Thursday, October 19, 2006

Musical Implications of the Harmonic Overtone Series: Appendix III


Melodic Musical Examples


This appendix example is also harmono-contrapuntal in conception, but I used a combination of large scale strategic and small scale tactical processes to create both the melody and the bass line that lend themselves to clear analysis, which is why I chose it.

Before I studied the Theory of Melody book of The Schillinger System of Musical Composition all of the melodies I wrote were almost completely the product of my intuition. Sure, I thought in terms of outlining the harmonies with melodic elements &c., but I didn't use large or small scale planning to organize them. This has been true for almost all composers of history, so far as the written theoretical record is concerned. If any composers did develop these kinds of tactics and strategies - which I suspect some like Bach and Brahms did - they were never recorded in a pedagogical manner.

Though the inherent genius of a gifted intuition has doubtless lead to some of the most sublime melodies ever written, Schillinger demonstrated that melodic elements could be categorized and analyzed to inform the intuitions and intellects of those of us who are not so trancendentally blessed. I know that my own work improved markedly after my exposure to Schillinger, so I'm sure this will work for others as well.

You'll notice in the analysis line of today's example that I have dispensed with the traditional degree analysis symbols. I believe my previous contrapuntal examples have amply demonstrated the shortcomings of traditional harmonic analysis in a contrapuntal context, so now I present a new alternative for the first time.

There are only five possible functions that harmonies can have: Tonic, subdominant, dominant, secondary dominant, and secondary subdominant. Every harmony on every degree of the intedrated tonal/modal twelve-tone system will fall into one of these categories. Tonic, subdominant, and secondary subdominant harmonies can imply either major or minor base triads, but dominant and secondary dominant harmonies will always imply major base triads because they are overtone chords or altered overtone chords (The real root may simply be absent in some cases).

Therefore, there are eight possible harmono-contrapuntal analysis symbols: T, t, SD, sd, D, SD2, sd2, and D2. These symbols correspond to tonic major, tonic minor, subdominant major, subdominant minor, dominant, secondary subdominant major, secondary subdominant minor, and secondary dominant.

Not only does this make more sense for contrapuntal textures, but it simplifies and streamlines the process as well. There is no substitute for degree analysis in harmonic textures because it is that very degree analysis that allows for root progression patterns to be created (Or detected, if you are analyzing the work of another composer: I use analysis in the compositional process, so that is the perspective I'll usually speak from). Beyond that, however, this functional analysis tells you how any particular harmony is actually working within the system implied by the harmonic overtone series, and so it is even more valuable than degree analysis in some ways.

With this system and in this context root progression and transformation indicators are also not needed (Traditional "common practice" composers did not use consistent transformation technology anyway, since it hadn't been figured out back then), which only makes the process less ponderous and frustrating.

There are instances of ambiguity, of course: No system is perfect, but there are certainly less ambiguities with figuring out functional categories than degree orientation in two part counterpoint.


This piece is the thirteenth of the Eighteen Axial Studies I wrote for solo guitar between 1987 and 1997: It dates from 1993. The idea for these came directly from Joseph Schillinger and his concept of the zero-axes of melodies. He used as one of his examples the subject from the D Minor Organ Fugue, which has a zero-axis that is actually played, versus being only implied (This piece is almost universally attributed to J.S. Bach, but I can tell you with 100% certainty after analyzing it and just a ton of Bach's music that he did not compose it. Beyond that, as a guitarist I can tell you with virtual certainty that the original must have been for the lute There is the remote possibility that it could have been for the violin). Modern scholars are now beginning to realize these things).

Anyway, I realized that the played zero-axis concept could lead to a nice series of idiomatic guitar pieces, and since the zero axis could be the root, third, or fifth of a tonic major or minor triad, there were six possibilities for each axis. For the high E string that worked out to E major, E minor, A major, A minor, C major and C-sharp minor. For the B string that lead to B major, B minor, E major, E minor, G major, and G-sharp minor. Finally, the G string lead to G major, G minor, C major, C minor, E-flat major, and E minor. This one, therefore, is the first of the G-Axis Studies in G major.

