Why Music Works: Chapter Eight

Root Motion and Transformation Types in the Delta, Epsilon, and Zeta Systems

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 Eight:

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 then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. Previously, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Last time, in chapter seven, we went back a bit and looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and now in chapter eight we'll look in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.

*****

CHAPTER EIGHT:

EXAMPLE 40:



Listen to Example 40

Here we have the progressions and regressions in Delta Prime, using the same end-contextualized musical proofs I've presented earlier for Alpha, Beta and Gamma. One nice thing about using musical proofs that the reader can listen to, is that I really don't have to explain very much now that all of the elements of the contextual system concept have been previously presented. All of these transformations are direct, as the point now is to hear the various harmonies in isolation, so as to hear their uniqueness. The bVI(A5M7) is a hot sonority, and being entered via quadra-tone and exited by tritone - or the other way around in the case of the regressions - really puts it in stark relief here.

EXAMPLE 41:



Listen to Example 41

It really is difficult to even notice the augmented second movements in the transformational stratum unless you are careful to listen for them.

EXAMPLE 42:



Listen to Example 42

As I pointed out above, unusual harmonies more or less alternate with more normal seventh chords in this example. That's a nice resource. Now, on to the Epsilon system.

*****

EXAMPLE 43:



Listen to Example 43

This is the system normally referred to as harmonic minor, so some of these effects may be familiar to you. Since this should all be old hat now, I'll firmally dispense with the observations, unless something truly unique arises.

EXAMPLE 44:



Listen to Example 44

EXAMPLE 45:



Listen to Example 45

Now, on to the Zeta system, which is like melodic minor with a Phrygian minor second.

*****

EXAMPLE 46:



Listen to Example 46

NOTE: The D in the penultimate measure of the top system appears as a D-natural instead of the D-flat it ought to be. I corrected this when I was proofing the audio examples, but I'd already uploaded the JPEG files by then. So, it looks wrong, but it sounds right. A big part of this post series is to work the kinks out of the examples and more perfectly define the presentation order (More on that in a few minutes).

Note also that in the progressive root motion from the bIII(A5m7) - normally called an augmented seventh chord - that there is the augmented second in the transformation as the B-natural moves down to A-flat: This is why the so-called augmented seventh chords do not fit into the secondary dominant galaxy of sonorities - the augmented fifth has to move an augmented second down to get to the new root.

There are some very wicked sounding sonorities in this system. The vii(d3d5d7) is particularly cool.

EXAMPLE 47



Listen to Example 47

EXAMPLE 48



Listen to Example 48

Lots more interesting sonic resources in the exotic systems, but wait until we get to the alien systems (Those with two augmented seconds). I worked those out today - I could swear I did that before - and they are really, really creepy. One of the sub-contexts is the so-called Arabian or snake charmer scale, and that system creates some very bizarre sonorities. Next time, however, we are going to look at harmonic canons.

I think I have the final chapter outline done for the book now. Instead of presenting the secondary dominant sub-system and the secondary subdominant sub-system together, as I've done in this series, I'm going to break them up like so.

01] The Harmonic Series: Its Structure, Forces, and Primordial Resolution
02] Genesis of the Native Diatonic Contextual Systems: Alpha, Beta, and Gamma
03] Root Motion and Voice Transformation in the Native Diatonic Contextual Systems
04] Sonorities of the Secondary Dominant Contextual Sub-System
05] Genesis of the Exotic Diatonic Contextual Systems: Delta, Epsilon, and Zeta
06] Root Motion and Voice Transformation in the Exotic Diatonic Contextual Systems
07] Sonorities of the Secondary Subdominant Contextual Sub-System
08] Genesis of the Alien Diatonic Contextual Systems: Eta, Theta, and Iota
09] Root Motion and Voice Transformation in the Alien Diatonic Contextual Systems
10] Harmonic Canons, Musical Escher Morphs, and Musical Mobius Loops
11] Genesis of the Hybrid Nonatonic Contextual Systems: Kappa, Lambda, and Mu
12] The Integrated Chromatic Contextual Systems: Chi, Psi, and Omega

One thing I wanted to avoid, was putting all of the contextual systems together at the beginning. Not only can it get tedious that way, but spacing them out lends itself to the built-in review device that I like to use when teaching.

*****

Why Music Works: Chapter Seven

The Exotic Diatonic Contextual Systems: Delta, Epsilon, and Zeta

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 Seven:

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 then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. Previously, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

At this point, in chapter seven, we will go back a bit, in a way, by looking at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and then in chapter eight we'll look in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.

*****

CHAPTER SEVEN:

The Delta diatonic contextual system is created by an overtone sonority resolving to a major tonic, and then to a minor subdominant. Since the rules for diatonic resolutional genesis call for retaining the inflection of notes that appear in previous harmonies, that means that the subdominant minor triad carries a hotly dissonant major seventh (m3, P5, and M7 create an augmented triad in this chord).

EXAMPLE 34:



Since resolutional genesis is now a familiar concept, audio examples will be saved for the progressive orientation examples from here on out.

OBSERVATIONS:

1. Augmented seconds are perfectly acceptable in harmonic transformations.

2. A minor, major-seventh chord makes a perfectly acceptable subdominant harmony.

3. Delta Prime is often called, "harmonic major" but Ionian minor sixth is more descriptively accurate.

4. The Delta Prime tonic scale is, 2, 2, 1, 2, 1, 3, 1: Two whole steps are adjacent at the beginning of the mode.

*****

The Epsilon diatonic contextual system is created by an overtone sonority resolving to a minor tonic, and then onto a minor subdominant.

