Monday, September 20, 2010

Why Music Works: Chapter Five

The Harmonies of the Secondary Dominant Contextual Sub-System

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Five:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems.

Chapter five will take us out of the diatonic harmonic world and into the chromatic realm as we discover the origins of the secondary dominant sub-system sonorities.



As the intuition of composers began to exhibit a more complete understanding of the overtone sonority's implications, they began to employ it for effect targeting degrees other than the tonic as secondary dominants. This process started with the nearest secondary dominants - V(m7)/V and V(m7)/IV - and progressed roughly a step at a time through V(m7)/ii and V(m7)/vi until finally the most remote secondary dominant, V(d5m7)/iii was reached. As we shall see, that V(d5m7)/iii - functioning first as V(d5m7)/V in the minor mode - unleashed entirely new classes of secondary dominant sonorities, many of which have not been properly described until I began to organize and classify them about five years ago (Though I figured out that the French Augmented Sixth was a V(d5m7)/V in minor when I was a doctoral candidate circa 1995).


Listen to Example 20

To turn a I(M7) chord into a V(m7)/IV secondary dominant, all that is required is to lower the seventh by a semitone from a major seventh to a minor seventh, and to change a ii(m7) chord into a V(m7)/V secondary dominant, all that is required is to raise the third by a semitone from a minor third to a major third. This second formula - raising the third from minor to major - also works for V(m7)/ii and V(m7)/vi, but a funny thing happens if you apply it to the vii(d5m7) chord: You end up with a V(d5m7)/iii.

Though I have never been able to nail this down with any degree of certainty, since the common practice minor mode was based on Alpha 6, - the Aeolian mode, where the vii(d5m7) from Ionian is the ii(d5m7) chord - it seems most likely that the historical origins for this sonority were as a V(d5m7)/V - or v - in minor.

Since in the evolution of western art music counterpoint preceded harmony, the augmented sixth effect targeting the dominant degree that results from the major third above a diminished fifth was already known, so the orientation of the V(d5m7)/V when it first appeared was in second inversion. This is the so-called classic French Augmented Sixth sonority. Unfortunately, this historical baggage combined with a ridiculous and utterly non-descriptive name relegated the V(d5m7) sonority to the level of an obscure and difficult to understand curiosity. The same is true to an even greater degree with the so-called German Augmented Sixth sonority, because it actually has the intervallic structure of an overtone chord with enharmonic notation. As I shall demonstrate in this chapter, both of these sonorities are just altered secondary dominants, and their historically limited use is unfortunate and was unnecessary: Understanding this chapter will give any composer vastly increased sonic resources.


1. This is example ten, which was example seven contextualized, with secondary dominants added.

All of these are the standard secondary dominants except for the V(d5m7)/iii. That chord could also be made into a standard secondary dominant by raising the fifth a half-step, but for the genesis of the other secondary dominant sonorities, I left it in its most natural state.

2. The V(d5m7)/iii is the point of origin for the altered dominant that generated Gamma Prime.

Remember, Gamma Prime is best described as a Phrygian mode with a major sixth and a major seventh, and Phrygian is the Alpha 3 mode here.

3. The V(d5m7)/iii is also the point of origin for the so-called "French Augmented Sixth" sonority.

4. The traditional so-called "French Augmented Sixth" chord is just a V(d5m7) in second inversion.

5. Historically, the V(d5m7)2nd probably first appeared in minor, where the ii(d5m7) became V(d5m7)2nd/V.

As you've certainly guessed by now if you've followed this series from the beginning, I have attempted to keep as much terminology and symbology from traditional classical and jazz theory as possible to make these concepts accessible to anybody trained in those disciplines, only modifying them and adding to them enough to properly describe the musical phenomena I'm defining. One set of modifications to the symbology is that I've replaced any arcane symbols with what can easily be found on a QWERTY keyboard, and another has been to at long last eliminate the figured bass formulations from inversions, as that old nomenclature is ponderous and confusing, even to me sometimes: It is much easier to understand a French sonority as V(d5m7)2nd - the 2nd meaning second inversion - than as a V(4/2/b).

6. The V(d5m7)2nd is just a naturally occuring altered dominant, available on any degree that can support a secondary dominant, and in any inversion.

We will see this in example twenty-one.

7. The diminished fifth in the V(d5m7) is an active leaning tone, replacing the passive perfect fifth in the overtone sonority.

8. Adding this additional active tone increases both the tension, and the resolution effect.

Once the intuition of composers lead them to discover that they could increase the resolutional desire of the overtone sonority by replacing a passive tone with another active tone, several new sonorities were created. In this case, the resolutional impetus is at 150% of normal, whereas later examples will completely double it.

9. To avoid parallel octaves, the root of a chord must always be a passive tone.

That is the case with the traditional French chord, but the upcoming fully-diminished seventh and German chords have no real root present in them, as we shall see.



Listen to Example 21

This is example twenty with secondary V(d5m7) on every degree until the final cadence, so as you can see, there are far more options for employing these chords than any of the traditional composers ever realized.


1. This is example twenty with diminished fifths added to the secondary dominants.

2. A V(d5m7) can reside on any degree that can host a secondary dominant.

3. The traditional "French" terminology does not properly describe the function - or the origin - of these chords.

4. The traditional "French" terminology limits these chords to only one of four possible orientations, the second inversion.

5. The traditional "French" terminology is ridiculous, and must be abandoned.



Listen to Example 22

This is example twenty with minor ninths added to the secondary dominants.

