
that RA = {3, 4, 12} while RA ′ = {3, 4, 6, 12}.
6 For larger numbers, a recursive construction can be envisaged.

NEW PERSPECTIVES ON RHYTHMIC CANONS AND THE SPECTRAL CONJECTURE 7
Solutions of A ≡ A ′ 1 mod 12 are just copies of A ′ 1 with every element translated by some multiple of 12: A = {12k0, 1 +
12k1, 5 + 12k2, 6 + 12k3, 7 + 12k4, 11 + 12k5}. k0 can be taken =0 up to rotation, it remains to compute 12 5 = 248, 832
possible A’s. Among these, there are 50,000 aperiodic tiles up to rotation, like {0, 1, 5, 6, 7, 35} or {0, 1, 5, 6, 31, 47}. But
they come in only 6 classes of values of RA, which are
{3, 4, 12}, {3, 4, 12, 15}, {3, 4, 12, 20}, {3, 4, 12, 20, 60}, {3, 4, 12, 15, 20, 60}
Notice that 24, 40 and 120 are always missing. As it happens, this precludes any associated outer rhythm B from being
aperiodic. This is the trickiest part (see Thm. 9) but a valuable one, as it prunes off some cases that might prove lengthy
to compute. Indeed, let us consider a hypothetical outer rhythm B for either of these A’s. As RA ∪ RB = Div(120),
it means that RB must contain at least 2, 8, 5, 6, 24, 40, 120. I pick some of those cyclotomic factors and compute their
product:
Φ8Φ24Φ40Φ120 = X 4 + 1 X 8 − X 4 + 1 X 16 − X 12 + X 8 − X 4 + 1 X 32 + X 28 − X 20 − X 16 − X 12 + X 4 + 1
= 1 + X 60 must divide B(X)
and this implies that B is 60-periodic: for every power X k featuring in B(X), there is also X k±60 , meaning that k ∈ B ⇒
k ± 60 ∈ B.
On the other hand, if A ≡ A ′ 2 mod 12, we get only 16,663 aperiodic solutions for A, with 10 possible RA values. Seven of
those are excluded for the same reason as above, e.g. RA = {3, 4, 6, 12, 20}. Others are plausible, and indeed 18 aperiodic
outer rhythms are found, e.g. {0, 1, 12, 18, 21, 24, 25, 36, 45, 49, 60, 69, 72, 73, 78, 84, 93, 96, 97, 117}. This is more than the
8 solutions provided by Vuza’s algorithm. By completion, we find that they tile with 8 different A’s – exactly the 8 ones
that we already knew from Vuza’s algorithm.
3.2. The wheels behind the works. Several interesting features of rhythmic canons are involved in the last example.
Mostly they can be found in the seminal paper [11], but they had seldom been noticed before Matolcsi and Kolountzakis
made use of them. A few useful rules of thumb are taken from the forthcoming collective book on rhythmic canons [6].
Let us take them in order, begininning with the run of the mill. The first three are taken from [11].
Theorem 5. (1) ♯A = A(1) =
p (T1).
p α ∈SA
(2) If A ⊕ B = Zn then SA ∩ SB = ∅, RA ∪ RB = Div(n) (the divisors of n, 1 excepted).
Theorem 6. If A tiles with some period n, then A also tiles with period m = lcm SA, i.e. (more precisely) the projection
A ′ of A in Zm tiles too. Also SA ′ = SA.
This is musically interesting, as it enables to find a smaller period RC. Computationally speaking, the best working
method, as examplified above, consists mostly in stressing the difference between all incarnations in higher periods of a
tile of a smaller group. In a way, this embodies the idea of a finer perception of complex rhythms. Alternatively, this
could be seen as a vindification of the notion of Vuza canon, as it is the smaller period of a RC (and the irreducible motif)
that is perceived by the listener.
Theorem 7. If A satisfies conditions (T1) and (T2), then A tiles with the standard CM complement B which is built from
SA alone.
