Computational Music Science
Series editors
Guerino Mazzola
Moreno Andreatta
More information about this series at http://www.springer.com/series/8349
Gabriel Pareyon Silvia Pina-Romero
Octavio A. Agustín-Aquino
Emilio Lluis-Puebla
•
Editors
The Musical-Mathematical
Mind
Patterns and Transformations
123
Editors
Gabriel Pareyon
CENIDIM-INBA
Centro Nacional de las Artes (CENART)
Coyoacán, Distrito Federal
Mexico
Silvia Pina-Romero
División de Electrónica y Computación
(CUCEI)
Universidad de Guadalajara
Guadalajara, Jalisco
Mexico
ISSN 1868-0305
Computational Music Science
ISBN 978-3-319-47336-9
DOI 10.1007/978-3-319-47337-6
Octavio A. Agustín-Aquino
Universidad de la Cañada
Teotitlan de Flores Magón, Oaxaca
Mexico
Emilio Lluis-Puebla
Departamento de Matemáticas
UNAM Facultad de Ciencias
Coyoacán, Distrito Federal
Mexico
ISSN 1868-0313
(electronic)
ISBN 978-3-319-47337-6
(eBook)
Library of Congress Control Number: 2017942977
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To the memory of Julián Carrillo
(1875–1965) and
Alexander Grothendieck (1928–2014)
Foreword
It is my great honour and pleasure to introduce you to this book which focuses on
fundamental challenges and issues in the relatively new field of Mathematical
Music Theory, in turn able to be translated into computational practice.
This book, under the title The Musical-Mathematical Mind: Patterns and
Transformations, collects the efforts of specialists who participated in the four-day
International Congress on Music and Mathematics (ICMM, which took place in
Puerto Vallarta, Jalisco, Mexico, November 26–29, 2014). Its contents reflect the
maturing of a variety of new conceptualisations on music and mathematics. This
congress was organised by the Mexican mathematicians, musicians and musicologists Octavio A. Agustín-Aquino, Juan Sebastiàn Lach Lau, Emilio Lluis-Puebla
(Congress Head), Roberto Morales-Manzanares, Pablo Padilla-Longoria, and
Gabriel Pareyon (Program Chair and Main Editor).
Mexican scholars have been uniquely proactive in the propagation and support
of the mathematical aspects of music in theory and practice, in creativity and
epistemology. Already in 2000, the First International Seminar on Mathematical
Music Theory took place in Saltillo, on the occasion of the annual congress of the
Mexican Mathematical Society, and the Fourth International Seminar on
Mathematical Music Theory took place in Huatulco, again in Mexico, respectively
organised by Lluis-Puebla, and by Agustín-Aquino.
It is remarkable that these Mexican conferences took place in the years when the
Society for Mathematics and Computation in Music (SMCM) had no conference:
its conferences are biannual and have taken place in the odd years since 2007. It is
also remarkable because the Mexican initiative proves that there is an increasing
intensity of scholarly and artistic work centred around mathematics and music. It
gives us a model of how the future of this mathemusical enterprise could look.
The program of the congress in Puerto Vallarta is not only a testimony of the
high level of scientific research achieved in the early years of the 21st century, it
also proposed a deep spectrum of musical, mathematical, physical, and philosophical perspectives that have emerged in this field of cultural and scientific
integration since its Pythagorean origins. The big difference that we observe when
comparing the state of this art to the achievements in the 20th century is the
vii
viii
Foreword
involvement of advanced techniques and concepts of modern mathematics and
physics, relating for example to Grothendieck’s topos theory and physical string
theory. It is not astonishing that the mathematician and philosopher of modern
mathematics, Fernando Zalamea, has—among other authors in this book—contributed a beautiful perspective on the philosophy that lies inside the efforts to
reunite mathematics with music as approaches to a unified universal knowledge.
Minneapolis, USA
January 2016
Guerino Mazzola
(ICMM 2014, Honorary President)
Preface
Proficiency and enthusiasm are gathered in this volume, as the fruit of a long-awaited
conference of international specialists who devote their lives to connect, exchange
and mutually involve music with mathematics and mathematics with music. We
celebrate this publication at the moment of Julián Carrillo’s (1875–1965) one
hundred and fortieth anniversary, to whom we also dedicated a special panel (with
results to be published separate from this book) during our International Congress on
Music and Mathematics (ICMM) held at Puerto Vallarta, Mexico (November, 2014).
