How to Compose Counterpoint (Imitation and Modulation 3)
This is the seventh and final installment of a series of posts that began with Where to Begin and progressed through How to Progress, Using Larger Forms, Using Three Voices, Imitation and Modulation 1, and Imitation and Modulation 2.
In this third and final post on Imitation and Modulation we are going to look at a piece I call Extempore in A minor, and with this we will have looked at all five movements of my Sonata Zero in A Minor for solo guitar. This Extempore in A Minor is the first movement, the Menuetto in B Major from the first post in this series is second, the Ricercare in C Major from the post prior to this is the third movement, the Scherzo in B Minor from Using Larger Forms is fourth, and the Fugue in A Minor from Imitation and Modulation 1 is the finale. So, the five movements of Sonata Zero are, Extempore, Menuetto, Ricercare, Scherzo, and Fugue.
The piece is, in reality, an imitative prelude, but I am already over half way through composing twenty-four homophonic preludes in all the keys for the guitar, so to avoid confusion with that Prelude in A Minor, I gave this a different designation. I got the idea from the fact that the compositional process for this piece was quite extemporaneous, and besides, Extempore, Menuetto, Ricercare, Scherzo, and Fugue rolls off the tongue nicely and has a cool ring to it. Hey, this stuff is important! LOL!
In this movement's brief fifty measures, it traverses sixteen - a full two-thirds - of the twenty-four possible major and minor keys, so it is just a riot of modulation. That's why I saved it for last: I came up with a very nice modulation scheme that allowed me to organize the key regions so that every key signature had both its major and minor appear. IOW, sixteen keys are travelled through, but only eight key signatures are used.
The subject, such as it is, is just the do, ti, do head figure from the upcoming ricercare and fugue themes, so this exposition is imitation boiled down to nearly its irreducible essence. At the answer we get the desirable entrance over a "dissonant fourth" and the resolution to a third, and then the final entry of the "subject" also happens as the desireable 6/4 sonority and the G-sharp in the bass gives a momentary diminished triad before the resolution to the tonic at measure four.
In the second half of measure four, a dominant function harmony is created - a vii(d6/3) - and then a i(6/3) is reached at the beginning of five. Note that to this point there have been no leaps at all; the voice leading has been totally smooth and stepwise. This exposition is another one of those things that sounds positively primordial in its logical inevitability.
There is another dominant function harmony in the second half of measure five - a root position vii(d5) this time - and then the development begins. For the development, I used the sixteenth-note diminution of the subject in sequence first to get things started. After tonicizing the tonic, it tonicizes the mediant of C and then the dominant of E. Notice, however, that at the "resolution" to E an augmented triad is created: G-sharp, C, and E. I pull out all of the stops with the dissonant counterpoint tricks in this piece.
Into measure eight the bass tonicizes the tonic again, so we really haven't modulated anywhere yet. The descending sixteenth-note figure in eight then descends to the subdominant note, making this a 2.5 measure phrase. Quite unusual.
The sequence then repeats into measure nine by first tonicizing the subdominant, then the submediant, and finally the tonic again into measure ten. Look at the sonority at the second beat of nine: A, F, B-flat. Since we are, in fact, in the subdominant minor key now, this would have to be an incomplete bVI(M7) in third inversion, and with the resulting minor ninth between the outer voices it is very hotly dissonant. The augmented triad at the beginning of ten sounds quite tame in comparison. At the end of ten, the second descending sixteenth-note run gets us to the subtonic minor of G, so we've accumulated two flats so far.
In eleven I invert the sequence, but the phrases are an even two measures now, so the dominant level of G minor is tonicized on D, then the mediant on B-flat, and finally the tonic on G. The final resolution is no longer to an augmented triad in this formulation. Notice how I used parallel movement into a DINO (Dissonance In Name Only, and augmented second in this case) at the end of twelve? This gives an incomplete fully diminished seventh on B - in second inversion - that takes the piece all the way back to C major, where the formula is repeated. And, with the appearance of C major, both of the natural keys have now appeared. I change the C major to C minor in fourteen, which allows the parallel movement into another DINO augmented second at the last eighth note. This produces an incomplete fully diminished seventh on E - again in second inversion - and so we're going to F.
