Fugue Subject and Answer: This is just... weird!
After I wrote the four voice fugue for string quartet years ago, I decided the next step would be - naturally - to write a five voice fugue. I was still in the phase where my subjects had elements in common with the Art of Fugue/Musical Offering subjects, and I wanted a five voice canonic stretto, so I composed the subject below. It makes a five voice canon at one measure of delay, which you will see later. Looking at the subject, you can see that it has the leap of a diminished seventh and the decending chromatic figure in common with the Musical Offering's "Royal Theme". In the fifth measure, I used triplets to get further chromatic motion. It makes an effective, if heavy, canon.
I had a simply horrific time coming up with the answer, but a few months after writing the canon, I finally had it: by replacing the chromatic motion with diatonic motion, the answer smoothly modulates back to the tonic. The answer has a stupid wide range, but in an orchestral piece, that is of no consequence (Well, it is of some consequence, but it's easily workable). I can't think of any subject/answer combination from the historic literature where this particular device is used, so if it's not unique, it certainly is rare. I like it a lot.
What interested me more about this answer, though, is that all of it except the last measure works as a canon at one-half measure of delay (Once you teach yourself canon technique, you will forever after be looking for canonic possibilities in everything).
So, it sat. For about three or four years. Then I wrote an exposition out. Big, heavy, ponderous, but "OK". While I was working on the exposition, I noticed that the head of the answer transposed to the tonic level, and in augmentation, worked in canon with the subject at one measure of distance (Like I said, it's an addiction). But, I didn't realize the full magnitude of the possibilities.
So, it sat again. For about two years... then. Bingo! The result is flabberghasting (Well, to a canon wonk like me it is): A five voice perpetual canon at the octave!
First, here it is:
As you can see, the subject is written out in simple canon technique, and since it is five measures long, the fifth measure is all five of the subject's measures in simultanaity. Then, in measure six, the answer's head figure (Well, all but the fifth measure of it, actually) transposed to the tonic level comes in in augmentation and dovetails perfectly with the previously written canon!
Now, if you look at this in terms of modal counterpoint, you'll be forced to say "no way!", but because I also think harmonically when I write, the dissonant sonorities are perfectly rational. On the downbeat of measure nine, the harmony is simply a bVI(M7) in first inversion: Not particularly exciting... Until you consider the voicing, that is.
Note that there is a minor ninth as the top interval: This is very hotly dissonant, and when combined with the doubled third and the rest of the voicing, and absolutely gorgeous series of harmonies is set up. The second half of that measure "resolves" the dissonant tonic-function harmony to a less dissonant diminished seventh, which becomes a V(m9) momentarily, and then the next measure has a major seventh on top, a minor ninth as the second interval, and the target chord is again more dissonant than the dominant is! This is a really cool effect, and it was a happy accident, but "Luck favors the prepared", as they say.
I wonder how many of Palestrina and Bach's more riotous contrapuntal combinations were "discoveries" versus intentionally forethought? We'll never know, but the more of this I do, the more I think "one thing leads to another" in unexpected ways a lot of the time.
On the second page, I show how the canon can dovetail back - which is the last necessary expositional requirement - and the last measure of page two is all five measures of the subject again: The perpetual nature of the canon is proven, and so it can come to an end.
On the last page, I let the augmented answer come in again - because those harmonies are certainly worth hearing one more time - and fade it out over an ostinato of the subject's final measure in the bass. When the answer statements reach the tonic, they then sustain to the end. In the final measure, I resolve to a major tonic by using a tierce de Picardie momentarily, and then use the triplet figure to get that voice too, down to the tonic. There is more than the touch of humor in this figure: The B-flat is the only tone not used in the canon, so it completes the set of twelve (Which is also humorous, of course, so it's "a riddle, wrapped in a mystery, surrounded by an enigma" in a funny kind of way).
On five voice harmony and counterpoint: Have you ever read something like "and composers relaxed certain rules when writing for more than four voices" in a counterpoint or harmony text? Doesn't that chap your cheeks? I mean, which rules did they relax? How did they relax them, and what rationale did they apply? I can only assume that the theorists couldn't answer these questions, because they never do (That I know of).
Well, here's one rationale that should help. In four voices, the largest complete sonority that can be expressed is a seventh chord, while in five voices, it is a ninth chord. In four voices, the next smaller chord is considered the perfect expression of the tonic degrees: The triads. The outer limits of the triads are the fifths, and unequal fifths can move in parallel in four voices (I use them in three voices, but I'm a Philistine).
In five voices, the most perfect expression of the tonic degrees are considered to be the seventh chords (Though this approach was not taken in common practice music - for the tonic especially - it is nonetheless undeniably true according to the implications of the overtone series, and I think "jazz major", "jazz minor", and da blues prove that well enough), so unequal sevenths can also move in parallel in five voices.
I use that in the fadeout of the canon where the bass has F, there is an E in the chord, and they move together to E and D respectively: The D is the seventh of the E dominant chord, and so is completely consonant in five voices. Sevenths must however be treated as are perfect consonances: Only parallel unequals are acceptable (In counterpoint: Pure Harmony using voice transformations is another matter entirely).
I'm not sure if I'll actually ever do anything with this, but I sure learned a lot from the experience. As usual, there will be PDF and MIDI files of this on my .Mac FileShare page as WIP_PC.pdf/.mid for those interested.
