Introduction to Mathemusical Thought:
meet the players
Aaron Greicius
Loyola University Chicago
c 2014 Aaron Greicius
All Rights Reserved
What mathematicians say
What mathematicians say
I “Mathematics and music, the most sharply contrasted fields of
intellectual activity which can be found, and yet related, supporting each
other, as if to show forth the secret connection which ties together all
activities of the mind...”
–Hermann von Helmholtz
What mathematicians say
I “Mathematics and music, the most sharply contrasted fields of
intellectual activity which can be found, and yet related, supporting each
other, as if to show forth the secret connection which ties together all
activities of the mind...”
–Hermann von Helmholtz
I “It is in its performance that the music comes alive and becomes part of
our experience; the music exists not on the printed page, but in our
minds. The same is true for mathematics; the symbols on a page are just
a representation of the mathematics. When read by a competent
performer...the symbols on the printed page come alive–the mathematics
lives and breathes in the mind of the reader like some abstract symphony.”
–Keith Devlin
What mathematicians say
I “Mathematics and music, the most sharply contrasted fields of
intellectual activity which can be found, and yet related, supporting each
other, as if to show forth the secret connection which ties together all
activities of the mind...”
–Hermann von Helmholtz
I “It is in its performance that the music comes alive and becomes part of
our experience; the music exists not on the printed page, but in our
minds. The same is true for mathematics; the symbols on a page are just
a representation of the mathematics. When read by a competent
performer...the symbols on the printed page come alive–the mathematics
lives and breathes in the mind of the reader like some abstract symphony.”
–Keith Devlin
I “A mathematician, like a painter or a poet, is a maker of patterns. If his
patterns are more permanent than theirs, it is because they are made of
ideas. His patterns, like the painter’s or the poet’s must be beautiful; the
ideas, like the colors or the words, must fit together in a harmonious way.”
–G. H. Hardy (1877-1947)
What musicians say
What musicians say
I “Music is the arithmetic of sounds as optics is the geometry of light.”
–Claude Debussy
What musicians say
I “Music is the arithmetic of sounds as optics is the geometry of light.”
–Claude Debussy
I “The fugue is like pure logic in music.”
–Frederic Chopin
What musicians say
I “Music is the arithmetic of sounds as optics is the geometry of light.”
–Claude Debussy
I “The fugue is like pure logic in music.”
–Frederic Chopin
I “Despite all the experience that I could have acquired in Music, as I had
practiced it for quite a long time, it’s only with the help of Mathematics
that I have been able to untangle my ideas, and that light made me
aware of the comparative darkness in which I was before.”
–Jean-Philippe Rameau
What musicians say
I “Music is the arithmetic of sounds as optics is the geometry of light.”
–Claude Debussy
I “The fugue is like pure logic in music.”
–Frederic Chopin
I “Despite all the experience that I could have acquired in Music, as I had
practiced it for quite a long time, it’s only with the help of Mathematics
that I have been able to untangle my ideas, and that light made me
aware of the comparative darkness in which I was before.”
–Jean-Philippe Rameau
I “I am not saying that composers think in equations or charts of numbers,
nor are those things more able to symbolize music. But the way
composers think–the way I think–is, it seems to me, not very different
from mathematical thinking.”
–Igor Stravinsky
What musicians say
I “Music is the arithmetic of sounds as optics is the geometry of light.”
–Claude Debussy
I “The fugue is like pure logic in music.”
–Frederic Chopin
I “Despite all the experience that I could have acquired in Music, as I had
practiced it for quite a long time, it’s only with the help of Mathematics
that I have been able to untangle my ideas, and that light made me
aware of the comparative darkness in which I was before.”
–Jean-Philippe Rameau
I “I am not saying that composers think in equations or charts of numbers,
nor are those things more able to symbolize music. But the way
composers think–the way I think–is, it seems to me, not very different
from mathematical thinking.”
–Igor Stravinsky
I “Music is not to be decorative; it is to be true.”
–Arnold Schoenberg
Definitions: meet the players
The following represent a reduction of many carefully constructed
definitions of mathematics and music found in the philosophical
literature.
Definitions: meet the players
The following represent a reduction of many carefully constructed
definitions of mathematics and music found in the philosophical
literature.