With the appearance of F major at fifteen, both mode genders of the one flat key signature have now appeared. Here I speed up the pace another notch by combining the sixteenth-note tonicization figure with the descending sixteenth-note run. The bass line gets a chromatically introduced leading tone, so at sixteen D minor is tonicized. I pass the figures between the voices, as you can see, so at seventeen G minor is tonicized, and then at eighteen we're back to C major. This allows for a varied form of the formula to return us to A minor at the beginning of nineteen, where yet a faster formula with a constant sixteenth surface rhythm begins.
Now, this piece is very much a prototype, or an "Alpha Test Version," so I know that the eighth notes will have to be changed to sixteenths in the middle voice, and the notes in the top voice of measure twenty may have to have their durations shortened as well. I call these versions "idealized" and I set them up this way so I can hear the MIDI playback the way I want it. When I get around to doing the fingering, I'll create a "performance" version. This piece isn't even anywhere near the top of my to do list yet.
The new constant sixteenth note figure modulates up a perfect fourth every iteration, so we arrive at E minor in twenty, and B minor in twenty one. Since the formula continues on the third system, we arrive at F-sharp minor in twenty-two, and C-sharp minor in twenty three. I'm keeping the sixteenths out of the bass for idiomatic reasons, by the way.
When we arrive at G-sharp minor in twenty-four, I use yet another formula that modulates up by semitone every cycle: A minor into twenty-five, B-flat major into twenty-six, and then B minor into twenty-seven. B-flat major completes the two flats pair of keys, as we have heard G minor previously. The variation on the current formula in twenty-seven takes us back to E minor in twenty-eight, and I use twenty-eight to re-launch into the constant sixteenth note formula, only this time we will go through major keys.
Twenty-nine starts out on C major, and since the formula takes us up a perfect fourth every cycle, thirty has the arrival to G major, and thirty-one has the arrival to D major. G major and D major complete the key pairs of one sharp and two sharps respectively.
Since this is the second time we've heard this formula, I shorten the number of iterations, even though we're traversing major keys this time. Thirty-two has the arrival to A major, and that completes the three-sharps key pair. Since this formula takes us up by semitone again, we again traverse B-flat major in thirty-three - a key we've heard before - but this time we hit B major in thirty-four: This completes the five sharps pair. Now, the only incomplete key pair is four sharps: We haven't heard E major yet. The last beat of thirty-four intimates that E minor is coming, what with the C-natural and G-natural, but this is a deception, as the resolution into the sequential homophonic episode at thirty-five arrives at an E major chord.
This episode is again a harmonized augmentation of the subject element, and it is longer than the corresponding episodes in the ricercare and fugue because of this. The extra length is also required to shed all of these accumulated sharps. We get D-natural in thirty-seven, G-natural in thirty-eight, and then C-natural in thirty-nine. I retain the F-sharp until forty-two, however, after I can put the la, ti figure in forty-one (lowest voice on the top system), and so the return to A minor is accomplished.
The recapitulation here is a real one, it being just a restatement of the exposition with a final concluding cadence added to get to the close position A minor in fifty.