Our conference was a unique feast of mind and feelings, sound and meaning,
imagination and empiricism, as the continuation and synthesis of a long tradition.
The link between music and mathematics is a notorious intersection at a common
origin of human civilisation embracing aesthetics, pragmatics and abstract thought.
As a matter of fact, aesthetics, pragmatics and abstraction arise as human practice
deeply rooted in a primary notion of repetition, rhythm, comparison, measurement,
spacialization and transformation, all of them common grounds for music and
mathematics.
In every part of the world, “civilisation” is a social complexity that seems to need,
from its early sources, the sprout of music and mathematics. Thus, in the context
of the original civilisations of Mesoamerica, music and mathematics are also
strongly associated. I should mention—at least briefly—some milestones in the long
history binding music and mathematics in ancient and modern Mexico: the Olmec
and the Maya peoples, so admired today for their architectural, astronomical and
mathematical achievements, must also be acknowledged for creating original
instruments, orchestras and choirs, as well as for developing their own graphic
representation of human sounds and sounds from nature. Thereafter, among the
Aztec people, the patron of poetry, symmetry, music and numbers is
Xochipilli-Macuilxochitl, a name that relates the number five with the symbolisation
of colour, abstraction, geometry, ratio and proportion.
Later, in the Spanish colony, Sister Juana Inés de la Cruz (1651–1695) developed
her own research about the connections between harmony, numbers and geometry.
Even today Sor Juana’s conceptualisations are still valid for the philosophical study
of music, such as the study of spirals for harmonic modelling. In 19th century
ix
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Preface
Mexico, Juan N. Adorno (1807–1880) published his treatise Harmony of the
Universe, based on principles of physical and mathematical harmony. Later, the
Porfirian thinker Juan N. Cordero (1851–1916) in his book Examen de los acordes
de transformación tonal (Examination of the Chords of Tonal Transformation)
proposed a principle of musical transformation based on logical axioms. A few
decades after, in the 20th century, José Vasconcelos (1882–1959) claimed that “only
the musical study of mathematics, and the rhythmic comprehension of numbers,
could be useful as effective forms of thought and discovery of the human nature”.
In the same epoch, another Mexican thinker, Samuel Ramos (1882–1959) wrote that
“All kinds of perturbation in the Universe are of a rhythmic nature. The fluency of
changes cannot be unarticulated among them; therefore the rhythm of changes is
accumulative”. Quoting Sor Juana, Adorno, Cordero, Vasconcelos, and Ramos
are part of what semiotician Mauricio Beuchot (1950–) —a contemporary of
us—acknowledges as “the Mexican devotion of Pythagoreanism and related
doctrines”.
Indeed, the orientation of Mexican cultures seems to be magnetised by the
intuitions of ratio, proportion, analogy, metonymy, and geometrical and algebraic
transformation. We may trace this influence in the most famous composers and
music theorists of modern Mexico, namely Augusto Novaro, Conlon Nancarrow,
Ervin Wilson, Julio Estrada, Manuel Enríquez, Antonio Russek, Roberto MoralesManzanares, Víctor Rasgado, and Hebert Vázquez, among others. Indeed, they
influence nowadays Mexican studies on music and mathematics as a new mixed
discipline. This transdisciplinarity also flourished thanks to the effort of mathematician Prof. Emilio Lluis-Puebla, who graduated an internationally active group
of specialists.
As I mentioned before, our meeting also devoted a special panel to the discussion of mathematics applied to music, in honour of the great violinist, conductor,
composer and maker of new musical instruments, Julián Carrillo, who through a
long and very productive life achieved the invention of music that transcended the
traditional Western principles of consonance and harmony, as he foresaw a
“universe of endless musical scales and chords”. Carrillo’s project in the domain of
physics and mathematics, and its musical output, is an inspiration for current discussion on these subjects, addressed from different viewpoints during our congress.
