Surprise Developments

There are many cool and fun things about the adventure of being a composer, but one of my favorite things is how stubborn thematic material can break at any time, usually out of the blue and for no apparent reason.

Back in about 2003 I came up with an epic five-measure fugue subject that I composed as a five-part canon (This as a logical development of the five-measure subject that works as a four-part canon that the Fugal Science and Freestyle Convertible Counterpoint series are based on). For months, and then years, I tried to write a five-voice fugue out of it, but I could never get it off the ground. Couldn't even get the exposition done.

What did happen, though, is that I discovered a dovetail with an augmented form of the subject that made a five-part perpetual canon happen. You can see and hear that piece, which I realized for string choir, here.

Well, since I came up with the austere two-part style that makes stately subjects work for solo guitar, I decided to see if I could make a two-voice fugue at the octave out of that subject (Two-part invention format). "Viola."

It isn't finished yet - there is one more canonic stretto at two measures of overlap I want to present - but I got the exposition, the conclusion, and a, "magic" sequential episode together, and it makes a very concise piece just like that. That is the criteria I always use, by the way: Is it magical? If you want to be more objectivist about it: Does it seem like more than just the sum of its constituent parts?

So, as a respite from the Freestyle Convertible Counterpoint series, here's this cool little ditty.

MPEG 4 Audio: Imitation Study Number 3



The subject is, as stated, five measures long, and it manages to present eleven of the twelve pitch classes. It works as a five-part canon, but we're only dealing with two here, obviously. The Perpetual Canon uses a dovetail with the augmented head of the subject, and that's what ended up being the breakthrough here: In the lead for voice one in measures six and seven is the head of the subject; do, re, me, fa, sol, ti, le. Then, in the bass starting in measure eight, is the augmented form of the head.

This dovetails marvelously with the rising chromatic sequential episode that begins in eleven. Tres cool, non?

Fifteen through nineteen are just the contrapuntal inversion of the subject and countersubject...



... and then twenty through twenty-three constitute the inversion of the episode. Note how the straight quarter note line is an inverted palindrome from the downbeat of twenty-one to the last quarter of twenty-three: It reads upside-down from back-to-front compared with front-to-back. It is simple, but very elegant. Kind of like some of the simple musical structures that Mozart used so often.

Twenty-four, then, is the concluding stretto. As I said, I have at least one more element to come up with before I can present the intermediate stretto, but this merge works almost too well: I haven't been able to come up with any other continuation after twenty-three yet.

After the recapitulation stretto is the concluding ostinato, which I also used in the five-part canon for string choir, but here it accompanies the complete augmented head/modified tail version of the subject. Also a, "magical" element, IMO: Twice it's interrupted by the sequential episode, and then the third time it is presented in its entirety.

Allright then, back to the Freestyle Convertible Counterpoint project.

Freestyle Convertible Counterpoint, Part 2

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.

Freestyle Convertible Counterpoint, Part 1

I first read Sergi Taneiev's treatise, Convertible Counterpoint in the Strict Style, in 1987, I believe. It was a seminal moment for me, as it opened my mind up to the staggering possibilities of vertical, horizontal, and combination shifts with respect to contrapuntal combinations. I couldn't wrap my brain around the formulas, though, because like the overwhelming majority of musically minded people, I don't have any talent for math.

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.