Take a good look around at all the different structures you see in the world. Not only
the physical structures, like buildings and mountains and dogs and trees, but the
conceptual structures, like language and government and the Dewey decimal system and
the management hierarchy at your place of work. Some of these structures seem to
exist independently of humans, but others seem to be completely invented, made up
from whole cloth, so to speak. But how can that be? Can you really create something
from nothing? I wonder.
I just got back from a sort of educational/recreational summer camp for adults (I'm
writing this in early August). It was the California Coast Music Camp, a week of
intensive classes and refreshing musical single-mindedness. There were no cars, no
computers, no worries -- nothing to do but play and sing and learn (and swim and eat
and hike).
The camp itself was classic: squat brown bunkhouses scattered in small groups like big
hollow dice; long low latrines, redolent with that cloyingly sweet chemical peculiar to
outdoor bathrooms; enormous vats of jello and drowned salad at the ends of the long
serving tables in the echoing dining hall; bug bites, bug spray, and bugs -- you get the
idea. And even though it was strictly adults, all the same feelings I remember from my
one camp experience as a kid were there in the wide and ragged spectrum of emotions it
generated. I had the same scary and uncertain feelings at the beginning, wondering if I
really should be there at all; the same gradual discovery that it was all OK, and in fact
was fantastic; and, at the end, the same bittersweet longing to start it all over again.
Music really is a lovely thing. One of the classes I took was called "Theory, Scales, and
Chord Construction on the Guitar" and it was on the third day of class, as I was
plunking away at yet another arpeggio, that I began to get my first glimpse of the
underlying structure of the notes on the guitar fretboard. (And I do mean glimpse: it's
something that will probably take a year or more to see clearly, and five or ten years
to really feel -- and that's if I practice every day.) I was suddenly struck by the
notion that it's the discontinuities in musical scales that make them both difficult and
interesting, that it's the complications in the structure of music that give it an
interesting shape. If you've ever taken a music theory class, or even a piano class, you
might know what I mean: if only there were a black key between every pair of white
keys, things would be much simpler. It's that damned half step between B and C and
between E and F that screws things up. But it also seems that those discontinuities --
those bumps and dips in an otherwise smooth, even progression -- are what give rise
to all the beauty and complexity and subtlety. It's precisely those "flaws" in the
structure that lend it an interesting texture.
However, the structure itself is artificial. Underneath, the range of musical tones is
actually continuous, as any slide whistle demonstrates. It's a spectrum, a continuously
varying quantity, in this case the frequency of vibration of a material. So the set of
notes we use -- our tuning system -- is externally applied, a necessarily arbitrary
set of discrete slots pasted onto an underlying continuum. It's a sort ofquantization of
something inherently smooth that lends it a tractable structure, that gives us a handle
with which to manipulate it and a context in which to make sense of it. By conceptually
making the smoothly varying curve into a step function, by arbitrarily chopping the
continuous line into discrete chunks, we can somehow work with it more manageably
and think about it more clearly.
If you look around a little, you'll find examples of this kind of artificial structuring of
continuums all over the place. The computer in front of you (or wherever it is) is an
excellent example. Many early computers were analog; they dealt with smoothly
varying quantities (usually voltages in circuits). But they turned out to be too hard to
program -- in effect you had to create a physical circuit that modeled the problem you
wanted to solve. By making the computer purely digital, we abstracted its operation
away from the physical realm into the realm of pure logic, and that really opened up a
lot of doors. Another much simpler example is a radio dial; it actually represents a
continuous spread of frequencies, but we've arbitrarily divided it into bands so that we
can parcel it out to those who want to use pieces of it.
More abstractly, in mathematics one often takes a continuously varying function and
"pretends" for a moment that it's a step function, simply because it makes things
easier to deal with. Then, when you've got a handle on the step function, you can use a
nifty trick (called calculus) to sort of extrapolate what you've discovered about the
steps and apply it to the whole curve. Time itself (which sure feels like a continuum,
whatever it is) is conveniently chopped into bits by humans to make it easier to keep
track of and to talk about.
But there are other quantities in our world that seem to come to us already divided up
into discrete chunks, already structured. The periodic table of the elements certainly
isn't a continuum. There's no smooth transition between sodium and magnesium,
though they sit next to each other in the table. The distinction between the phases of
matter -- solids, liquids, and gases -- seems pretty clear, too. (Well, OK, there are
some bizarre in-between states you learn about in college, but they're encountered
only in extreme conditions, usually artificially induced in laboratories and definitely
inhospitable to mammals.) Living things appear to be made up of lots of discrete
functional blocks -- organs and cells and organelles and protein molecules and such --
and the interactions of these discrete parts are what makes them "go." DNA itself, the
structure that stores the instructions for building, is just a binary (or rather
quaternary) string, with each discrete position capable of storing only four possible
values. Even the seeming continuum of a fluid like water is an illusion. In reality it's
made of discrete particles, and it's their interactions with one another that give rise to
"fluidness."