We may mention some recurring concepts and theoretical approaches that
motivated us during our meeting: tessellation in topological-musical spaces, scaling
and even distribution, diatonicity, algebraic transformations, networks and geometry, partitions and well-formedness theory, theories of gestures, morphisms, set
theory and fuzzy logic, as well as a new discussion on elementary particles and
quantum symmetry as interests of systematic musicology. Despite this variety, all
our mathematical proposals fell into five general areas: I. Dynamical Systems,
II. Logic, Algebra and Algorithmics, III. Gestural Theories, IV. New Methods for
Music Analysis, and V. Modern Geometry and Topology. Although we followed
this thematic division during our congress, this book is classified by alphabetical
order of authors, for the sake of practical consultation and because most of the
contributions present developments in more than one subject.
Preface
xi
I wish to end this Preface emphasising the fact that the international President of
our Congress, Prof. Guerino Mazzola, is one of the leading thinkers in the field
of the Mathematical Theory of Music; and our national Head of Congress, Prof.
Emilio Lluis-Puebla pioneered systematic musicology in Mexico and Latin
America, organising the Seminars on Mathematical Theory of Music in previous
years. We completed our group of national and international guests with the best
and more original proposals received after almost two years of organisation that
reached its climax during the four days of ICMM 2014. We remain grateful to all
our contributors.
Guadalajara, Mexico
December 2015
Gabriel Pareyon
(ICMM 2014, Program Chair and Editor)
Acknowledgements
This book would not have been possible without the generous support of many
wonderful people and organisations. We would like to thank all authors contributing their knowledge and expertise during our International Congress on Music
and Mathematics (ICMM), held in Puerto Vallarta, Mexico, November 26–29,
2014.
We are particularly grateful to Prof. Alonso Castillo-Pérez, Head of the
Department of Computer Science and Electronics, University of Guadalajara (UDG,
Mexico), for his kind and decisive support of our congress, and for his generous and
wise guidance giving rise to the construction of more modern facilities and optimal
conditions of research at CUCEI-UDG and CU-Costa-UDG, key to the preparation
and development of our conference.
Our ICMM would neither have been possible without the efforts of Dr. Yael
Bitran-Goren, head of the National Centre for Music Investigation, Documentation
and Information (CENIDIM–INBA, Mexico), who provided institutional support
and kept faith with our project.
We also owe our gratitude to our Scientific—Organizing Committee, and to
Silvia Pina-Romero, Ph.D. Math., for her generous assistance. We also thank our
panel chairs’ generous cooperation (R. Brotbeck, D. Clampitt, J.S. Lach-Lau,
M. Montiel, S. Pina-Romero).
We acknowledge the support of the Office of International Affairs of the State of
Jalisco (Dirección de Relaciones Internacionales, Secretaría de Educación,
Gobierno del Estado de Jalisco), particularly the efforts of Prof. Jorge Alberto
Quevedo-Flores, that made possible the visit of several of our international guests.
Finally we want to express our most special gratitude to Clarence Barlow,
D. Gareth Loy, Guerino Mazzola, and Fernando Zalamea, whose passion and wisdom connecting music and mathematics has been luminous for most of us. Our
congress in Puerto Vallarta would not have been possible without your inspiration.
xiii
Contents
Extended Counterpoint Symmetries and Continuous Counterpoint . . . .
Octavio A. Agustín-Aquino
1
Gödel-Vector and Gödel-Address as Tools for Genealogical
Determination of Genetically-Produced Musical Variants . . . . . . . . . . . .
Carlos de Lemos Almada
9
A Survey of Applications of the Discrete Fourier Transform
in Music Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Emmanuel Amiot
17
Gestures on Locales and Localic Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . .
Juan Sebastián Arias
29
On the Structural and the Abstract in My Compositional Work . . . . . .
Clarence Barlow
41
A Proposal for a Music Writing for the Visually Impaired . . . . . . . . . . .
Teresa Campos-Arcaraz
53
Group Theory for Pitch Sequence Representation:
From the Obvious to the Emergent Complexity . . . . . . . . . . . . . . . . . . . .
