Sonata One in E Minor IV: Axial Fugue in E Minor
If I was to go strictly by the fugal categorizations as I learned them from analyzing J.S. Bach's music, this movement would have to be called a ricercare if for no other reason than the number and remoteness of the key regions it traverses. But there is much more that is unique to this piece, as it is a combination of fugal and sonata processes, it uses vertical-shifting counterpoint a la Sergi Taneiev, and the subject-answer theme appears in several modified forms. No other composer has ever used this subject modification scheme before, though Palestrina did present modal variants of some of his thematic material in a more casual way. I chose to call it a fugue because the constructivist approach I took to it is so rigorous.
Since the subject uses a zero axis that is played - and that zero axis can be the root, third, or fifth of a tonic major or minor triad - there are six rectus forms available for each open string of the guitar that is used as a zero axis: The high E, the B, and the G strings, respectively. The inversus forms only work with the root as the zero axis, so the low E and A strings offer an additional four possibilities. That makes for a total of twenty-two (!) possible permutations of the subject/answer theme. Obviously, presenting all of these would lead to a gargantuan and unwieldy work, so I had to come up with an organizational strategy that would present only the best possibilities in a logical order.
Obviously, the subject here bears more than a passing resemblance to the subject from the Organ Fugue in D Minor from the Toccata and Fugue in D Minor which has long been attributed to Bach, but which I'm 100% certain Bach did not compose. The Wikipedia entry on that work is a good place to start, if you want to research this for yourself. As for myself, I transcribed that fugue for the guitar, and I was amazed by how well it sits on the instrument in E minor: The answer at the fourth above uses the open B and E strings, just as I do here. The only problem is that of range in the bass, as a seventh string would be necessary - a low B - to get the organ transcription onto the guitar. I say "organ transcription" because I'm also 100% certain that the D Minor Organ Fugue was originally a piece for Baroque lute, which had many more courses than the guitar has strings. Not only that, but there is no counterpoint in the organ fugue, as the counter-subject, such as it is, merely doubles the melodic trajectory of the subject or answer in thirds or sixths. bach would never have done that, but this is exactly the sort of thing you'd expect of a lute virtuoso who had a shaky understanding of counterpoint, and who wanted to simply indulge in idiomatic fugue-like playing. As for the episodic material, I'm not sure: The provenance of the piece is via one of Bach's students who had a less than honorable reputation, so almost anything is possible.
In any event, doing that transcription exercise gave me the idea for this piece, as the idea is a good one, but the execution of that idea was very ham-fisted and lame. Yet another reason I'm sure Bach is not the author of that "fugue." Convinced I could do much, much better justice to the zero axis fugue theme concept, I looked through the Six Studies on an E Axis that I wrote back in 1987 to get some ideas. While doing that, I noticed that the melody and bass line of E-Axis Study No. 2 in A Minor contained nothing but contrary and oblique motion: If there is nothing but contrary and oblique motion between a melody and it's counterpoint, then both trajectories can be inverted at the octave, and they can be doubled in thirds or sixths! This is Sergi Taneiev's Convertible Counterpoint. So I was in business.
Here is an MP3 of me playing the E-Axis Study in A Minor.
The score is below.
As you can see, the zero axis E here is functioning as the fifth of the tonic minor triad, and there is nothing but contrary and oblique motion between the trajectory of the melody and the trajectory of the bass line, so either can be doubled in thirds and all resulting contrapuntal relationships will be technically correct. This is exactly what became the answer and counter-answer in the final fugal exposition, as you'll see. The problem is, using those doublings is not technically executable on the guitar in the key of A minor, but it is in the key of A major! This set up the fugue as a battle between minor and major, with major winning out at the end.
Here's page two of the study so you can follow the whole thing as you listen if you wish.
What was the melody and bass line of the study became the answer and counter-answer of the fugue, as I said.
Here is a MIDI to MP3 conversion I made in iTunes of the Axial Fugue in E Minor that you can play while reading the score.