EXAMPLE 35:



OBSERVATIONS:

1. Epsilon Prime is often called, "harmonic minor" but Aeolian major seventh is more descriptively accurate.

2. The Epsilon Prime tonic scale is, 2, 1, 2, 2, 1, 3, 1: Two whole steps are adjacent in the middle of the mode.

*****

The Zeta diatonic contextual system is created by an overtone chord with a diminished fifth resolving to a minor tonic, and then into a minor subdominant.

EXAMPLE 36:



OBSERVATIONS:

1. Zeta prime is often called, "Phrygian harmonic" but Phrygian major seventh is more descriptively accurate.

2. The Zeta Prime scale is, 1, 2, 2, 2, 1, 3, 1: The three whole steps are adjacent.

*****

Here are the displacement modes of the Delta contextual system.

EXAMPLE 37A:



Four harmonies of the Delta system can be arranged in progressive order.

EXAMPLE 37B:



Listen to Example 37B

As you can hear, the augmented second in the final transformation does not sound overly strange.

OBSERVATIONS:

1. Delta or Epsilon can be considered as a point of origin for the V(m7m9) - vii(d5d7) sonorities.

Plainly, these harmonies are an artifact created through the genesis of these systems.

2. As with Beta, the Delta system has three independent and three dependent sub-contexts.

3. A m(d5d7) sonority can function perfectly well as a dominant with progressive root motion.

That would be as in Delta 3 above.

*****

Here are the displacement modes of the Epsilon contextual system.

EXAMPLE 38A:



Four harmonies of the Epsilon system can be arranged in progressive order.

EXAMPLE 38B:



Listen to Example 38B

If you were into 80's and 90's virtuoso rock guitar, you might even detect a little of the feel of the background for some of Yngwie Malmsteen's stuff in that, as I do. He was very fond of Epsilon Prime aka "harmonic minor."

OBSERVATIONS:

1. Like Beta and Delta, Epsilon has three independent and three dependent sub-contexts.

*****

Here are the displacement modes of the Zeta contextual system.

EXAMPLE 39A:



Only three harmonies of the Zeta system can be arranged in progressive order.

EXAMPLE 39B:



Listen to Example 39B

OBSERVATIONS:

1. Delta, Epsilon, and Zeta each have three independent and three dependent sub-contexts.

2. The Delta, Epsilon, and Zeta systems add an additional 21 modes to the 21 that resulted from the genesis of the Alpha, Beta and Gamma systems.

So, we're up to a total of 42 diatonic modes now: 21 normal modes and 21 exotic modes.

3. Three additional contextual systems are available with two augmented seconds.

As I mentioned previously, I thought I had worked these out, but I can't locate them now, so I'm not positive; there may be only two more. In any event, there will be a brief intermission as I create some more examples, because this is the end of the examples I created back in 2008 before I moved from Alpine to San Antonio. When we do continue, it will be to look at and listen to the various root motion and transformation types that Delta, Epsilon, and Zeta exhibit, using the same proofs we used for Alpha, Beta, and Gamma.

Since this post will put chapter one off of the home page, there is now a section in the sidebar for this series.



It's all drama with some girls.

Why Music Works: Chapter Six

The Harmonies of the Secondary Subdominant 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 Six:

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 then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities.

Now in chapter six, we will look at the secondary subdominant sub-system of harmonies. Whereas in the secondary dominant sub-system the harmonies were all overtone chords or altered overtone chords, in the secondary subdominant sub-system the harmonies are actually different genders of chords: Major sevenths, minor sevenths, and dominant sevenths.

*****

CHAPTER SIX:

Secondary dominants are more well known than secondary subdominants, but both contextual sub-systems offer sonic resources that the composer ought to be aware of. The intuition of many rock music writers has lead them to the secondary subdominant major triads over the years - most notably to me, Pete Townshend of The Who - and this is partly explained by the fact that major triads sound so good with overdriven guitar amps. But, secondary subdominant major triads also produce a unique sonic environment that can't be duplicated in any other way. For example, The Real Me from The Who's Quadrophenia album is just loaded with them - as are many of the songs in that rock opera concept - and this is exactly where I started to figure them out when I was back in high school: I was positively addicted to those records.

Traditionally, the secondary subdominant major triads and major sevenths have been justified through the concept of modal interchange, which is borrowing harmonies from modes parallel to whichever one you have nominated as home. While this is a very useful compositional concept - I use it all the time - in this chapter we will see how the origin of these harmonies is better explained by the progressive resolutional paradigm established by the overtone chord.

In the modal interchange model, if we are in Alpha Prime - traditional major, or Ionian - the idea is to borrow the subdominant Lydian chords from the other Alpha System parallel modes. With I(M7) as the Ionian tonic, we get the bII(M7) Lydian chord from the parallel Phrygian, the bIII(M7) from the parallel Dorian, the IV(M7) is the native Ionian subdominant, bV(M7) comes from Locrian, the bVI(M7) comes from Aeolian, and finally, the bVII(M7) comes from Mixolydian. This creates a hybrid system of all subdominant chords surrounding a tonic much like the secondary dominants create a hybrid system of all dominants surrounding a tonic (And you can just as easily justify secondary dominants through modal interchange too). Later, when we look at integrated chromatic contextual systems, we will find that the secondary subdominants progressively leading away from the tonic create a grand subdominant preparation for the most remote of the secondary dominants, which then lead back to the tonic. Taking musical gravity into account, this creates a descending barber pole loop that spirals ever downward.

