Fugal Science: Vol. 1, No. 1 - Two-Part Fugue for Solo Guitar

EDIT: This is part 1 of 8. Here are the links to the entire series:

Index of Fugal Science, Volumes 1 and 2

I am now finishing up the final versions of the four fugues of Volume 1, which are a two-part fugue for solo guitar, a three-part fugue for string trio, a four-part fugue for string choir (minus contrabass), and a five-part grand fugue for symphony orchestra. The collection is designed to be performable as a concert, and I will also make a written treatise out of it. This post is the beginning of organizing my thoughts for that.

The subject of this set is a five-measure melodic trajectory that works in five-part canon at the octave and at one measure of delay, so that in the fifth measure, all five measures of the subject are heard simultaneously. The volume is set up to progressively reveal that.

This two-part fugue for solo guitar has the answer at the octave, so it is a real answer in the strictest sense. Bach called these octave fugues two-part inventions, but an answer at the octave is the most natural for two voices, just as the most natural two-part canon is at the octave. A tonal or real answer at the fifth implies more voices to come, and is very awkward to pull off. Many examples in the historical literature actually have a third entry on the tonic to rectify this deficiency. Interestingly, an answer at the fourth is perfectly natural for two voices, because it is sort of like starting with the answer, and having the subject enter second.

A note about objective renditions: I am routinely criticized for the mechanically perfect computer realizations of my music, but I don't think that is in any way valid. First of all, I love to hear my music performed perfectly by the computer, because it is precise enough to stand up to that, and if it moves me without any subjectivity, it is a successful piece of music. Of course I will put performance indications in the final versions, but we're not anywhere near that yet (Though they are developing in my head now).

So, without further delay, here is the recording for today's piece. It is just a recording of the sound fonts I compose with, and it is a CD quality AIFF file, so you'll need to have Quicktime activated in your browser.

Two-Part Fugue for Solo Guitar

I suggest opening a second tab so you can listen and follow the score.

One of the scientific features of these fugues is that as many voices resolve to the tonic as possible at every seam between sections. This is not the way fugue writing has ever been taught, but it is the best way to give maximum flexibility in the application and arrangement of the thematic and accompanying material. This will become clear as we progress through the volume.

The five measure subject starts in the bass, but it could just as easily have started in the soprano. I chose the bass first version because of a subjective reason, which is the idiom of the guitar, and also that the final canon is bass first, for reasons we will address when we get there.

The countersubject, then, first appears in the bass, which I subjectively like better anyway.

The six measure episode - and it is the only episode needed for this fugue, as will be revealed - is what I call a hard part: It cannot be inverted because the voices get separated by two octaves, which would be a very poor sounding unison if inverted.

The first middle entries are in the same arrangement as the exposition, with the bass entering first, and this is a hard part, because the voices become separated by two octaves at the end. This is also a stretto at one measure of overlap, which would be trivial except for the fact that this is, structurally, a perfect dovetail, requiring no modifications other than truncation of the countersubject.

That means the thematic section is nine measures this time - versus ten in the exposition - and the episode is a measure shorter this time too, due to the modulation to the dominant. So, as is desirable, the pace is quickening.

For the second middle entries on the dominant level - third thematic statements - we have a reversal, with the soprano entering first. This is a hard part, so that is unavoidable, but it fits into the plan for the fugue perfectly, as we'll see.

Part of the mystery of the one-and-only episode is revealed here: The bass line can support the subject. The treatment of dissonance is quite modern, and I'll have more to say about that in later elaborations on this piece. The stretto is eight measures, so one shorter than previously, but the episode is the same five measures as before, so still a bit of an increase in pace.

As you can see, the end of the episode very beautifully and dramatically modulates the piece to the relative major.

Here the bass comes in first, which is perfect after the episode/subject combination with the subject in the soprano! But note that this section is not a hard part, and could be inverted. The stretto overlap is now three measures, so the plan of the fugue is obvious: Ever closer stretti until the recap. That makes for seven measures in the thematic section, followed by a four measure version of the episode, so things continue to move more quickly.

The end of the episode modulates us to the subdominant in a very elegant way. This sets up a sort of false recapitulation on the subdominant, which is in the form of the first dovetail middle entries, but here I use the tonal answer version of the head figure to remodulate us back to the tonic! Very beautiful and dramatic.

Once back on the tonic level, the final version of the episode on the tonic appears, here extended with the pedal section. After the half-cadence in measure seventy-two, we get the recapitulation, which is the canon at one measure of delay/four measures of overlap. Note that the bass enters first, and this is the most desirable configuration, as the effect of the soprano handing off to the soprano is an objectively inferior arrangement. This fugue is the most perfect possible arrangement of the material, and I did experiment with all of the possibilities to arrive at it.

Note that I extended the canon to six measures with the addition of the trill figure, and this leads to a perfect cadence for the false ending. After that, we get, with the coda, the crowning contrapuntal combination of the piece: A hyper-stretto in which the soprano has the subject, and the bass has the subject in augmentation. This was a magical discovery, and it came to me way back in 1993! It is even more elaborate in three and four parts.

The ending is on a flourish with a secondary version of the thematic trill figure, and a strummed out six-part final barre chord.

Perhaps I'll have the three-part fugue ready for next month.