There is a reason for this, by the way: The musical mind works in terms of sound and sight - this is why composers need to see the music on paper - and numbers are mute and invisible: If I can't hear it and see it in my mind's ear and eye, it is not possible for me to manipulate it in any way. As I said, this is the case for most musicians, with Taneiev being a spectacular exception.
Nonetheless, I attempted to go through the treatise again back in 2005 when I started this weblog - links in the sidebar - but I abandoned the project when I realized that the solution for me - and those musicians like me - was to simply do the calculations mechanically (While my math aptitude scores are at a dismal 42nd percentile, my abstract reasoning and mechanical reasoning scores are in the top one percent, so I can rock with this approach).
The other problem with Convertible Counterpoint is, of course, that it only applies to the strict style. Unless you want to sound like Palestrina then, it's truly useless, even if you can wrap your brain around the formulas.
Once I had the idea for a mechanical approach, the only thing left to do was to develop the methodology, and of course that was a monumental organizational challenge, and composers tend to be better at organizing things than most. So, here we are.
What I did was to break the process down into tasks that were logically ordered, and that covered the most valuable and easy to employ combinations first, and saved the more esoteric devices for later. The process itself establishes a pattern at the outset with the more obvious possibilities, and that pattern is repeated with the latter and more remote concepts. Also, I organized things to take advantage of Encore's copy-and-paste abilities, which makes this monumental task quite quick and painless.
Without further ado then.
Step 1 is to write out the subject and answer - or countersubject, or counter-answer, or sonata theme, &c. - in both modes and in all four orientations. Note that with tonal answers on the dominant level, inversion makes them go to the subdominant. I shall exploit this. lol.
Some may wish to omit the retrograde forms, but I find that working with them is great mind exercise.
Simplest of all combinations are the pro forma vertical-only shifts, because there are only two possibilities, and they always work: Lines can be doubled at thirds above/sixths below, or vice versa.
To take advantage of Encore's copy-and-paste capabilities, you should always make the thematic elements Voice 1, and the shifted elements Voice 2, whether the shifted elements are above or below. That way, you can just solo Voice 1 and copy the four versions for the next operation. Needless to say, a massive time-saver which makes this epic project much more bearable.
This now explains itself.
After the vertical-only shifts have been done, the next step is the horizontal-only shifts, and not just any horizontal-only shifts, but those that begin on the strong beat (The beginning of the measure in 2/2 here). Those are called the thesis points, versus those starting in the middle of the measure, which are the arsis points (Yes, I'm aware those terms have been reversed in the past, but this is the original - and technically correct - way these terms were employed).
Since I composed this subject as a four-voice canon, all of the thesis points work, and there are two ways of looking at the shift: How many measures from the beginning, and how many measures from the end (Or, how many measures overlap). With a five measure subject, the absolute value will always equal five.
And again, things become self explanatory.
I also note potential traps, as I've done here. The minor version of the canon uses unequal parallel fourths and fifths to maintain invertibility, but in the major mode the fourths are both perfect, so inverting them would give parallel fifths.
And so here is the closest canon. Of course, you can also solo the parts and listen to the combinations.
The tonal answer also works at every thesis point, but with caveats.
What you have to do at this point is to disregard any conflicts of mode, key, or accidentals - those things can be resolved when you get ready to employ the combinations - and just look at what I call the absolute value of the counterpoint. Some of these combinations may have to be, "modalized" to work, and those modal resultants can be spectacularly colorful (They can also suck royally. lol). In any event, don't get bogged down with any consideration that you don't have to.
Things again become obvious.
When different elements are combined, the possibilities usually drop dramatically. In fact, this fugue subject and answer are "insanely great" as Steve would have said, lending themselves, as they do, to so very many combinations. But, for subject-answer combinations here, only two can be made to work.
And for the answer-subject combinations, only one will work.
So, here are all of the possibilities for vertical-only and horizontal-only shifts at the thesis points for this subject and its tonal answer. At this point there is a choice as to how to proceed: One can either repeat the process for horizontal-only shifts at the arsis points - which would certainly yield very limited results, if any - or proceed to combination shifts at the thesis points. To me, it seems more logical to go ahead with the combination shifts at the thesis points, as that will finish those up entirely, and then go on to the horizontal-only and combination arsis shifts, so that they will also be grouped together.
Part 2 then will be the combination vertical-horizontal shifts.
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