Tuesday, October 10, 2006

Musical Implications of the Harmonic Overtone Series: Chapter II


Harmonic Motion Dynamics


When I was attending Berklee back in the early 80's, some of the jazz theorists I studied with used to put brackets under ii to V progressions in order to make them easier to make out in the analysis. This was the first time I was exposed to an incipient system for labelling root progression types - and it was helpful in detecting root progression patterns - so I not only adopted it, but I also expanded it to label each and every kind of root motion.

My first guide in this effort was Arnold Schoenberg, who in his Structural Functions of Harmony classified root motion types using terminology like strong, super-strong, ascending, and decending, and ranked them by how many common tones the adjacent chords shared and whether the targeted root was a new tone, or one present in the preceeding sonority. While this system did enable me to write much better harmonic continuities due to greatly facilitated pattern recognition - and good and bad continuities are almost always separated by whether the root progression patterns are logical or not - Schoenberg's system did not relate in any but the most casual and incidental ways to the implications inherent in the harmonic overtone series.

Over the course of a couple of decades of private teaching, I began to revise his terminology until I had replaced it en toto with my own system which related all root motion types to the primordial resolution of the overtone chord. This root motion, the falling perfect fifth/rising perfect fourth - as I have mentioned previously - is Progressive, and it gets a capital "P" in the analysis. Calling the root motion progressive is not arbitrary, rather it reflects what the series is actually doing in this root motion/transformation type: It is progressing according to the inherent desires of the musical force present in the series.

The contrary root motion to this, the rising perfect fifth/falling perfect fourth, would therefore be Regressive, and it gets a capital "R" on the analysis line. If the progressive root motion is the ticking of the watch, then the regressive motion is the winding of the spring. The regression from the tonic to the dominant, for example, is just the windup for the pitch: The following dominant to tonic progression is the actual delivery of the ball.

It takes two falling thirds to equal one falling fifth, so a decending third root motion is Half-Progressive in nature. It takes two falling thirds to get the voices to the same point as a single falling fifth does as well, as I shall demonstrate shortly. In the analysis a half-progression is labelled as ".5P." Its complimentary opposite, the rising third, is therefore a half-regression, and so is ".5R" in the analysis layer.

Another set of root motions in the diatonic system are the rising and falling seconds. With these root motions we will get parallel perfect fifths or parallel perfect fourths (Depending on the voicing and inversion) in the upper voices during any of these transformations which have perfect fifths (Or, perfect fourths) between the root and fifth and the third and seventh of the adjacent sonorities. This is not a problem: These parallisms are simply the natural result of these root motion/transformation combinations. Parallel perfect fifths are only "wrong" if the voices involved do not transform: In other words, if the root remains the root and the fifth remains the fifth, or alternately, if the third remains the third and the seventh remains the seventh. In these instances the voices move in a clockwise or counterclockwise circular transformation (Depending on the root motion direction), so they are technically correct. These root motions really mimic leading-tone and leaning-tone resolutions, except that the root is not treated as an active tone (Much more on this when we get to the secondary dominant galaxy of sonorities). For this reason I call the rising second Super-Progressive and the falling second Super-Regressive. They get "SR" and "SP" respectively in the analysis.

The final root motion type is that by tritone. Obviously, the tritone can mimic progressive motion or regressive motion depending upon whether the motion is from the subdominant to the leading tone, or vice versa. In the former instance it mimics a progressive motion and gets "Ptt" (For Progressive tritone) in the analysis; in the latter it mimics regressive root motion and receives an "Rtt" (For Regressive tritone).

Instead of demonstrating these root motion types in sterile isolation, I have created some simple musical continuities. I will develop these throughout the course of this series until some of them will eventually wind up in the simple composition examples at the conclusion.

Whoever said, "Writing about music is like dancing about architecture." was a fool, by the way. I have found that the better I can explain music, the better I can compose music. The reason is simple: Being able to explain music clearly means you have a better understanding of it. Better understanding of music leads to better compositions.

In the first five examples we will look at triadic continuities, and in the second five tetradic continuities over the same root progression patterns, as the transformations are different depending upon how many voices are present in the upper stratum. For triadic transformations there are only two possibilities: Clockwise and counterclockwise. In a clockwise transformation the root of the first chord becomes the third of the second, the third of the first chord becomes the fifth of the second, and the fifth of the first chord becomes the root of the second. A counterclockwise transformation is simply the opposite of this.

In Example I I have arranged all of the chords in the diatonic system in progressive relationships with each other. After a super-regressive motion from I to vii(d5), all of the remaining root motions are progressive until the super-progressive motion from IV to V in the final measure. Note that progressive root motions result in counterclockwise transformations. This is always the case in a triadic context. Super-regressions transform clockwise, while super regressions transform counterclockwise: Opposite root motion types always result in opposite transformational directions as well.

