The Extempore in A minor is all polished up now, and I discovered some interesting things as I analyzed it. What I found lead me to make a relatively minor modification to it that added much more than the sum of the changes would suggest. Before I can explain what I did, however, I'm going to have to lead you quickly back through the piece.
For those not familiar with this from earlier postings, I'll offer a scant review. The piece uses the fugal process but boiled down to it's irreducible essence: Instead of a proper subject, it has the tiny do, ti, do
motif that it starts out with. It is set up with immitative entrances on the tonic and dominant levels, so it has an exposition, but it's a very short one.
Without any stretto possibilities, suspension chains, or other common fugal techniques to work with, I employed a free fractal technique based on the diminution of the motif that naturally appeared in measure four. All of the elements in the rest of the piece are either this motif, or they are related to it as spin-offs in the counterpoint. All of the variations of the motif and it's contrapuntal spin-offs were recombined to create the body of the piece, which has a very well detailed and highly adventurous modulation scheme: The piece goes to regions that are far more remote than any regular fugue would.
If you want to see a motivic analysis, just scroll down a few posts and you'll find it.
What I did this morning was to analyze the keys that the piece went through to wrap my brain around the internal relationships. On the first page here, you can see that after starting out on the tonic minor, the first fractal episode of 2.5 measures length modulates to the subdominant minor region
in the middle of measure eight. The progression in harmonic terms is Progressive
(Root progresses by falling fifth or rising fourth), hence the large "P"
The following 2.5 measures are a repeat of the previous phrase on the new tonal level (But in three voices), so it naturally progresses to the minor subtonic region
at measure eleven. The new phrase here is only two measures in length, but it also moves progressively to the major mediant region
(The relative major key) at measure thirteen. To this point the piece has been accumulating flats, but now with the relative major region, both of the modes of the natural key signature have been traversed: The whole modulatory scheme is organized on getting the relative major and minor modes of each key signature "taken care of": I'll put a breakdown of how it pans out later.
Aside: I was just interrupted my my next door neighbor, a very sweet little old lady, who brought pumpkin bread! YUMMY!
I love living here. Anyway...
Measure fifteen hits the major submediant region
, but it immediately returns to the tonic in the middle of the measure to set up the rest of the invertible counterpoint episode. This movement from bVI
is a Mild Ascending
progression in terms of root motion, so that is the reason for the "M+"
. The rest of the top system retraces the previous steps (Same keys as the beginning of the piece, but gone through more quickly) - again through progressive modulations - before arriving at the tonic yet again in measure nineteen: This repeat of the first regions solidifies the piece and makes the following launch to the edges of the stratosphere more followable. Note that the mild ascending
motion of measure fifteen is answered - or rather balanced out - with a mild decending
motion into nineteen: These kinds of meta-patterns in modulation schemes seem to me to add a lot to the cohesiveness of the whole of a piece.
The modulation scheme starting in measure nineteen is the exact mirror image of the earlier episodes: Here we have Retrogressive
modulatory motions, and are accumulating sharps instead of flats. Note, however, that it is still the minor genders of the regions that this episode is dealing with. By the end of this passage at measure twenty-four, five sharps
have added up: Far more remote an area than you would see in a traditional fugue (I'm going to go through this quickly now, and dig into the details of the regional relationships later, so hang on).
At twenty-four a chromatically modulating episode begins that traverses the tonic again. All of the chromatic motions are ascending, therefore the "C+"
. Measures twenty-seven and twenty-eight are variations of seventeen and eighteen, and land the piece back on the realtive where the episode of measure nineteen resumes, but now major gender modes are traversed.
This go-round both the retrogressively moving episode and the chromatically moving episode are foreshortened, and the supertonic major region
is arrived at in measure thirty-four. Measure thirty-four is a varied version of measure twenty-seven, and it also targets the dominant tonal level. Though it appears to suggest that the dominant minor region
will appear, it is in fact the dominant major region
which appears in measure thirty five. One reason for this is that I wanted C-natural as the highest note in the piece: It continues the chromatic line that started at G-sharp back in measure thirty one, and it's also the highest note in the final fugue of Sonata Zero
The reason I changed the dominant pedal episode to begin in E major instead of E minor, however, goes back to "taking care of" all of the key signatures. The rest of the piece explains itself, so let's get to a better look at the regional relationships.
I find it useful to put analyses out of the context of the musical score so that they can bee seen all at once: It's much easier to see patterns and relationships that way. Here is the piece without the music:
The numbers at the left are the numbers of the keys as they make their appearances; then those keys are defined, their key signatures are indicated, and the gender of the mode and it's place in the duet of relatives is indicated. So, for example, 01) is the first key to appear, it is the tonic minor, it has a natural key signature, and the minor mode (Small "m") is the first of the natural keys to appear.
When key 04) appears, it is the mediant major (relative), and it also has a natural key signature, so it is the major gender mode of that signature, it appears second, and it completes the duet of keys for that signature. Out of the first five keys to appear, four are relative major/minor pairs that are completed: Only the subtonic minor is "left hanging" without it's mate up to this point. Then, as I mentioned previously, the first four keys are revisited: The (C) indicates the key pair was completed previously, and the (1) indicates that it is a member of an incomplete pair that has previously appeared once.
After the third appearance of the tonic minor, the episode of accumulating sharps begins, and after the tonic's fourth appearance, the subtonic minor, 03), finally get's it's mate with the leaning tone major. As you can see, there are eight total key signatures and sixteen total keys: All pairs are complete.
But, it wasn't that way when I started this analysis: Where 16) appears - the dominant major - I originally had another appearance of the dominant minor (This is the pedal point episode). When I realized it was the only incomplete duet, I decided to try changing it, and BAM!
(As Emeril would say): The missing ingredient was finally found. This change is the thing that adds more than the sum of it's parts.
Since the pedal point episode is closely related to the corresponding episode in the concluding fugue of the sonata, it now works better as a contrast
, and that this episode has a lot more sharps to "shed" in it's modulation is also much more fitting considering how far afield the piece ventures. It's just oodles and gobs better, if you'll pardon the technical terminology.
I don't consider a piece finished anymore until I've analyzed it like this, and I didn't used to retroactively analyze anything at all (For many years). I believe the longer and/or deeper a piece is, the more important it is to do this.
The updated versions of the PDF and MIDI files are now on my FileShare page
Merry Christmas, everybody!