For now, we will simply look at the different genders of secondary subdominant, and where they came from.

EXAMPLE 28:



Listen to Example 28

As you can now see, the secondary subdominants are also generated by the resolutional paradigm established by the primordial resolution of the overtone sonority. Previously, with the secondary dominants, we were seeing overtone chords and altered overtone chords on the diatonic degrees progressively moving toward the tonic, whereas here, we see Lydian harmonies progressively leading away from the tonic, and so into the chromatic realm, at bVII(M7) and on. Where the connection between the secondary dominant sub-system and the secondary subdominant sub-system occurs is at the enharmonic progressive root motion that would be from the bV(M7) - the most remote of the secondary subdominants - to the V(d5m7)/iii - the most remote of the secondary dominants. Since G-flat is the enharmonic of F-sharp, this would lead to B-natural in the Alpha Prime on C here. As I mentioned above, we will look at this when we get to integrated chromatic contextual systems.

OBSERVATIONS:

1. The secondary subdominant major seventh chords are Lydian sonorities extending progressively away from the tonic.

2. This is opposed to the secondary dominant Mixolydian sonorities and altered Mixolydian sonorities, that approach the tonic progressively.

3. Upper structure tones for secondary subdominant major sevenths are: Major ninth, augmented eleventh, and major thirteenth.

4. As the iv(m7) chord in the penultimate measure shows, secondary subdominants can also me minor seventh chords.

Obviously, the minor subdominant was a better choice here because of all of the flatted notes in the bV(M7), which would have produced two augmented seconds going into a major chord on the fourth degree with the normal clockwise transformation that a super-regression carries.

5. Secondary subdominant major sevenths are generated by Alpha Prime (Ionian/Pure Major).

6. So, secondary subdominant minor sevenths will be generated by Alpha 6 (Aeolian/Pure Minor).

7. Therefore, secondary subdominant minor sevenths will be Dorian sonorities.

*****

EXAMPLE 29:



Listen to Example 29

This is example twenty-eight in the Alpha 6 independent sub-context.

OBSERVATIONS:

1. Secondary subdominant minor seventh chords are Dorian sonorities extending progressively away from the minor tonic.

2. This is opposed to the secondary dominant Mixolydian sonorities, which approach the tonic progressively.

3. Upper structure tones for secondary subdominant minor sevenths are: Major ninth, perfect eleventh, and major thirteenth.

4. The direct half-step resolution of the enharmonic V(d5m7m9)/V justifies the jazz subV7 practice.

Just as I was providentially able to present a minor subdominant in example twenty-eight, I was able to work in a V(d5m7m9)/V harmony here (The point of origin for the German Augmented Sixth in paleo-terminology). That sonority is enharmonic because of the tied G-flat, which would otherwise be an F-sharp. The resulting structure in the transformational stratum therefore reads like an A-flat dominant seventh chord, which jazz theorists call a subV7/V (Which shares its tritone with the dominant on D-natural), and all four notes move down by semitone into the primary dominant in this direct transformation. So, a jazz substitute secondary dominant resolving down by semitone in parallel is actually a V(d5m7m9/0) notated enharmonically and transforming directly. Pretty funny.

5. However, that direct resolution of the enharmonic V(d5m7m9)/V also nullifies jazz subV7 theory.

6. Additionally, that resolution of the V(d5m7m9)/V also nullifies traditional "German Sixth" theory.

If you could even say that the traditional German Augmented Sixth nomenclature rises to the level of a theory.

7. Both traditional and jazz theories are wrong here, but at least the traditional notation is correct.

8. Secondary subdominants can also be overtone sonorities generated by Beta Prime.

This is, in a way, out of bounds for the pure secondary subdominants generated by Alpha Prime and Alpha 6, but since Alpha 6 is combined with Beta Prime to create the traditionally so-called melodic minor nonatonic hybrid contextual system, I thought I'd go ahead and present them.

*****

EXAMPLE 30:



Listen to Example 30

This is example twenty-eight with secondary subdominant overtone chords.

OBSERVATIONS:

1. Secondary subdominant M(m7) chords are Mixolydian Augmented-fourth sonorities.

2. These are the chords actually generated by the harmonic series to partial eleven.

3. Secondary subdominant M(m7) chords are not considered dominant because they don't target degrees of Alpha Prime.

4. In context, if these chords target degrees of Alpha 6 or Beta Prime they may be considered dominant.

This is the way many paleo-theorists have done it in the past, but since Alpha Prime is, well, Alpha Prime, no overtone chord that doesn't come from a natural degree of Alpha Prime targeting a natural degree of Alpha Prime is a functional dominant harmony.

5. All of the altered dominant forms are available as subdominants as well.

6. The secondary dominant and secondary dominant sub-systems join to create a downward spiral of overtone sonorities.

7. That downward spiral of overtone chords covers the entire circa 144 semitone range of human hearing.

8. That pattern is imprinted in the subconscious of every human being, probably while they are still in the womb.

This why in absolute music - where music creates its own context - there has to be a musical contextual system or sub-system present that is independently functional. I'll present an example demonstrating this at the end of the first part of this book that describes the complete harmonic system. It really does sound like something that is deeply engrained, perhaps even at the genetic level.