Note also that the voice leading is not totally smooth in the SR and SP motions: There are skips of thirds present. Pure transformational harmonic voice leading will always be totally smooth and stepwise if the implications of the series are followed, so obviously the series implies a four part transformational texture where either the root of triads will be doubled, or seventh sonorities will be complete.

I have demonstrated the complimentary opposite correlation in Example II, where after two half-progressive motions all the rest of the root movements are regressive. Also note that, in a triadic texture, progressive root motions move the voices higher in pitch through time, while in regressive root motions the voices subside over time. If you have a keyboard handy, you ought to play these examples without the ties (Which are only present to demonstrate the number of common tones between sonorities).

With our admittedly incomplete triadic upper stratum (According to the wishes of the series), Super-progressive and Super-regressive root motions result in zero common tones, Progressive and Regressive root motions result in a single common tone, and Half-progressions and Half-regressions result in two common tones. The percieved abruptness or smoothness of the root motion/transformation combination depends heavily on this, naturally.

Example III and Example IV demonstrate the same yin and yang principle for the half-progressive and half-regressive root motions respectively. As I said, not only does it take two half-progressive motions to move the root down by a progressive fifth, but it takes those two steps to transform the voices as far as a single falling fifth as well. If you'll note, the I vi IV motion in Example III would result in exactly the same voicing if it went directly from the tonic to the subdominant.

The final continuity, Example V, contains every possible root motion type available in the diatonic system with the exception of a Regressive tritone movement. From measure four to five is the Progressive tritone, and that divides the continuity into a pair of four measure antecedent/consequent phrases. Note that the root motion types are mirrored between the antecedent and consequent: Retrogression is answered by Progression; Super-progression is answered by Super-regression; the Half-progression is answered by a Half-regression; and finally, the Super-progression of the final measure turns the phrase around on itself. This is a perfectly balanced phrase, as the voice leading makes an uninterrupted loop. The previous phrases - with their preponderance of the same types of root motion - are very unbalanced. Obviously, with unbalanced phrases you can run into problems of range if you are not careful or do not know how to manage the rising or falling tendencies. The solution is to alternate between three and four voices in the upper stratum (Triadic and tetradic episodes, respectively), as these also yeild a yin and yang correlation: If the triadic transformation raises the voices, the tetradic transformation will lower them, and vice versa.


In Examples VI through X I have presented the previous root progression continuities with a four voice upper stratum. Whereas with three voices the transformations are always circular - either clockwise or counterclockwise - with four voices crosswise transformations become possible. I have made all of these transformations directly with no interrupting triad with doubled root to demonstrate the numbers of common tones between the adjacent sonorities. With the fourth voice, SR and SP motions now have a single common tone, P and R motions have a pair of common tones, and in .5P and .5R motions only one voice moves in the transformation. This is the actual texture that is implied by the harmonic overtone series itself, and you can use the intermediary triads with doubled root at your discression: It is only really imperitave to use them when the progressive motion is from an overtone sonority, and even then it is not positively mandatory (As you can see from the sixth to the seventh measure of Example VI).

Earlier I alluded to the fact that some aspects of super-progressive and super-regressive root motions might be considered problematic. In three voices, treating the root as a real root (Versus treating it as an active third over an absent real root - more on this concept later) results in the skip of a third in the transformation with either of these root motions. In four voices we get parallel perfect fifths or perfect fourths, depending on the voicing and inversion. In these close position strata, parallel perfect fifths are produced, as you can see in the first and last measures of Example VI. As I said before: Since the voices are transforming, this is not a problem: It would only sound "crude" if the voices did not transform. Please understand that I am not intentionally dissing jazz voice leading, in which non-transformational progressions are an integral aspect of the style, but I can not help but point out that non-transformational progressions are not the most attuned to the implications of the series.

I would be remiss if I did not note here that I discovered the elegant simplicity of this transformational logic in The Schillinger System of Musical Composition, and some of the latter ideas in this series come from Joseph Schillinger as well. Though quite controversial - and for many good reasons - Schillinger was one of those flawed geniuses whose work has never been either properly presented, or properly understood. Since some of his students compiled The System after his death, I have no doubt but that he would have presented the material differently, but the real problem with Schillinger's ideas is that, while a few of them (Like this transformational logic) are sublime, many of them are simply ridiculous: There is a lot of crap one must wade through in The System in order to discover the pearls. The reason for this is simple: Schillinger often lost sight of the implications of the harmonic series and went off, wild eyed, into nonsensical and anti-musical speculation. His musical examples also sucked in the worst way (Unless they were not his, of course).

Again, you ought to play these examples without the ties on a keyboard and compare them with the triadic versions. I can explain most of the key features, but there is no substitute for educating your own intuition through actual interaction with the implications of the series.


Yes: Give me a girl who likes to fish.


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