So, the appearance of the keys - and yes, deciding between what is a modulation and what is a tonicization here is difficult, to say the least - goes like this:
01) i= Tonic Minor-------------Natural Key-----(m=1)
02) iv= Subdominant Minor------One Flat--------(m=1)
03) bvii= Subtonic Minor--------Two Flats-------(m=1)
04) bIII= Mediant Major (Relative)--Natural Key-----(M=2/Complete)
05) bVI= Submediant Major------One Flat--------(M=2/Complete)
01) i= Tonic Minor-------------Natural Key-----(C)
02) iv= Subdominant Minor------One Flat--------(C)
03) bvii= Subtonic Minor--------Two Flats-------(1)
04) bIII= Mediant Major (Relative)--Natural Key-----(C)
01) i= Tonic Minor-------------Natural Key-----(C)
06) v= Dominant Minor---------One Sharp------(m=1)
07) ii= Supertonic Minor--------Two Sharps------(m=1)
08) vi= Raised Submediant Minor--Three Sharps----(m=1)
09) iii= Raised Mediant Minor-----Four Sharps-----(m=1)
10) vii= Leading Tone Minor------Five Sharps-----(m=1)
01) i= Tonic Minor--------------Natural Key----(C)
11) bII= Leaning Tone Major (N)----Two Flats-----(M=2/Complete)
07) ii= Supertonic Minor----------Two Sharps ---(1)
06) v= Dominant Minor----------One Sharp-----(1)
04) bIII= Mediant Major (Relative)---Natural Key----(C)
12) bVII= Subtonic Major---------One Sharp-----(M=2/Complete)
13) IV= Subdominant Major-------Two Sharps----(M=2/Complete)
14) I= Tonic Major--------------Three Sharps---(M=2/Complete)
11) bII= Leaning Tone Major (N)----Two Flats-----(C)
15) II= Supertonic Major----------Five Sharps----(M=2/Complete)
16) V= Dominant Major----------Four Sharps----(M=2/Complete)
13) IV= Subdominant Major-------Two Sharps----(C)
04) bIII= Mediant Major (Relative)---Natural Key----(C)
The numbers at left are the order of the keys as they appear, then the roman numeral designations for them, and the functional relationship they have to the tonic. The middle column has the key signature, and the right column has the check-off list.
The "(m=1)" for A minor means it is a minor key, and this is the first appearance of a key with that signature. Then, the "(M=2/Complete)" means it is a major key, it is the second appearance for that signature, and so the pair is complete. Subcequent "(1)" entries mean that the key has already appeared, but the pair isn't complete, and subsequent "(C)" entries means the key pair for that signature was completed previously. This is how you organize a modulation scheme.
In this third and final post on Imitation and Modulation we are going to look at a piece I call Extempore in A minor, and with this we will have looked at all five movements of my Sonata Zero in A Minor for solo guitar. This Extempore in A Minor is the first movement, the Menuetto in B Major from the first post in this series is second, the Ricercare in C Major from the post prior to this is the third movement, the Scherzo in B Minor from Using Larger Forms is fourth, and the Fugue in A Minor from Imitation and Modulation 1 is the finale. So, the five movements of Sonata Zero are, Extempore, Menuetto, Ricercare, Scherzo, and Fugue.
The piece is, in reality, an imitative prelude, but I am already over half way through composing twenty-four homophonic preludes in all the keys for the guitar, so to avoid confusion with that Prelude in A Minor, I gave this a different designation. I got the idea from the fact that the compositional process for this piece was quite extemporaneous, and besides, Extempore, Menuetto, Ricercare, Scherzo, and Fugue rolls off the tongue nicely and has a cool ring to it. Hey, this stuff is important! LOL!
In this movement's brief fifty measures, it traverses sixteen - a full two-thirds - of the twenty-four possible major and minor keys, so it is just a riot of modulation. That's why I saved it for last: I came up with a very nice modulation scheme that allowed me to organize the key regions so that every key signature had both its major and minor appear. IOW, sixteen keys are travelled through, but only eight key signatures are used.
The subject, such as it is, is just the do, ti, do head figure from the upcoming ricercare and fugue themes, so this exposition is imitation boiled down to nearly its irreducible essence. At the answer we get the desirable entrance over a "dissonant fourth" and the resolution to a third, and then the final entry of the "subject" also happens as the desireable 6/4 sonority and the G-sharp in the bass gives a momentary diminished triad before the resolution to the tonic at measure four.
