Wednesday, December 14, 2005

Extempore in A minor: Applied Fractal Self-Similarity I

Nice weekend. I ate until it hurt every day, all day, and mom and I had a splendid time. On with the show...


Writing the previous guitar fugue made me hungry to do a follow-up, and since the subject of that fugue was a decorated 5-4-3-2-1 Shenker line, I wondered how much further I could reduce it. I decided that by taking the do-ti-do head figure, I would imply an irreducibly primordial 3-2-1, and so that's what I decided to do.

Since the "subject" here is really nothing more than a motif, I didn't have any stretto possibilities or suspension chains to work with. In lieu of those kinds of techniques - since I have been thinking a lot about fractals, chaos, and self-similarity lately - I decided to create a new (For me) technique of letting the piece self-propagate from nothing but the motif. This is almost exactly the same process Beethoven used in the opening Allegro of the Fifth Symphony with his Fate Motto, only here the technique is more closely related to the polyphonic fugal process than the homophonic sonata process.



As you can see, the piece starts out with a traditionally laid out but much-truncated exposition of four bars. The main motif is labelled A. Note that the voice leading is totally smooth and stepwise until the bass leaps down a sixth in the transitional fifth measure. It was this desire to keep the lines smooth that lead me to the diminished version of the motif at the end of measure four, which is labelled A': I originally had two eighths there progressing from re to ti.

This diminution of figure A has basically generated the entire rest of the piece. Beginning in measure six, the diminished version of the motif tonicises the three degrees of the tonic triad in an ascending direction, which allows the bass line to progress smoothly by step for an octave and a perfect fourth. This leads to some nice effects, as the chord of "resolution" in the second half of measure seven is an augmented triad: bIII+(6/4). The progression of fourths involving the bass is not a problem because they are perfect, diminished, and perfect. This phrase breaks the "Tyranny of the Bar Line" by being 2.5 measures of length, and it modulates one flat in the subdominant direction, after which it repeats on the new tonic level of D minor, but with a third voice over the beginning of the motif figures. Note that if you consider the leap of a seventh in the bass as a simple octave displacement, the bass line has only one real leap in it until measure thirteen. Measure nine has some interesting and strange effects with the outer lines moving in stepwise contrary motion, and a second augmented triad appears on the downbeat of measure ten.

The decending sixteenth note figure was spontaneously generated, and is labelled B, and the chromatic figure that effects the modulation is labelled C.

Starting in measure eleven, the motif tonicises the degrees of the tonic triads of the moment in decending order, and the phrases are foreshortened to two measures in length by omitting the main motif in the bass voice. The parallel movement into seconds at the end of measures twelve and fourteen are OK, because they are augmented seconds: It is important to note the difference between intervals which are dissonant in name but not sonority and vice versa, as much freedom in counterpoint can be gained by following the rules for consonances where name-only dissonances can appear. I did the same thing with the diminished fourth (= major third) earlier to get smoothly into the augmented triads.

I called this an Extempore - Ex tempore: Literally "out of time" - because it's really kind of a written-out improvisation: I had no set plan, but rather allowed the piece to organically generate itself out of the sub-motifs that happened to spin off of the main one in the counterpoint. As a result of this, the modulations go very far afield compared to a real fugue, which I like very much. It's kind of like a fugal process Fantasia.

In measure fifteen, the third section starts, which is an immitative treatment of the A', B combination that first appeared in the bass (Sorry for the awkward page break, but this is still in the sketch phase).



This immitative section spins off figure D, and it's variants become quite important. After the bass gets the A', B figure, measure eighteen is transitional to the next section, which has a constant sixteenth note surface rhythm. Note the suddenness of some of the modulations: G minor to C major from seventeen to eighteen is particularly nice.

The figure of B+A' followed by B' that the section from measures nineteen through twenty three uses modulates up a fifth each measure: The sequence starts on the home key of A minor and goes all the way to G-sharp minor: Traditional fugues do not stray anywhere near this far from the tonic.

Starting in measure twenty-four, I re-interpret the roots in the lead as leading tones in the second half of each measure to get a chromatically ascending modulation sequence, which gets us up to the note B, which is the highest note in the piece thus far. Measures twenty-seven and twenty-eight are just repeats of measures seventeen and eighteen - slightly varied and on a different level - which returns us to the constant sixteenth note sequence, only this time it's going through major keys starting from the relative, rather than minor keys starting from the tonic (So, there is structural organization going on, but it's very fluid and... well... extemporized.

The piece continues for four more measures after the page break so far, and it's been one of those pieces where I write a few measures, sleep on it, and then write a few more. In that regard, it's much more like writing a prelude than writing a fugue.

I was thinking of combining this with the earlier fugue to make a double fugue, but the characters of the pieces have become too highly disparate as I have progressed through this one, so I dropped that idea.



"Did someone say 'desperate'?!"

No... "disparate"...

I'm going to wait to post this on my Fileshare page, as I anticipate finishing it in the next day or two.

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