Wednesday, October 18, 2006

Musical Implications of the Harmonic Overtone Series: Appendix II


Contrapuntal Musical Examples


Again, real living music combines aspects of all five of the musical elements, but today's examples display a conception which is biased in the direction of counterpoint.

Before composers had intuited out the harmonic system, stepwise contrapuntal effects positively dominated contrapuntal melodies and disjunct harmonic effects were rare, primarily because the methods for handling them hadn't been worked out back then. J.S. Bach was the earliest composer to offer both a summation of almost all that had come before him in compositional history, and a perfect blending of counterpoint and harmony.

I have read some theorists who think that much was lost when harmony invaded counterpoint, but my own view is that the older modal style was simply not fully evolved music. Besides, there is nothing preventing composers today from writing music with a very pure contrapuntal conception if they so choose, and the entire chromatic system is now available to them. Some of Penderecki's music displays these features, for example. As for myself, I tend toward preferring the rhythmic drive that well ordered root motion patterns provide, so my contrapuntal style is biased in the direction of harmonic effects, as you'll see.


This example is the Menuetto Sans Trio from my Irreducible Sonatina of 1992. The piece actually dates from 1986 though, and is in fact only the second piece I ever composed in the contrapuntal style. It is the second oldest piece in my set too, behind the Six Variations in A minor that conclude the sonatina. I understand that Scherzo-type menuettos are supposed to be notated in 3/4 time - and that it is a part of the "joke" of the scherzo - but I prefer the ease of reading and understanding, as well as the reduced measure count, that 6/8 time provides.

The simple and streamlined harmono-contrapuntal style here is not Bachian at all. In fact, the direct inspiration for this piece was the Scherzo of Beethoven's Ninth Symphony, which is my single favorite piece in the entire symphonic literature. This piece is not immitative, however, and is in the simple ternary form of A, A, B, A. The piece also does not modulate, remaining in B minor throughout.

While two-part counterpoint tends to sound empty and incomplete on keyboard instruments - especially the piano, which is kind of hostile to counterpoint in general - on the guitar it is quite idiomatic and fully satisfying. The problem and the advantage with two-voice contrapuntal textures, of course, is that the harmonies implied tend to be nebulous - often offering more than one possible interpretation - and the freedom that the melodies have is very great. Keep this in mind as we go through the analysis.

Beginning simply on the tonic of B minor with the interval of a minor tenth, the second half of the first measure proceeds to the subdominant. I put a iv(m7) symbol in the analysis, but what is happening in the melody is actually a contrapuntal effect: The third from the tonic harmony is held over as a syncopation - or a suspension if you prefer - and as a dissonant seventh it then resolves down to become a major sixth. As I say, this is a purely contrapuntal effect. This resolution could also be harmonically interpreted as a ii(6/3) chord, which points out the nebulousness of applying harmonic terminology to counterpoint - especially in two voices - but I chose to call it a subdominant chord because root position inferences take precedence, and the sixth proceeds further to the perfect fifth before the end of the measure due to the triple time.

This subdominant also proceeds to the dominant, as expected, at the beginning of the second measure, and the lower neighbor major second over the root of the dominant is another vigorous contrapuntal effect. In the second half of the second measure, however, a purely harmonic effect is introduced: A rising tonic arpeggio. This is over a third in the bass line - and in harmony we would not want to be doubling the thirds of chords - so this is what I call a harmono-contrapuntal effect: It has properties of each element.

Measure three begins with an implied bVI(6/3), and this is a great example of that harmony functioning as a tonic substitute. The folowing implied bIII can also be thought as a tonic substitute, but again, the eleven-ten resolution above the bass note is a purely contrapuntal suspension-resolution effect. The major ninth at the end of the measure is quitted by leap in the bass, and this is not an issue when the top voice continues by step and the bass arrives at a consonant relationship with it.

The bVI in the beginning of measure four is a subdominant substitute, and the five-four-three progression over the bass note is another highly contrapuntal effect. This prepares for the dominant harmony that concludes this antecedent phrase, which I have labelled as a V(4/2), but this is yet another harmono-contrapuntal effect, as the minor sixth there can also imply a harmony on the supertonic - or even an expected six-five over a subdominant harmony - so this again points out some of the limitations inherent in using purely harmonic analysis in a contrapuntal context. The fa sol there does imply to the ear that it is a kind of IV V with a harmono-contrapuntal syncope present in it. It is quite a nice effect.

