In the wake of centuries
of effort to seek deep connections between music and mathematics,
a team today concludes that music does have geometry.
More than 2000 years ago, Pythagoras discovered that pleasing
musical intervals could be described using simple ratios.
And the idea of the so-called musica universalis or "music
of the spheres" emerged in the Middle Ages which
said that the proportions in the movements of the celestial
bodies - the sun, moon and planets - could be viewed as
a form of music, inaudible but perfectly harmonious.
Now, three music professors - Clifton Callender at Florida
State University, Ian Quinn at Yale University, and Dmitri
Tymoczko at Princeton University - have devised a new
way of analysing and categorising music to reduce musical
works to their mathematical essence, suggesting that mathematics
is a more fundamental language of nature than music.
The two-note chords form a
Mobius strip, whose boundary
is a 'trefoil knot'.
Following a pioneernig effort by Prof Tymoczko in 2006,
the trio has now outlined a method called "geometrical
music theory" in the journal Science that they say
can turn music into shapes.
"To me," Prof Tymoczko says "the most satisfying
aspect of this research is that we can now see that there
is a logical structure linking many, many different musical
concepts. To some extent, we can represent the history
of music as a long process of exploring different symmetries
and different geometries."
"Our methods are not so great at distinguishing Aerosmith
from the Rolling Stones," Tymoczko said. "But
they might allow you to visualise some of the differences
between John Lennon and Paul McCartney. Paul McCartney's
tunes tend to look more traditional, John Lennon's tunes
tend to be a little more "rock" - violating
more of the traditional rules.
And they certainly help you understand more deeply how
classical music relates to rock or is different from atonal
music.
"The team can take sequences of notes, like chords,
rhythms and scales, and categorize them so they can be
grouped into "families."
They have found a way to assign mathematical structure
to these families, so they can then be represented by
points in complex geometrical spaces, much the way "x"
and "y" coordinates, in the simpler system of
high school algebra, correspond to points on a two-dimensional
plane.
This achievement, they expect, will allow researchers
to analyse and understand music in much deeper and more
satisfying ways. "The music of the spheres isn't
really a metaphor - some musical spaces really are spheres,"
said Tymoczko, who like Callender is also a composer.
"The whole point of making these geometric spaces
is that, at the end of the day, it helps you understand
music better. Having a powerful set of tools for conceptualizing
music allows you to do all sorts of things you hadn't
done before."
"You could create new kinds of musical instruments
or new kinds of toys," he said. "You could create
new kinds of visualisation tools - imagine going to a
classical music concert where the music was being translated
visually. We could change the way we educate musicians.
There are lots of practical consequences that could follow
from these ideas."
The work represents a significant departure from other
attempts to quantify music, according to Rachel Wells
Hall of the department of mathematics and computer science
at St. Joseph's University in Philadelphia. In Science,
she writes that their effort, "stands out both for
the breadth of its musical implications and the depth
of its mathematical content."
MORE
INFORMATION
This article first appeared
in the Telegraph UK. Their website can be found at www.telegraph.co.uk
Music without words means leaving behind the mind. And leaving behind the mind is meditation.
Meditation returns you to the source. And the source of all is sound. — Kabir
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