*****

EXAMPLE 31:



Listen to Example 31

Of course, these examples can also transform directly.

OBSERVATIONS:

1. This is example twenty-eight with the harmonies transforming directly.

2. The continuous high surface tension provided by constant M(M7) chords is quite dissonant.

*****

EXAMPLE 32:



Listen to Example 32

OBSERVATIONS:

1. This is example twenty-nine with the harmonies transforming directly.

2. The increased surface tension provided by continuous m(m7) harmonies is much more mellow.

*****

EXAMPLE 33:



Listen to Example 33

OBSERVATIONS:

1. This is example thirty with the secondary overtone chords transforming directly.

2. While the M(M7) and m(m7) transformations were diatonic from chord to chord, these include chromaticism.

As the major thirds descend to minor sevenths in the target chords, as we also saw with direct transformations of secondary dominants.

3. The constant chromatic side-slipping creates a vaguely jazzy effect.

Though Mozart was no stranger to side-slips.

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.

*****

CHAPTER FIVE:

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).

EXAMPLE 20:



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.

OBSERVATIONS:

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.

*****

EXAMPLE 21:



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.

OBSERVATIONS:

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.

*****

EXAMPLE 22:



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).

OBSERVATIONS:

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.

*****

EXAMPLE 23:



Listen to Example 23

OBSERVATIONS:

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.

*****

EXAMPLE 24:



Listen to Example 24

OBSERVATIONS:

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.

*****

EXAMPLE 25:



Listen to Example 25

OBSERVATIONS:

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

*****

EXAMPLE 26:



Listen to Example 26

OBSERVATIONS:

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

EXAMPLE 27:



Listen to Example 27

OBSERVATIONS:

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.

Why Music Works: Chapter Four

Root Motion and Voice Transformation Types in the Beta and Gamma Contextual Systems

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 Four:

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 is dedicated to examining how Beta Prime and Gamma Prime compare to Alpha, using the same musical proof formats.

*****

CHAPTER FOUR:

EXAMPLE 14:



Listen to Example 14

Example fourteen is the same as example eleven, but with the inflections necessary to put it into the Beta Prime system. Obviously, these are end-contextualized, and are comparing the progressive resolutions to their opposite regressive versions.

The first of these root motions - the one from vii(d5m7) to bIII(A5M7) - is labelled PA5, for Progressive Augmented Fifth (This could also conceivably be labelled a Pd4 for Progressive Diminished Fourth, but since falling fifths are the most natural progressive root motions, I've decided to stick with fifths here). [As you'll see, my terminology eventually evolved into calling these quadra-tones. - Geo] This is a surprising and uncanny effect, because of both the root motion and the structure of the target harmony. Immediately following that - into the vi(d5m7) - we get a Progressive Tritone, which is labelled Ptt in the analysis. This root motion can actually be found in the Alpha system when moving from IV(M7) to vii(d5m7), but since the original intent of this example was to put all seven harmonies of Alpha into normal progressive order, we have not seen that yet.

After these initial histrionics - a very nice resource to affect the listener into the realm of the uncanny - the rest of the progressive relationships are relatively normal.

Since the second system is essentially the first system in reverse, the strangeness occurs near the end there. When I created this example back in 2008, I had still not completely nailed down the way I wanted to treat the progressive and regressive augmented fifths, so there is an Rd4 in the analysis, but the final version will say Rqt there for regressive quadra-tone. Note that the Ptt in Alpha is IV(M7) progressing to vii(d5m7) while the Rtt there is moving from vii(d5m7) to IV(M7): Both root motions are by tritone, but one is progressive and the other is regressive. The same thing applies in PA5 versus RA5.

OBSERVATIONS:

1. Overtone chord progressions imply falling fifths, hence, Progressive Augmented Fifth, Crosswise.

2. Overtone chord regressions imply rising fifths, hence, Regressive Augmented Fifth, Crosswise.

3. Progressive and regressive tritone root motions are perfectly usable in proper context.

4. Progressive and regressive augmented fifth root motions are perfectly usable in proper context.

5. Since augmented and diminished fifth root motions yield uncanny effects, employ accordingly.

6. The tonic minor/major seventh also evokes the uncanny.

7. The lowered mediant degree augmented/major seventh sonority also evokes the uncanny.

8. the harmonies and root motions possible in the Beta System provide new expressive resources.

*****

EXAMPLE 15:



Listen to Example 15

Example fifteen is the same as example twelve, but inflected to put it into Beta Prime.

OBSERVATIONS:

1. Beta system .5P's and .5R's contain less dramatically uncanny effects than the P's and R's did.

2. This is because the descending and ascending thirds smooth out the tritones and quadra-tones.

*****

EXAMPLE 16:



Listen to Example 16

This is example thirteen inflected to put it into Beta Prime

OBSERVATIONS:

1. Beta Prime SP's and SR's are again less uncanny than the system's P's and R's are.

In this case, all root motion and transformation is stepwise, hence the smoothness.

*****

EXAMPLE 17:



Listen to Example 17

Now we are ready to use example eleven/fourteen, example twelve/fifteen, and example thirteen/sixteen inflected into the Gamma System. By this point in creating these musical proofs, my terminology for the augmented fifth progressions and regressions had evolved to the point of referring to them as quadra-tones, which compares better to the tritone analysis symbols: Pqt and Rqt respectively, to fit in better with Ptt and Rtt. by this point, you should understand the analysis symbols well enough that the proofs become self-explanatory.