Emilio Erándu Ceja-Cárdenas
Mazzola’s Escher Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Yemile Chávez-Martínez and Emilio Lluis-Puebla
The Mechanics of Tipping Points: A Case of Extreme
Elasticity in Expressive Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elaine Chew
Lexicographic Orderings of Modes and Morphisms . . . . . . . . . . . . . . . . .
David Clampitt
61
71
79
89
xv
xvi
Contents
Music of Quantum Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Micho Đurđevich
99
Partitiogram, Mnet, Vnet and Tnet: Embedded Abstractions
Inside Compositional Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Pauxy Gentil-Nunes
Algebraic Combinatorics on Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Franck Jedrzejewski
Proportion, Perception, Speculation: Relationship Between
Numbers and Music in the Construction of a Contemporary
Pythagoreanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Juan Sebastián Lach Lau
Topos Echóchromas Hórou (The Place of the Tone of Space).
On the Relationship Between Geometry, Sound and Auditory
Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Jaime Alonso Lobato-Cardoso
Models and Algorithms for Music Generated
by Physiological Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Jaime Alonso Lobato-Cardoso and Pablo Padilla-Longoria
Music, Expectation, and Information Theory . . . . . . . . . . . . . . . . . . . . . . 161
D. Gareth Loy
Gestural Dynamics in Modulation: (Towards) a Musical String
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Guerino Mazzola
Manuel M. Ponce’s Piano Sonata No. 2 (1916): An Analysis Using
Signature Transformations and Spelled Heptachords . . . . . . . . . . . . . . . 189
Mariana Montiel
Textural Contour: A Proposal for Textural Hierarchy Through
the Ranking of Partitions lexset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Daniel Moreira de Sousa
The Sense of Subdominant: A Fregean Perspective
on Music-Theoretical Conceptualization . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Thomas Noll
How Learned Patterns Allow Artist-Level Improvisers
to Focus on Planning and Interaction During Improvisation. . . . . . . . . . 217
Martin Norgaard
Tuning Systems Nested Within the Arnold Tongues: Musicological
and Structural Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Gabriel Pareyon
Contents
xvii
Wooden Idiophones: Classification Through Phase Synchronization
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Gabriel Pareyon and Silvia Pina-Romero
A Fuzzy Rule Model for High Level Musical Features
on Automated Composition Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Iván Paz, Àngela Nebot, Francisco Mugica and Enrique Romero
The Musical Experience Between Measurement and Computation:
From Symbolic Description to Morphodynamical Unfolding . . . . . . . . . . 253
Mark Reybrouck
Generic Additive Synthesis. Hints from the Early Foundational Crisis
in Mathematics for Experiments in Sound Ontology . . . . . . . . . . . . . . . . 263
Julian Rohrhuber and Juan Sebastián Lach Lau
Dynamical Virtual Sounding Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Edmar Soria, Roberto Cabezas and Roberto Morales-Manzanares
Melodic Pattern Segmentation of Polyphonic Music as a Set
Partitioning Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Tsubasa Tanaka and Koichi Fujii
Diagrams, Games and Time (Towards the Analysis of Open Form
Scores) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Samuel Vriezen
On Minimal Change Musical Morphologies . . . . . . . . . . . . . . . . . . . . . . . 309
Michael Winter
Restoring the Structural Status of Keys Through DFT Phase Space. . . . 329
Jason Yust
Mazzola, Galois, Peirce, Riemann, and Merleau-Ponty: A Triadic,
Spatial Framework for Gesture Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Fernando Zalamea
Contributors
Octavio A. Agustín-Aquino Universidad de la Cañada, Teotitlan de Flores
Magón, Oaxaca, Mexico
Emmanuel Amiot Institut de Recherche et Coordination Acoustique/Musique,
Paris, France
Juan Sebastián Arias Universidad Nacional de Colombia, Bogotá, Colombia
Clarence Barlow Department of Music, University of California, Santa Barbara,
CA, USA
Roberto Cabezas Music Technology Graduate Program, UNAM, Coyoacán, Del
Carmen, D.F., Mexico
Teresa Campos-Arcaraz Facultad de Música, UNAM, Coyoacán, Del Carmen,
D.F., Mexico
Emilio Erándu Ceja-Cárdenas Departamento de Matemáticas, Centro