Another set of secondary dominant function sonorities that are not taught properly are the so-called secondary fully-diminished seventh chords. What is usually called the root of these sonorities is actually a leading tone, so it's active and can't be the real root. The real root is always a major third below the leading tone, and so it's missing if all you are presented with is the symmetrical structure that consists of nothing but minor thirds (It's still OK to describe it as a fully-diminished seventh in that situation, so long as you realize that's just describing the structure, and not the function). The true function of the secondary fully-diminished seventh chord is as a secondary dominant with a minor ninth and no root: (Root), M3rd, P5th, m7th, and m9th. Since the root has to be a passive tone, the 9th in the transformational stratum is replacing the root with an active leaning tone. When looked at this way, the normal delayed crosswise transformation that secondary dominants make is not changed: 9 > 5, 5 > R, 7 > 3, and 3 > 7 after the resolutional interruption.

As with the V(d5m7) - the sonority formerly known as French - there are now three active tones in the upper stratum instead of two: The fifth is still passive... except for in the case of the V(d5m7m9)/iii that starts things off here. That sonority is the one traditionally described as a German Augmented Sixth, and all four voices in the transformational stratum there are active: (Root), M3rd (leading tone), d5th (leaning tone 1), m7th (leaning tone 2), and m9th (leaning tone 3).


1. This is example twenty with minor ninths added to the secondary dominants.

2. The V(d5m7m9)/iii is the point of origin for the so-called "German Augmented Sixth" sonority.

3. The traditional so-called "German Augmented Sixth" sonority is just a V(d5m7m9) without a root, with the diminished fifth under the leading tone (To get the augmented sixth interval instead of the diminished third heard here), and the minor ninth in the bass (To lead into a second inversion sonority, and so avoid the parallel perfect fifths that result from the normal transformation of this chord, as we have here).

4. Historically, the V(d5m7m9/0) probably appeared first in minor, where the ii(d5m7) became V(d5m7m9/0)/V.

5. The V(d5m7m9/0) has the same intervallic structure as an overtone chord, except it is spelled enharmonically.

This has caused tons of confusion about the nature and function of this sonority. Basically, if you notate the D-sharp in the second measure above enharmonically as an E-flat, the transformational stratum is an F(m7)3rd chord. That coincidence - happy though it may be - has no bearing whatsoever on the functions of the notes in the chord: The F isn't a passive root, it's an active diminished fifth.

Nonetheless, generations of jazz musicians have been taught that errant way of looking at the chord through the so-called "substitute secondary dominant" theory: I know, because I was one of them. In that theory, to cite a single example, the V(m7)/I in C - a G(m7) sonority - can be replaced by a subV(m7)/I - which is a Db(m7) chord. Though expedient and simple - and certainly superior to the German terminology - this just isn't the way in which the overtone sonority implies that these chords are generated. The classical notation is correct, but its description is useless, while the jazz notation is incorrect, but at least its terminology is useful.

6. None of the notes in the traditional spelling can be the real root, however, because all of them are active.

7. The V(m7m9) chords also often appear without roots as so-called secondary fully-diminished seventh chords.

8. (If you're keeping up, you know this is not correct above: The notated third in a fully-diminished seventh is a passive 5th in the V(m7m9) chord from which it comes. - Geo) The notated root in a fully-diminished seventh chord is an active leading tone, so it can't be the real root.

9. A secondary fully-diminished seventh chord is properly understood as a V(m7m9/0): the root is simply missing.

10. Since real roots must be passive, minor ninths replace roots with active leaning tones.

11. With the minor ninth as a root substitute, the interrupted crosswise transformation is normal.

As with the V(d5m7) sonorities, the V(d5m7m9) chords can live on any degree that can carry a secondary dominant too.



Listen to Example 23


1. This is example twenty with diminished fifths and minor ninths added to the secondary dominants.

2. A V(d5m7m9) can reside on any degree that can host a secondary dominant.

3. The traditional "German" nomenclature does not properly describe the function of these chords.

4. The traditional "German" nomenclature limits these chords to only one or two of four possible inversions.

5. The traditional "German" nomenclature is ridiculous, and must be abandoned.

6. Parallel perfect fifths result from the transformations of these chords: This is normal.

That is why there was often a so-called I(6/4) chord between the V(d5m7m9/0) and the tonic triad in common practice music; to avoid the parallel perfect fifth.

All of these secondary dominant types can also transform directly. While this maintains surface tension by always presenting a seventh chord, it sounds slippery and strange because the chromatically inflected leading tones are thwarted, and return to their non-inflected diatonic state instead of resolving. For me, it's an effect best used sparingly, but the rest of the examples in this chapter are 20-23 above with direct transformations.



Listen to Example 24


1. This is example twenty with direct transformations of the secondary dominants.

2. Direct transformations are less natural sounding, as the leading tones are not resolved.

3. Direct transformations maintain greater surface tension, since a seventh chord is always sounding.

4. Choosing between directs or interrupts comes down to the effect/affect desired.



Listen to Example 25


1. This is example twenty-one with the V(d5m7) chords transforming directly.



Listen to Example 26


1. This is example twenty-two with the V(m7m9) chords transforming directly.


Listen to Example 27


1. This is example twenty-three with the V(d5m7m9) chords transforming directly.

I have managed to work all of the secondary dominant types into this systematic presentation with the exception of the so-called Augmented Seventh chords, where the passive perfect fifth is replaced by a minor sixth. I have that worked out, but not in this example set, so we'll now go on to secondary subdominants.


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