Geometry of Music…,9171,1582330,00.html

Friday, Jan. 26, 2007

The Geometry of Music

When you first hear them, a Gregorian chant, a Debussy prelude
and a John Coltrane improvisation might seem to have almost nothing in
common–except that they all include chord progressions and something
you could plausibly call a melody. But music theorists have long known
that there’s something else that ties these disparate musical forms
together. The composers of these and virtually every other style of
Western music over the past millennium tend to draw from a tiny
fraction of the set of all possible chords. And their chord
progressions tend to be efficient, changing as few notes, by as little
as possible, from one chord to the next.

Exactly how one style
relates to another, however, has remained a mystery–except over one
brief stretch of musical history. That, says Princeton University
composer Dmitri Tymoczko, “is why, no matter where you go to school,
you learn almost exclusively about classical music from about 1700 to
1900. It’s kind of ridiculous.”

But Tymoczko may have changed
all that. Borrowing some of the mathematics that string theorists
invented to plumb the secrets of the physical universe, he has found a
way to represent the universe of all possible musical chords in graphic
form. “He’s not the first to try,” says Yale music theorist Richard
Cohn. “But he’s the first to come up with a compelling answer.”

answer, which led last summer to the first paper on music theory ever
published in the journal Science, is that the cosmos of chords consists
of weird, multidimensional spaces, known as orbifolds, that turn back
on themselves with a twist, like the Möbius strips math teachers
love to trot out to prove to students that a two-dimensional figure can
have only one side. Indeed, the simplest chords, which consist of just
two notes, live on an actual Möbius strip. Three-note chords
reside in spaces that look like prisms–except that opposing faces
connect to each other. And more complex chords inhabit spaces that are
as hard to visualize as the multidimensional universes of string theory.

But if you go to Tymoczko’s website
you can see exactly what he’s getting at by looking at movies he has
created to represent tunes by Chopin and, of all things, Deep Purple.
In both cases, as the music progresses, one chord after another lights
up in patterns that occupy a surprisingly small stretch of musical real
estate. According to Tymoczko, most pieces of chord-based music tend to
do the same, although they may live in a different part of the orbifold
space. Indeed, any conceivable chord lies somewhere in that space,
although most of them would sound screechingly harsh to human ears.

discovery is useful for at least a couple of reasons, says Tymoczko.
“One is that composers have been exploring the geometrical structure of
these maps since the beginning of Western music without really knowing
what they were doing.” It’s as though you figured out your way around a
city like Boston, for example, without realizing that some of your
routes intersect. “If someone then showed you a map,” he says, “you
might say, ‘Wow, I didn’t realize the Safeway was close to the disco.’
We can now go back and look at hundreds of years of this intuitive
musical pathmaking and realize that there are some very simple
principles that describe the process.”

likely to help both scholars and teachers, he argues. By showing how
compositions of various styles move through his orbifold spaces, says
Tymoczko, you can see how different styles of Western music relate to
each other and evolve. Tymoczko’s maps can also be an aid to composers,
says Cohn. Most have a favorite corner in orbifold space, a set of
related chord types that they tend to explore over and over in
different ways. Venturing into a different part of space can be tough;
you have to learn your way around a whole new auditory neighborhood.
You can do that intuitively by wandering around and seeing where you
get to. But with the maps, you can plot a route that you know in
advance will make some sort of sense.

That doesn’t mean you can
program a computer with Tymoczko’s orbifold maps and have it spit out
beautiful compositions. “I don’t want to sell these maps as the royal
road to composition,” he warns. “They don’t substitute for the hard
work of learning how to move notes around.” But they can help show when
a new idea is promising and when it will probably lead to a dead end.
“They might make an O.K. composer good,” says Tymoczko, “but they won’t
make a good composer great.”


The Truth About Interest Rates…

Sacrificing the Dollar to Save the Banks…

World Affairs Brief, March 14, 2008. Commentary and Insights on a Troubled World.

Copyright Joel Skousen. Partial quotations with attribution permitted. Cite source as Joel Skousen’s World Affairs Brief.


We hear a phrase all too often nowadays: “The dollar sank to a new record low this week.” Get used to it.
It’s going to continue its steady decline as the Fed continues to
bailout the major banks and financial institutions whose solvency
depends on a package of paper assets that no longer has a verifiable
value. As currency traders react to each new injection of liquidity and
the lowering of US interest rates, the dollar keeps losing value.
It lost 1% this past week compared to the Euro. It has gone down 6% in
the past month, 17% in the last year, and a whopping 31% since late 2005
The debasement of our currency is even worse than that, however. The
markets are comparing the dollar to the Euro, or the Yen, or the
British Pound. Each of those currencies is also being debased, but not
quite as much as the dollar. Real inflation of the money supply is
higher than the percentage fall of the dollar since all comparative
currencies are moving downward as well
. This week we’ll talk
about who are the winners and who will be the losers in this frantic
game of maintaining US buying power.

Read complete article at wordpress blog tobefree

Occult Meaning of Three flags flying together…

Three flags together comes from the Egyptian hieroglyphic for ‘neter’.

‘Neter’ is the word for gods.

Three flags flying mean those behind the flags think of themselves as ‘Gods’. You will find three flags are flown in front of businesses and also in front of Government buildings as well as other places these occultist work or reside.

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