For this melody's organizational strategy I used two structures: A directional unit and a rotational unit. The directional unit is a series of four notes which all progress in the same direction, while the rotational unit is a series of four notes which make upper and lower neighbor relationships between a repeated pitch.

For the directional units, there are only two possible forms: They either go up, or they go down. The rotational units, however, can appear in four forms: Original, inverted, retrograde, or inverted retrograde.

Both directional and rotational units can appear in conjunct contrapuntal forms or disjunct harmonic forms, though I do not exhaust all of those possibilities in this particular piece.

The directional units are labelled with the letter "A" in the melodic analysis, and the rotational units are labelled with the letter "B." The two possible orientations of the directional unit are labelled A and A', while the four forms of the rotational unit are labelled B, B', B'', and B'''. I have written which orientation is which in the melodic analysis.

The first section of this piece is a ten measure phrase that repeats. The bass line is interesting in that, after a tonic half note, it progresses stepwise up the entire diatonic scale to the seventh degree in a series of quarter notes, and then proceeds in alternating falling fifths and rising fourths through all of the diatonic degrees again, but now in half notes. The phrases turn-around is then an implied four to five. If you sing this bass line, it is actually quite infectiously catchy (I have it as an "earworm" just thinking about it).

Over this bass line, the melody descends through six degrees of the diatonic system before turning around in the second half of the first rotational unit. So, the first two measures are the decending version of the directional unit, and the second two are the original form of the rotational unit.

Though the nature of the directional unit is clear - they always are, by definition - the merge into the first form of the rotational unit makes its nature unclear. So, even though it is a rotation on the subdominant level - fa, mi, fa, sol - I immediately repeat it on the tonic level above in it's original form as do, ti, do, re. These are also intervallically strict in relation to each other.

Now I can use the other orientations of the rotational unit and the mind's ear will follow, at least intuitively. The first variation I present is the inversion labelled B'. Notice that as mi, fa, mi, re it is an intervallically strict inversion as well. Immediately after the inversion, I use the retrograde of the original as do, ti, la, ti to turn the phrase around.

This entire first section is completely diatonic to the major mode, and so it sounds quite happy. In fact, diatonic major sounds happy because it conforms most perfectly with the implications of the series for a diatonic system! I have saved the first appearance of chromaticism and the final form of the rotational unit for the repeat. The other form of the directional unit appears in the repeat as well.

As you can see, I use a D2 in the form of an augmented sixth interval targeting the fifth degree at the beginning of the third system. Along with the resolution to the D this creates a chromatically altered form of the inverted retrograde of the rotational unit, which is highly effective and satisfying.

The dominant targets the tonic as expected, and then another measure of dominant functionality prepares for the next section. The final two measures present the ascending form of the directional unit for the first time, only now it is intervallically expanded to a series of thirds, whic create, en toto a tonic major seventh arpeggio with the fifth and seventh being reinterpreted as the root and fifth of the D-function chord.

After all the sweet happiness of the first section I throw a couple of wrenches into the works. The second section - which has the function of an interlude - metrically modulates to 3/4 time, and the dominant sonority at the end of the previous phrase "resolves deceptively" in traditional parlance to the sixth degree tonic substitute.

The bass part of the first phrase of the interlude allows for a secondary dominant targeting the primary dominant, while the melody above has created a secondary dominant targeting the preceeding subdominant degree. This combination of the broadening out of the pace combined with the more expressive implied harmonies give this interlude a plaintive, yearning quality. That the expected primary major tonic never appears adds significantly to this.

In this repeat as well I have employed some extended harmonic implications of the series: In the first measure of the fifth system I have used a double chromatic approach to the dominant degree, the final of which creates another augmented sixth. It is important to note that the progression from c-sharp to e-flat is a diminished third, and so the ear hears this as a contrapuntal whole step and not a harmonic third. Kind of a nifty effect.

This primary dominant harbors an augmented triad in the melody, which is kind of unusual in a major key piece, and it sets up another measure which makes the repeat a five-measure phrase. You can see from the functional analysis that a momentary tonic inference is made at the beginning of that final measure, and then a dominant series which is simply a degree exchange between the upper and lower voices.