This endless interplay between the continuous and the discrete, between discovering
structure and creating it, seems to be at the heart of many (maybe most) human
endeavors. On the one hand, we often labor mightily to reveal structures that are
somehow already there. Many of the sciences, in particular, are precisely an attempt
to make clear the underlying structure of the universe, to peel away the layers of
obfuscation that our senses have piled on. But it's not limited to science: Michelangelo
spoke of sculpting not as inventing the shape of the statue, but rather as freeing it
from the stone in which it was imprisoned.
On the other hand, many human activities are all about applying structure to
something formless, or about creating structure from nothingness. Again, the arts
spring to mind. A painting, a poem, a story, a song -- all these begin from nothing:
from a blank canvas, from an empty sheet of paper, from silence. Computer programs,
those awesomely complex logical constructions we devote ourselves to so slavishly,
seem to be created from thin air, and serve as the structure for an otherwise
"formless" machine. Business contracts, sheet metal ventilation ducts, bingo games,
steering committees, and acoustic guitars -- these are all structures that we've
created from, essentially, nothingness.
Ah, but there's the real question: are the structures we build really new? Do we really
just invent them, whole, from nothing? Or do they grow from and reflect other
underlying structures that are already there? The latter seems much more likely to
me. If you look closely, even a structure that at first blush seems really new turns out
to be a recombination or an extension or a reworking of some existing structure.
Sometimes I think of evolutionary processes this way, as a sort of extrapolation, a
patient elaboration, of a very few essential, innate principles that lie buried far
beneath the surface, an endless cycle of structure standing on the shoulders of what has
gone before. Perhaps the macroscopic shapes of living things hint at the underlying
nature of matter: quantum reality writ large, for all to see.
So, getting back to music, does the tuning system that we use reflect some underlying
structure, some relationship among the frequencies, or is it truly arbitrary, decided
by some bewigged old coot in the dim and dusty past? To find out, I did a little snooping
around in the local library and on the net. The answer turns out to be complex (no
surprise there), and it has as much to do with accidents of history, people's
personalities, and the practicalities of tuning instruments as it does with mathematics.
There is an underlying structure beneath the western tuning system and many others
(the relationships among the harmonics of a vibrating string or column of air, first
elaborated by -- you guessed it -- the Greeks). But it also turns out that most music
has evolved away from those "ideal" frequencies for a variety of reasons that have
little to do with mathematics or physics. Here's what I think: I think the structure of
music is largely determined by the structure of us.
Sunburst shapes -- mandalas -- appear over and over in pictures drawn by children,
no matter what their culture or language or what part of the world they live in. You
could argue that it's simply because there are lots of radially symmetric things in the
world, and that children are simply drawing what they see. But I prefer to think that
mandalas somehow mirror the internal structure of the human mind, that they are, in
a sense, pictures of humanity.
Humans spend enormous amounts of time shaping things, refining things, expressing
things, creating. Where does all the structure come from? I think it's simply an
outgrowth, an elaboration, a reinterpretation and repackaging, of the structure inside
ourselves, which in turn reflects the structure of the universe we live in. Taken all
together -- all the songs, all the buildings, all the stories, all the social groupings and
computer programs and bad jokes and trash and art -- the structures we humans
reveal and create form a churning, turbulent, clouded mirror, a mirror that
occasionally, if we look very closely, may afford us a glimpse of who and what we
really are.
DAVE JOHNSON has for years had the same favorite quote, from Albert Einstein: "He
. . . who can no longer pause to wonder and stand rapt in awe is as good as dead; his eyes
are closed." But recently he encountered another, in a book by Primo Levi, that came
very close to unseating Einstein's: "It is enough to think of intestinal worms: they feed
themselves at our expense with a food so perfect that, unique in creation, together
perhaps with the angels, they have no anus." Although Dave howled uncontrollably at
that one, Einstein still wins in the end. *
Thanks to Lorraine Anderson, Jeff Barbose, Brian Hamlin, Mark "The Red" Harlan,
Bo3b Johnson, Lisa Jongewaard, and Ned van Alstyne for their always enlightening
review comments. *
Dave welcomes feedback on his musings. He can be reached at JOHNSON.DK on
AppleLink, dkj@apple.com on the Internet, or 75300,715 on CompuServe.*