Wednesday, October 11, 2006

Musical Implications of the Harmonic Overtone Series: Chapter III


The Secondary Dominant System


Today's music theory hat tip goes out to
Dr. Gene Cho, who I studied under while I was a Doctoral Candidate at The University of North Texas in the early to mid 1990's. Dr. Cho's slim monograph, entitled Theories and Practice of Harmonic Convergence is, in my opinion, one of the greatest music theory books of all time. Under Dr. Cho, I finalized my conception of integrated modality, though it took about ten more years for me to work out my present terminology, and to relate it all back to implications present in the inherent desire of the musical force in the harmonic series.

Dr. Cho also gave me the spark of insight which lead directly to my understanding of the secondary dominant galaxy of sonorities, which I will present in this chapter. Though I had begun to gain a grasp on the standard secondary dominants, the so-called substitute secondary dominants, and the so-called secondary diminished seventh chords as far back as the late 70's while at The Guitar Institute SW under Jackie King and Herb Ellis, it was Dr. Cho who made me realize that the more exotic so-called French and German Augmented Sixth chords were part of the secondary dominant system as well. Though I could never get Dr. Cho to agree with me that these sonorities originated with implications present in the harmonic overtone series - he stuck with his notion that they were purely contrapuntal in origin (Which is correct in the historically specific sense, but incorrect in the theoretical sense) - it was nonetheless Dr. Cho who is directly responsible for my ability to present these sonorities and their origin clearly.


Example I is the original continuity proof again, only this time it contains all of the standard secondary dominant seventh chords. In fact, I wrote that phrase with this and some of the following demonstrations in mind.

Just as harmony itself was intuited backwards from cadential points, so were the secondary dominants. Since the primary dominant is an overtone chord resolving to a tonic target, it was a simple step for composers to decide to target that primary dominant with an overtone chord of its own, and so the secondary dominant concept was born out of intuited implications present in the harmonic overtone series.

Over time, overtone chords appeared on all of the diatonic degrees which could move in a progressive manner. So, from the original V(m7)/I the series progressed to the V(m7)/V, V(m7)/IV, V(m7)/ii, V(m7)/vi, and finally, the V(m7)/iii. Note that for the V(m7)/iii I have left the f-natural so that the sonority is an overtone chord with a diminished fifth. This is the actual point of origin for the so-called French Augmented Sixth sonority, though it actually first appeared in the guise of a V(+4/3/b)/V in the minor mode (Where the vii(d5m7) is the ii(d5m7) chord). As I said, I could never get Dr. Cho to recognize this, but this is what it is: The traditional "French" sonority simply being the second inversion which yeilds the augmented sixth interval.


By understanding these sonorities as simply being overtone chords with diminished fifths (In a particular inversion per traditional use), they become available as sonic resources on any degree which can have a secondary dominant. I have demonstrated this in Example II.

Now, by starting out in closed root position, these modifications yeild diminished thirds versus augmented sixths: This is not an issue. Both inversions are available, and though they have decidedly different sonic effects, both are usable depending on the sonority and result the composer desires. This gives much more freedom and many more resources than the stilted and clumsy traditional way of looking at these chords.

But, what is it that is actually happening here? By that question I mean, what feature of the overtone chord is it which allows and even encourages this modification of the diminished fifth? The answer to that goes back to the active tone/passive tone dichotomy within the overtone structure: By diminishing the fifth, you are changing it from a passive tone which neither desires to rise, nor desires to fall, into an active leaning-tone which desires to resolve down by semitone to the root of the target chord. This second diminished fifth only magnifies the energy present in the natural version of the overtone chord (But, it does not double the energy, because the root remains a passive tone, and therefore a real root).

Finally, it should be noted that the French-derived sonority consists of two tritones a whole step apart, and therefore, harmonically speaking, it generates a hexa-tonic whole tone scale.


The so-called secondary diminished seventh chords present a new problem: Here, the lowest note in the chord spelling is an active tone - a leading-tone - and therefore it cannot be the real root of the chord: Roots must always be passive tones. The lowest tone in the spelling of a fully diminished seventh chord is what is called a theoretical root. The real root of the chord is a major third below the leading tone, and its presence will create an major overtone sonority with a minor ninth added to it.

It should be noted that, because the fully diminished seventh sonority is a perfectly symmetrical structure (It is a circle of minor thirds), without the presence of the real root, any one of its tones can function as the leading tone, or theoretical root. In context, it is easy to tell which tone ought to be the theoretical root, but in actual point of fact, it is not until the resolution that the suspicion is confirmed or denied. This gives the fully diminished seventh two admirable properties: 1) It momentarily suspends the tonal/modal system's functionality and sense of direction (Or, it at least puts it in some doubt, depending on the prolongation of the sonority), and 2) it allows for some unexpected modulations. Composers have been taking advantage of these properties since just before Bach's time.

