Fugal Science, Volume 1, Number 4

You will have to read the first three installments in this series to get much out of this post.

To review, we started out with a fugue at the octave (Two-part invention format), turned that into a fugue with a tonal answer at the fifth, and then introduced some character to finish it as a piece of music. That's as far as I could take it in two voices with the elements I limited myself to.

So, this is the initial three-voice fugue, and I had to open it up an octave to make things work, so it's for string trio.

Here's the M4A audio: Fugue Number 4



I had to open the exposition up an octave, otherwise the answer and counter-answer would have converged on a unison. This worked fine in the four-voice string quartet version of the fugue I wrote first, but it didn't float my boat in only three. As a result, the answer and counter-answer begin a twelfth apart, just as in the guitar versions.

For the three-voice subject statement, I had to create a new element, which is a second countersubject. Many of the old school fugal theorists didn't like third or fourth lines that had so much the character of harmonic fill-ins - I'm thinking of Andre Gedalge - but if you do them right, they can have plenty enough character as independent lines. And, when you have an interesting and active primary countersubject, they are often the only kind you can write. This second countersubject is so distinctive that it would actually make a very good fugue subject. And yes, that gives me some ideas going forward.

Since this is just the initial three-voice version, I again have taken the approach of changing as little as possible to bring it into perfect balance. That means that the plan is arranged around the idea of presenting all of the possible two-voice duets in a logical order. As a result, the piece is merely competent and not killer. It does contain some killer new elements, however.

Once the viola part rests after 16 - first in/first out - episode one proceeds with the outer voices and is just like all of the first episodes that have come before.



I've probably read 99.9% of all of the old treatises and textbooks on counterpoint and fugue that have been translated into English over the years, and almost invariably they take the position that three-voice fugues offer the best combination of harmonic fullness and lineal independence. I, however, find them difficult and lacking: Four voices is easiest for me, which is why that's usually where I start.

A great example of the difficulty and awkwardness that can crop up in three voices is the transition from the end of the first episode to the first middle entries here: The only way you can change the thematic entrance order in three voices is to begin a statement in a voice that is already active, versus a more effective entrance from a voice that is resting. At least this one comes after the bass has had a whole note, which is the next best thing to following a period of rests. I had to do this, because without this particular arrangement - the outer voices - the entire rest of the fugue would be impossible. I'll explain how it came to this in a minute.

Musically, by the way, this entire section is exactly as it was in the previous guitar version. However, the section stirkes the listener differently now, as the countersubject is not a new element (It was in the exposition).

A larger problem is that the second countersubject is not employed here: It would work perfectly with a truncated first statement and then a complete second statement, but that will have to await the first four-voice version (Where I'll also be able to rotate the elements between voices), or a later three-voice version that includes more structural and formal elements.



Look first at the new element that starts in forty-four: In order for the thematic entrance to be on top with the violin, this section must be a duet between the viola and cello.

A word about voice exits: If the voice has a whole note during the dominant chord before the cadence - that would be on either sol or re. - then it does not have to participate in the actual resolution. If, however, the voice is on an active tone and/or is less than a whole note, it does (This would be fa or ti.).

Here, in this MIDI/Soundfont version, the violin seems to obscure the beginning of the duet, but in a live performance the instrumentalists would fix that quite easily and tactfully.

At forty-four is the old, "interlude" that has the subject over the bass line of the primary episode, but I found that the lead voice of the old episode also works in this combination... by only changing one note. That would be the fourth note in the viola part at forty-six, which is now a D but which used to be a G. Needless to say, a wonderful discovery. I went back and changed that note in the versions without the subject too.

When you discover combinations like this post facto, oftentimes some interesting modal harmonic successions occur: In this case, there is a D minor seventh sonority in a 4/2 arrangement in forty-six, which doesn't sound particularly strange here, but just wait until I invert it.



Note first that the viola had to participate in this resolution because it is a fa-mi formula, and also the previous note was a half. Also note that the viola's first cadence was to the tonic, and this one is to the dominant (Even if it is functioning as the major third in the relative). Otherwise, the music in this duet is unchanged from the previous guitar version, with the exception of the single note in the episode mentioned earlier.

That leads to a duet between the upper voices at sixty, which must be that way because of the next section. The music itself is again unchanged from the previous version.



At sixty-nine is the second three-voice combination of the episode and the subject, and in the previous fugue this lead into the pedal section, which it also does here, only in a different manner. Now, the pedal is introduced by the cello, who just holds the last note of the subject over. During the pedal point proper, the viola has a new part that perfectly fleshes out the combination and which also increases the energy of the final half-cadence magnificently. This produces an air of invincible inevitability as the viola descends through the cadence toward what can only be the beginning of the recapitulation canon.

Note that in measure seventy-one there is a root position G minor seventh chord. Groovy, baby.

Since we have three voices to work with, we get a three voice concluding canonic stretto (I composed the subject as a four-voice canon, remember).



One of the elements that is improved over the two-voice extraction is the final hyper-stretto between the subject in the lead, and the augmented subject in the bass. First of all, the pickup figure in eighty-six is now to a non-thematic part, so the thematic entrances stand out better. Secondly, the third voice mediates effectively by more completely defining the harmonic context. Sure, the listener is going to be at least subconsciously aware that the minor-ninth-plus-octave in eighty-eight belongs to the V(m7m9) sonority, but having the minor seventh present makes things much clearer. Also, at the beginning of ninety, we get a complete augmented triad that is just awesomely bad ass.

The fun continues as the third part adds to the power of the already powerful conclusion too.

So then, the plan is: Exposition, outer voice duet, lower voice duet, three voice interlude, outer voice duet, upper voice duet, three voices from the concluding episode on. Nicely structured and balanced. How I finally arrived at this arrangement solution is a great old trick: Work from the end of the fugue back to the beginning.

I had the exposition and the hyper-stretto - alpha and omega - and the three-voice recap canon was obvious too. Then, the pedal section was obvious, and only a duet between the upper voices could lead into it. An outer voice duet was the best preparation for the upper voice duet, and the three-voice interlude could only be preceded by a lower voice duet, so that lead back to the first outer voice duet. That's when I found the exposition and first middle entries could only be joined as they are: The exposition-following episode can only be between the two outer voices.

I'm not at all convinced that this fugue will survive long without modification, but I am convinced this is the best arrangement for this limited number of elements. I'm currently working on an exhaustive analysis of all of the possible combinations this subject and countersubject can possibly make - I finally cracked a mechanical solution to Sergi Taneiev's Convertible Counterpoint formulas - and that may very well redefine these fugues all the way back to Number 1. That will be next, but it's going to be a while.