For the consequent phrase beginning on the second system, the analysis is the same for the first two measures as the antecedent's was. The third measure of the consequent phrase, however, contains a contrapuntal effect I have labelled as a ii(d5) to i(6/3), even though it is quite apparent to the ear that the supertonic to tonic motion is implying some kind of dominant to tonic progression. From this point on the two melodies are moving in a purely contrapuntal contrary fashion, and this provides a powerful conclusion to the phrase.

In the last measure of the phrase is the alternate to the previously implied V(4/2) effect, and with this version's stepwise movement, it really is more like a subdominant to dominant progression, and then the tonic appears to end the phrase with a purely harmonic effect: A descending tonic arpeggio.

The second section begins exactly as the first sections antecedent and consequent phrases did, but any possible enticipation of yet a third repeat is dashed with a very strange - and I mean highly unusual from the historical perspective - effect which I have simply labelled as (4/2). The subtonic rarely appears in this fashion in tonal counterpoint - it is usually a raised leading tone, especially when it proceeds upward as this one does - but the seven-six-five contrapuntal effect over it is easy to understand, and I can tell you that I was thinking of a compound axial arrangement: The subtonic is actually targeting the dominant degree, while the intervening supertonic is actually targeting the third degree. This gives the bass part here a nice driving effect.

The two sequential and sequencing phrases of the second section alternate between contrapuntal and harmonic effects, but the phrases are very clearly biased toward harmonic effects, and the abundance of progressive root motions adds serious propulsion to the music. The fully diminished sonority at the end of the first of these phrases also turns the phrase around on itself in a powerful manner with the diminished seventh proceeding to the perfect fifth as it does (In an added octave separation, of course).

Where there is a turn-around at the end of the first of these two phrases, the second has a half-cadence effect which allows for the third, and final, phrase. This phrase is almost completely harmonic, or harmono-contrapuntal in nature, and has a nice sol fa me re do gravitational return to the top contained in it.

With the repeated eight measure first section and the twelve measure second section, there is a 16:12 proportion to the form. This reduces to the perfect fourth's ratio of 4:3, which is pretty common in music, and yet another example of the series at work in the intuition of musicians: I certainly didn't plan it, but then, I wasn't surprised to discover it either.

A lot of audience members ask me who wrote this piece, and when I tell them it's one of mine, most of them seem surprised. I must admit that it does sound like it has existed forever, and that I simply discovered or rediscovered it. I love simple little gems like this.


"And now for something completely different" as the old Monte Python routine went. This is a jazz swing tune with the swing written out in 12/8 time. I wanted to show this piece off as an example of countrapuntal composition in the jazz idiom. Though I am aware that others have come up with jazz counterpoint styles, I'm not really familiar with most of them, and the pieces I have heard, like Steve Reich's piece that Pat Metheny used to perform, haven't really impressed me in the least, to be blunt.

The problem has been, from my perspective, that the purity of the straight ahead swing style has not been preserved in the application of the contrapuntal concepts, and much was lost in the rhythmic drive department. I solved that problem by writing the piece using the actual technique that jazz composers use. Since I was a jazz guy long before I moved toward traditional techniques, this was totally natural for me.

I think the history of this piece is kind of humorous, as I wrote the melody back in 1979 when I was studying at the Guitar Institute SW under Jackie King and Herb Ellis. This was an actual assignment, in fact. I was given the harmonic continuity from an old standard - I wish I could remember which one it was - and told to write a swing or bebop style melody to it: The melody you see here is exactly what I came up with when I was twenty-one years old.

Jackie loved this tune, as did I, so I kept it around: I'm glad I did. Last year I was doing arrangements of some of my jazz compositions for a guitar duo who are students of mine, and when I dug this piece out of the archive I realized I could make a solo guitar arrangement of it in the jazz contrapuntal style I'd developed, so composing this piece "only" took me twenty-seven years.

This is actually the concluding "menuetto" of the Scherzo of my first guitar sonata now: There is an opening version of this piece followed by a "trio" variation preceeding this. I figured this was enough for the demonstration.

