In the first post in this series,
I demonstrated a method for finding all possible vertical-only and horizontal-only shifts at the thesis
points for a five-measure fugue subject and its tonal answer, and in all four quadrant orientations (Original, inverted, retrograde, and inverted retrograde). As I intimated then, this is much more practical than Taneiev's theoretical treatise, it has the advantage of not relying on mathematical formulas, and it also has applicability beyond the strict style. Additionally, the method takes advantage of the copy-and-paste and playback abilities of Encore, so it's far more useful for the composer. I can't stress too much that this method goes beyond fugal composition to any type of contrapuntal writing: Sonata themes can be subjected to the same treatment.
Today, we will continue the process for combination shifts. This will use the same method as for the horizontal-only shifts, only now at all of the intervals other than unison/octave. Just as I went through the most distant canons to the closest canons previously, I'll do the same thing now working logically through the vertical shifts in order:
V= +2/(-7), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +3/(-6), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +4/(-5), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +5/(-4), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +6/(-3), H= +4/-1, +3/-2, +2/-3, +1/-4
V= +7/(-2), H= +4/-1, +3/-2, +2/-3, +1/-4
The top line would read, then, Vertical shift of a second up, or a seventh down, with Horizontal shifts of plus four/minus one measure, plus three/minus two measures, plus two/minus three measures, and plus one/minus four measures: Simple process of elimination. Repeat this with thirds, fourths, fifths, sixths, and sevenths, and you're done.
Here's how it worked out with my, "super subject." I chose this subject to develop this process with, by the way, because it is a best-case scenario with respect to all of the combinations it makes (Or, it's a worst-case scenario with respect to the amount of labor involved! lol).
None of the plus second/minus seventh combinations worked in the first subject-subject step, so we start with plus third/minus sixth, and one measure of overlap.
Note first that here in minor, the +3/(-6) shift is to the relative major.
Note secondly that, since the subject makes a four-voice canon at the octave, that these +3/(-6) shifts could be used to double the subject in thirds or tenths above, or sixths or thirteenths below. This will be a nice orchestration possibility when I get to a fugue of that magnitude.
The closest +3/(-6) shift did not work, so we're on to the +4/(-5) shifts.
And that was the only one of those that panned out, so here we are at +5/(-4) shifts.
Note that +5/(-4) shifts are at the answer's level, and we already know that two of them work, so this is really no biggie.
This +6/(-3) shift is to the relative minor from the major mode perspective, so that's kind of interesting.
You'd have to double the leading tone to get this one to work out, but with three or more voices and an irregular resolution, it is possible. In fact, one of the things that makes freestyle counterpoint generate so many more viable possibilities than the strict style, is that you can make a lot of combinations work out with creative use of harmony.
Here we are at +7/(-2) then.
This offers some interesting modal and modulation possibilities, and that finishes up the subject-subject combination shifts.
Since there are really only two intervals different between the subject and answer, this will seem somewhat like a rehash. Again, none of the +2/(-7) shifts work.
These do have a different flavor and function though.
And again, all but the closest possible arsis
point works. Oh, by the way: Pay no attention to the page numbers. I had to break the Encore files up into smaller chunks after the fact to get them to convert to PDF reliably. Not sure what that bug is, but beyond about thirty pages the conversions are not sized properly.
The +5/(-4) shifts put the answer form on the tonic level, which is unusual, but only this one measure overlap works.
And on to the +5/(-4) shifts.
And these are quite interesting. A composer would likely never stumble upon all of these through casual experimentation, which is the point of this exercise.
And the +6/(-3) shifts.
From the minor, this combination provides a modulation from the dominant directly to the relative major. Very hip.
Now the +7/(-2) shifts.
The way this has worked out, I'll write modulating thematic phrases with the combination shifts that overlap for two measures. These are to be highly prized because it's much easier to write a modulating episode or interlude than it is to modulate with strictly thematic material.
And now for the subject-answer combinations. Here - and in the upcoming answer-subject combinations - you have to keep in mind that the subject and answer are already shifted in relation to one another,
so here the answer is a step higher than it already was, answering on the sixth instead of the fifth.
More unusual modulation possibilities.
Here's the first instance where one measure of overlap doesn't work, but two does. Some remote modulation possibilities here.
These combinations are actually like the original canon, but with the answer form following on the tonic level. No big deal, IOW.
Yadda, yadda, yadda/blah, blah, blah. lol.
A perpetually modulating canon could be made with this type of arrangement: Up a step every iteration.
These are so close to the +3/(-6) subject-subject combinations that the difference is trivial.
Still nice modulation possibilities though, and that ends the subject-answer possibilities.
And the last thing to cover in Part 2 are the answer-subject combinations.
Still, some nice subtle variations on the modulation possibilities.
But plenty of trivialities too.
Obviously, very similar to the same subject-subject and subject-answer combinations. I'll group them and compare them later.
With most subjects and answers - those not composed from the outset as canons, like this one was - the composer is looking for whatever works. With the embarrassment of riches here, I need to figure out what to discard.
I'll trash the single measure overlaps, use the two measure overlaps for modulations, and the three measure overlaps - there are only a couple so far - I'll put in places of special prominence.
This rocking combo enters on a v(m7)
sonority in the minor and a V(m7)
chord in major. Won't find those in Palestrina. lol.
Obviously, I could have skipped all of these trivial one measure overlaps if I weren't doing this as a demonstration, and just hunted down the best stuff.
This is a cool combination. After the answer on the dominant level, you get the subject on the dominant level, which confirms the modulation. Precede this with a subject statement in the tonic, and you'll have some nice modulating stretto possibilities.
Because I originally wrote this as a canon, this one works too.
This is the only non-original subject-subject combination that works with four measures of overlap, but it's not really a big deal because only the head figures differ between the subject and answer. Still, I'm glad I made myself think of this angle.
Another modulation candidate.
And one last unusual combination with three measures of overlap.
I'm keeping the systems large and not worrying about the clashes because it's easier to enter the notes on the largest size staves. When I get to the audition and construction phase, I'll reduce their size and clean them up.*****
So then, here are all of the combination shifts at the thesis
points to add to the vertical-only and horizontal-only results. Next step is to do the horizontal-only and combination shifts for the arsis
points. Off-beat entrances can be very sublime, so I'm hoping a few nice ones turn up, but I'm also hoping for less than forty-four pages of results! lol.