So, the subject starts out using the open B string as the fifth of the tonic E minor triad on the top system - I used a system for a solo guitar piece because the music is far to much for a single stave, obviously - and that subject is 7.25 measures in length. As I've said before, odd and fractional bar lengths for fugue subjects are highly desirable.
The answer then comes in on the last eighth of measure seven over the desirable "dissonant fourth," and notice how the sixteenth ornamentation makes the previous zero axis on B "disappear" into the new melodic trajectory: I am very scrupulous and exacting in how I handle the axes and melodic trajectories in this fugue. Likewise, the melodic trajectory of the subject goes smoothly into the counter-answer from measure seven to measure eight.
On the third system down is a brief episode that I call a "release area" that sets up an inverted statement, still in A minor, on the fourth system down. Note how the melodic trajectory on the top staff in the release area and the bass line converge on the note A in measure nineteen, and the zero axis of E falls smoothly into the counter-answer of the inverted statement as well: No loose ends.
This inverted statement is one of the most difficult sections to perform in the fugue, because the E axis here is not an open string on the guitar. I toyed with the idea of using a scordatura tuning with the D raised to E, but that screws up some of the following material, so I'll just have to grin and bear it. In any case, this statement is positively required to get the structural architecture of the piece off the ground.
Starting in measure 27 there is the first of many sequential episodes that modulates the piece to C major.
Now, the open E string zero axis is functioning as the major third of the tonic C major triad: See how cool this device is? I'm able to get many modulations and still use the open strings of the guitar, which is the only thing that makes this fugue possible: It is quite idiomatic, for the most part. This is the subject form of the theme, by the way: The subject's trajectory descends at the end, while the counter-subject's trajectory rises at the end (The opposite is the case with the counter-subject and counter-answer, obviously). Again, there are no loose ends with the trajectories: The G above and the E below in measure forty converge on F-natural in forty-one, and this continues throughout the episode's sequences.
The second system, then, is another sequential episode that is organically spun out from the tail of the subject/counter-subject combination. This one doesn't modulate, however, it only changes mode genders by introducing the E-flat in measure forty-seven.
On the third system is another inverted statement, only this one is is the key of C minor, the G axis is an open string, and it is functioning as the fifth of the tonic triad again. In 55 is a new kind of episode, which leads to the counter-exposition. So, we've already been through the keys of E minor, A minor, C major, and C minor, and we're just getting out of the exposition (Remember, this is a mono-thematic sonata process exposition, and not a strictly fugal one). Note that this episode's ending sounds rather "incomplete" ending on the C in the bass as it does: This is intended, and I'll "fix it" next time we hear this episode, which will be leading into the recapitulation.
OUr counter-exposition begins in measure sixty-one, and we're back in E minor. The subject is exactly like it was at the beginning, but now the main counter-subject is below it, and it has a drone above. Since both the zero axis B and the drone E are open strings, this is not overly difficult.
Whereas the exposition's answer was two voices, here in the counter-exposition it is three voices. Though counter-answer two, the new element, crosses the melodic trajectory of the answer, it effectively doubles it in thirds, which is part of the progressive uncovering of the "original combination" which will appear in the recap.
The 'release area" episode is the same as before, only now I introduce a C-sharp at the end to make the next inverted statement in A major, versus the previous A minor. These sharps will help to affect the next modulation to C-sharp minor, as you can see in the sequential episode that begins at measure eighty-seven.
So the bottom system here is exactly the same as the C major statement in the same place from the exposition, but now we're in C-sharp minor with the zero axis open E string functioning as the minor third of the tonic triad. This is really, really cool, if I do say so myself.
This means that the following sequential episode has a sharp to shed, and so it is more interesting as well. At the end of this episode, I use some chromatic motion in the bass to lead into an entirely new element, which is a longer episode based upon an ascending chromatic bass line. The C-sharp in measure 107 appears to be "left hanging" - the first instance of this in the fugue - but it is all part of the plan of the episode. That C-sharp finds it's home as the D descends to C natural at the very end of the section in measures 124 and 125.
