Reductio ad Absurdum
I consider myself a minimalist, but not in the usual way that minimalism is thought of vis-a-vis the music of Phillip Glass et al. Rather than taking a small musical idea and repeating it ad nauseum as Glass is wont to do, I am always striving to find the irreducible musical truth at the root of whatever musical subject I'm dealing with: For harmonic voice leading that is circular and crosswise transformations of the chord tones, for root progression patterns it is the relationship with the primordial falling fifth that the overtone series gives us, for counterpoint it is the set of laws disallowing paralel intervals that are superparticular ratios in both inversions, also from the overtone series. You get the idea.
*****
Fugue has recieved the same treatment from me, and after fifteen years during which my subjects have become increasingly elemental (For the most part), I have reached the ultimate reduction of the subject to a single note, and the four-voice exposition to four measures. This may not seem particularly significant to you the reader, but for me as a composer and contrapuntist it is like stumbling onto some sort of ultimate musical truth. It's really kind of earth shattering for me.
As an initial experiment with this idea I decided to write as short a fugue as possible and make it... a guitar piece, natch: Exposition, transition 1, counterexposition/middle entry in the relative, transition 2, recapitulation, and a codetta is all there is to it. It comes to just twenty measures of 2/4 and takes only 30 seconds to play.
We're in A minor here, and as you can see the top system is the exposition, and the subject is a single half note which is answered at the fifth. The answer is obviously real (ha, ha). Note also that up until the middle of measure five, the exposition is a double canon at the octave. I plan on following that possibility in an upcoming exercise.
If the subject is the irreducible minimum, then it follows that the counterpoint should be the most obvious as well. Under the first answer the counterpoint is a 2:1 ratio that makes a 4-3 resolution: There is no more hallowed entrance setup for an answer than a prepared fourth.
Over the second subject in measure three there are both 4-3 and 6-5 resolutions, and the harmony is a plagel iv to i progression. Finally, in measure four, there is a 7-6 and 5-4 over the answer and a 4-3 under it, which creates a modal progression ending on a i(6/3) chord. It's a lot like a v(4/3) to i(6/3), but the v has no third. The killer is the downbeat of measure five: It's a vii(d4/3).
Measures five and six were originally composed in the version you see in measures eighteen and nineteen which end the piece. By simply allowing the G in the bass to remain natural I was able to effect the modulation to C, and the counterexposition/middle entries are essentially the same except for the mode.
To add to the absurdity of the piece, I had voices one and two exchange places, as well as voices three and four. This must be reflected in the notation from a theoretical standpoint, but it isn't something my ear seems concerned about.
The added part which voice four has in the middle maintains the basic 1/4 note pulse, and also allows for the dissonance climax to coinside with the pitch climax in measure nine (The 50% point, excluding the measure of repose after the final resolution).
For the second transition back to the tonic an extra measure was required, and the sixteenth note figuration was allowed a bit of development.
*****
Once the initial idea struck me (And, struck is the operative word), it took all of two to three hours to put this together. After pondering it for the afternoon, however, I came up with the following slightly elaborated version.
This is better in almost every way: The subject begins on an upbeat, which makes the 4-3 resolution (And the rest of the compound resolutions) both better prepared, and also more effective. There is also much more rhythmic drive to this version of the piece, and that's part of the problem. The eighth note articulations create an expectation of a continuance, which is not delivered when the quarter notes reappear. There is something less than perfect about this, but the articulation idea itself is OK: It just needs a better execution.
*****
So, I came up with a third version.
Adding the leading tones provides some colorful contrapuntal effects, as well as some directional impetus to the piece. The lilting dotted-eighth/sixteenth rhythm eliminates the expectation of continuous eighths, which allows the piece a range of rhythmic space in which to breathe. This is a really cool little ditty.
*****
Obviously, this idea will eventually lead to a much larger piece - I can already see it modulating to E minor to outline the tonic minor triad - but for now this will be enough food for thought to keep me occupied.
*****
Reductio ad Absurdum, indeed™.
*****
Fugue has recieved the same treatment from me, and after fifteen years during which my subjects have become increasingly elemental (For the most part), I have reached the ultimate reduction of the subject to a single note, and the four-voice exposition to four measures. This may not seem particularly significant to you the reader, but for me as a composer and contrapuntist it is like stumbling onto some sort of ultimate musical truth. It's really kind of earth shattering for me.
As an initial experiment with this idea I decided to write as short a fugue as possible and make it... a guitar piece, natch: Exposition, transition 1, counterexposition/middle entry in the relative, transition 2, recapitulation, and a codetta is all there is to it. It comes to just twenty measures of 2/4 and takes only 30 seconds to play.
We're in A minor here, and as you can see the top system is the exposition, and the subject is a single half note which is answered at the fifth. The answer is obviously real (ha, ha). Note also that up until the middle of measure five, the exposition is a double canon at the octave. I plan on following that possibility in an upcoming exercise.
If the subject is the irreducible minimum, then it follows that the counterpoint should be the most obvious as well. Under the first answer the counterpoint is a 2:1 ratio that makes a 4-3 resolution: There is no more hallowed entrance setup for an answer than a prepared fourth.
Over the second subject in measure three there are both 4-3 and 6-5 resolutions, and the harmony is a plagel iv to i progression. Finally, in measure four, there is a 7-6 and 5-4 over the answer and a 4-3 under it, which creates a modal progression ending on a i(6/3) chord. It's a lot like a v(4/3) to i(6/3), but the v has no third. The killer is the downbeat of measure five: It's a vii(d4/3).
Measures five and six were originally composed in the version you see in measures eighteen and nineteen which end the piece. By simply allowing the G in the bass to remain natural I was able to effect the modulation to C, and the counterexposition/middle entries are essentially the same except for the mode.
To add to the absurdity of the piece, I had voices one and two exchange places, as well as voices three and four. This must be reflected in the notation from a theoretical standpoint, but it isn't something my ear seems concerned about.
The added part which voice four has in the middle maintains the basic 1/4 note pulse, and also allows for the dissonance climax to coinside with the pitch climax in measure nine (The 50% point, excluding the measure of repose after the final resolution).
For the second transition back to the tonic an extra measure was required, and the sixteenth note figuration was allowed a bit of development.
*****
Once the initial idea struck me (And, struck is the operative word), it took all of two to three hours to put this together. After pondering it for the afternoon, however, I came up with the following slightly elaborated version.
This is better in almost every way: The subject begins on an upbeat, which makes the 4-3 resolution (And the rest of the compound resolutions) both better prepared, and also more effective. There is also much more rhythmic drive to this version of the piece, and that's part of the problem. The eighth note articulations create an expectation of a continuance, which is not delivered when the quarter notes reappear. There is something less than perfect about this, but the articulation idea itself is OK: It just needs a better execution.
*****
So, I came up with a third version.
Adding the leading tones provides some colorful contrapuntal effects, as well as some directional impetus to the piece. The lilting dotted-eighth/sixteenth rhythm eliminates the expectation of continuous eighths, which allows the piece a range of rhythmic space in which to breathe. This is a really cool little ditty.
*****
Obviously, this idea will eventually lead to a much larger piece - I can already see it modulating to E minor to outline the tonic minor triad - but for now this will be enough food for thought to keep me occupied.
*****
Reductio ad Absurdum, indeed™.
posted by Hucbald at 12:27 AM
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