OBSERVATIONS:

1. Gamma System progressions and regressions contain even more uncanny sounding effects.

2. The areas of alternating quadra-tone and tritone root motions sound particularly sinister.

Obviously, these are gnarly sonic resources.

*****

EXAMPLE 18:



Listen to Example 18

Here are the half-progressions and half-regressions in Gamma. Note that I neglected to parenthetically denote the diminished thirds for the vii(d3d5m7) sonorities here. They are still notated properly - and so they sound correct - so that is just an oversight on my part.

OBSERVATIONS:

1. Gamma System .5P's and .5R's again contain fewer dramatically uncanny effects then the system's P's and R's do.

2. This is again, as before, because the .5P's and .5R's smooth out the tritones and quadra-tones.

*****

EXAMPLE 19:



Listen to Example 19

OBSERVATIONS:

1. Gamma Prime SP's and SR's are again less dramatically uncanny sounding than the system's P's and R's are.

2. Again, the fact that all root motion and transformation is stepwise aids smoothness.

3. Nonetheless, the Gamma System is filled with unsettling harmonic effects, which is an effective and affective resource.

Now that we have seen and heard the basic resources of the normal diatonic Alpha, Beta and Gamma contextual systems, it is time to enter the chromatic realm with secondary dominants and secondary subdominants as derived from the Alpha system.

Why Music Works: Chapter Three

Root Motion and Voice Transformation Types in the Alpha Contextual 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 Three:

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.

Here we will look at the contextualization of Alpha Prime, the various different root progressions types it can exhibit, their various transformation types, and through this we will also begin to peer into the world of musical affect and effect.

*****

CHAPTER THREE:

EXAMPLE 10:



Listen to Example 10

Example ten is the contextualization of example seven, which was simply the ordering of the seven harmonies present in the Alpha System into a progressive order. To contextualize example seven into Alpha Prime, all that is required is to put a tonic harmony at the beginning, and a V(m7) P_ I at the end. The root motion form I(M7) to vii(d5m7) at the beginning - down by step - is called a Super-Regression, which gets the symbol SR in the analysis, and the voices transform in a clockwise - > - manner: R > 3 > 5 > 7 > R. I will present these in detail later in this chapter.

After that beginning contextualization, the harmonic continuity is just like example seven until measure eight, where the IV(M7) moves up into a V(m7) before the ending progressive resolution. The subdominant to dominant root motion - up by step - is the opposite of that at the beginning, and is called a Super-Progression. it gets SP in the analysis, and the voices transform in a counter-clockwise - < - manner: R > 7 > 5 > 3 > R. Opposite root motion types will always have opposite transformation types if they are circular and not crosswise.

Note also that Super-Progressions and Super-Regressions can produce parallel perfect fifths, as they do here into measure two, and into the final measure: This is normal. The biggest problem with traditional voice leading as it has been historically taught is that it is an amalgam of harmony and counterpoint, and not harmony isolated into its pure state, as we have here. In pure harmonic transformations, parallel perfect fifths sometimes result, and they are not a problem. In fact, it is the most organic and natural way of things, as the smooth transformational logic proves. Once you know this, the centuries of agonizing over whether to allow parallel perfect fifths in homophonic music becomes positively funny.

Another thing to notice is that in these root motions that are not Progressive, the transformations are direct, with no interruption as the Progressive root motions produce. The fact is, Progressive root motions can support direct transformations too, which I have demonstrated with the same continuity rendered that way on the second system: All of the Progressive transformations are direct until the final resolution to the tonic triad. Remembering that "P_+" is a Progressive Interrupted Crosswise Transformation, that becomes simply "P+" for Progressive Crosswise Transformation (The "P_" at the end is simply Progressive Interrupted Transformation, which is what you'll always see at perfect endings).

OBSERVATIONS:

1. In root motions other than progressive, transformations are direct, with no interruptions.

2. Progressive root motions may also support direct transformations.

3. The instant the overtone chord resolution varies, the realm of musical affect and effect is entered.

Affect and effect are nominally breached with all non-dominant progressive root motions that we've seen so far in the diatonic systems. One overtone sonority transforming to another through the chromatic system would be the default pure natural succession, and we'll look at that later. But really, when you allow for direct transformations over progressive root motions and especially non-progressive root motions, that is where the manifold harmonic effects that can effect the listener arise, and those are what we will be looking at for the rest of this chapter and the next.

4. Super-Regressions transform in a clockwise, circular manner: R > 3 > 5 > 7 > R.

5. Super-Progressions transform in a counterclockwise, circular manner: R > 7 > 5 > 3 > R.

6. Opposite root motions will always have opposite transformational directions, unless crosswise.

7. Direct transformations maintain surface tension by always presenting complete seventh chords.

This is one of the best features of direct transformations, as continuously interrupting the transformations sounds like a series of final-type resolutions, even if they are less perfect modal variants of the primary overtone sonority's resolution. Deft use of direct versus interrupted resolutions is one of a composer's basic resources for producing expressive effect.

8. musical contextualization is provided by beginnings and endings, or at least endings.

9. Omitting a contextualizing beginning can be an effective resource for affecting the listener.

Even in these examples, the beginning tonic harmony is a seventh chord, and so a stable context is not initially provided. In the following examples we will see how it is really the ending that provides a pure musical contextual definition. Obviously, starting the listener out in a foreign land, so to speak, and bringing them home can be a great musical plot device. This can be done by providing no initial context, or a false one.

