Monday, July 14, 2008

How to Compose Counterpoint (How to Progress)

In my first post in this series, I gave an example of where to begin composing counterpoint from scratch after you have learned the basic rules in the classroom (Or in an autodidactic manner, as I did it). This time I'll be giving examples that show the progress I made in composing two-voice counterpoint for solo guitar in the seven year period from 1987 to 1994.

The first thing you need is a vehicle that will allow you to compose several pieces within a relatively short time frame. You may want to take Bach's miniatures as an example and compose suites, for example. As for myself, I knew I didn't want to do that, rather I wanted to write idiomatic pieces for the guitar, because my overarching long term goals were, 1) To develop a personal contrapuntal style of my own on the guitar, 2) to learn how to compose very formal fugues on the guitar (I'm talking stately Art of Fugue type of deals in miniature here), and, 3) to eventually get to the point where I could write a multi-movement sonata for the guitar.

Fortunately, I was studying Schillinger in 1987, and his Theory of Melody book gave me a very wonderful idea for three sets of idiomatic guitar pieces using his concept of the zero axes of melodies. In short - all melodies have a zero axis - or a set of them if the melody modulates - but most of the time these axes are not played and are only detectable through analysis. In some cases, though, the zero axis is played, as is the case with the fugue theme from the Toccata and Fugue in D Minor for organ, which is usually attributed to Bach, but which Bach in fact did not compose (You can start with the Wikipedia entry if you'd like to research this for yourself).

In any event, that fugue subject gave me the idea to write three sets of Axial Studies using the open high E string, B string, and G string of the guitar as every other note in the melody, just like that fugue subject had. Since the zero axis can be the root, third, or fifth of a tonic major or minor triad, this yielded six studies in each set for a total of eighteen: Just the vehicle I needed.

I started composing the E-Axis Studies in 1987 immediately after writing the menuetto that was the example for the previous post. Here's the one in C Major that will demonstrate the first goal you'll want to tackle, and that is implying secondary dominants in your counterpoint.

First, a couple of observations about the form. I am a big fan of odd numbers of measures for sectional pieces, and here the A section is eleven measures in length, and it also has a first and second ending. In contrast to the odd number of measures of A, I have used an even number for what looks like B there at measure thirteen. This is actually an interlude between the A and B, and it also has two endings. The B section proper begins at measure twenty-three, and it is sequential, has four measure phrases, and is sixteen measures in length. I felt the need for the interlude to relieve the texture, which becomes monotonous without this relief.

So, with the repeat scheme the form is, A, A', I, I', B, A, A', I, I', B, A''. Just in terms of formal conception, then, I had already made significant progress.

Here's the second page.

I was still writing the melody out first and then adding the bass line at this time - cantus firmus technique - but I got the idea to imply a V(6/3)/V at the end of B there in measure thirty-seven. It's the only accidental in the entire piece, so it stands out and provides a kind of a climax. Once I had this insight, it was a small step to get the idea to add secondary leading tones in the melody.


I had all six of the E-Axis Studies done before I started working on my master's degree, so for my master's degree lecture recital, which I did in lieu of a thesis, I presented, performed, and discussed the B axis studies. I composed these between 1989 and 1990, and here you can see just how far I advanced in this short time frame.

Here's No. 2 in E minor.

Formally, this is the same as the previous example right down to the eleven measure A sections, but the B sections have odd measure phrases. That's a bit of progress in itself, but notice how chromatic it is?

After the open B pickup note, and the B-axis is the fifth of the tonic triad here, I ascend through the chromatic tetrachord from sol to do. Then, I continue with chromatic figures in measures five and seven: The augmented eleventh created in measure five implies a V(4/2)/iv, measure six has a c-sharp in the bass, so that's an implied V(6/3)/bVII, and the A-sharp in measure seven creates an augmented sixth targeting V. This is a really gnarly sounding A section, so I knew I'd have to pull out all the stops for the B section.

The first ending of A returns home via an arpeggiated primary diminished seventh chord, and the second ending leads into the interlude with another chromatic tetrachord to the tonic an octave higher. In the interlude, I use the descending form of the chromatic tetrachord broadened out, and the do, ti, te, la, le, sol, la, ti line created in the lead is just like the same figure Beethoven uses at the end of the first movement of the Ninth. This interlude is way cool, and it provides a nice and needed lessening of pace and tension between the A and B sections... but not too much.

Here the B section is based on odd five measure phrases, as I said, and I take the ascending chromatic line idea to another level. The G-sharp in twenty-five again implies a V(4/2)/iv and the A-sharp in twenty-six makes an augmented sixth targeting V.

