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       M U S I C          T H E O R Y         O N L I N E

                     A Publication of the
                   Society for Music Theory
          Copyright (c) 1995 Society for Music Theory
| Volume 1, Number 1     January, 1995     ISSN:  1067-3040   |

  All queries to: mto-editor@husc.harvard.edu
AUTHOR: Clampitt, David
TITLE:  Report: An International Symposium on Music and Mathematics
        (Bucharest, Romania)
KEYWORDS:  mathematics

David Clampitt
SUNY, Buffalo
Department of Music
Buffalo, NY

[1] An International Symposium on Music and Mathematics, 
organized by Anatol Vieru, John Rahn, and John Clough, 
was held in Bucharest, Romania on May 29-30, as part of 
the 1994 International Week of New Music.  Thirteen 
composers, mathematicians, and music theorists from 
Romania, Israel, France, and North America contributed 
papers.  The papers presented fell into three 
categories: compositional theory, extensions from 
diatonic set theory, and computer models and formal 
logic.  A range and variety of interests were 
represented; nonetheless, papers from one area sometimes 
overlapped those of another in interesting ways.  

[2] The various events of the Week of New Music took 
place in the heart of Bucharest, within a few blocks of 
Revolutionary Square, around which one still sees 
buildings scarred by the fighting in 1989.  The 
architectural melange includes elements from the time of 
Vlad the Impaler, cheek by jowl with gray Ceausescu-era 
edifices.  The Symposium itself was held in the marbled 
Hall of the Union of Composers and Musicologists, which 
is in the Georges Enesco Museum, formerly the home of 
that multifaceted musician.  There were about thirty 
interested listeners, in addition to the participants, 
at the sessions. 

[3] Anatol Vieru, one of the leading Romanian composers 
and really the tutelary genius of the Symposium, offered 
some brief opening remarks and the first paper, "The 
Musical Significance of Multiplication by 5 and 7: 
Diatonicity and Chromaticity."  A composer and theorist 
on the faculty of the Bucharest Academy of Music, 
trained in Moscow in the 1950s, Vieru is the author of 
*Cartea modurilor*, of which an expanded English version 
has been published as the *Book of Modes*, (Vieru 1993).  
This work includes both a version of what we call set 
theory, arrived at by the author largely independently 
of American theory, and various compositional theories.  
The flavor and some of the substance of this work has 
been available to a North American audience through two 
articles in *Perspectives of New Music* (Vieru 1985, 
1992).  Vieru's Symposium paper elaborated a topic 
treated in the *Book of Modes*, in which diatonicism is 
associated with the multiplicative units 5 and 7, while 
chromaticism is associated with the units 1 and 11.  
Vieru measures the diatonic-chromatic content of any 
modal structure, i.e., a set-class determined by 
transpositional equivalence, by comparing its ic-1 
fragmentation to its ic-5 fragmentation.  He views the 
diatonic-chromatic duality as a fundamental aspect of 
the 12-pc universe. 

[4] Mathematician Dan Vuza gave an introduction to 
supplementary sets, the subject of his four-part study 
in *Perspectives of New Music* (Vuza 1991, 1992, 1993).  
Vuza's paper, "Supplementary Sets: Theory and 
Algorithms," traced the development of his work from its 
genesis in the pitch domain to its complex unfolding in 
the rhythmic domain, giving a summary of some of his 
results.  Vuza has been a significant resource for some 
Romanian composers, among them Vieru, who remarked of 
him: "Il n'est pas musicien, mais il a des reves 

[5] Stefan Niculescu, in "Quelques aspects de la 
relation entre musique et mathematique dans mes 
oeuvres," presented a notion of complementary rhythmic 
counterpoint related to Vuza's approach.  Niculescu, 
whose music was the focus of the Week's opening concert, 
categorizes rhythmic textures in terms of symbolic logic 
and the relations *simultaneity* and *succession*. 

[6] Eytan Agmon, from Bar-Ilan University, spoke of 
"Diatonicism and the Farey Series."  The Farey series of 
order N consists of the fractions in reduced terms 
between 0 and 1 whose denominators do not exceed N.  In 
a development not unrelated to Vieru's opening paper, 
Agmon investigated the "chromatic and enharmonic 
consequences of the Farey property," as well as the role 
the property plays in his own model of diatonicism, as 
set forth in Agmon 1989. 

[7] John Clough, from the State University of New York 
at Buffalo, and mathematician Jack Douthett, from 
Albuquerque Community College and Technical-Vocational 
Institute, continued their investigations into the 
nature of diatonicism with "Hypertetrachords."  The 
class of scales Clough and Douthett defined as 
*hyperdiatonic* in their 1991 *Journal of Music Theory* 
paper, "Maximally Even Sets," also generalizes the 
tetrachordal structure of the usual diatonic.  Their 
work treated the Indian *gramas* as well, and gave scale 
axioms for diatonic sets and for the *gramas*. 