Universitario de Ciencias Exactas e Ingenierías (CUCEI), Universidad de
Guadalajara, Guadalajara, Mexico
Yemile Chávez-Martínez Universidad
Coyoacán, D.F., Mexico
Nacional
Autónoma
de
México,
Elaine Chew School of Electronic Engineering and Computer Science, Queen
Mary University of London, London, UK
David Clampitt The Ohio State University, Columbus, OH, USA
Carlos de Lemos Almada Federal University of Rio de Janeiro, Centro, Rio de
Janeiro, Brazil
Daniel Moreira de Sousa Federal University of Rio de Janeiro, Rio de Janeiro,
Brazil
Micho Đurđevich Institute of Mathematics, UNAM, Mexico, Mexico
xix
xx
Contributors
Koichi Fujii NTT DATA Mathematical Systems Inc., Tokyo, Japan
D. Gareth Loy Gareth Inc., San Rafael, CA, USA
Pauxy Gentil-Nunes Federal University of Rio de Janeiro, Rio de Janeiro, RJ,
Brazil
Franck Jedrzejewski French Atomic Energy Commission (Commissariat à
l’énergie atomique et aux énergies alternatives), Gif-sur-Yvette, France
Juan Sebastián Lach Lau Conservatorio de las Rosas, Morelia, Mexico
Emilio Lluis-Puebla Universidad Nacional Autónoma de México, Coyoacán, D.
F., Mexico
Jaime Alonso Lobato-Cardoso SEMIMUTICAS-IIMAS, UNAM, Circuito
Escolar 3000, Ciudad Universitaria, Coyoacán, D.F., Mexico
Guerino Mazzola School of Music, University of Minnesota, Minneapolis, MN,
USA; Institut Für Informatik, Universität Zürich, Zürich, Switzerland
Mariana Montiel Georgia State University, Atlanta, GA, USA
Roberto Morales-Manzanares Facultad de Música, DAAD, Universidad de
Guanajuato, Guanajuato, Mexico
Francisco Mugica Departament de Llenguatges i Sistemes Informátics, Soft
Computing Research Group, Technical University of Catalonia, Barcelona, Spain
Àngela Nebot Departament de Llenguatges i Sistemes Informátics, Soft
Computing Research Group, Technical University of Catalonia, Barcelona, Spain
Thomas Noll Escola Superior de Música de Catalunya, Departament de Teoria,
Composició i Direcció, Barcelona, Spain
Martin Norgaard Georgia State University, Atlanta, GA, USA
Pablo Padilla-Longoria IIMAS, UNAM, Cd. Universitaria, Coyoacán, D.F.,
Mexico
Gabriel Pareyon CENIDIM-INBA, Torre de Investigacion, CENART, Coyoacán,
D.F., Mexico
Iván Paz Departament de Llenguatges i Sistemes Informátics, Soft Computing
Research Group, Technical University of Catalonia, Barcelona, Spain
Silvia Pina-Romero División de Electrónica y Computación, CUCEI –
Universidad de Guadalajara, Guadalajara, Jalisco, Mexico
Mark Reybrouck University of Leuven, Leuven, Belgium
Contributors
xxi
Julian Rohrhuber Institute for Music and Media, Robert Schumann Hochschule,
Duesseldorf, Germany
Enrique Romero Departament de Llenguatges i Sistemes Informátics, Soft
Computing Research Group, Technical University of Catalonia, Barcelona, Spain
Edmar Soria Music Technology Graduate Program, UNAM, Coyoacán, Del
Carmen, D.F., Mexico
Tsubasa Tanaka IRCAM, Paris, France; Tokyo University of the Arts, Tokyo,
Japan
Samuel Vriezen Independent Researcher & Composer, Amsterdam, Netherlands
Michael Winter Independent Researcher, Los Angeles, CA, USA
Jason Yust School of Music, Boston University, Boston, MA, USA
Fernando Zalamea Universidad Nacional de Colombia, Bogotá, Colombia
Acronyms
BWT
CC
CSV
DFT
DV
FFT
GMC
Gv/Ga
HSA
ICMM
LMC
MCT
NDFSA
OGMC
PA
PYL
RTM
SAT
TIP
TM
UDP
Burrows-Wheeler Transform
Combinatorial Constraints
Comma-Separated Variables
Discrete Fourier Transform
Developing Variation
Fast Fourier Transform
Global Morphological Constraints
Gödel-vector/Gödel-address
Hypothesis of Self-Similarity in Euclidean Axiomatics
International Congress on Music and Mathematics
(Puerto Vallarta 2014)
Local Morphological Constraints
Musical Contour Theory
Non-Deterministic Finite State Automata
Optimal Global Morphological Constraints
Partitional Analysis
Partitional Young Lattice
Rhythm in arrays notation (from RTM-notation to ENP-score-notation)
Self-referential Abstract Thought
Theory of Integer Partitions
Tonal Music
User Datagram Protocol
xxiii
Introduction
Emilio Lluis-Puebla
For those who read for the first time or inquire about music and mathematics, let me
tell you that this field is both a recent area of study and also a very old one. At the
beginning of history, there was a connection between numbers and music. Later,
Pythagoras made a mathematical effort to say things about music with a certain
foundation. The names Descartes, Galileo, Kepler, Leibniz, Euler, d’Alembert,
Helmholtz, and some others are relevant here.
