Vuza Canons At Their 20 th Birthday A Not Happy Anniversary

Vuza Canons At Their 20 th Birthday A Not Happy Anniversary Dan Tudor Vuza Dedicated to the memory of my mother Dr. Valentina Vuza, deceased on 17 June 2012 D. T. Vuza worked in what is called today the mathematical theory of music between years 1980 1990 while employed by the mathematics department of INCREST, under the harsh conditions imposed by communism 1. Stimulated by discussions with Romanian musician Anatol Vieru with which he had a long-standing collaboration, Vuza developed in 1981 the algebraic formalism for Vieru s modal calculus [2]. This led him naturally to his model of periodic rhythms [3] of 1985, the first stage towards the theory of rhythmic canons that was to be constructed later. The group of pure mathematicians working in the institute by that time received his work mostly with skepticism and criticism. Even the early recognition of the rhythmic model by Russian linguist M. G. Boroda, who asked for republication of [3] in his own Musikometrika series and presented the paper as coming from a leading expert, did not improve the situation. Vuza was even obliged by the editorial board to withdraw a paper submitted to Revue Roumaine de Mathématiques Pures et Appliquées, which caused the study [2], supposed to have six parts, to be forcibly terminated after the fifth. Between 1985 1989, Vuza worked on his theory of periodic rhythmic canons that constituted the subject of the paper submitted in 1990 to Perspectives of New Music, which published it in several installments [4]. This was his last contribution to mathematical theory of music, before he quit the field and the pure mathematics in favor of other research subjects 2. The paper was soon remarked by the young and enthusiastic researcher Moreno Andreatta who brought it to the attention of his colleagues from IRCAM. Since that moment, the subject flourished. Today, a whole research topic is active around what have been called Vuza canons. In the following we shall consider the set Q of all rationals. A rational r is said to divide a rational s if s/r is an integer. According to the rhythmic model [3], a periodic rhythm is a subset R of the rationals Q satisfying the following conditions: i) R [a, b] is finite for every bounded interval [a, b]; ii) there is t Q\{0} such that t + R = R. The smallest positive t satisfying ii) is called the period of R and denoted Per R. The group Q acts on the set of all periodic rhythms by (s, R) s + R, the orbits of this action being called rhythmic classes and denoted with italic letters. Rhythms in the same rhythmic class R have the same period, which is therefore called the period of the class R and denoted Per R. The elements of the set R are to be thought as the beats of a rhythmic pattern that repeats itself indefinitely 3. A type of rhythm particularly important for the model is the regular rhythm, that is, any set of the form {s + nt n Z} for some s Q and t Q\{0}. The unique regular rhythmic class of period t will be denoted by Reg(t). Given the rhythmic classes R, S and the representants R R, S S, the set R + S is a rhythm whose class, denoted by R + S, depends only on R and S. In this way the set of rhythmic classes is endowed with an additive law turning it into a semigroup. Its idempotents are precisely the regular rhythmic classes. Two rhythms R, S are said to be intervalically disjoint if every rational belonging to (R R) (S S) is divisible with both Per R and Per S. The definition is extended to rhythmic classes, via representants.

In the framework of the theory of [4], a rhythmic canon on n voices is defined as a collection = {R 1,..., R n } of periodic rhythms belonging to the same rhythmic class, called the ground class of. By definition, the period of is the period of its ground class. The action of Q is naturally extended to the set of all canons, its orbits being called canon classes. We may think of a rhythmic canon as of an ensemble of performers that deliver indefinitely the same rhythmic pattern while maintaining constant temporal shifts between each couple of voices. The rhythmic class of the union R 1... R n is called the resultant class of ; it is the rhythmic pattern perceived when hearing the performance as a whole, by superposing the beats from all voices. The ground pattern represents the most obvious aspect of the periodicity of a canon, which Andreatta calls the inner periodicity [29]. However, there is a less obvious aspect, the outer periodicity, which equally has a musical meaning. In order to model it, start with the rhythmic classes R, S and pick representants R R, S S. The ensemble {s + R s S} is a collection of rhythms which turns out to be finite, hence is a canon whose class depends only on R and S and is denoted Can(R, S). The pair (R, S) is called normal if Per S divides Per R. The map (R, S) Can(R, S) establishes a bijection between normal pairs and canon classes. In this representation, the class S, which was called in [4] the metric class of the canon, is the one that describes the outer