10. Super-progressions and super-regressions can result in parallel perfect fifths, which is normal.

It is humorous, in retrospect, to view the conniptions some composers went to in order to avoid this effect in eras past.

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EXAMPLE 11:



Listen to Example 11

In these examples, I have only contextualized the endings to demonstrate observation nine above. Here, we also begin to look in detail at the types of root motion other than Progressive. The opposite of a progressive root motion is regressive, and it gets an "R" in the analysis (Many theorists have called these retrogressions in the past, but I prefer the simple yin/yang of two tri-syllabic words). This is what I have presented on the second system. There is also the root motion of a descending third into the penultimate measure of the second system, which is called a half-progression, and it gets .5P in the analysis. We'll see these in isolation in example twelve.

OBSERVATIONS:

1. Progressions move the transformational stratum lower.

2. Regressions move the transformational stratum higher.

3. Half-progressions and half-regressions result in three common tones between harmonies.

4. Progressions and regressions result in two common tones between harmonies.

5. Super-progressions and super-regressions result in one common tone between harmonies.

Composers need to know this, because the number of common tones between harmonies - note I'm referring to the transformational stratum - is what gives the effect of smoothness versus abruptness in the various root motion types.

6. The leading tone cannot be treated as a real root in a final resolution to a tonic triad.

7. The leading tone can be treated as a real root in an intermediate resolution to a tonic seventh chord.

If we were to attempt to move from the vii(d5m7) to I at the end of the second system, parallel octaves would result because both leading tones would move to the root of the tonic. While parallel fifths are fine in transformations that produce them in the upper stratum, parallel octaves are not between the two strata, for the simple reason that two voices are transforming the same; 7 > 1. Therefore, the leading tone cannot be treated as a real root in a final resolution (In an intermediate super-progression, of course, the upper stratum leading tone is held as a common tone, so it is 1 > 7). Discovering this was a much bigger deal than I initially thought, as it allowed me to figure out so-called secondary diminished seventh chords and also "German" augmented sixths, neither of which contain a real root, because all four tones are active: The root must be a passive tone.

8. Progressions (Not regressions as the observation mistakenly says) result in an incremental decrease of intensity, akin to musical gravity.

9. Regressions result in an incremental increase of intensity, akin to musical anti-gravity.

The phenomena of musical gravity and anti-gravity are quite real, as these examples demonstrate, and along with musical gravity comes a decrease of intensity as the pitch level of the transformational stratum lowers, while musical anti-gravity - or propulsion - brings with it an increase in perceived intensity as the pitch level of the transformational stratum rises. These are also effects the composer must be aware of.

10. Context can be nebulous or even missing at the beginning, so long as it is present at the end.

Again, unless an extra-musical context is provided, such as in a film score, where the scene creates the context.

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EXAMPLE 12:



Listen to Example 12

Now were ready to look at the isolated half-progressions and half-regressions. As you can see and hear, these root motion types are very smooth sounding due to all of the common tones involved.

OBSERVATIONS:

1. It takes two half-progressions to move the bass as far as one progression.

2. Two half-progressions transform the voices exactly the same as one progression.

3. It takes two half-regressions to move the bass as far as one regression.

4. Two half-regressions transform the voices exactly the same as one regression.

I did not give the root motion types arbitrary names. Rather, starting with the resolution of the overtone sonority as a normative progression, I compared all other types to it, and named them logically: The opposite of a progression is a regression, so half of a progression or a regression is exactly that.

5. Half-progressions are musical gravity moving at half speed.

6. Half-regressions are musical anti-gravity moving at half speed.

7. Direct octaves occur between the bass and a transforming voice in half-progressions: This is normal.

Half-regressions do not have this feature, because the bass moves into a voice that is tied in the upper stratum. This is important to note, as in a half-progression, the root of each new chord is a new note that did not exist in the previous harmony, while in a half-regression the bass is not a new note, but one already established in the previous harmony. This is one of the features that produces the different effects between the two root motion types.

8. Half-progressions transform clockwise, and half-regressions transform counter-clockwise.

Missing above, but I'm going to recreate these in Sibelius anyway for the final version, I think.

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EXAMPLE 13:



Listen to Example 13

Finally for this chapter, here are the super-progressions and super-regressions. Despite the stepwise smoothness of the bass, these root motion types sound quite abrupt because there is only a single common tone between the adjacent harmonies.

OBSERVATIONS:

1. Two super-progressions move the bass as far as three progressions.

2. Two super-progressions transform the voices down as far as three progressions.

3. Two super-regressions move the bass as far as three regressions.

4. Two super-regressions transform the voices up as far as three regressions.

5. Super-progressions are musical gravity moving down at 1.5 times normal speed.

6. Super-regressions are musical anti-gravity moving up at 1.5 times normal speed.

For 5 & 6, this despite the direction of the bass line.

7. Super-progressive transformations can result in parallel perfect fifths: This is normal.

8. Super-regressive transformations can result in parallel perfect fifths: This is normal.

9. Super progressions transform counterclockwise, super-regressions transform clockwise.

We have now seen all of the root motion types in the Alpha Prime system with the exception of the tritone root motion that occurs between IV(M7) and vii(d5m7). This root motion type will be encountered in Beta Prime and Gamma Prime - where they are unavoidable if we follow the pattern of the examples presented in this chapter - so we will examine those, and more, in chapter four.