In the twentieth century, acoustics and its technology were very successful
applying mathematics to music, as well as computer science and some other fields
like linguistics. Later, the work of Clough in 1979, Lewin in 1982, and Mazzola in
1985 inspired both music-inclined mathematicians and mathematics-inclined
musicians to continue working in mathematics and music.
A big trend in the last three decades in mathematics was to do not only applications but to do new mathematics in a variety of fields of knowledge, and the field
of music has been no exception.
So, mathematical music theory is both a recent area of study and also a very old
one. From Pythagoras until the 1980s, very little and not very sophisticated
mathematics was employed in music. When sufficiently powerful mathematical
machinery became available and talented mathematicians used it, modern mathematical music theory was born.
One of the main goals of mathematical music theory (I will state some of
Guerino Mazzola’s thoughts mainly from [1] and from personal conversations with
him) was to develop a scientific framework for musicology. This framework had as
its foundation, established scientific fields. It included a formal language for
musical and musicological objects and relations. Music is fundamentally rooted
within physical, psychological and semiotic realities. But the formal description of
musical instances corresponds to mathematical formalism.
Mathematical music theory is based on category theory, algebraic topology, in
particular, topos theory, module theory, group theory, homotopy theory, homology
theory, algebraic geometry, just to name some areas, that is, on heavy mathematical
machinery. Its purpose is to describe musical structures. The philosophy behind it is
xxv
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Introduction
understanding the aspects of music that are susceptible to reason in the same way as
physics does it for natural phenomena.
This theory is based in an appropriate language to manage the relevant concepts
of the musical structures, in a group of postulates or theorems with regard to the
musical structures subject to the defined conditions, and in the functionality for
composition and analysis with or without a computer.
Mazzola also says that music is a central issue in human life, though it affects a
different layer of reality than physics. The attempt to understand or to compose a
major work of music is as important and difficult as the attempt to unify gravitation,
electromagnetism, and weak and strong forces. For sure, the ambitions are comparable and hence the tools should be comparable too.
It is only in the last three decades that there is consistent work in mathematical
music theory. Thus I will address this period of time in Mexico’s history on this
subject, since Gabriel Pareyon [2] summarises the time span before 1980. I will
write about this in a personal way.
When I was 21 years old, in 1974, I was listening to the station Radio
Universidad (University Radio Station), to a low, magnificent voice that was talking
(in Spanish) about the application of finite group theory to the musical analysis of
Bach’s music, etc. This caught my attention and I went to see the owner of this
voice. I located him in the old building of the Escuela Nacional de Música de la
UNAM (UNAM Faculty of Music) and this young thin man kindly showed me a
bunch of papers he had. I read them for half an hour or so and got the idea of what
he was doing. This young man was Julio Estrada, a distinguished Mexican composer and musicologist.
