THE GEOMETRY
OF MUSICAL CHORDS
Composer
reveals musical chords' hidden geometry
July 6, 2006. Source:
News@Princeton by Chad Boutin
Composers
often speak of fitting chords and melodies together, as
though sounds were physical objects with geometric shape
- and now a Princeton University musician has shown
that advanced geometry actually does offer a tool for
understanding musical structure.
In an attempt to answer age-old questions about how
basic musical elements work together, Dmitri Tymoczko
has journeyed far into the land of topology and
non-Euclidean geometry, and has returned with a new --
and comparatively simple -- way of understanding how
music is constructed. His findings have resulted in the
first paper on music theory that the journal Science has
printed in its 127-year history, and may provide an
additional theoretical tool for composers searching for
that elusive next chord.
"I'm not trying to tell people what style of music
sounds good, or which composers to prefer," said
Tymoczko (pronounced tim-OSS-ko), a composer and music
theorist who is an assistant professor of music at
Princeton. "What I hope to do is provide a new way to
represent the space of musical possibilities. If you
like a particular chord, or group of notes, then I can
show you how to find other, similar chords and link them
together to form attractive melodies. These two
principles -- using attractive chords, and connecting
their notes to form melodies -- have been central to
Western musical thought for almost a thousand years."
Tymoczko's findings appear as a report in Science's July
7 issue.
Making graphical representations of musical ideas is not
itself a new idea. Even most nonmusicians are familiar
with the five-line musical staff, on which the notes
that appear physically higher represent sounds that have
higher pitch. Other common representations include the
circle of fifths, which illustrates the relationships
between the 12 notes in the chromatic scale as though
they were the 12 hours on a clock's face.
"Tools like these have helped people understand music
with both their ears and their eyes for generations,"
Tymoczko said. "But music has expanded a great deal in
the past hundred years. We are interested in a much
broader range of harmonies and melodies than previous
composers were. With all these new musical developments,
I thought it would be useful to search for a framework
that could help us understand music regardless of
style."
Traditional music theory required that harmonically
acceptable chords be constructed from notes separated by
a couple of scale steps -- such as the major chord,
whose three notes comprise the first, third and fifth
elements in the major scale, forming a familiar harmony
that most audiences find easy to enjoy. Many
20th-century composers abandoned this requirement,
however. Modern chords are often constructed of notes
that sit right next to one another on the keyboard,
forming "clusters" -- dissonant by traditional standards
-- that to this day often challenge listeners' ears.
"Western music theory has developed impressive tools for
thinking about traditional harmonies, but it doesn’t
have the same sophisticated tools for thinking about
these newer chords," Tymoczko said. "This led me to want
to develop a general geometrical model in which every
conceivable chord is represented by a point in space.
That way, if you hear any sequence of chords, no matter
how unfamiliar, you can still represent it as a series
of points in the space. To understand the melodic
relationship between these chords, you connect the
points with lines that represent how you have to change
their notes to get from one chord to the next."
One of Tymoczko's musical spaces resembles a triangular
prism, in which points representing traditionally
familiar harmonies such as major chords gather near the
center of the triangle, forming neat geometric shapes
with other common chords that relate to them closely.
Dissonant, cluster-type harmonies can be found out near
the edges, close to their own harmonic kin. Tymoczko
said that composers have traditionally valued a kind of
harmonic consistency that does not require that the
listener jump far from one region of the space to
another too quickly.
“This idea that you should stay in one part of space,”
he said, “is an important ingredient of our notion of
musical coherence.”
To bring these ideas to life, Tymoczko has created a
short movie that illustrates the chord movement in a
piece of music by 19th-century composer Frederick
Chopin. His E minor piano prelude (Opus 28, No. 4) has
charmed listeners since the 1830s, but its harmonies
have not been well explained.
"This prelude is mysterious," Tymoczko said. "While it
uses traditional harmonies, they are connected with
nonstandard chord progressions that people have had
trouble describing. However, when you plot the chord
movement in geometric space, you can see Chopin is
moving along very short lines, staying primarily within
one region."
The movie is available at
http://www.princeton.edu/pr/media/chopin/chopin3_350k.mov
Tymoczko said that the geometric approach could assist
with our still-murky understanding of music ranging from
the mid-1800s through the contemporary period, including
the cluster-based compositions of Georgi Ligeti, whose
work formed a dramatic part of the soundtrack to the
film "2001: A Space Odyssey."
"What all this implies is that you can begin with any
sort of harmony your ear enjoys, whether it's a familiar
chord from a 300-year-old hymn or the most avant-garde
cluster you can imagine," he said. "But once you have
decided where to start from and what region of space
your harmony inhabits, very general principles of
musical coherence suggest that you stay close to that
region of space."
Tymoczko, whose compositional influences include
classical music, rock and jazz, said he does not expect
people will start writing music by "connecting the dots"
as a result of his research. But he hopes it will at
least provide a new tool for understanding the
relationships behind music.
"Put simply, I'm a composer and I like to write and play
music that sounds good," he said. "But what does it mean
to 'sound good'? That's a question that the musical
community has grappled with for centuries. Our
understanding of the Chopin piece, for example, had
previously been very local -- as if we were walking in a
heavy fog and could only see a few steps in front of our
feet at any one time. We now have a map of the whole
terrain on which we can walk, and can replace our
earlier, local perspective with a much more general
one."
Commenting on the significance of the work, Yale's
Richard Cohn said that Tymoczko has made a useful
contribution to a fundamental problem in music theory.
"Dmitri's solution is exhaustive, original, and
expressed clearly enough to be meaningful even to those
musicians and scholars who do not have Dmitri's
mathematical abilities," said Cohn, who is the Batell
Professor of the Theory of Music at Yale. "His work
leads to a deeper understanding of why composers in the
European tradition favor certain types of scales and
chords, and it suggests that melody and harmony are more
fundamentally intertwined than has been previously
thought. His achievement will become central to future
work in the modelling of musical systems."
Abstract
THE GEOMETRY OF MUSICAL CHORDS
Dmitri Tymoczko, Princeton University
Musical chords have a non-Euclidean geometry that has
been exploited by Western composers in many different
styles. A musical chord can be represented as a point in
a geometrical space called an orbifold. Line segments
represent mappings from the notes of one chord to those
of another. Composers in a wide range of styles have
exploited the non-Euclidean geometry of these spaces,
typically by utilizing short line segments between
structurally similar chords. Such line segments exist
only when chords are nearly symmetrical under
translation, reflection, or permutation.
Paradigmatically consonant and dissonant chords possess
different near-symmetries, and suggest different musical
uses.
Department of Music,
Princeton University, Princeton, NJ 08544, USA, and
Radcliffe Institute for Advanced Study, 34 Concord
Avenue, Cambridge, MA 02138, USA.
E-mail:
dmitri@princeton.edu
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