Since I wrote this piece starting with a standard jazz technique - write the harmonic continuity first, and then the melody - I have put the chord symbols for that progression above the staff as you would see in a regular lead sheet. This way, you can compare the original progression the melody was composed to with the bass line, which I composed using good old fashioned cantus firmus or cantus prius factus technique.

For that bass part, I decided to think just like a jazz bassist does, but with the added dimension of creating the bass line in non-real time and concieving of it as real counterpoint against the melody, versus a simpler improvisational approach. So, I'm holding down the primary structure tones there - the root, third, fifth, and seventh - and I'm adding directionality with series-implied directional units such as chromatic passing tones and chromatic approach tones, but I'm also applying the three immutable rules of contrapuntal motion, almost without exception (There are a couple of exceptions of artistic license).

The pickup is actually an augmented sixth interval, which fits with the joking nature of this piece as a Scherzo. I mean, how much bigger of a joke in a "classical" multi-movement sonata can you get than to make the third movement a swing tune? This interval is actually quite common in jazz music in an incidental way, due to all of the chromaticism jazz bassists use, but I'm applying it in the traditional manner here (Except that it is a pickup).

In the first measure you can see that the bass part is just like what a jazz cat might play - 1, 5, 1, M7, m7 - but I'm being careful not to imply parallel perfect consonances or parallel dissonances in stepwise motion. Leaping into dissonances is not an issue in this style, however, as the effects are harmono-contrapuntal in nature. The second E in the bass part in the second measure is an example of this: It is just an octave movement of the first E, so the minor seventh is percieved by the mind's ear as a harmonic effect. There is another augmented sixth at the end of the second measure, and employed as a descending chromatic passing tone, it is entirely stylistically consistent with what a jazz bassist is likely to do.

After the three preceeding descending chromatic passing notes, I used an ascending version at the end of the third measure. This creates a minor ninth to minor seventh progression across the barline - something you wouldn't expect to find in traditional counterpoint, but in this idiom is is really quite cool, and is yet another contrapuntal effect, the result of which is interpreted harmonically.

The 3, 1, 3, 5, d5 progression in the bass in measure four is also a typical jazz bassist kind of a deal, and the following long ascending line with chromatic passing embellishments is also quite idiomatic. This results in parallel minor sevenths into the second beat of measure six, but I wanted the line, and so took the license. I'm allowed.

Once the line has risen for two measures in a slick and jazzy way, it turns around at the beginning of measure seven and descends stepwise for two measures. This is a really swinging little episode.

You can see in the analysis that I am analysing key changes before the tonic appears, and this is a stylistic feature of the ii V I mentality of traditional swing era composers: They thought like this.

Note also that the first four measures of the bass line is a compound axial combination: There is a lineal progression going on at the lowest level, but the D's above, and the approaches to them, create another melodic axis for the bass line. Jazz guys do not organize on this level, that I am aware of, but it does add quite a lot to the quality of the bass part. I bring this up now because I flirt with the same concept in measures ten through thirteen: It adds balance with the linear episode in between and then returns to being a single harmono-contrapuntal line leading to the turn around, which is made via the original augmented sixth, of course.

The second ending starts out the same as the first, but then goes it's own way. Notice that I have labelled the E-flat majorminor-seventh chord in the second measures as a tonic, even though it is an overtone (dominant) sonority. This kind of thing goes on in jazz all the time, and relates back to the blues-based origins of the style.

There is a lot of modal interchange going on in the second ending as well. The ii(d5m7) V(m7m9) of minor targets a major tonic at measure twenty, and then this is followed by a minor-origin iv(m7) bVII(m7) progression. I am following the same stylistic and harmono-contrapuntal protocols in the bass part throughout these areas, and the result has a high level of consistency to it. I am still intermittently implying compound axes in the bass as well.

At the end there you can see that the final motion to the tonic is by an augmented sixth, as the first was, but this one isn't syncopated and it is an octave further apart. This provides an effective close, and the stylistically appropriate I(sus6add9) sonority is the Scherzo's final punch-line.


Yet another old appliance-related incident. I'm sure household embarrassments like this are less common today. Shame.


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