UPDATE 09/18: As with the first post in this series, I found after re-reading it a few times that some clarifications are in order.
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In the first post on this topic, I followed the harmonic implications of the overtone series from the primordial falling fifth progressive dominant resolution, through diatonic triadic and tetradic tonality, and all the way to the complete series of secondary overtone chords (dominant seventh chords). I also showed the harmonic origins of the so-called French Augmented Sixth chord as a vii(dm7) chord with a raised third in second inversion (In actuality, the first derivation of this chord was from the ii(dm7) of the minor modality, but it is exactly the same paradigm). By freeing this chord from it's ridiculous nationalistic name, it can be employed on any of the degrees of the diatonic system as a V(d5m7) (And, in any inversion), which is where we'll start off today.
Example IV is again the original continuity study, but now the secondary dominants have had their fifths diminished. Note that this results in alternating secondary dominants containing diminished thirds and augmented sixths. There is no problem with this, and it sounds excellent, with or without the constant root bass part. This concept is so simple I'm going to leave it at that.
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Another set of chords which have been widely misunderstood, and therefore poorly taught, are the so-called secondary diminished seventh chords. One of the problems with this lack of understanding is, again, ignorance of the implications of the overtone series. What these chords actually are is a set of dominant seventh chords with minor ninths and missing roots. In the voice leading schemata, the ninths will simply temporarily take the place of the roots, as can be seen in the following examples.
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Example V - after an initial progressive root movement from I(M7) to IV(M7) - presents some of these "passing diminished sevenths" and, as you can see, there is no way the momentary leading tones can be considered as roots. To be considered as a root of a chord, the transformation must allow for a constant root bass part without running into parallel octaves. As you can see, the transformation does not allow for that. These momentary leading tones are exactly that, leading tones, and so are functioning as the thirds of modified overtone chords.
Therefore, to explain these sonorities with a logical unified theory, a root must be assumed a major third below these secondary leading tones. As is the case with the French Augmented Sixth-derived secondary V(d5m7) chords, this brings these fully diminished chords within the easily understandable realm of the secondary dominant system: Any fully diminished entity targeting any diatonic degree is simply a secondary dominant seventh with a minor ninth and a missing root.
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UPDATE 09/18: During the parenthetical half-progressive root motions, the clockwise transformations in the tetradic continuity go thusly: Root becomes third, third becomes fifth, fifth becomes seventh, and seventh becomes ninth, and that is indicated with the 7-> 9 symbol. That ninth, which momentarily replaces the root in the upper strata, functions as the root would in it's return to the fifth in the crosswise resolution after the momentary target triads, and that is indicated by the <-9/5-> symbol after the parenthetical progressive root motions.
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So, in measure three of example V there is the origin of this chord with the V(m7m9/0)/V. The degree targeting vi is presented in measure five, and then this example turns around with some good,old fashioned secondary dominant action.
Example VI introduces the V(m7m9/0)/ii and the V(m7m9/0)/iii before it too, returns to the tonic via "normal" secondary dominants. Obviously, these chords can be introduced chromatically anyplace there is a whole step, or modified versions of iii(m7) and vii(dm7) may be used as well.
Note that in the root motion analysis some of the symbols are parenthetical: This is where the roots are assumed, or are "theoretical." You can also see that in the voice leading continuity, the ninth temporarily replaces the root, and takes the root's function and returns to the fifth of the target chord, thus "saving" the crosswise transformation.
Now it's time to address that whole German Augmented Sixth/Substitute Secondary Dominant "thing."
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These are the same examples as before, but where the turn-arounds are, I have now introduced the so-called German Augmented Sixth/Substitute Secondary dominant sonorities.
These chords are highly problematic from a theoretical standpoint because structurally, they are overtone chords, but the roots are active tones, and therfore they cannot be the real roots. Once I figured out the fully diminished seventh deal, I applied the same process to these sonorities, and bingo: They also have a missing theoretical root a major third below the momentary leading tone. The resultant nomenclature gets a little ponderous, but they nonetheless integrate smoothly and inevitably into the secondary dominant galaxy.
The original "German" chord targeted the primary dominant, and you can see it in measure seven as V(d5m7m9/0)/V. In English that is a rather formidable mouthfull: "Five major, diminished-fifth, minor-seventh, minor-ninth, missing-root of five." Whew. But it is actually what the chord is as implied by the series.
Example VIII presents some so-called German chords targeting other degrees.
So as of just recently, I have a completely unified theory of secondary dominant function chords - regular secondary dominants, secondary dominants with diminished fifths (French), secondary dominants with minor ninths without roots (Fully diminished sevenths), and secondary dominants with diminished fifths and minor ninths and missing roots (German).
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The atomic energy at the heart of the overtone chord that creates the primary motivational force in music is the tri-tone. Both the leading-tone impetus and leaning-tone impetus are contained within this interval. Far from being the diabolus en musica, the tri-tone is God.
This pantheon of secondary dominant-function sonorities all "work" according to the implications of the overtone series, because what is happening in them is that the natural leading-tone/leaning-tone tendencies of the tri-tone are being amplified or even compounded. The regular secondary dominants intoduce tri-tones onto degrees which are naturally without them, which increases the resultant energy of the resolution over the diatonic versions. The French-derived secondary dominants increase the leaning-tone energy with the second diminished fifth they introduce (Only the tri-tone between the third and seventh is fully active though, as the root is a real root), and the secondary dominant function chords which are fully diminished seventh sonorities actually contain double tri-tones (which are both active), and that really makes the resolution pungent. Finally, the German-derived secondary dominant chords are counterfeit overtone chords in and of themselves, and the wildly "deceptive" nature of their resolution creates surprise as well as drive (It also introduces the oportunity for "normal" modulation a tri-tone away from the intended target).
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UPDATE 09/18: I should have pointed out more clearly that the French-derived secondary dominant chords have two tri-tones a whole step apart (And so they harmonically generate a whole-tone scale), and the fully-diminished seventh chords contain two tri-tones a minor third apart (And so they generate a dimished octatonic (1+2) scale).
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I hope you've enjoyed my "Grand Unified Theory of the Harmonic Implications of the Overtone Series." I've been working on it on-and-off for thirty years now, and I about have it perfect... but not quite.
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Now, about those Neapolitan Sixth chords...
We'll save Secondary Subdominants for another time.
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