On the bottom system is the inverted statement in C again, but this time it's in C major instead of C minor, and the counter-answer is doubled in thirds for the first time.
That then leads to a gnarly episode with thirds in the lead, and that returns us to A major for the answer section in the major mode with the counter-answer below doubled in thirds. I needed a quarter rest in 143 because a unison is not physically possible there: I needed to re-attack the same E as is in the bass part. By the way, this is hard as hell to play.
At 147 the release area reappears, only this time in the major mode, and that leads to the first of the inversus statements, and it is also the first time that the zero axis open A is functioning as the root of the tonic triad, while the inverted counter-subject is doubled in thirds. This isn't exactly easy to execute either, but you ain't seen nothin' yet.
The inverted sequential episode is really weird and wonderful: The converging trajectories make it triad, diad, monad every iteration, and the whole of the thing modulates us back to E minor for the development sections. Yeah, it's a bitch to play.
Here in the development we start out with an answer-form variant on the original pitch level of E with the countersubject in the lead and a pedal point below. The only things that make this possible are all of the open strings involved.
At 173 is a new form of the release area, and 177 is the answer and counter-answer two over an alternating pedal point. Both the A and E are open strings, so this isn't that difficult to play, actually.
This leads to what at first sounds like the original release area episode, but this modulates to G major at the end in a startling way. Note the chromatic line from F-natural to E, and then D-sharp on the top staff which becomes D-natural on the bottom staff. I'll reverse the D-natural and D-sharp to get to G-sharp minor next time. Don't remember how I thought of this, but it's a nice effect.
The statement in G major is pretty tricky, but is is possible to keep the high G drone going with some fancy finger-work. note here that the open B zero axis is functioning as the major third of the tonic triad.
This sequential episode - notice how they are all the same and yet different: I like fractal self-similarity principles, and use them to get unity in variety throughout this piece - leads us back to E minor, and the first of two episodes based upon an ascending chromatic line in the bass that are the central pillars that this piece balances on.
The end of that episode introduces a sixteenth note run that is ridiculously difficult to execute with all of the other stuff going on, and we are back in E minor. I'll change that sixteenth note figure to lead into E-flat major next time.
From 216 into 217 is the only illegal intervallic sequence in the piece. Between the F-sharp in the bass and the G-natural above it in 216 is a minor ninth. That moves in stepwise parallel motion into a major ninth between the low E and F-sharp above in 217. Since the parallel ninths are unequal and the G is the beginning of the establishment of a new zero axis, I decided to allow myself this license. Plus, it doesn't sound in any way bad or wrong. It's one of those things that works, despite being technically illegal.
Our E minor statement at 217 is new, as the open G is functioning as the minor third of the tonic here, the open B above that is a drone, and the counter-subject is in the lead. All of that over the open low E string's pedal point. Yeah, it's a nightmare, and using the c finger of the right hand is positively required.
As a result, final sequential episode on the page is the most vigorous one yet, and this leads to the exuberance climax on the next page.
The second half of the development starts in the major mode, and now counter-answer two is added. Then, the newer release area episode is just a major mode variant.
OK, now for the third system. Stay with me here. We're in A major now, and we have the answer, counter-answer one as the top line of the lower stave, counter-answer two adding the thirds to the answer's trajectory, and a syncopated E pedal point in the bass. This is not impossible, but it is very difficult, and I actually anticipate using the bass player's technique of thumb slaps on that low E. If I can ever pull it off, it will be totally awesome, dude. LOL!
At 249 is the modulatory release area again, and as promised, I reverse the D-sharp and D-natural from before to modulate to G-sharp minor here.