periodicity, which is dual to the inner periodicity described by R. Imagining that each performer enhances with a metric accent the beginning of each repetition of the pattern he plays, then S is the class of the rhythm obtained by collecting the accents from all voices (hence the term metric class ). The semigroup structure of the set of rhythmic classes proves to be a useful tool in the study of canon, as the resultant class of Can(R, S) is precisely R + S. Given a normal pair (R, S), we may invert it to (S, R) and then normalize by replacing R with Reg(Per S) + R. This suggests the following procedure. Start with a normal pair (R 0, S 0 ) and form recursively (R n, S n ) by R n = S n-1, S n = Reg(Per S n-1 ) + R n-1. The even indexed canon classes Can(R 2n, S 2n ) are condensed versions of the initial class Can(R 0, S 0 ). We stop the recursion when we have R 2n+2 = R 2n and S 2n+2 = S 2n, which is always the case after a finite number of steps. This happens precisely when Per R 2n = Per S 2n. Canons belonging to classes Can(R, S) for which Per R = Per S were termed by Vuza canons of maximal category. They have equal degrees of inner and outer periodicities. They can be inverted to form Can(S, R), in which the ground and metric classes have changed places. The canon of maximal category formed by the above-described recursion is the least condensed one that can be obtained from a given canon; this statement can be given a formal sense by introduction of an appropriate order relation on the set of canon classes. A regular complementary canon as defined in [4] is a canon {R 1,..., R n } for which the sets R i are pairwise disjoint and whose resultant is a regular class. Their musical significance comes from the union of two apparently contradictory features. On one hand, voices are merged together into a uniform repetition, since the resultant rhythm is regular; but on the other hand, each voice can still be distinctly heard since each beat in the resultant rhythm comes from exactly one voice, the simultaneity of beats from different voices being not allowed 4. In the representation with normal pairs, a canon class Can(R, S) is regular complementary if and only if R and S are intervalically disjoint and R + S is regular. The modulus of a regular complementary canon is the ratio between the period of its ground class and the period of its resultant class; the modulus is always an integer. Regular complementary canons of modulus n are in correspondence with tilings of the cyclic group Z n, that is, pairs of subsets M, N with the property that every z Z n can be uniquely written as m + n with m M and n N. Tilings in which none of the sets M, N is periodic 5 correspond precisely to regular complementary canons of maximal category. It is the latter canons that have been termed Vuza canons by the authors who continued the research. In the following they will be called RCMC canons. It is far from obvious whether such canons exist. One of the main results in the paper [4] that started their study was that RCMC canons do exist: their minimal modulus is 72 and at least six voices are needed for their construction. 2

Regular complementary canons, and in particular RCMC canons, have exerted a fascination on both theorists and musicians. Much has been done [5-21, 23-25, 27-38] for the development of their theory, especially at IRCAM in Paris but also in other places, such as the Graz University. Today RCMC canons of moduli 72 and 108 are completely classified: it came rather as a surprise that there are only two RCMC canons of modulus 72, up to affine transformations. Among the rich collection of new interesting music-theoretical models introduced by Vuza, it is undoubtedly the concept of Regular Complementary Canons of Maximal Category which has proven to be a central construction being capable of intersecting a variety of different areas in mathematics. (Andreatta [29]). Today it is known that the mathematics of RCMC canons is strongly linked to Galois theory, as pointed out by E. Amiot [5-7] a fact that could be foreseen even since the original paper [4], where the proofs made heavy use of the properties of irreducible cyclotomic polynomials. As H. Fripertinger has shown [13-15], enumeration of RCMC canons is connected to Polya s combinatorial methods. Those mathematicians in the former INCREST department, that were inclined to perceive Vuza s activities as a threat to the purity of mathematics, may now sleep in peace Amiot was able to establish that RCMC canons may be relevant even for problems that can be phrased in pure mathematical terms. Once again, Vuza canons are precisely the musical constructions that could help mathematicians to give an answer to this open problem and, as a consequence, to Fuglede Conjecture in the onedimensional case. More precisely, Emmanuel Amiot showed that if a subset R of a cyclic group Z/nZ exists which tiles the space by translation without being spectral, then the Tiling Rhythmic Canon generated by R either is a Vuza Canon or can be reduced to one whose inner voice is not