I then sequentially repeated the previous phrase starting on the subdominant level in all details. Here, the C-sharp over G in measure thirty implies a V(4/2)/bVII and the D-sharp over F-sharp in thirty-one creates a V(6/4) targeting the tonic.

The pitch climax in thirty four is the A at the seventeenth fret of the nineteen fret classical guitar, so it is quite high for the instrument, and thirty four and thirty five imply a iv to v progression that arrives at an upper neighbor diminished chord in thirty-six before returning to v, and then a return to the A section, this time via an apreggiated augmented triad. This is really neat, and the resulting final phrase is six measures in length.

The A section and the interlude are repeated, and then we return to the B, but this is a true B' at forty, and now both voices move in contrary chromatic motion! I read in some rather lame counterpoint book or other that this was not good practice, so - of course - I had to do it. LOL! I found that if you present the version with the diatonic bass line first, then the repeat with the chromatic bass is very highly effective, and the intervallic sequences are highly colorful and unusual.

From forty to forty-three the intervals are: 8ve, M9th, d4th, A4th, m6th, A6th, and 8ve: Bad-ass, if I do say so myself. Since forty-five to forty-eight are just a sequential transposition, the interval sequence is the same there, only on a higher level. the final six measures are the same as before, which leads to the ending statement of the A section.

Yeah, I'd gotten a pretty good handle on chromaticism by this point, and it was all because I had a vehicle that allowed me to compose quite a few pieces with limited variables in relatively short order.


By the time I got to UNT in the fall of 1991, I was ready to tackle the G-Axis Studies. We'll look at No. 1 in G major, which shows several more evolutionary traits.

I wrote this one in 1994, and if I recall correctly, I had all six finished in just a few weeks: My facility was pretty decent by this time, and I was also writing both voices simultaneously.

The A section here is ten measures in length, and the first statement of it is totally diatonic. Notice that the bass line first goes up all seven notes of the scale stepwise, and then goes through all seven again (counting the F-sharp in four twice as the pivot) in alternating descending fifths and ascending fourths.

In the second ending at eleven, however, I introduce an augmented sixth targeting the dominant, and thirteen and fourteen give us a I to V leading into the interlude, which starts on VI via the traditional deceptive motion. Notice that I change time signature to 3/4 for the interlude - more progress - and while the first phrase is four bars, the repeat is five - yet more conceptual growth.

The F-natural in sixteen implies a V(4/3)/IV, the C-sharp in the bass at seventeen implies a V(6/3)/V, and then the interlude repeats. In the second ending at nineteen, I use a double chromatic neighbor pair to target V, and the last C-sharp over E-flat creates an augmented sixth, of course. The descending line in twenty is an augmented triad - rare in a major key - and then the final measure gets us to a V(6/3) to set up the B section.

After all of the histrionics of the interlude, I returned to 2/4 time, four measure sequential phrases, and broadened things out to 4:1 counterpoint to begin the B section.

Here in the third phrase I introduced what should be a V(4/3)/V over the C-sharp in the bass, but this is spectacularly interrupted by the fourth phrase, which starts on an implied tonic - a really, really deceptive motion - and proceeds to the climax of the piece. The interval of A-sharp over E-flat in thirty-five is a doubly-augmented fourth, which I was saving for just this moment, and the B in measure thirty-six is the highest note on the standard nineteen fret classical guitar. Sweet, no? Of course, doubly-augmented fourths resolve most typically to I(6/4) sonorities, which is the case here: That chord then generally resolves its suspensions to a root position V, which is again the case here.

Thirty-eight is actually a deceptive motion to vi, and then a ii, V under a descending diminished seventh leads to the second interlude: Yes, there are two interludes here.

This second interlude is an octave lower, and it starts on I instead of vi. There's the V/IV in forty-three, and then the V/V in forty-four which leads to the repeat, but again, no deceptive motion this trip.

The second ending is further extended this time, making the phrase six measures versus the previous five of the first interlude repeat. In this phrase I managed to work a traditional so-called Neapolitan Sixth in at forty-eight, and I think that's the first, and still the only, time I've ever used that sonority in the traditional manner, but it was a specific goal I had to use it here.

So, you can see that by limiting my variables and writing these eighteen pieces over the course of seven years, I totally mastered two-voice counterpoint and made great overall progress as a composer. There are countless paths you can take, but the steps will be nearly the same on each one, you just need to find a path. Of course, this wasn't all I was writing during this time period - I had gotten pretty good with fugue by the time I wrote the G-Axis Studies - but these pieces allowed me to work out all of the elements for myself in the comfort of my own idiom, the guitar. It was a long time after this before I finally figured out how to write fugue subjects that would work for the guitar, however.

Whew! Long-ass epic post. I'm pooped.


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