Sunday, October 15, 2006

Musical Implications of the Harmonic Overtone Series: Chapter IX


The Series' Implications for Melody


Schillinger defined melody as a trajectory of pitch through time, and that is about as concise a definition as one can get. Looking at the big picture in such a general way is quite useful, as one of the things that separates student musical exercises from the work of a competent composer is the student's lack of direction over the long term. In other words, the counterpoint and harmony in student exercises might be technically perfect, but the exercise as a whole might suffer because only the local mechanics were addressed, and not the larger scale axial and trajectorial issues. This is also true, I might add, for harmonic continuities written by students: The voice leading might be technically correct, but in many instances the root motion patterns are not well thought out, or they may be even totally lacking in any organizational structure whatsoever over the course of the phrase.

As good as Schillinger's definition is, and as useful as his axial concepts are, without a thourough understanding of the mechanics of melody on the notational scale, writing them is left almost enturely to the intuition (Which has been the case for most of the history of Western music). So, before we start thinking in the grand schema, we need to focus on the micro-mechanics of melodic movement.

There are two kinds of melodic motion: Conjunct and disjunct. Conjunct motion is stepwise (Either diatonic or chromatic) and it implies contrapuntal effects, while disjunct motion proceeds by leap (Again, either diatonic or chromatic) and implies harmonic effects (Now you can begin to se why melody is best addressed last: It has all of the features of harmony, counterpoint, and rhythm combined in it).

An old axiom among jazz improvisers goes something along the lines of, "There are no wrong notes, only awkward phrasing." This is true: Any of all twelve chromatic tones are available over any harmony, so long as they are employed in accordance with the implications of the harmonic overtone series. In order to learn how to do that, we need to take a look at how notes can function over the various harmonic structures, and how the leading-tone/leaning-tone impetus present in the series can be harnessed and even amplified to get highly directional melodic effects.


On the top system, in Example 1, I have presented Schillinger's concept of the zero axis. The zero axis is basically the gravitational locus of a melody, and it can be the root, third, or fifth of a tonic major or minor triad.

In the second system are the unbalancing axes, which lead directionally away from the zero axes above or below (Gravity in music can be real or artificial: Gravity or anti-gravity), increasing tension as they go. Then, the third system has the balancing axes, which lead back to the zero axes and release tension.

All of the axes in these examples, except for the zero axes, of course, proceed by step and therefore they imply contrapuntal effects. In a nutshell, contrapuntal effects in melodies are the alternation of harmonic and extra-harmonic tones in stepwise melodic progressions. The first of the unbalancing axis examples has the melodic progression D, E, F, G leading away from the zero axis on the tonic of C. So, that axis displays the so-called accented passing tone concept with D and F being extra-harmonic tones between the harmonic tones of E and G (With C as the axis underneath).

The second of the unbalancing axis examples has the zero axis on C again, only this time it is functioning as the fifth of the subdominant chord residing on F. So in this instance, the B and G are the extra-harmonic tones as the augmented eleventh and major ninth respectively. The balancing axis examples are just the other way around over the same harmonies, and so they demonstrate the so-called unaccented passing tone paradigm.

On the bottom system we begin to get to the meat of the matter with two demonstrations of the different types of disjunct harmonic motion within melodies: Primary Structure Harmonic Motion and Secondary (Upper) Structure Harmonic Motion.

Primary structure harmonic motion is simply outlining the triad or tetrad of the moment, or making use of two or more of its tones in a disjunct manner. Secondary structure harmonic motion, on the other hand, makes use of the harmonic tones the series implies which are beyond the seventh: The ninths, elevenths, and thirteenths.

In the first measure of Example 4, the melody simply outlines the dominant tetrad over a zero axis at its root. In the next measure, however, the upper structure major triad consisting of the major ninth, augmented eleventh, and major thirteenth is used. These same tones, when surrounded by chord tones and entered and exited in conjunct motion, imply contrapuntal effects. In this context, however - a context in which they are entered and exited by disjunct motion - they imply higher order harmonic effects.

If you play through the dodecaphonic system at a high enough velocity, you can manage to squeeze all of the notes in the chromatic scale in over any kind of harmonic entity - Chopin made a career out of this - but the moment you start leaping into notes and exiting them by leap, the ear will try to interpret them harmonically instead of contrapuntally.

Since contrary stepwise motion can legitimize any melodic sequence in counterpoint, but harmonic requirements are more selective, greater care must be exercised when you start making melodic leaps.


Since the overtone sonority is in and of itself unstable - containing as it does a tri-tone - it is also the most accomodating to upper structure tones. The series itself gives a major ninth, but so the series-generated minor mode provides a minor ninth. The major ninth is not only from the series, but is also out of the diatonic major system, and then the augmented ninth comes from the nona-tonic blues modality. The series also gives us the augmented eleventh, but almost every series-generated modal system gives the perfect eleventh. The perfect eleventh is the single most problematic tone over any major triad because it creates a minor ninth against the major third below it. This undermines the stability of the major triad because it is a simultaneous clash of wills between a leaning-tone and its target. Not to mention that it sounds like crap. In exceptional harmonic instances I have used the perfect eleventh in a semitone relationship with the major third (tenth) to good effect, but in melodies making harmonic motions it is best avoided. The minor thirteenth and major thirteenth come from the minor and major series-generated modalities, respectively, and both have a very pleasant sound which is much used in jazz music.

Due to the problems inherent with the perfect eleventh over major triads, the tonic function major seventh almost never gets any melodic effects on that degree. For that reason, I put the subdominant major seventh in Example 6, which does have a complete upper structure triad.

Minor sevenths kind of split the difference between dominant and subdominant harmonies, as they can have four - or in some cases five - tones available for making upper structure triads. Perfect elevenths make major ninths with the minor thirds of minor seventh chords, so they are not only workable, they sound very good. The minor or major thirteenths are almost equally workable, and they derive from the Aolean and Dorian modes respectively, but the minor thirteenth is quite a dark effect due to the minor ninth it makes with the triad's fifth. This is not nearly as bad as a minor ninth against a major third, however. I saved the ninth for last, because while the major ninth of Dorian and Aolean can be used with impunity, there is also a minor ninth on the Phrygian chord which can sometimes be used to good effect if you are very careful: Using a minor ninth in a harmonic context over a Dorian or Aolean chord, however, will get you an invitation to the clam bake.

Just as the leading-tone and leaning-tone effects of the overtone chord were amplified or doubled in the harmonic system with the development of the various secondary dominant sonorities, they have also been amplified in the melodic system, and quite remote from the host harmonies from a functional standpoint at that. In Examples 8 and 9 I have demonstrated how artificial leading-tones and leaning-tones can be lept into in a harmonic manner, even though some of them are extremely dissonant over the host harmony, and as long as they target either a primary or a secondary harmonic tone, they work fine. In classical music theory you get appogiaturas, and escape tones and all this other crap - frankly I've forgotten most of that ponderous terminology - and never do you get an explaination as to why they work, and why they carry the effects they do: All you have to do to work out what will and will not work, and why, is to relate it back to the harmonic overtone series and all that it implies.

Another couple of melodic resources to mention are the octa-tonic diminished scale and the hexa-tonic whole tone scale. The octatonic works over fully diminished seventh chords, or their dominant with the real root V(m7m9) chords, and the one that you use begins with a semitone between the leading tone of the moment and the target degree. Whole tone scales work over augmented triads or the French-derived V(d5m7) sonorities. Just start on the root.

In order to explain these concepts fully, we'llhave to look at some actual melodies, which we'll do in the appendix.


Praise the Lord for modern appliances, right ladies?


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