A universe begins with a pedal E-flat

Over the last two days, I've worked my way slowly through Das Rheingold, the first opera in Wagner's Der Ring des Nibelungen. I'm not doing it particularly well, actually: I'm listening to one disc at a time, and I'm not bothering to listen to them all in one sitting. I know that Wagner would probably not approve, but c'est la vie.

After all the superlatives that have been heaped upon this tetralogy over the years, it's probably useless for me to try to add to them. Suffice it to say that Der Ring does what few other works of art do, at least for me: it transports me to someplace else. Many works of art leave me wondering what they are saying about the world in which we live, and how the artist's vision relates to my world, but with Der Ring I always get the sense that, at least while I'm listening to it, there is no "me". Wagner is pulling me into his world, and since Wagner's world is a world of stunning musical architecture and orchestral tone-painting and high drama, leaving my world for a time to enter his seems fair. If leaving Der Ring makes me feel anything, it's the desire to leave some token on Wagner's doorstep as a way of saying, "Thanks for letting me in there for a while."

(And no, I don't care that Wagner-the-man was a colossal shit. I'm no expert, but thus far in life, all of my efforts to find the evidence of Wagner-the-shit in his operas have come to naught, and I just don't care anymore. It's just a fact of our world that great works can be produced by lousy people.)

Anyhow, I was listening to the prelude to Rheingold, and again I felt the thing just pulling me in, first with that pedal E-flat that can almost be felt rather than heard. I remember the first time I heard this prelude, back when I checked the opera out of the library on LP when I was in tenth grade, and I recall being flummoxed by the structure: no real "tune", just a single tone that soon gives way to undulating chords and then to iterations of a theme that's almost nothing more than a major arpeggio and then more and more iterations of same, crescendoing impressively before the Maidens start singing. As I read the story, I realized that this was the basic depiction of the Rhine itself; but it goes beyond mere depiction. Something else is going on here, something more fundamental than just giving us a picture of a river. We're not being shown, so much as we're being transported. Wagner is inducting us into a world where, at first, all is void until we become aware of water, and then of main current, and then of swirling undercurrents and eddies, and only then of the beings living here.

This, of course, came as quite the revelation to me in a time when I thought of opera beginnings in terms of traditional overtures.

Anyhow, I was listening to this prelude yesterday, and another thought occurred to me. I'm not sure why, but as I listened to the way Wagner structured it, I thought of a mathematical construct called a Koch curve. This is a fractal curve that begins as a straight line, which is then segmented into three equal portions with the middle portion being taken as the base of an equilateral triangle, and then the same steps being applied to the resulting curve. (A handy Java doohickey for manipulating a Koch curve manually can be found here, although I didn't test its warning that it would crash my computer after a sufficient number of iterations.)

The prelude to Rheingold is almost a musical version of a Koch curve, starting with the straight line of that pedal E-flat, and then proceeding through more iterations as each voice is heard, constructing the curve from the arpeggiation of the Rhine motif. Maybe that's why this prelude is so magically elemental: its musical construction springs from tonal relationships that in turn reflect some very basic mathematical underpinnings of our universe.

(I have the suspicion that this is one of those lightning bolts of revelation that, to myself as a layman, seems so utterly profound, but to someone really well-versed in the subject at hand would have them rolling their eyes and sighing, "Well, everybody knows that. Duhhh!")

BTW: I'd love to see the "Native American" version of Der Ring being produced in Washington, DC. As ACD notes, the idea has lots of mythic potential. But then, I'd love to see any version of Der Ring, really. My familiarity with the cycle comes exclusively via recordings, I'm sad to say. Maybe we can do a "Buffalo" version, with the Niagara standing in for the Rhine....