The third section is the "real" second section: The interlude was designed to be a pause in the texture, which would have become oppressive if it had continued unabated. Here, the two directional units are used in a sequential section, with the original stepwise descending version now following the ascending intervallically expanded form.

The bass part has broadened out further into tied half notes. There is a rest in the second measure of this secion because the guitarist must physically release the low G in order to reach the upper notes: It remains implied.

The disjunct motion followed by conjunct motion implies alternating harmonic and contrapuntal effects in the texture, and it is quite sweetly diatonic again for these eight measures. As you can see, the initial ascending tonic triad and the following ascending supertonic tetrad both reach the same G. This is at the fifteenth fret on the guitar, and so is quite high.

Oh yeah: Restrictions on the distances between voices in counterpoint are poppycock for instrumental music. As you were.

The overall harmonic motion is four measures of tonic, two measures of subdominant, and then the dominant for two measures.


The continuation of the second section - now in its third phrase - offers another deceptive motion from the preceeding dominant to the submediant tonic substitute. This is followed by two measures of a secondary dominant function, which leads to the expectation of a resolution to the dominant, but this is spectacularly thwarted.

On the second system down is the climax of the piece. Remember that ascending scale in the bass of the first section? Though it rubbed up against the zero-axis it never actually made it to the tonic degree. That was a setup, and here is the fulfillment. The G in the bass part - as opposed to the G in the zero-axis - is the resolution of that preparation of long ago: It is also the highest note in the bass part, and since the axial G is an open string, this real unison is played on the guitar (Take that, you single-manual keyboard players).

The bass part descends in a chromatically altered form of the original directional unit into a D2-function sonority in the form of a doubly-augmented fourth. This resolves across the barline into the primary dominant, satisfying both the e-flat and the previous c-sharp that was leapt out of. Remember the c-sharp to e-flat bass motion from the first interlude? Another setup for this climactic resolution. The b-natural in the melody, by the way, is at the nineteenth fret on the guitar: That is the highest note on the traditional classical instrument. This climax is only possible because the D in the bass part is an open string.

In the final four measures of this section we have the denouement of the piece, and again, this only works because the E, A, and D in the bass are open strings on the guitar. I remember when I wrote this how amazed I was that it all worked out so perfectly: It's a great combination of pure music and an idiomatic guitar piece.

As I used an augmented triad earlier, I have answered that with a fully diminished descending directional unit to end the section. This gives a whistful end, and the overall effect of the piece is sort of Romantic, even though it's contrapuntal in conception. This diminished tetrad is also the first intervallic expansion of the descending directional unit to appear.

Following this section is a second interlude. This is almost exactly like the earlier one, except for the fact that the melody is an octave lower, and the resolution is as expected to the tonic, versus the sixth degree tonic substitute.

The repeat is quite different, however: Since the peiece will come to an end after the second time through this interlude, I do not want it to set up the leading-tone/leaning-tone dual target to the dominant degree again. For this reason I have allowed the c-sharp to resolve at the end of the first measure of the repeat, but I immediately introduce the sixth degree tonic substute to set up the final element at the ending of the piece.

In the penultimate measure of the second interlude's repeat is the only secondary subdominant in the piece, and it is the traditional so-called Neapolitan Sixth, but instead of preparing the dominant it resolves to the tonic (I could have put a small letter "t" there, but I was factoring in for the G-axis, which I usually ingnore in the functional analysis, naturally).

This sets up the traditional plagel "Amen" in the last three measures of the piece.


This concludes the appendices for the present version. I suppose I ought to have some examples to demonstrate rhythmic and formal aspects of the implications of the series, but to be honest, I don't concieve of pieces from those perspectives. The musical material I develop combined with the local tactics and regional strategies I employ are what determine the rhythmic and formal properties of the pieces I write. I know of composers who say the first decision they make is what the duration of the piece will be, but that is almost impossible for me to imagine. The possibilities of the material I come up with determine the length for me.

I will write a final epilog with some philosophical conclusions, and then I'll be done with this for the time being.

I suppose that is technically possible.


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