In the second half of measure four, a dominant function harmony is created - a vii(d6/3) - and then a i(6/3) is reached at the beginning of five. Note that to this point there have been no leaps at all; the voice leading has been totally smooth and stepwise. This exposition is another one of those things that sounds positively primordial in its logical inevitability.
There is another dominant function harmony in the second half of measure five - a root position vii(d5) this time - and then the development begins. For the development, I used the sixteenth-note diminution of the subject in sequence first to get things started. After tonicizing the tonic, it tonicizes the mediant of C and then the dominant of E. Notice, however, that at the "resolution" to E an augmented triad is created: G-sharp, C, and E. I pull out all of the stops with the dissonant counterpoint tricks in this piece.
Into measure eight the bass tonicizes the tonic again, so we really haven't modulated anywhere yet. The descending sixteenth-note figure in eight then descends to the subdominant note, making this a 2.5 measure phrase. Quite unusual.
The sequence then repeats into measure nine by first tonicizing the subdominant, then the submediant, and finally the tonic again into measure ten. Look at the sonority at the second beat of nine: A, F, B-flat. Since we are, in fact, in the subdominant minor key now, this would have to be an incomplete bVI(M7) in third inversion, and with the resulting minor ninth between the outer voices it is very hotly dissonant. The augmented triad at the beginning of ten sounds quite tame in comparison. At the end of ten, the second descending sixteenth-note run gets us to the subtonic minor of G, so we've accumulated two flats so far.
In eleven I invert the sequence, but the phrases are an even two measures now, so the dominant level of G minor is tonicized on D, then the mediant on B-flat, and finally the tonic on G. The final resolution is no longer to an augmented triad in this formulation. Notice how I used parallel movement into a DINO (Dissonance In Name Only, and augmented second in this case) at the end of twelve? This gives an incomplete fully diminished seventh on B - in second inversion - that takes the piece all the way back to C major, where the formula is repeated. And, with the appearance of C major, both of the natural keys have now appeared. I change the C major to C minor in fourteen, which allows the parallel movement into another DINO augmented second at the last eighth note. This produces an incomplete fully diminished seventh on E - again in second inversion - and so we're going to F.
With the appearance of F major at fifteen, both mode genders of the one flat key signature have now appeared. Here I speed up the pace another notch by combining the sixteenth-note tonicization figure with the descending sixteenth-note run. The bass line gets a chromatically introduced leading tone, so at sixteen D minor is tonicized. I pass the figures between the voices, as you can see, so at seventeen G minor is tonicized, and then at eighteen we're back to C major. This allows for a varied form of the formula to return us to A minor at the beginning of nineteen, where yet a faster formula with a constant sixteenth surface rhythm begins.
Now, this piece is very much a prototype, or an "Alpha Test Version," so I know that the eighth notes will have to be changed to sixteenths in the middle voice, and the notes in the top voice of measure twenty may have to have their durations shortened as well. I call these versions "idealized" and I set them up this way so I can hear the MIDI playback the way I want it. When I get around to doing the fingering, I'll create a "performance" version. This piece isn't even anywhere near the top of my to do list yet.
The new constant sixteenth note figure modulates up a perfect fourth every iteration, so we arrive at E minor in twenty, and B minor in twenty one. Since the formula continues on the third system, we arrive at F-sharp minor in twenty-two, and C-sharp minor in twenty three. I'm keeping the sixteenths out of the bass for idiomatic reasons, by the way.
When we arrive at G-sharp minor in twenty-four, I use yet another formula that modulates up by semitone every cycle: A minor into twenty-five, B-flat major into twenty-six, and then B minor into twenty-seven. B-flat major completes the two flats pair of keys, as we have heard G minor previously. The variation on the current formula in twenty-seven takes us back to E minor in twenty-eight, and I use twenty-eight to re-launch into the constant sixteenth note formula, only this time we will go through major keys.
Twenty-nine starts out on C major, and since the formula takes us up a perfect fourth every cycle, thirty has the arrival to G major, and thirty-one has the arrival to D major. G major and D major complete the key pairs of one sharp and two sharps respectively.