Why Music Works: Chapter Two

Analysis of the Modal Sub-Contexts of the Alpha, Beta, and Gamma Systems

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 Two:

In chapter one, we looked at the structure of the harmonic series and found that it is what theorists call, for good reason, a dominant seventh chord: A major triad with a minor seventh. After removing the fundamental generator and its superfluous perfect twelfth, I identified the basic musical forces that reside within the tritone - the leading tone and leaning tone impetuses - and demonstrated how these forces imbue the overtone sonority with a desire for resolution. We found that with the five part texture of pure harmony, the resolution creates a delayed or interrupted crosswise transformation in the upper stratum, and with a single further diatonic resolution the Alpha Prime contextual system is created. Then we saw how with a resolution to a minor triad the Beta Prime diatonic contextual system results, and finally, how by starting from a V(d5m7) - the point of origin for the ridiculously so-called French Augmented Sixth sonority - that the Gamma Prime contextual system results.

Here in chapter two, we will look at the independent and dependent sub-systems of Alpha, Beta, and Gamma. Furthermore, I will demonstrate the primacy of Alpha Prime, the comparatively limited nature of Beta, and the outright flawed character of Gamma.

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CHAPTER TWO:

EXAMPLE 7A:



On the top system, I have presented all of the harmonies of Alpha Prime in standard notation. Underneath, I have put the analysis symbols in the format I will be using. A capital Roman numeral will represent a major triad, and a small case Roman numeral will be assumed to be a minor triad, unless there is a further indication for the fifth, as is the case for vii(d5m7) in the final measure, where the fifth is diminished. The parenthetical indicators are for alterations to the triad, descriptions of the seventh, and also any upper structure tones that may be present: M, m, A, d represent Major, minor, Augmented, and diminished, respectively. This results in more detail than is often presented: For example, what is usually called a V7 in most other systems is a V(m7) here. I prefer this more perfect logic and level of detail. Finally, under the chord analysis symbols is the tone/semitone pattern for Alpha Prime: 2, 2, 1, 2, 2, 2, 1. Obviously, this is the traditional major mode specifically called Ionian.

When I was coming up, diagrams almost precisely like example 7a were given at or near the beginning of harmonic theory. This really does a disservice to the student, as it can lead to the incorrect notion that scales generate harmony, when the truth - as I demonstrated in chapter one - is the other way around: Harmonic continuities generate the scales or modes.

With this in mind, I have demonstrated how the displacement modes of Alpha Prime are best represented by their less perfect resolutional paradigms that imitate the original: Alpha independent sub-context 2: Dorian is defined by v(m7), i(m7), and IV(m7); Alpha independent sub-context 3: Phrygian is defined by v(d5m7), i(m7), and iv(m7); Alpha independent sub-context 4: Lydian is defined by V(M7), I(M7), and #iv(d5m7); Alpha independent sub-context 5: Mixolydian is defined by v(m7), I(m7), and IV(M7); Alpha independent sub-context 6: Aeolian is defined by v(m7), i(m7), and iv(m7); and finally, the Alpha dependent sub-context 7: Locrian is defined by bV(M7), i(d5m7), and iv(m7). The Locrian mode is the only dependent sub-context in the Alpha system because it does not contain a perfect fifth, and so it has no proper dominant function harmony, and the i(d5m7) is not stable enough to provide a proper conclusion. In a chromatic versus a diatonic context, it is perfectly possible to target a (d5m7) harmony with an overtone sonority - or any other a perfect fifth above - so it is primarily the dissonant nature of the sonority that renders it unable to function on its own. Within a larger independent context, of course, Locrian effects are a perfectly fine resource.

EXAMPLE 7B:



Listen to Example 7B

The Alpha System allows for all seven harmonies to be put into a progressive order. As we shall see later, the construction of the Beta and Gamma systems do not allow for this. Here is one factor that explains the primacy of Alpha. Additionally, all of the diatonic degrees from vii(d5m7) up to and including the tonic can carry secondary dominant harmonies, as I shall demonstrate later. Furthermore, from the primary subdominant of IV(M7) on, the secondary subdominants can continue the cycle into the chromatic realm. Ultimately, a barber pole loop consisting of the primary subdominant and secondary subdominants leading away from the tonic into the chromatic realm will prepare for the most remote of the secondary dominants leading back to the primary dominant and then the tonic. That will be the ultimate musical proof, whereas this is the initial.

OBSERVATIONS:

1. All master contexts will have functional dominant, tonic, and subdominant harmonies.

2. All independent sub-contexts will have functional dominant and tonic harmonies.

3. A non-functional subdominant harmony does not destroy the independence of a sub-contextual mode.

4. A dominant harmony with a diminished fifth is functional.

5. Only one sub-contextual mode of the Alpha System is contextually dependent: The Locrian mode.

6. A string of harmonies in progressive order creates a double harmonic canon in the upper stratum.

This isn't glaringly obvious without some elaboration of the parts, but - within the upper stratum - the soprano voice follows the tenor by a measure at the fourth above, and the alto follows the bass at the same distance and interval. This is such a cool phenomenon, that I'll devote an entire future chapter to it.

