Axial Fugue
J.S. Bach's fugue from the Tocatta and Fugue in D minor for organ inspired a set of eighteen Axial Studies for solo guitar that I wrote years ago. The way that happened was that Joseph Schillinger quoted the subject from that fugue in his System of Musical Composition as an example of a melody in which the zero axis was played, instead of being only implied. I realized that the repeated note zero axis would be a very idiomatic device for the guitar if I employed the open strings, and so I wrote those studies as a way to practice writing two voice counterpoint for the guitar. Since the zero axis can be the root, third, or fifth of the tonic major or minor triad, that gave six studies each for the open high E-string, the B-string, and the G-string. I wrote the first of those back in 1988 or 1989, and had them finished up by about 1990 or 1991. They are not immitative, but I consider them my guitarist's answer to the Two-part Inventions. They still comprise a significant part of my set to this day.
Over they years, as I have listened to Bach's original fugue more, I am beginning to thing dangerous thoughts about it. This may seem like heresy, but I don't think Bach wrote it, and if he did, I think it was originally written for the lute. The reason that it seems unlikely to me that he wrote it is that the subject's concept is very nice, but the execution is rather clumsy. As a result, it is not possible to write any countersubjects to it which have very much contrapuntal vigor. In fact, the solutions as they exist in the fugue consist of almost exclusively parallel movement in thirds and sixths or their compounds. This seems rather lame for Bach, but I'm certainly no musicologist.
I downloaded the music for it a few nights ago, and as I started to look at it closely it occured to me that transposed to the key of E minor it might transcribe nicely to the guitar. The fugue famously has the answer at a fourth above the initial statement of the subject, which is another reason I think this was originally a lute piece that Bach arranged for the organ: The open strings of the lute would make such a thing natural, and not peculiar in the least. And as I mentioned before, there is the fact that the technique used in the subject is completely idiomatic for plucked string instruments. So, I transcribed it for the guitar. All I had to do to make it "fit" was to lower the low E down to D and octave-transpose a few passages, and it really was no work at all. Piece of cake, in fact. So easy I can't believe that nobody has done this before.
Well, it's a serious virtuoso piece for solo guitar, and the clunky subject and lame counterpoint bothered me, so I just didn't think it would be a worthy investment of my time to learn it (Some of the episodes are very cool, and some of them totally bite). The problem is, the Baroque lute had, like, thirteen courses of strings, so it could play things that are just not even conceivable on the guitar. So, since I had written a "repaired" or "improved" version of the subject anyway - which allows for much more varied and interesting counterpoint - I decided to just write my own piece. It will be much more idiomatic for the guitar this way, and it will be... mine.
It's going to be a monster. Since I have done so many experiments with axes, I have a lot of ideas for it. Here's the first page:
I've seen various attempts to rationalize the answer at the fourth, but the way I'm writing this piece, it's in A minor, and the first statement of the subject is... the answer. I also changed the texture to eighth notes and the time signature to 2/4. You can see the differences between Bach's subject and mine easily enough. By using the growth series-related rotational figures I employed, I was able to write the subject so that in it's initial form, it modulates all on it's own and deposits the lowest voice on the real tonic.
That series of rotational figures also allowed for a countersubject in half notes that has a very interesting property to it: It has a very "hollow" sound. The initial intervals in evey measure are perfect - P11, P8, P5, P8, P5, P8, P11 - and they are palindromic as well. Only the very last quarter note has an imperfect interval on the attack. Some counterpoint books say you shouldn't alternate octaves and fifths, and those counterpoint books are full of shit: It's a cool sound. You just know this fugue is "going to get all medieval on yo' ass" when you hear it.
You may think this is going to be a two voice fugue. You would be wrong. It's going to be three voices, but the initial subject/countersubject combination is going to be presented in both octave inversions and both modes before the second countersubject appears (That's why I have it on two staves).
After the first two appearances of the subject (Which use the guitar's open B and E strings respectively), there is an episode of the same length as the thematic statements before the lower voice gets the subject again. It is important to note that I usually try to write subjects that are and odd number of measures in length, because the resulting fugue flows better that way. Even measure lengths for subjects often get you into that Tyrrany of Four bind, and the phrases end up sounding too regular.
The third statement at measure twenty-two does not have an open string for the zero axis, and as a result it's fairly difficult to execute. Any time the bass part moves fast on the guitar it's a problem, and this is moving quite briskly here (Quarter= 180). Both this statement and the following episode are octave-inverted from their original positions, and as a result the perfect fifths and perfect elevenths have exchanged places. This is just way freaking cool (Note: I treat the fourth as a consonance or a dissonance, depending.... just depending).
The end of the inverted first episode modulates the piece to the relative major, where I have a real cool trick waiting up my sleeve:
This is the subject, but it isn't in its original form: Originaly the fifth was the zero axis, but now it's the third that takes over that function. This not only allows me to use the open E-string again, but the first four measures now have imperfect intervals on the attacks - M9, M6, M3, m6 - before going back to the P5/P8 alternation. The countersubject is exactly the same until the end, where it turns the other direction so I can get a new episode spun out of it. That episode ends up being an intevallically inverted variant of the previous two episodes.
When the lower voice gets its statement in the relative, the subject is back in its original form, and this allows for the open G-string to be used. With that, the two-part presentation of the initial materials is completed, and so a fresh episode takes us back to the "beginning" - as it were - with the statement appearing under the B-axis. The repeated open E's above make a kind of inverted pedal, which increases the intensity level significantly after all of this time spent in two voices. Of course, they also prepare for the upcoming full three-voice statement of the subject and countersubjects one and two as well.