spectral either. In other words, (...) a possible counterexample of the spectral conjecture may already exist within the yet unwritten pages of the catalogues of all possible (and still unheard) Vuza canons. (Andreatta [29]). A sustained effort has been done by C. Agon and his colleagues at IRCAM for developing computational models and algorithms under the platform of the OpenMusic programming language [8, 19, 30]. Not only they made available to the musician packages for the construction and manipulation of regular complementary canons, but they also contributed to the discovery of aspects that could not be easily foreseen from pure theory. As an example mentioned by Andreatta [29], computation has shown that at least one of the terms in the Can(R, S) representation of a large fraction of RCMC canons has a structural symmetry of palindrome type, which established an unexpected insight into their musical significance. The series of papers published by Dan Vuza in Perspectives of New Music (...) not only constitute a milestone in the development of the mathematical theory of tiling canons but also offer new possibilities for composers to free themselves from the constraint of regularity with respect to the entries of the voices in a canon. (Andreatta [29]). Among the musicians who applied Vuza canons in their compositions one may quote Fabien Lévy (Où niche l'hibou?, 1999; Coïncidences, 1999), Georges Bloch (Empreinte sonore pour la Fondation Beyeler, 2001), Mauro Lanza (La descrizione del diluvio, 2007), Tom Johnson [27]. There were doctoral and master degree thesis [33-36] that continued the research on Vuza canons, there were courses at the Chicago University 6 on the topic of Vuza canons. The monumental monograph The Topos of Music [37] (over 1300 pages) included an account of Vuza s rhythmic model of 1985 and canon theory of 1990. What his colleagues mathematicians who criticized his work were not able to understand, was that Vuza, in his time, was among the people who contributed to the foundation of a new research field. Vuza's series of articles and his use of non-elementary mathematical concepts, like characters and the discrete Fourier transform applied to locally compact groups, still remains one of the most significant mathematical contribution to the musicological community. (Andreatta [29]). A field well established today under the name of mathematical theory of music and to which a scientific publication is dedicated Journal of Mathematics and Music, listed among the ISI publications. As recognition of his contribution, Vuza was elected in the first editorial board of the named journal. In 2009, the journal edited a special issue on canons and related problems, comprising the contributions [23-25] and dedicated to 3

Dan Tudor Vuza, with profound gratitude for providing a timeless subject around which mathematics and music naturally meet [22]. Today, at the celebration of 20 years since the publication of the last installment of [4] in Perspectives of New Music, the journal that published the paper decided to edit in its turn a special issue, comprising the contributions [27-32]. In his foreword [26], professor John Rahn, the chief-editor, has written: Perspectives of New Music is uniquely appropriate for such a feature, because all this mathematical work on tilings in music derives from a very long, unprecedentedly difficult (for musicians) article by a young Romanian mathematician named Dan Tudor Vuza. (...) Twenty years later, this is the fruition. This feature in Perspectives of New Music on musical tilings is dedicated to Dan Tudor Vuza. And Tom Johnson, in his contribution [27] to the special issue, wanted to confess the significance that the encounter with Vuza s paper had for him. It must have been 1999 when Moreno Andreatta gave me a copy of Dan Tudor Vuza s landmark essay (...). I recently reread the Vuza essay, and it seems all the more clear to me that this text has been important for all the mathematicians and music theorists cited above (...). I now consider this work the most important music theory treatise of the last 20 years, particularly since it is one of those rare cases where music theory has preceded musical practice. Harmony and counterpoint books, essays on serial techniques, manuals for figured bass, and music analysis texts have generally dealt exclusively with musical procedures already practiced by composers. Only in rare cases like Vuza, Leonhard Euler, and Hugo Riemann have theorists preceded composers, though Henry Cowell s 1930 book New Musical Resources should also be mentioned. (...) Cowell s book has left a significant mark on music history, as Vuza s ideas are just beginning to do. With such recognition from abroad, has the situation of Vuza canons changed in their native country? Not at all. In Romania it is generally believed that in order to get recognition inside the country, one should first get recognition abroad. In this respect, Vuza canons represent the exception in that even after getting the recognition from many experts and publications