[8] David Clampitt, graduate student in music theory at 
SUNY at Buffalo, presented "Some Refinements of the 
Three Gap Theorem, with Applications to Music."  The 
Three Gap Theorem is a result proved by the Hungarian 
mathematician Vera Sos and others in the 1950s that 
bears on the class of scales called *well-formed* (Carey 
and Clampitt 1989), and gives information about any set 
generated by a single interval.  The continued fractions 
that arise in this approach are related to the fractions 
of the Farey series discussed by Agmon. 

[9] On Monday six more papers were given, beginning with 
Jay Rahn, from York University, Ontario.  Rahn's paper, 
"A Non-Numerical Predicate of Wide Applicability for 
Perceived Intervallic Relations," was an extension of 
his 1992 *Perspectives* article, applying symbolic logic 
in the tradition of Benjamin Boretz. 

[10] "Abstract Machines and Music" was the subject of a 
talk by Marc Chemillier, a computer scientist situated 
in Paris and Caen.  Given his title and Ligeti's 
predilection for the *meccanico*, it was not entirely 
unexpected that Chemillier would offer an "analysis and 
computer reconstruction of a musical fragment of 
Ligeti's 'Melodien.'" 

[11] John Rahn, from the University of Washington at 
Seattle, followed Chemillier's discussion with a more 
general paper on formal models, "Remarks on Network 
Models for Music," stemming from Rahn's long-standing 
interest in neural nets and parallel processing.  Like 
Stefan Niculescu and Aurel Stroe, Rahn participated in 
the conference both as theorist and as composer, and the 
Symposium had its origins in his association with Anatol 

[12] The meetings concluded with three papers 
representing the diversity of Romanian theoretical work.  
Mihai Brediceanu presented examples of topological 
transformations in music with "Topology of Sound Forms 
and Music."  The above-mentioned Aurel Stroe discussed 
the mathematical formulation of some of his own 
compositional methods in "Classes de composition 
musicales."  Dinu Ciocan brought the session to an 
appropriately open-ended close with analyses of Bach and 
Schoenberg that at once stemmed from and called into 
question linguistic approaches to analysis with 
"Quelques problems de modelisation semiotique et 
computationelle du langage musical." 

[13] There were ample opportunities for informal 
exchanges among the participants that were of 
considerable interest.  Among the ceremonial occasions 
was a lunch hosted by Octavian Cosma, Vice-President of 
the Union of Romanian Composers and Musicologists and 
editor of Muzica.  This was just one example of the 
impressive marshalling of resources by the organizing 
committee chaired by Anatol Vieru.  Both practically and 
substantively, this multi-disciplinary, multi-lingual 
Symposium was highly successful. 

[14] Reference List 

     Agmon, Eytan. 1989. "A Mathematical Model of the 
Diatonic System." *Journal of Music Theory*, vol. 33, 
no. 1: 1-25. 
     Carey, Norman and David Clampitt. 1989. "Aspects of 
Well-Formed Scales." *Music Theory Spectrum*, vol. 11, 
no. 2: 187-206. 
     Clough, John and Jack Douthett. 1991. "Maximally 
Even Sets." *Journal of Music Theory*, vol. 35, nos. 1 & 
2: 93-173. 
     Rahn, Jay. 1992. "An Advance On *A Theory For All 
Musics*: At-Least-As Predicates For Pitch, Time, And 
Loudness." *Perspectives of New Music*, vol. 30, no. 1: 
     Rahn, John. 1994. "Some Remarks on Network Models of 
Music." In *Musical Transformation and Musical 
Intuition: Essays in Honor of David Lewin*. Edited by 
Raphael Atlas and Michael Cherlin. Roxbury, MA: Ovenbird 
Press, 1994. 
     Vieru, Anatol. 1985. "Modalism--A 'Third World'. " 
*Perspectives of New Music*, vol. 24: 62-71. 
     ________. 1992. "Generating Modal Sequences (A 
Remote Approach to Minimal Music)." *Perspectives of New 
Music*, vol. 30, no. 2: 178-201. 
     ________. 1993. *The Book Of Modes*. Translated by 
Yvonne Petrescu. Bucharest: Editura muzicala. 
     Vuza, Dan Tudor. 1991, 1992, 1993. "Supplementary 
Sets and Regular Complementary Unending Canons." 
*Perspectives of New Music*, (Part One) vol. 29, no. 2: 
22-49; (Part Two) vol. 30, no. 1: 184-207; (Part Three) 
vol. 30, no. 2: 102-125; (Part Four) vol. 31, no. 1: 


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