How to Compose Counterpoint (Imitation and Modulation 2)
In part one of Imitation and Modulation we looked at a strict fugue, which is technically limited to modulations to the close keys - those differing from the tonic by one accidental in either the sharp or the flat direction. In that fugue were examples of normal modulations achieved by introducing a new dominant harmony, modulations achieved by deceptive motions from a dominant, and a modulation achieved through a sequential homophonic episode.
Today we will look at a ricercare, which is a looser fugal idiom that is not limited as to which regions it can modulate to. While the previous Fugue was the finale of Sonata Zero, this Ricercare is the middle movement (of five).
The key is C major and the subject is just an ascending form of the same subject used for the fugue. This creates a very interesting exposition, as the subject begins on the subdominant degree, and while it is at the very bottom of the guitar's range, it is actually the middle voice that speaks first. The answer, then, is a perfect fifth higher than the previous statement of the subject, even though it is now the lowest voice that states the thematic element.
Since the subject started out on the subdominant and its tail figure tonicized the tonic, the answer starts on the tonic and tonicizes the dominant. This allows the final statement of the subject to be presented over a vi triad, and then all three voices can move in parallel into the vii(d5) due to the unequal fifths involved. Note that the last statement of the subject is a full two octaves higher than the original statement. The original entering middle voice also has the range of a full two octaves in the exposition. These features are so unusual that they very well may be unique.
At the end of the exposition the harmony arrives at the tonic in thirteen. This modulatory episode is yet a third variation on the two heard in the fugue, and it brings the piece to a D(m7) chord, which will take the piece to the dominant region.
The incongruous first system is actually a major key restatement of the exposition of the first movement Extempore in A Minor (I composed this piece last of the imitative trio), and it ends on an implied I(6/3) of the new dominant level major key at measure twenty. I did this so that I could actually start out with the answer in this new key area - which presents an ascending series of 4-3 suspensions - and that leads to a tonicization of "the dominant of the dominant" at twenty-four, where an inverted form of the subject (the actual original fugue answer, to be precise) - along with another 4-3 suspension chain - brings the piece back to the dominant of the key of G at the end of twenty-seven.
Into the second modulatory episode I use a deceptive motion to an E minor chord, the vi of the key here, and this is yet a fourth variation on that original episode I came up with for the fugue. Notice how this time the sequence returns to the tonic of the moment, G major, at the beginning of measure thirty, and ends with an F#(m7) at the end: We're going to the key of B minor, which is the leading tone minor of the original key of C major. This is something you wouldn't want to do in a strict fugue, but in a ricercare just about anything goes.
Now in B minor, we get the answer along with 2-3 suspension resolutions, and it is the answer, so into thirty-five, F-sharp minor appears to be tonicized. However, since I use the answer form of the original fugue subject there, this is actually a modulation: We are now three sharps away from where we started - quite remote! The rest of this set stays in F-sharp minor to present the rectus and inversus forms of the 7-6 and 2-3 suspension-resolution chains. In fact, the last three systems here are just a half step lower and in the minor modal gender compared to the corresponding section of the fugue! That means, of course, that the final sonority at the end of measure forty-six is a C-sharp dominant chord. What to do with this?! What to do, what to do...
Hey, I got an idea: Why not treat that C-sharp overtone sonority as a so-called German Augmented Sixth chord (A subV7/I in jazz-speak, and a V(d5m7m9/0)/I in my modern terminology)? Then, we can just arrive at the original tonic! So, that's just what I did.
The sequential harmonic episode is the ricercare's subject (answer, actually) in augmentation, and therefore it arrives at a dominant-level triad.
Any time you want to modulate and you arrive an a dominant, there are eight resolutional possibilities: 1) Treat it as a dominant and resolve to normally to a new major tonic, 2) Treat it as a dominant and resolve it normally to a new minor tonic, 3) Treat it as a dominant and resolve it deceptively up by semitone to a major tonic, 4) Treat it as a dominant and resolve it deceptively up by whole tone to a new minor tonic, 5) Treat it as a dominant and resolve it deceptively up by semitone to a new minor tonic, 6) Treat it as a dominant and resolve it deceptively up by whole tone to a new major tonic, 7) Treat it as a German Augmented Sixth/subV7/V(d5/m7/m9/0) and resolve it down by semitone to a new major tonic, or, 8) Treat it as a German Augmented Sixth/subV7/V(d5/m7/m9/0) and resolve it down by semitone to a new minor tonic.
Likewise, arriving at a fully diminished seventh chord yields eight possible modulations treating each tone as a possible leading tone into a major or minor chord, and arriving at an augmented triad yields six possibilities through the same means. Knowing these possibilities and being aware of them is the major part of the process of learning to structure effective and varied modulations.
Now, the recapitulation is an inversion of the exposition, right down to mode gender, and I was able to accomplish this using harmonics for the first statement of the theme. I ornamented it with 4-3's where the answer comes in, and again over the last subject statement. I also modified the inversion step by step so that the last statement is the subject of the final fugue. At the ending I worked in some humorous chromaticism, and the ending to a C major chord comes as a funny surprise. Ricercares are inherently fun.