Then I went to Canada to do a Ph.D. on algebraic K-theory. I was in love with
pure mathematics like homological algebra, algebraic topology, algebraic geometry,
homotopy theory, etc. Nobody could have ever told me that these marvellous pieces
of pure mathematics were ever to appear more than thirty years later in the other
field of my passion: music.
When I came back to México, in the early 1980s I wanted to do some work in
mathematics and music, in particular to guide an undergraduate thesis for a student,
but the angry face and terrible gesticulations of a colleague who was in charge of
some high position at the department demoralised me. Does this sound familiar to
anyone?
Some years later, in the 1990s, a lady from the mathematics undergraduate
program at UNAM with a piano background, with great conviction, full of energy,
appeared in my office, completely determined to do an undergraduate thesis in
mathematics and music, particularly based on the ideas of Julio Estrada which
turned into a book that he published in the 1980s [3]. I gave her more papers and
books and she started to look for more bibliography. The librarian got some references of Guerino Mazzola. Particularly his book Gruppen und Categorien in der
Musik, some articles by him, and others, including Chemiller’s papers, plus some
from the American School. This lady was Mariana Montiel. Now she is a full
professor in the United States.
Introduction
xxvii
Mariana decided also to do a master’s thesis on mathematical music theory,
especially on denotator theory. I invited Guerino Mazzola to México for the first
time in 1997 and we began a wonderful friendship.
In 2000, when I was President of the Sociedad Matemática Mexicana
(Mathematical Society of Mexico), I dared to organise the First International
Seminar on Mathematical Music Theory which took place simultaneously at the
Facultad de Ciencias (Faculty of Sciences) and the Escuela Nacional de Música
(School of Music) both from UNAM. Thomas Noll and Guerino Mazzola attended,
among others.
Some days before the first international seminar, we had a previous special
session on mathematical music theory at the annual Congreso Nacional de la
Sociedad Matemática Mexicana in Saltillo which had an attendance of about 2000
persons, with great success. As a frame to both meetings we had concerts by
Guerino Mazzola in Saltillo, Sala Carlos Chávez and at the Sala Xochipilli in
Mexico City which turned into a delightful free jazz recording called Folia:
The UNAM Concert with Guerino Mazzola playing Rachmaninoff’s Corelli: La
Folia theme as motive.
At both meetings, many mathematicians and musicians attended with surprise on
their faces. The proceedings of the seminar were published by the Sociedad
Matemática Mexicana Electronic Publications and lately were unified with the
proceedings of the Second International Seminar which took place in Germany in
2001 and with the third one which took place in Switzerland in 2002 and was
published by Epos Music of the University of Osnabruck in 2004 [4]. (I almost did
not see this publication because I almost died. I was very ill for six months with an
unknown disease which was later believed to be a viral meningitis, for which there
was no cure!)
After not dying, six years later, in 2009, a student of mine, a young, impetuous
and talented mathematician and musician, Octavio Agustin-Aquino, convinced me
to organise the fourth seminar. It took place in Huatulco, Oaxaca, in 2010 as the
Fourth International Seminar on Mathematical Music Theory [5]. By the way,
Octavio became the first Ph.D. in mathematics graduated in Mexico at UNAM in
mathematical music theory in 2011 with a thesis on microtonal counterpoint. He is
now a full professor at the Universidad de la Cañada which belongs to the SUNEO
in Oaxaca State, Mexico.
Finally, in November 2012 another very talented man (musicologist, also doing
systematic musicology) which I admire the most because of his vast culture, ability,
organisational capabilities, enormous memory and many other wits, contacted me
in order to organise a sequel of the international seminars which turned out to be the
International Congress on Music and Mathematics, 2014. This great man is Gabriel
Pareyon.
Through the years there were also some more students who did some work with
me but they did not continue in this field due to economic or vocational reasons. In
2013 and 2014, two of my students (Yemile Chávez and Santiago Rovira, both with
music backgrounds) approached me like Mariana and Octavio before. They
xxviii
Introduction
presented a lecture at ICMM 2014, and I hope they continue to work in this
marvellous field.
Of course there are some other colleagues who have worked in mathematics and
music in a rather isolated way, but now we had the opportunity to collect their
efforts in this book, and made the connections to have a stronger unified community
worldwide.