Since this is the second time we've heard this formula, I shorten the number of iterations, even though we're traversing major keys this time. Thirty-two has the arrival to A major, and that completes the three-sharps key pair. Since this formula takes us up by semitone again, we again traverse B-flat major in thirty-three - a key we've heard before - but this time we hit B major in thirty-four: This completes the five sharps pair. Now, the only incomplete key pair is four sharps: We haven't heard E major yet. The last beat of thirty-four intimates that E minor is coming, what with the C-natural and G-natural, but this is a deception, as the resolution into the sequential homophonic episode at thirty-five arrives at an E major chord.
This episode is again a harmonized augmentation of the subject element, and it is longer than the corresponding episodes in the ricercare and fugue because of this. The extra length is also required to shed all of these accumulated sharps. We get D-natural in thirty-seven, G-natural in thirty-eight, and then C-natural in thirty-nine. I retain the F-sharp until forty-two, however, after I can put the la, ti figure in forty-one (lowest voice on the top system), and so the return to A minor is accomplished.
The recapitulation here is a real one, it being just a restatement of the exposition with a final concluding cadence added to get to the close position A minor in fifty.
So, the appearance of the keys - and yes, deciding between what is a modulation and what is a tonicization here is difficult, to say the least - goes like this:
01) i= Tonic Minor-------------Natural Key-----(m=1)
02) iv= Subdominant Minor------One Flat--------(m=1)
03) bvii= Subtonic Minor--------Two Flats-------(m=1)
04) bIII= Mediant Major (Relative)--Natural Key-----(M=2/Complete)
05) bVI= Submediant Major------One Flat--------(M=2/Complete)
01) i= Tonic Minor-------------Natural Key-----(C)
02) iv= Subdominant Minor------One Flat--------(C)
03) bvii= Subtonic Minor--------Two Flats-------(1)
04) bIII= Mediant Major (Relative)--Natural Key-----(C)
01) i= Tonic Minor-------------Natural Key-----(C)
06) v= Dominant Minor---------One Sharp------(m=1)
07) ii= Supertonic Minor--------Two Sharps------(m=1)
08) vi= Raised Submediant Minor--Three Sharps----(m=1)
09) iii= Raised Mediant Minor-----Four Sharps-----(m=1)
10) vii= Leading Tone Minor------Five Sharps-----(m=1)
01) i= Tonic Minor--------------Natural Key----(C)
11) bII= Leaning Tone Major (N)----Two Flats-----(M=2/Complete)
07) ii= Supertonic Minor----------Two Sharps ---(1)
06) v= Dominant Minor----------One Sharp-----(1)
04) bIII= Mediant Major (Relative)---Natural Key----(C)
12) bVII= Subtonic Major---------One Sharp-----(M=2/Complete)
13) IV= Subdominant Major-------Two Sharps----(M=2/Complete)
14) I= Tonic Major--------------Three Sharps---(M=2/Complete)
11) bII= Leaning Tone Major (N)----Two Flats-----(C)
15) II= Supertonic Major----------Five Sharps----(M=2/Complete)
16) V= Dominant Major----------Four Sharps----(M=2/Complete)
13) IV= Subdominant Major-------Two Sharps----(C)
04) bIII= Mediant Major (Relative)---Natural Key----(C)
The numbers at left are the order of the keys as they appear, then the roman numeral designations for them, and the functional relationship they have to the tonic. The middle column has the key signature, and the right column has the check-off list.
The "(m=1)" for A minor means it is a minor key, and this is the first appearance of a key with that signature. Then, the "(M=2/Complete)" means it is a major key, it is the second appearance for that signature, and so the pair is complete. Subcequent "(1)" entries mean that the key has already appeared, but the pair isn't complete, and subsequent "(C)" entries means the key pair for that signature was completed previously. This is how you organize a modulation scheme.
posted by Hucbald at 12:27 PM
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