Countersubject two - which is the middle voice - crosses the subject a couple of times, but I was able to make it work out so that the only unison is on B, which is an open string, so it's perfectly playable (One of the reasons the guitar is in some ways better for counterpoint than a piano: Unisons). Anyway, I wrote in a concluding chord, but it's far from finished: I want to get inverted froms and use the tonic as zero too. This could end up over two-hundred measures easy.
Good Lord, it's light outside! I need to get some sleep.
I'm feeling artistic: give me that palette and I'll paint you. No, I'll paint you.
Over they years, as I have listened to Bach's original fugue more, I am beginning to thing dangerous thoughts about it. This may seem like heresy, but I don't think Bach wrote it, and if he did, I think it was originally written for the lute. The reason that it seems unlikely to me that he wrote it is that the subject's concept is very nice, but the execution is rather clumsy. As a result, it is not possible to write any countersubjects to it which have very much contrapuntal vigor. In fact, the solutions as they exist in the fugue consist of almost exclusively parallel movement in thirds and sixths or their compounds. This seems rather lame for Bach, but I'm certainly no musicologist.
I downloaded the music for it a few nights ago, and as I started to look at it closely it occured to me that transposed to the key of E minor it might transcribe nicely to the guitar. The fugue famously has the answer at a fourth above the initial statement of the subject, which is another reason I think this was originally a lute piece that Bach arranged for the organ: The open strings of the lute would make such a thing natural, and not peculiar in the least. And as I mentioned before, there is the fact that the technique used in the subject is completely idiomatic for plucked string instruments. So, I transcribed it for the guitar. All I had to do to make it "fit" was to lower the low E down to D and octave-transpose a few passages, and it really was no work at all. Piece of cake, in fact. So easy I can't believe that nobody has done this before.
Well, it's a serious virtuoso piece for solo guitar, and the clunky subject and lame counterpoint bothered me, so I just didn't think it would be a worthy investment of my time to learn it (Some of the episodes are very cool, and some of them totally bite). The problem is, the Baroque lute had, like, thirteen courses of strings, so it could play things that are just not even conceivable on the guitar. So, since I had written a "repaired" or "improved" version of the subject anyway - which allows for much more varied and interesting counterpoint - I decided to just write my own piece. It will be much more idiomatic for the guitar this way, and it will be... mine.
It's going to be a monster. Since I have done so many experiments with axes, I have a lot of ideas for it. Here's the first page:
I've seen various attempts to rationalize the answer at the fourth, but the way I'm writing this piece, it's in A minor, and the first statement of the subject is... the answer. I also changed the texture to eighth notes and the time signature to 2/4. You can see the differences between Bach's subject and mine easily enough. By using the growth series-related rotational figures I employed, I was able to write the subject so that in it's initial form, it modulates all on it's own and deposits the lowest voice on the real tonic.
That series of rotational figures also allowed for a countersubject in half notes that has a very interesting property to it: It has a very "hollow" sound. The initial intervals in evey measure are perfect - P11, P8, P5, P8, P5, P8, P11 - and they are palindromic as well. Only the very last quarter note has an imperfect interval on the attack. Some counterpoint books say you shouldn't alternate octaves and fifths, and those counterpoint books are full of shit: It's a cool sound. You just know this fugue is "going to get all medieval on yo' ass" when you hear it.
You may think this is going to be a two voice fugue. You would be wrong. It's going to be three voices, but the initial subject/countersubject combination is going to be presented in both octave inversions and both modes before the second countersubject appears (That's why I have it on two staves).
After the first two appearances of the subject (Which use the guitar's open B and E strings respectively), there is an episode of the same length as the thematic statements before the lower voice gets the subject again. It is important to note that I usually try to write subjects that are and odd number of measures in length, because the resulting fugue flows better that way. Even measure lengths for subjects often get you into that Tyrrany of Four bind, and the phrases end up sounding too regular.
The third statement at measure twenty-two does not have an open string for the zero axis, and as a result it's fairly difficult to execute. Any time the bass part moves fast on the guitar it's a problem, and this is moving quite briskly here (Quarter= 180). Both this statement and the following episode are octave-inverted from their original positions, and as a result the perfect fifths and perfect elevenths have exchanged places. This is just way freaking cool (Note: I treat the fourth as a consonance or a dissonance, depending.... just depending).
The end of the inverted first episode modulates the piece to the relative major, where I have a real cool trick waiting up my sleeve:
This is the subject, but it isn't in its original form: Originaly the fifth was the zero axis, but now it's the third that takes over that function. This not only allows me to use the open E-string again, but the first four measures now have imperfect intervals on the attacks - M9, M6, M3, m6 - before going back to the P5/P8 alternation. The countersubject is exactly the same until the end, where it turns the other direction so I can get a new episode spun out of it. That episode ends up being an intevallically inverted variant of the previous two episodes.
When the lower voice gets its statement in the relative, the subject is back in its original form, and this allows for the open G-string to be used. With that, the two-part presentation of the initial materials is completed, and so a fresh episode takes us back to the "beginning" - as it were - with the statement appearing under the B-axis. The repeated open E's above make a kind of inverted pedal, which increases the intensity level significantly after all of this time spent in two voices. Of course, they also prepare for the upcoming full three-voice statement of the subject and countersubjects one and two as well.
Countersubject two - which is the middle voice - crosses the subject a couple of times, but I was able to make it work out so that the only unison is on B, which is an open string, so it's perfectly playable (One of the reasons the guitar is in some ways better for counterpoint than a piano: Unisons). Anyway, I wrote in a concluding chord, but it's far from finished: I want to get inverted froms and use the tonic as zero too. This could end up over two-hundred measures easy.
Good Lord, it's light outside! I need to get some sleep.
I'm feeling artistic: give me that palette and I'll paint you. No, I'll paint you.
posted by Hucbald at 7:27 AM
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