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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
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| Volume 1, Number 1 January, 1995 ISSN: 1067-3040 |
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All queries to: mto-editor@husc.harvard.edu
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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
musicaux."
[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
Vieru.
[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:
158-183.
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:
270-305.
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