The recipe for the standard CM complement B (sketched in the example above) is given in [11, 4] and is used for ‘cyclotomic
canons’ in the OpenMusic visual programming language. The interesting point is that (compare with previous theorem)
only SA is relevant. In the same light, we have already mentioned Thm. 2, stating that sharing the same RA as a known
tile is enough to ensure the tiling property. It actually holds when RA ⊂ RA ′, but not when we only assume SA = SA ′.
Another interesting issue is the difficult theorem B2 of [11], completed by myself (see proof in [3, 4]):
Theorem 8. If A tiles Zn with Zn being either a ‘good group’ (Hajós group), OR n having at most two prime factors,
then (T1) and (T2) are true.
This provides a starting point for building tiles with a given SA in a ‘good group’ where the hoped-for tile is projected:
this is originally an idea of Matolcsi, to find the outer rhythm B as a universal CM complement of the putative A; then
completing B and selecting all aperiodic A’s in the list of solutions.
Now for the periodicity criteria, also used in [19]:
Theorem 9. The subset A ⊂ Zn is periodic (meaning A + τ = A for some 1 < τ < n) iff for some maximal prime power
factor p α of n, all multiples of p α are in RA.
Proof. We prove that this means exactly that the polynomial ∆n,p = Xn − 1
X n/p − 1 = 1 + Xn/p + X 2n/p + . . . divides A(X).
The roots of this polynomial are the n th roots of unity which are not (n/p) th roots of unity, i.e. whose order divides n

8 EMMANUEL AMIOT
but is a multiple of pα , the largest power of p in n: this is because, out of the cyclotomic factors Φd of Xn − 1, d running
over all divisors of n = pαqβ . . . , we have cancelled out all Φd with d a divisor of n/p = pα−1qβ . . . . All that remains are
the Φd, d = pαqβ′ . . . , β ′ ≤ β, and so on. A small check is still needed to see that divisibility by ∆n,p is the same thing as
n/p periodicity. Clearly, if A(X) = ∆n,p × A(X) where A(X) is a polynomial with coefficients 0 or 1, and A is a subset of
Z n/p (meaning its elements are distinct modulo n/p) then A is just the concatenation of p copies of A and hence periodic.
Conversely, we must be sure that if A is n/p−periodic, then A(X) = A(X)/∆n,p is 0-1. Firstly, it is well defined modulo
X n/p −1 as A(X) is defined modulo X n −1. Next, for any element a ∈ A, the whole orbit {a+k n/p} is part of A. Taking
in each orbit the one and only element that lies in [0, n/p − 1] we define the required A ⊂ Z n/p satisfying the polynomial
identity.
Another way to see it is that the Discrete Fourier Transform 7 of A is nil except on a subgroup of Zn, a classical characterization
of periodicity.
This enables to discard a number of cases, either because A would be periodic, or because a complement (outer rhythm)
B would.
This is a nice feature of tiles of Vuza canons: their RA’s must be neither too large nor too small, but ‘just right’. Though
VC may appear as extreme cases in terms of canonical reduction, their aperiodicity is in fact a perfect balance, seldom
achieved, between the natural propensities to periodicity of both factors A and B.
Theorem 10. If A tiles a cyclic group Zn but lcm SA = m < n, then A ′ = A mod m tiles Zm.
Along the proof (see [11]) is seen that SA = SA ′. This enables to construct VC from smaller RC, but only for some sets
SA (those without all the largest prime powers dividing n).
3.3. Coming to grips. Using the techniques explained in [19], we managed to find all VC for n = 120, just like they did
for n = 144. The code and the full lists of the A’s and B’s found are in a Mathematica notebook online: [5]. The table
in fig. 6 provides the number of different solutions, classified according to the repartition of the divisors of n between RA
and RB. The most interesting case is the last but one (discussed in [19]).
Let it be mentioned that for n = 72, 108 or 144, it is known with Thm. 3.3 that condition (T2) is satisfied for any RC.