"What are you looking at?!"
UPDATE: Concerning the four versus five voice counterpoint/harmony "thing": I failed to make clear that the reason triads are the perfect expression of completeness in four voices and that seventh chords are the perfect expression of completeness in five voices is because of the harmonic principle of root doubling. As you were.
I had a simply horrific time coming up with the answer, but a few months after writing the canon, I finally had it: by replacing the chromatic motion with diatonic motion, the answer smoothly modulates back to the tonic. The answer has a stupid wide range, but in an orchestral piece, that is of no consequence (Well, it is of some consequence, but it's easily workable). I can't think of any subject/answer combination from the historic literature where this particular device is used, so if it's not unique, it certainly is rare. I like it a lot.
What interested me more about this answer, though, is that all of it except the last measure works as a canon at one-half measure of delay (Once you teach yourself canon technique, you will forever after be looking for canonic possibilities in everything).
So, it sat. For about three or four years. Then I wrote an exposition out. Big, heavy, ponderous, but "OK". While I was working on the exposition, I noticed that the head of the answer transposed to the tonic level, and in augmentation, worked in canon with the subject at one measure of distance (Like I said, it's an addiction). But, I didn't realize the full magnitude of the possibilities.
So, it sat again. For about two years... then. Bingo! The result is flabberghasting (Well, to a canon wonk like me it is): A five voice perpetual canon at the octave!
First, here it is:
As you can see, the subject is written out in simple canon technique, and since it is five measures long, the fifth measure is all five of the subject's measures in simultanaity. Then, in measure six, the answer's head figure (Well, all but the fifth measure of it, actually) transposed to the tonic level comes in in augmentation and dovetails perfectly with the previously written canon!
Now, if you look at this in terms of modal counterpoint, you'll be forced to say "no way!", but because I also think harmonically when I write, the dissonant sonorities are perfectly rational. On the downbeat of measure nine, the harmony is simply a bVI(M7) in first inversion: Not particularly exciting... Until you consider the voicing, that is.
Note that there is a minor ninth as the top interval: This is very hotly dissonant, and when combined with the doubled third and the rest of the voicing, and absolutely gorgeous series of harmonies is set up. The second half of that measure "resolves" the dissonant tonic-function harmony to a less dissonant diminished seventh, which becomes a V(m9) momentarily, and then the next measure has a major seventh on top, a minor ninth as the second interval, and the target chord is again more dissonant than the dominant is! This is a really cool effect, and it was a happy accident, but "Luck favors the prepared", as they say.
I wonder how many of Palestrina and Bach's more riotous contrapuntal combinations were "discoveries" versus intentionally forethought? We'll never know, but the more of this I do, the more I think "one thing leads to another" in unexpected ways a lot of the time.
On the second page, I show how the canon can dovetail back - which is the last necessary expositional requirement - and the last measure of page two is all five measures of the subject again: The perpetual nature of the canon is proven, and so it can come to an end.
On the last page, I let the augmented answer come in again - because those harmonies are certainly worth hearing one more time - and fade it out over an ostinato of the subject's final measure in the bass. When the answer statements reach the tonic, they then sustain to the end. In the final measure, I resolve to a major tonic by using a tierce de Picardie momentarily, and then use the triplet figure to get that voice too, down to the tonic. There is more than the touch of humor in this figure: The B-flat is the only tone not used in the canon, so it completes the set of twelve (Which is also humorous, of course, so it's "a riddle, wrapped in a mystery, surrounded by an enigma" in a funny kind of way).
On five voice harmony and counterpoint: Have you ever read something like "and composers relaxed certain rules when writing for more than four voices" in a counterpoint or harmony text? Doesn't that chap your cheeks? I mean, which rules did they relax? How did they relax them, and what rationale did they apply? I can only assume that the theorists couldn't answer these questions, because they never do (That I know of).
Well, here's one rationale that should help. In four voices, the largest complete sonority that can be expressed is a seventh chord, while in five voices, it is a ninth chord. In four voices, the next smaller chord is considered the perfect expression of the tonic degrees: The triads. The outer limits of the triads are the fifths, and unequal fifths can move in parallel in four voices (I use them in three voices, but I'm a Philistine).
In five voices, the most perfect expression of the tonic degrees are considered to be the seventh chords (Though this approach was not taken in common practice music - for the tonic especially - it is nonetheless undeniably true according to the implications of the overtone series, and I think "jazz major", "jazz minor", and da blues prove that well enough), so unequal sevenths can also move in parallel in five voices.
I use that in the fadeout of the canon where the bass has F, there is an E in the chord, and they move together to E and D respectively: The D is the seventh of the E dominant chord, and so is completely consonant in five voices. Sevenths must however be treated as are perfect consonances: Only parallel unequals are acceptable (In counterpoint: Pure Harmony using voice transformations is another matter entirely).
I'm not sure if I'll actually ever do anything with this, but I sure learned a lot from the experience. As usual, there will be PDF and MIDI files of this on my .Mac FileShare page as WIP_PC.pdf/.mid for those interested.
"What are you looking at?!"
UPDATE: Concerning the four versus five voice counterpoint/harmony "thing": I failed to make clear that the reason triads are the perfect expression of completeness in four voices and that seventh chords are the perfect expression of completeness in five voices is because of the harmonic principle of root doubling. As you were.
posted by Hucbald at 5:06 AM
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