I love hanging out at Stately Smith Manor. Brian designed and built it himself (He restores historical landmarks for a living), and it really is magnificent.
Now, back to the matters at hand.
I have finally came to the ultimate conclusion of Fuga: Reductio ad Absurdum. If you'll recall, I reduced the fugue subject to a single half-note, and the four-voice exposition to four measures of 2/4. The first version was a simple fugue.
Though this was nice, there was an ultimate solution awaiting me, and this wasn't it.
The second idea I had was for an even simpler double canon.
This was less satisfying, but it lead to the breakthrough I was looking for.
Ta da! Using the simple process of quadrant rotation, I made the entire fugue out of nothing but the exposition: Original, Retrograde Inversion, Inversion, and Retrograde. If you number the quadrants clockwise in the usual manner, that means it is a 1, 3, 2, to 4 progression.
The piece is a double pallindromic canon as a whole, and each half is a double pallindromic canon in inversion (Double mirror crab canons).
I have color coded the voices to make the canon(s) easier to see. Red and blue voices are canonic, while the green notes are free additions to fill out the cadential elements. In measures one through nine, the leading blue voice and the folowing blue voice read the same forwards and backwards respectively, but they are in inversion (A mirror crab canon). Likewise, the leading red voice and the following red voice read the same forwards and backwards respectively, also in inversion (A second mirror crab canon). Therefore, measures one through nine are a double mirror crab canon.
Measures ten through eighteen are simply measires one through nine backwards, for a second double mirror crab canon. Finally, the entire piece creates a larger double crab canon, since it reads the same forwards and backwards (With the exception of the green notes, of course), but it is a true pallindrome and is not a mirror.
Humorously, I came up with this while driving east on I 40 between Memphis and Nashville Tennessee last Monday morning: That would be "Music Highway" in case you have never driven that route. OK fine, I guess you had to be there, but I thought it was funny.
There is something deeply and irresistably compelling about this musical structure. It is an irreducible musical truth. Since even before Bach, Western Art Music has reflected the so-called Golden Mean ratio in its idealized proportions, but research I did years ago intimated to me that there was a bi-fold symmetrical way to approach musical structure - for music in general, and fugue in particular - and this is it. I plan to use this structure as the framework for larger pieces, which I will realize by elaborating on it: After twenty years of reducing fugue down to this primordial kernel, I can now build it back out.
Imagine this as the classic Mandelbrot formation at the heart of this fractal image: The image has bi-fold symmetry running crosswise through the axis of the central Mandelbrot figuration, and everything exterior to that is self-similar elaboration. Now that I have the musical version of the Mandelbrot formula, I can now begin to add the fractal elaborations to it.
This is a huge breakthrough for me.
Vacation is over, so...
I can now concentrate on music again. Or something.