In the two examples above I have presented all of the so-called passing diminished sevenths. Example III presents those from the tonic to the mediant, and Example IV presents those from the subdominant to the leading tone: Anywhere there is a whole step in any mode of the moment, one of these chords can be inserted. The root motion indicators are in parentheses because the real roots would not be present except for the analysis.

Note what happens in the transformations: The seventh of the first chord becomes the ninth of the V(m7m9/0), thereby taking the place of the root momentarily. In keeping with its functional nature as a root-substitute, the ninth then resolves back to the fifth, and so the expected delayed crosswise transformation still takes place.

Also take note of the fact that - as with the French-derived sonority - the fully diminished sonority contains two tritones. However, with the fully diminished seventh the tritones are a minor third appart, and they are both fully active. Again, the harmonic series allows for these sonorities because it is only magnifying or doubling (In this instance) the natural leading-tone/leaning-tone force of the original tritone of the overtone chord.

This sonority also harmonically generates a non-diatonic scale: In this case the octa-tonic diminished scale, also called a 1 + 2 scale, because it consists of alternating semitones and tones.

The final little nugget in Example IV is that I have targeted the vii(d5m7) with a fully diminished seventh chord: Although the Locrian mode cannot function as an independent mode, there is nothing keeping the diminished minor-seventh sonority from being targeted by a secondary dominant function chord within a tonal or modal context. I don't recall ever seeing this in any Classical or Romantic works, though I'm sure I'm not the first composer to realize this possibility.


Once I had the theoretical root versus real root problem worked out, I decided to apply it to another set of secondary dominant function chords that have been historically problematic: The so-called German Augmented Sixth chords. Us jazzers were introduced to these sonorities as substitute secondary dominants, and their theoretical justification was that they shared the same tritone with the primary dominant, but their root was a tritone above. In jazz this chord does not transform in the manner predicted by the implications of the series - in fact, it does not transform at all since all nominal chord functions remain the same between the two sonorities, the first root simply sliding down by semitone to the target root - but in traditional music this sonority is problematic. First of all, the sonority in and of itself is an overtone chord, but it is usually spelled with some enharmonics, and it does not appear to resolve in the functional manner that you would expect an overtone chord to do. This appearance is an illusion: It actually does resolve perfectly in keeping with the implications of the series.

Key to understanding this chord is, again, understanding that the root of its non-enharmonic spelling is an active tone, and so it simply cannot be the real root: It must be handled as a theoretical root. The real root is again a major third below the leading tone of the non-enharmonic spelling of the sonority, which in this case is actually the minor seventh. Adding the real root reveals the true nature of the so-called German chord as a V(d5m7m9) secondary dominant (Or primary dominant).

In Example V I have presented some of these sonorities in a way which makes their origins as just another altered dominant sonority clear: All you have to do to produce one of these chords is to start with a fully diminished seventh chord, and then flat its theoretical third (Its real fifth if the real root is in the bass). Another way to look at it is as a French-derived sonority with a minor ninth replacing the root.

In the transformational analysis you can again see that the ninth is substituting for the root momentarily in the upper stratum, only this time it is from a crosswise motion versus the previous clockwise. The ninth still resolves properly back to the fifth after the delay.

For my final parlor trick, I have put German-derived sonorities on all of the degrees which can carry standard secondary dominants using the original harmonic continuity. With this root motion pattern - all strong motion between real roots - the root itself moves up to become the ninth. resolutions are still properly crosswise after the delay, of course.

I cannot stress enough how important it is for you to play these. With and without the bass parts.

The so-called doubly-augmented fourth augmented sixths are just a set of secondary dominants in another enharmonic spelling of the German-derived sonorities: They resolve to an intermediary major triad in second inversion (The tonic, traditionally, but it could be a major triad on any degree) before proceeding on to the dominant. I don't feel the need to present those in this series, but I will in the final book.

Also, the so-called Italian Augmented sixth is just a French-derived sonority with a missing root. It is usually found in triadic textures, and so is a bit of a sick man among the augmented sixths. I feel no particular need to present those at this point either.

Finally, Frederick Chopin was fond of an augmented sixth sonority which had a major ninth over the real root, which made for a 6-5 (Or, 13-12) resolution over the target. I've never used those myself, but he (And Wolf as well) got some good mileage out of them. The main point to reaize, of course, is that all of these types of sonorities are simply altered forms of the standard secondary dominants, and so are perfectly well predicted by implication from the overtone series.


I'm not sure which would be more dangerous: The undertow or those curves.


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