Music is the art of structured sound.
Mathematics is the science of abstract structure.
Definitions: meet the players
The following represent a reduction of many carefully constructed
definitions of mathematics and music found in the philosophical
literature.
Music is the art of structured sound.
Mathematics is the science of abstract structure.
The definitions reveal both a potential hurdle to making a
connection between the two fields (art/science), as well as a
potential avenue of attack: the idea of structure.
Vantage points
The course will examine three points of contact. I list them here in
order of increasing profundity (toward a deep connection), and
ornamented with some fancy philosophical terms.
Vantage points
The course will examine three points of contact. I list them here in
order of increasing profundity (toward a deep connection), and
ornamented with some fancy philosophical terms.
1. Ontological. Musical objects are very much like mathematical objects.
We will describe and define the main musical parameters (melody,
rhythm, harmony, timbre) in mathematical language (sets,
sequences,topological spaces, groups).
Vantage points
The course will examine three points of contact. I list them here in
order of increasing profundity (toward a deep connection), and
ornamented with some fancy philosophical terms.
1. Ontological. Musical objects are very much like mathematical objects.
We will describe and define the main musical parameters (melody,
rhythm, harmony, timbre) in mathematical language (sets,
sequences,topological spaces, groups).
2. Methodological. Mathematical thought, operations and objects are
frequently employed both in the analysis and composition of music. We
will look closely at examples of mathematical methods in both of these
areas of musical practice.
Vantage points
The course will examine three points of contact. I list them here in
order of increasing profundity (toward a deep connection), and
ornamented with some fancy philosophical terms.
1. Ontological. Musical objects are very much like mathematical objects.
We will describe and define the main musical parameters (melody,
rhythm, harmony, timbre) in mathematical language (sets,
sequences,topological spaces, groups).
2. Methodological. Mathematical thought, operations and objects are
frequently employed both in the analysis and composition of music. We
will look closely at examples of mathematical methods in both of these
areas of musical practice.
3. Epistemological. Music often bears a strong logical quality. We speak of
understanding a piece of music, of one passage of music following from
another passage. Can these activities be compared to understanding or
following mathematical arguments? We will explore these connections
with the aid of formal logic.
Goals
The following goals are listed in order of increasing ambitiousness.
1. Get to know some classics in both music and mathematics:
compositions, theorems, musical forms, proofs, etc.
2. Develop a short, “cocktail party” answer to the question:
What exactly is the connection between math and music?
3. Get comfortable reading both musical scores and
mathematical arguments. Come to understand better the
nature of music and mathematics as practices.
4. Improve upon our “cocktail party” answer and articulate a
deeper connection between music and mathematics.
Questions
Progress toward our last, most ambitious goal can be measured in
part by our ability to answer the following questions:
1. Does the connection between music and math actually extend
beyond the surface level, that is beyond the fact that works of
music can be seen as mathematical objects?
2. What is special about the math/music relation? Why is it any
deeper than the connection between say math and painting, or
math and improv comedy?
3. What precisely is the difference between the art of music, and
the science of mathematics?
Classic 1 (Musikalisches Opfer, Canon I. a 2 cancrizans, by
J.S. Bach)
Below you find a facsimile of J.S. Bach’s Canon I, from
Musikalisches Opfer (or The Musical Offering). As the
performance instructions indicate, this is an example of a crab
canon. Video link. Tim Smith’s overview of Musikalisches Opfer.
Performance instructions:
Instrument 1 plays through from left to right, then back.
Instrument 2 plays from right to left, then back.
Classic 1
The score you have shows both parts written out separately; in this
form, each instrument now performs the music from left to right,
then back.
Turn the score into a Möbius band. You should first fold the score
in half lengthwise, obtaining a strip with Instrument 1 on one side
and Instrument 2 on the other.
1. Describe Bach’s composition as a path along your Möbius
band. Make sure your path traverses the whole piece (36
measures in all)!
2. What properties of Bach’s composition are articulated by the
geometry of the Möbius band? What does the geometry say
about the role of the two different instruments?
3. We could have also made a simple cylinder (or hoop) out of
our score-strip; what advantage does the Möbius band
representation have (if any)?
4. Compare our Möbius band representation to the one in the
video. Which is better?