Wednesday, December 14, 2005

Extempore in A minor: Applied Fractal Self-Similarity II

As I anticipated, I finished the Extempore in A minor within an hour of waking up this morning... OK... within an hour of waking up this afternoon. What I didn't anticipate is that this piece would solve a "problem" I have had with Sonata Zero: Namely, that the first sonata process movement did not directly relate to the following Scherzo or Fugue. This Extempore, however, is very closely related to the concluding fugue, and it is also contrapuntal in nature - as the Scherzo is - so by simply replacing the Sonata with the Extempore, voila: Sonata Zero is now complete as Extempore, Scherzo, and Fugue. The sonata process piece is now a Work in Progress, and I expect that it will eventually become a movement of Sonata One (Either the first movement of a sonata in A minor, or the third movement of a sonata in C major, which now seems more likely).



There are no changes to the first page, other than the layout vis-a-vis the numbers of measures per system. On that note, I group measures per system to reflect the phrasing, and I do not allow the music to decide the layout based on what "fits" where. So, I don't follow precident as set by engraving history and practice, therefore the music may appear "weird" to you. There is a lot of stuff about music engraving I don't care for, so I just set it up the way I want to see it, tradition be damned.

The expositional section with it's transitional measure - I wouldn't call it an episode - makes up the first two systems of five bars, and the first fractally generated episodic passage makes up the second two systems, which is also a total of five measures (Two 2.5 measure phrases).

At the bottom of the first page is the second episode, which is two phrases of two measures each, so as in imaginary number based fractal images, the fractal elements are becoming more compressed with repetition.



I have repaired the awkward page break, and the third immitative episode based on invertible counterpoint is now clearly visible. Measure eighteen is a fractally related transitional measure into the next episode.

Episode four engages in a constant sixteenth-note surface rhythm, modulates up by fifth each measure, and is five measures long, which takes the piece to the remote region of G-sharp monor at measure twenty-four. There, the fifth episode begins, which is a chromatically ascending modulatory sequence, and the home key of A minor is traversed in measure twenty-five before arriving at B-flat major in measure twenty-six: I managed to get the leading-tone minor and leaning-tone major regions in surrounding the tonic (The leaning-tone major region is traditionally referred to as the Neapolitan region, but this is kind of ridiculous since that appelation does nothing to really describe the function or relationship to the tonic).

As I noted in the previous post, measures twenty-seven and twenty-eight are slightly varried duplicates of measures seventeen and eighteen, and they set up the reappearance of the constant sixteenth note episode, only this time in major keys starting in the relative region. I foreshortened this episode to three measures (Or rather, the piece shortened it for me), because the final measure of the sequence starts in the major subdominant region, and modulates to the tonic major region. The structural significance of this should require no explanation.



Here's the final page. I again employed the chromatically ascending sequentially modulating figure - also foreshortened - to re-arrive on the tonic level of B which first appeared in measure twenty-seven. This time, however, it is B major instead of B minor. Other than that, though, measures twenty-seven and thirty four are exactly the same.

With the second arrival to the lowest E on the guitar at measure thirty-five, I begin the dominant pedal episode that sets up the recapitulation. This episode is exactly like the pedal point episode in the concluding fugue of the sonata, except instead of being a harmonized version of that fugue's subject in augmentation, this episode starts out as a sequentially decending harmonization of the motif in augmentation. This only required the insertion of two new measures into the previously written analogous episode from the fugue. When I gave up an went to bed early this morning, I had, in fact, cut and pasted the fugue's episode in here. I actually leapt out of bed this AM... er... PM saying "of course: Just add a couple of harmonized sequential repetitions of the head figure!" I had the piece finished before the coffee finished brewing. I love it when stuff like that happens. Nobody can tell me we don't compose in our sleep, because I do it on a regular basis.

The fact that the motif of this Extempore is the head figure of the concluding fugue, combined with the strong similarity between the two concluding episodes they both have, creates a series of powerful bonds of self-similarity between them that makes the resulting Sonata Zero triptych very effective and "of a piece", which I am perfectly satisfied with. Not only that, but since the Extempore is about one-third the length of the previously written Sonata (It and the fugue are, in fact, of the same duration!), the pitch climax of the whole is now at or near the 66.6% point. Nifty. I played them all in order after I completed this piece, and they are perfect together.

The recap is a true recap, being as it is an exact repitition of the expo, only starting over a tonic pedal. I only had to add two concluding measures to get the final statement of the motif in the penultimate measure leading to the final cadence to the perfect triad of four voices in close position with the exterior octave doubling. Note that this penultimate measure displays a 3-2-1 Schenker line, which is literally the entire piece in a nutshell.

There may be some very minor polishing of the piece throughout the upcoming days, but the Extempore and Sonata Zero are both basically signed-off on now. Nice thing to finish up on the eve of my birthday.



"I'll sign-off on it for you, Huc!"

Thanks Molly.

EDIT: Oh yeah. The PDF and MIDI files of this are now on my .Mac FileShare page as Sonata_Zero_1.pdf/.mid for those who would like to take a listen.

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