from abroad, Romania continues relentlessly to ignore them. In 2002, when the canons celebrated their 10 th birthday in total anonymity, Vuza was announced in the morning that the newly established Foundation for Art and Science of the Romanian Academy decided to award him a prize for contributions to mathematics and music. From morning to the afternoon when the ceremony took place, the word music disappeared from the motivation, as if the Academy became meanwhile ashamed of that word, which left a price just for mathematics and nothing else. This was in contrast with the other persons who received awards for well-defined reasons and concrete achievements. Later on, the Academy heads thought it was better to add a reason to the award, so they chose debut in science. Given that Vuza had at that moment the age of 47, well suited for a debut in science, this really turned the award into a comedy à la Caragiale 7. Only professor Solomon Marcus has understood the canons and has tried to do something for them [39]. But so far, he is a voice alone in the desert. A desert that takes possession of a country eroded by political disputes and financial interests. Vuza tried to cross the interdisciplinary barriers, but interdisciplinarity is not at home in Romania. The leaders of interest groups establish strong barriers between fields of activities that are not to be crossed by outsiders. Vuza was always an outsider. He was not understood by his colleagues mathematicians. And of course he was an outsider for musicians as officially he does not belong to their profession 8. Vuza activated always alone, as even the people abroad that have read his musical theories acknowledged it. At their 20 th birthday, the canons continue to wander through the world, while still banned from their native country. And while the researcher who brought them to the world is now confronted with the legal issues of the Romanian state that infringes the Family and Human Rights. I wish I could tell the canons, Happy Anniversary. But it is not a happy one. 28 June 2012 4

Notes 5 1 An account of the difficulties and bad events he had to face by that time is presented in [1]. 2 Today he works in the field of electronic systems development. 3 The fact that rhythms are subsets of Q reflects the requirement that in European music all durations should be commensurable. 4 As shown in [4], the usage of regular complementary canons of simple structure can be traced up to Bach s work. 5 A subset M Z n is called periodic if there is t Z n \ {0} such that t + M = M. 6 Thomas M. Fiore, 2009 Chicago REU Lectures on Mathematical Music Theory, Lecture 4 on Vuza Canons, http://www-personal.umd.umich.edu/~tmfiore/1/music.html 7 This can be seen even today at www.acad.ro/fnsa/fnsa_evn_2004.htm. In 1987 Vuza received the Romanian Academy Prize for his works in Functional Analysis. Then how could he made his debut in science in 2002? 8 And of course, at present he is an outsider with respect to the groups that dominate the world of electronic development. References 1. D. T. Vuza, "The Loneliness of a Researcher - A paraphrase to Solomon Marcus The Loneliness of the Mathematician", in Meetings with Solomon Marcus, volume 2, Spandugino Publishing House, Bucharest, 2011, p. 753-774. 2. D. T. Vuza, "Aspects mathématiques dans la théorie modale d'anatol Vieru", Rev. Roum. Math. Pures Appl. Part I: 27(2), 1982, p. 219-248. Part II: 27(10), 1982, p. 1091-1099. Part III: 28(7), 1983, p. 665-673. Part IV: 28(8), 1983, p. 757-773. Part V: 31(5), 1986, p. 399-413. 3. D. T. Vuza, "Sur le rythme périodique", Revue Roumaine de Linguistique - Cahiers de Linguistique Théorique et Appliquée 22(1), 1985, p. 73-103. Reprinted in Musikometrika I (editor : M.G. Boroda), p. 83-126. Studienverlag Dr. N. Brockmeyer, Bochum 1988. 4. D. T. Vuza, "Supplementary sets and regular complementary unending canons", Perspectives of New Music. Part I: 29(2), 1991, p. 22-49. Part II: 30(1), 1992, p. 184-207. Part III: 30(2), 1992, p. 102-125. Part IV: 31(1), 1993, p. 270-305. 5. E. Amiot, "Why Rhythmic Canons are Interesting", in E. Lluis-Puebla, G. Mazzola et T. Noll (eds), Perspectives in Mathematical and Computational Music Theory, EpOs, p. 190-209, Universität Osnabrück, 2004. 6. E. Amiot, "À propos des canons rythmiques", Gazette des mathématiques, 106, Octobre 2005. 7. E. Amiot, "Rhythmic canons and galois theory", In H. Fripertinger and L. Reich (eds.), Proceedings of the Colloquium on Mathematical Music Theory, Grazer Mathematische Berichte, vol. 347, p. 1-25, Graz, Austria, 2005. 8. E. Amiot, M. Andreatta, and C. Agon, "Tiling the (musical) line with polynomials : Some theoretical and implementational aspects", Proc. International Computer Music Conference, p. 227-230, Barcelona, Espagne, September 2005. 9. M. Andreatta, "On group-theoretical methods applied to music: some compositional and implementational aspects", Perspectives in Mathematical and Computational Music Theory, ed. G. Mazzola, T. Noll and E. Lluis-Puebla. (Electronic Publishing Osnabrück, Osnabrück), 2004, p. 169-193.