Owing to the decomposition techniques developed in [1], n = 120 was the first period where it might have been possible
to find a counterexample to the “(T2) conjecture”. So the classification of all Vuza canons (which do satisfy (T2)) for this
period has theoretical interest: we can now claim that the “(T2) conjecture” is true for all periods lower than 168.
It is still mysterious how some RA’s are possible and some others are not. This is true in greater generality for ordinary
RC: the big mystery is the gap between SA (which perfectly determines SB) and RA, even in the cases when condition
(T2) is known to hold. What is one to make, for instance, of the products of elements of SA and SB? Even condition (T2)
tells nothing at all about these (we only have some exclusion conditions when looking for aperiodic tiles). But using the
toponymy of RA as a starting point enables to get an inkling of which sets of divisors will give rise to canons, and which
will not (see table in fig. 6). Proving such yet informal ideas (like ‘SA must have at least two coprime elements’) might
significantly reduce the computation times for the search of new VC. This toponymy is a slightly broader classification
than orbits under the affine group modulo n (see [4]), but a much more illuminating one. We may expect shortly a few
conjectures in this direction.
All but one case for SA had been seen before, either directly from Vuza’s algorithm or by transformational techniques; but
there are several new values for RA. Also it is a huge progress to know that only the values in the table are allowed for
VC, and especially to appreciate the few values of RA (often only one) that are allowed for each of these SA. For instance,
it is interesting to notice how the different powers of a given prime are always intertwined between SA and SB.
4. Comparison with previous algorithms for VC
4.1. Completion. One strange thing about the completion routine fully described in [19] is that, like the QuickSort
algorithm ([23]), its complexity (running time) is not precisely predictable. In practice, and for heuristic reasons, it
works fairly quickly, especially when the tile to complete is ‘irregular’ as it forces several ‘no choice’ situations instead
of a general combinatorial explosion. But for general cases, it fares no better than the brute-force-tree-search used by
Fripertinger. It must be mentioned that Fripertinger enhanced his algorithm with a trick or two, like assuming without
loss of generality that the first two beats in the pattern exhibit the largest gap, i.e. the program searches for a set
B = {b0, b1 . . . bk}, 0 = b0 < · · · < bk < n, completing A, with the condition that ∀i, bi − bi−1 ≤ b1 − b0. Hence the number
of cases studied is not worse than n
♯B
such formula is known for Matolcsi’s algorithm, though it runs spectacularly faster in some cases, and just as slowly in
others (like ‘metronomes’, e.g. A = {0, k, 2k, 3k, . . . }). It is thus particularly well-suited to Vuza Canons.
7 Related to the associated polynomial by FA(t) = A(e 2iπ t/n ).
♯B
, which is lower than e ♯B and often several orders of magnitude below that. No

NEW PERSPECTIVES ON RHYTHMIC CANONS AND THE SPECTRAL CONJECTURE 9
n RA RB number of A’s number of B’s
72 {2, 8, 9, 18, 72} {3, 4, 6, 12, 24, 36} 6 3
108 {3, 4, 12, 27, 108} {2, 6, 9, 18, 36, 54} 252 3
120 {2, 3, 6, 8, 15, 24, 30, 120} {4, 5, 10, 12, 20, 40, 60} 20 16
120 {2, 5, 8, 10, 15, 30, 40, 120 {3, 4, 6, 12, 20, 24, 60} 18 8
144
144
144
144
{2,8,9,16,18,24,72,144}
or {2,8,9,16,18,72,144}
{2, 4, 9, 16, 18, 36, 144} or
{2, 4, 6,9, 16, 18, 36, 144} or
{2, 4, 9, 12, 16, 18, 36, 144}
{3, 4, 6, 8, 12, 24, 48, 72} or
{3, 4, 6, 8, 12, 24, 36, 48, 72}
{2, 3, 6, 8, 12, 24, 48, 72} or
{2, 3, 6, 8, 12, 18, 24, 48, 72}
{3,4,6,12,24,36,48} 36 6
{3,6,8,12,24,36,72} 8640 3
{2,9,16,18,144} or
{2,9,16,18,36,144}
Figure 6. Classification of all Vuza canons up to n = 144.