Holding the G-sharp as a drone is at the edge of impossibility, so I made them eighth notes this time. I may do that with the earlier G's as well; we'll see when I start learning it... if that day ever comes. ;^)
We have a bunch of sharps to shed in this sequential episode to get us back to E minor for the upcoming chromatic bass line episode, so this one is more interesting, as it should be later in the work, and notice the F-natural at the very end of it in 265: This was an F-sharp last time, so now we have an augmented sixth with the D-sharp above. This is a setup for the modulation to E-flat major at the end of this next episode.
The second of our "pillars" is then exactly the same until the last measure where I change the sixteenth note run by introducing F-natural, A-flat, and B-flat in the run-up to E-flat in the lead, and the F-natural in the bass again. Since the guitar doesn't go down to E-flat, I was able to avoid the parallel ninths this time by simply having the bass drop out. LOL! That F-natural finally finds a home on the next page, however, when the low E picks it back up.
Since the B-flat drone is not an open string, obviously, I made the upper voices eighth notes for the E-flat major statement. Between this and the radically fast modulation down a semitone, this sounds "wicked pissah," as we used to say in Boston.
Finally, we get the second appearance of the episode that ended the exposition, but now it's completed and makes a proper modulation at the end. The high C-sharp in measure 294 is also the pitch climax of the piece: That's at the 73% point, which is virtually perfect for a piece based on the answering interval of a perfect fourth (a ratio of 4:3, i.e. 3÷4= .75).
So, here's our recapitulation, and now we are in E major. On the top system we have, top to bottom, a drone on the open high E string, the major mode variant of the subject with the open B zero axis functioning as the fifth of the tonic major triad, counter-subject two doubling the trajectory of the subject in thirds, and counter-subject one in the bass.
Then, the second system has the "original combination" in all of its glory: The answer using the open high E as the zero axis, which is the fifth of the tonic, of course, then counter-answer two doubling the answer's trajectory in thirds, and finally counter-answer one doubled in thirds. This is what I wrote first, and it's only executable in A major: The entire fugue is based on the possibilities of this combination. This is so highly virtuosic, I don't know if I'll live long enough to learn it. I'm not kidding. It would take someone like Kazuhito Yamashita to pull this off with the bravura I envision.
I then use the release area, which is stupid-simple after what just went before, to turn around to E minor again and the first of the inversus statements using the open low E string as the zero axis. It is again the root, of course.
At 320 is and inverted from of the release area episode, and that leads to...
... the first statement of the rectus form of the subject that uses the high E zero axis as the root of the tonic triad: Saving the best for last, as usual. A variant of counter-subject two is also present, but without the thirds in the bass this is much, much easier to play.
At 331 I introduce the ending episode, and it sounds like the minor mode is going to hijack the piece and end it early, but E major interrupts at the last possible moment in 342, and we get the major mode version of the inversus and its inverted release area episode.
As I said, save the best for last: These major mode variants using the zero axis as root work much better than their minor mode counterparts, which is the whole point.
At 361 the final episode based on a chromatic bass line interrupts the action, but this one is based upon a descending chromatic bass line, as is apropos nearing the end of the piece, and it balances out the one in the counter-exposition perfectly. Notice that I let placement of the sixteenth note ornaments from the subject clash with those in the episode this time: I've scrupulously avoided that up to now, because it is a really wild effect I wanted to save for the end.
The ending episode then returns at 373, but this time the bass gives mi, re, do after the descending chromatic tetrachord. Since the chromatic tetrachord implies minor and the bass lick implies major, this is the final clash of wills between minor and major.
After the point where major interrupted previously, minor finally surrenders as the chromatic line becomes diatonic and doubled in thirds on the second system. The final triumphant statement is the trajectory of the subject and the counter-subject in augmentation harmonized using all the strings of the guitar possible: Six voices except for measures 393 and 394, which are five voices. Ta da!
Probably needless to say, this piece is at the very bottom of my to-do list. LOL!
Of course, the series must end with a redhead.
That's a mirror, she's not twins. Otherwise, yeah, twin redheads would pretty much be my ultimate fantasy. LOL!