6 10. M. Andreatta, "De la conjecture de Minkowski aux canons rythmiques mosaïques", L Ouvert, n 114, Mars 2007, p. 51-61. 11. M. Andreatta et M. Chemillier, "OpenMusic et le problème de la construction de canons musicaux rythmiques", Actes des sixièmes Journées d Informatique Musicale, Paris 1999, p. 179-185. 12. M. Andreatta & C. Agon, "Elementi di teoria matematica della musica e problema della classificazione di canoni regolari complementari di categoria massimale", Il Monocordo, Vol. 6/7, 1999. 2001. 13. H. Fripertinger, "Enumeration of non-isomorphic canons", Tatra Mt. Math. Publ., 23, p. 47-57, 14. H. Fripertinger, "Remarks on Rhythmical Canons", in H. Fripertinger and L. Reich (eds.), Proceedings of the Colloquium on Mathematical Music Theory, Grazer Mathematische Berichte, vol. 347, p. 1-25, Graz, Austria, 2005, p. 73-90. 15. H. Fripertinger, "Tiling problems in music theory", in G. Mazzola, Th. Noll, and E. Lluis- Puebla, editors, Perspectives in Mathematical and Computational Music Theory, pages 153-168. epos Music, Osnabrück, 2004. 16. R. W. Hall and P. Klinsberg, "Asymmetric rhythms and tiling canons", American Mathematical Monthly, 113(10), p. 887 896, 2006. 17. F. Jedrzejewski, "A simple way to compute Vuza canons", séminaire MaMuX, janvier 2004. 18. T. Johnson : "Tiling the line (pavage de la ligne). Self-Replicating Melodies, Rhythmic Canons, and an Open Problem", Les Actes des 8e Journées d Informatique Musicale, Bourges, 2001, p. 147-152. 19. T. Noll, M. Andreatta, C. Agon, G. Assayag et D. Vuza, "The Geometrical Groove : rhythmic canons between Theory, Implementation and Musical Experiments", Actes des Journées d Informatique Musicale, Bourges, 2001, p. 93-98. 20. A. Tangian, "Constructing Rhythmic Canons", Perspectives of New Music, 41(2), 2003. 21. Georges Bloch, Vuza Canons into the Museum, in C. Agon, G. Assayag & J. Bresson (eds), The OM Composer s Book 1, Collection " Musique/Sciences ", Ircam-Delatour France, 2006. 22. M. Andreatta and C. Agon, "Special Issue: Tiling problems in music - Guest Editors Foreword", Journal of Mathematics and Music 3(2), 2009, p. 63-70. 23. E. Amiot, "New Perspectives on Rhythmic Canons and the Spectral Conjecture", Journal of Mathematics and Music 3(2), 2009, p. 71-84. 24. M. N. Kolountzakis and M. Matolcsi, "Algorithms for Translational Tiling", Journal of Mathematics and Music 3(2), 2009, p. 85-97. 25. F. Jedrzejewski, "Tiling the integers with aperiodic tiles", Journal of Mathematics and Music 3(2), 2009, p. 99-115 26. J. Rahn, "A Brief Guide to the Tiling Articles", Perspectives of New Music 49(2), 2011, p. 6-7.

7 27. T. Johnson, "Tiling in My music", Perspectives of New Music 49(2), 2011, p. 9-21. 28. F. Lévy, "Three Uses of Vuza Canons", Perspectives of New Music 49(2), 2011, p. 23-31. 29. M. Andreatta, "Constructing and Formalizing Tiling Rhythmic Canons: A Historical Survey of a Mathemusical Problem", Perspectives of New Music 49(2), 2011, p. 33-64. 30. C. Agon and M. Andreatta, "Modeling and Implementing Tiling Rhythmic Canons in the OpenMusic Visual Programming Language", Perspectives of New Music 49(2), 2011, p. 66-91. 31. E. Amiot, "Structures, Algorithms, and Algebraic Tools for Rhythmic Canons", Perspectives of New Music 49(2), 2011, p. 93-142. 32. J. P. Davalan, "Perfect Rhythmic Tilings", Perspectives of New Music 49(2), 2011, p. 144-197. 33. M. Andreatta, Gruppi di Hajos, Canoni e Composizioni, tesi di laurea, Dipartimento di matematica, Università di Pavia, 1996 34. M. Andreatta, Méthodes algébriques en musique et musicologie du XXe siècle : aspects théoriques, analytiques et compositionnels, thèse, École des hautes études en sciences sociales, Paris, 2003. 35. Giulia Fidanza, Canoni ritmici a mosaico, tesi di laurea, Università degli Studi di Pisa, Facoltà di SSMMFFNN, Corso di laurea in Matematica, 2008. 36. E. Gilbert, Polynômes cyclotomiques, canons mosaïques et rythmes k-asymétriques, mémoire de Master ATIAM, mai 2007. 37. G. Mazzola, The Topos of Music, Birkhauser, 2002. 38. F. Jedrzejewski, Mathematical Theory of Music, Collection " Musique/Sciences ", Ircam- Delatour France, 2006. 39. S. Marcus, "A fi vizibil în cultur" (To Be Visible in Culture, in Romanian), România literar 36, September 2009.