7. A string of harmonies in progressive order lowers the two strata.

As you can see, as the voices progressively transform, they get lower. This is what I call musical gravity, and it is the same phenomenon that Heinrich Schenker discovered, but he never figured out exactly what he was looking at: He was seeing an artifact of music that has a preponderance of progressive harmonic relationships in it. Since progressive root motions are statistically the most common type in western art music, you'll get the 3, 2, 1's; 5, 4, 3, 2, 1's &c. It's no big deal, really, and I can't think of too many musical exercises more futile than Schenkerian analysis: While mildly interesting, it's laborious and doesn't really teach the composer anything of much practical value.

The Natural Laws of Pure Harmony:

A slightly elaborated list. I'm still figuring out how best to list, word, and present these observations.

1. Pure harmony consists of five total voices.

2. These five voices are divided into a four-part, close-position transformational stratum above a constant-root bass part.

3. All chords are in root position in pure harmony.

4. The upper stratum consists of complete seventh chords, or triads with a doubled root.

5. The upper stratum transforms in a crosswise or circular manner, depending upon the root motion type.

Number five is a bit of a peek ahead, as I will show the other root motion types at a later point.

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EXAMPLE 8A:



This example of the Beta Contextual System is in the same format I presented for Alpha. The top system is a simple lineal presentation of the harmonies, and then the six displacement modes are on the following three systems. Alpha Prime - the Ionian mode - is considered the model for comparison, so any deviations from the pattern established there are represented in the analysis symbols. For example, the third degree of Alpha Prime is major, so the harmony residing on the minor third degree of Beta is described as a bIII(A5M7): the small case "b" standing in for the flat symbol. This convention will be followed throughout.

The proper way to name Beta Prime is as a Dorian mode with a major seventh. Independent sub-context 2, therefore, is described as a Phrygian mode with a major sixth. The third sub-context in Beta, Lydian augmented fifth, is a dependent sub-context, because the augmented triad cannot function as a tonic, and therefore there is also no dominant or progressive relationship between #v(d5m7) and the tonic.

Beta 4 is again an independent sub-context, and it is properly described as a Mixolydian mode with an augmented fourth. As I mentioned previously, this is the actual scale created by the harmonic series to P11.

Because the tonic harmony for Beta 5 is an overtone sonority, it also has to be compared to the Mixolydian mode, which means it is a Mixolydian with a minor sixth. Likewise, Beta 6 & 7 have (d5m7) chords on the tonic degree, so they are best compared to the Locrian mode: Locrian major second in the case of Beta 5, and Locrian diminished fourth in the case of Beta 6: Of course, both Beta 5 & 6 are dependent sub-contexts.

EXAMPLE 8B:



Listen to Example 8B

NOTE: I had to use separate example scores, so yes, the first harmony in 8B starts with a triad moving into a seventh. I just noticed that, sorry.

Whereas all seven harmonies in Alpha could be arranged in a progressive order, here in Beta only five of them line up that way.

OBSERVATIONS:

1. An overtone chord can be a functional subdominant.

2. A minor.major seventh can be a functional dominant.

3. The mode created by the harmonic series to P11 would not allow the overtone chord to function as a dominant.

This means that, as far as music is concerned, the harmonic series is complete at P7.

4. The Beta System has only three independent sub-contexts.

5. Fully three Beta System sub-contexts are dependent on outside contextual definition.

6. The primacy of the Alpha System is demonstrated by the fact that all seven of its harmonies can be ordered in progressive relationships.

7. Traditional so-called melodic minor is a bi-modal combination of Alpha 6 and Beta Prime.

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EXAMPLE 9A:



Here we have the Gamma contextual system, which is the last of the normal diatonic systems generated by the harmonic system, normal being defined as systems consisting of two semitones, and five tones. Gamma Prime is best described as a Phrygian mode with a major sixth and major seventh since the minor second degree is what distinguishes Phrygian from the other minor modes in Alpha. The Gamma system is interesting and difficult to navigate because only one of its sub-contexts, Gamma 4, is independent: All the rest are dependent. Gamma 7 is particularly bizarre, because the tonic triad has a diminished third, a the rest of the scale contains diminished fourth, and a diminished fifth.

EXAMPLE 9B



Listen to Example 9B

Due to this structure, only three of the harmonies in the Gamma system can be arranged in a progressive order.

OBSERVATIONS:

1. The Gamma System has only one independent sub-context.

2. Fully five of the Gamma System's sub-contexts are dependent on outside contextual definition.

3. Only three Gamma System harmonies occur in progressive order.

4. Due to 1-3, the Gamma system is the antithesis of the Alpha system, and the Beta system lies in between.

5. The Gamma System ammounts to a hexatonic whole tone scale with one of the tones filled in.

6. The augmented fifth/minor seventh on bIII - commonly called an augmented seventh chord - is a byproduct of the genesis of the Gamma System.

7. Both of the altered dominants on bIII and V can be used in non-diatonic contexts.

More on this later.

8. Between the Alpha, Beta, and Gamma Systems, all normal diatonic resources are present.

9. The combined diatonic contextual resources available - independent and dependent - totals 21 modes.

10. Twelve of these modes are independent, while 9 are contextually dependent.

11. Three additional contextual systems are possible allowing for one augmented second.

I've worked these out, and will present them later.

12. A further three contextual systems are possible allowing for two augmented seconds.

I thought I had worked these out, but I can't find them, so I may be mistaken here. In any event, this is a topic for much later. The next chapter will be looking at and listening to the various root motions in the Alpha system in context.

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