And well, what relationship does exist between music and mathematics? Or
equivalently what connection or correspondence exists? We know, for example,
that mathematical concepts were applied several years ago and recently (coming
after all from nature or from man’s abstract thought, etc.), just to mention four
examples I use in my lectures [6]: to the entertainment with a game of dice in
Mozart’s creations; to aesthetics, as in Birkhoff’s theory; to musical composition,
for example by Bartók; and to create a precise language for musicology and music
by Mazzola, among others. Certainly, there are many other music fields where
mathematics contributes to our understanding, like in performance or analysis, etc.
For me, the most important relationship between mathematics and music is that
both are “fine arts”. They possess similar characteristics. They are related in the
sense that mathematics provides a way to understand music, and musicology has a
scientific basis in order to be considered a science, not a branch of common poetic
literature.
I have worked since the 1970s on homotopy theory, cohomology theory, algebraic topology, homological algebra, among other fields of mathematics. As I wrote
before, at the time these were considered pure mathematics. However, thirty years
later, these wonderful pieces of mathematics came to be applied mathematics, and
guess where? It turned out to be (as I wrote before) in my other passion: music! But
not only as an application, you can do new mathematics as well!
Let me tell you an anecdote. In 2001, when I was president of the Sociedad
Matemática Mexicana, during a visit to Rio de Janeiro I called a friend of mine, the
president of the International Mathematical Union at that time, the Brazilian Jacob
Palis. We agreed to meet at the famous Copacabana Palace where I was going to
play Rachmaninoff's Second Piano Concerto as a soloist of the Rio de Janeiro
Philharmonic Orchestra. He did not know I was a pianist. When he got there, he
saw the president of the Sociedad Matemática Mexicana getting out on stage and
sitting down to play the concerto. He was thrilled and invited me to dinner. We had
a very long talk and having answered all his questions about me as a pianist and
about mathematical music theory, he told me almost the same phrase that Guerino
Mazzola got from Grothendieck: “the mathematics of the future!”. So, in brief
words, let me tell you that, for me, mathematics is one of the “fine arts”, the purest
of them, which has the gift of being the most precise of all sciences.
I was very honoured to meet all of the participants of ICMM in order to stimulate
the interchange of visions, thoughts and points of view on this fascinating subject in
a very friendly way. I am sure we all have profited from this interaction in such a
wonderful place.
As you know, not only in Mexico, the funding for meetings is practically
nonexistent. Many persons interested in coming could not join us because they did
Introduction
xxix
not have economic support from their universities. We thought we could obtain
some funding for it, but once more, as in the Fourth International Seminar, we had
to do it with our own personal budgets, energies and personal work and risk. We
proudly can say that once more we have done it by ourselves!
Besides the small support (for such a big meeting) of very few institutions (see
the acknowledgements in this book) we only had a small contribution from the
Sociedad Matemática Mexicana to partially finance two of my own students which
we, again, sincerely thank. The rest is exclusively ours and yours.
On Gabriel Pareyon’s behalf (I recognise all his tremendous work on the
organisation), the other organisers and myself, we thank all the participants of the
International Congress on Music and Mathematics. We had a wonderful conference!
References
1. Mazzola, G.: The Topos of Music. Birkhäuser (2002)
2. Pareyon, G.: A Survey on the Mexican Tradition of Music and Mathematics. Opening lecture
of ICMM 2014, Puerto Vallarta, Mexico (abridged in the Preface of this book)
3. Estrada, J., Gil, J.: Música y Teora de Grupos Finitos (3 variables booleanas). UNAM (1984)
4. Lluis-Puebla, E., Mazzola, G., Noll, T. (eds.): Perspectives in Mathematical and Computational
Music Theory. epOs, Osnabrück (2004)
5. Agustín-Aquino, O., Lluis-Puebla, E. (eds.): Memoirs of the Fourth International Seminar on
Mathematical Music Theory. Serie Memorias, vol. 4. Publicaciones Electrónicas de la Sociedad
Matemática Mexicana (2011)
6. Lluis-Puebla, E.: Music and Mathematics: Two Fine Arts. Perspectives in Mathematical and
Computational Music Theory. epOs, Osnabrück (2004)