156
+6
48
+12
{4, 9, 16, 18, 36, 144} 324 6
4.2. Going further. Fripertinger’s exhaustive search had left us at n = 108 (n = 120 would probably have been manageable).
With some ingenuity and tricks, which where developed in order to face the tougher cases, n = 144 was completely
and rather quickly solved by Matolcsi and Kolountzakis (see [19] in the same issue), and I followed with the case n = 120
– apparently easier, but complicated by the non assumption of condition (T2) – shortly thereafter. I also used Matolcsi’s
method for checking Fripertinger’s results for n = 72, thus getting acquainted with his ideas. Italian mathematician Giulia
Fidanza, in her tesi di laurea, had already improved Vuza’s algorithm, thus adding some more solutions to the database
for n = 144 and n = 216. See [16]. All in all, the computer time (once all the good moves are found) does not exceed a
few hours on a personal computer – less than one hour for confirmation of the n = 72 case. With a few improvements, all
Vuza canons up to n = 200 or maybe even 300 should be attained soon.
Alternatively, the nice 3D algorithm devised by Kolountzakis, apart from its theoretical consequences ([19]), actually
enables to compute VC for reasonable values of n. 8 This might offer an alternative for medium-sized values of n to the
previously known algorithms, if it yields new solutions, and hence wider possibilities for composers. It is particularly
well-suited to the computation of random VC by musical software.
4.3. Conjectures. The computations above do not address the question of whether a given subset of Zn is, or is not,
spectral. But the consideration of RA is directly connected with the slightly stronger ‘(T2) conjecture’. This is known to
be true in Zn when n has at most two prime factors ([11]) and in all Hajós groups ([3]). Here the computation for all
specific values of RA enables to focus on the cases where a counterexample might be found: factors of SA and SB have to
be exactly right for it to be simply possible. 9 In such a light, it might be possible to find counterexamples by researching
some very specific values of SA. . . at any rate if the conjecture is false. It could be a safe bet, as all settled conjectures on
this subject have been wrong so far.
In the same direction but with a dual approach, it is interesting to enlarge the database of cyclic groups where ‘(T2) ⇐⇒
tiling’ is known to be true, n = 120 being a first step in that direction. One good reason for this is that it enables to build
larger VC recursively, using tilings of smaller groups like it was done in the example: if the (T2) conjecture is known to
8 After Kolountzakis exposed his algorithm during a MaMux session at IRCAM, we managed to produce VC in the comparatively small
group Z180 with his method; unfortunately they were all previously known, as members of the cyclotomic class, SA = {2, 3}.
9 For instance, for n = 120 most partitions in SA, SB yield factors that are known to satisfy (T2). An exception is SA = {3, 5, 8}: it took
extra care to check that this case yields no Vuza Canons without assuming that any RC satisfied (T2), starting from the other side SB = {2, 4}.
Luckily, it also made for a shorter computation.

10 EMMANUEL AMIOT
be true in the smaller group, then the ‘Universal CM Complement’ can be used there, speeding up the process. This is
useful both for exhaustive catalogues of Vuza canons, and for trying to build counterexamples.
Finally, though the Universal CM Complement is only supposed to work when (T2) is satisfied, paradoxically it might
be the best way to construct counterexamples to the (T2) conjecture, as such complements B for a given SA are by
construction overloaded with superfluous cyclotomic factors; hence it may be hoped that some complements A of the
complement B will lack one or two products of elements of SA in their RA, i.e. (T2) might be false though A ⊕ B = Zn.
Moduli below 900 are unlikely to be productive in that respect, so we will need finer programs, or faster computers; 10 but
using this wealth of new ideas, this now seems well within reach.
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10 A personal computer was unable to give the aperiodic complements of a CM Universal Complement in Z900 in a fortnight.