Why Music Works: Chapter Eleven

Root Motion and Transformation Types in the Eta, Theta, and Iota 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 Eleven:

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. After the secondary dominants, 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.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

In the previous chapter, chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and with today's chapter eleven we will finish examining all of the nine master diatonic contextual systems and the total of sixty-three independent and dependent diatonic modes. All that is left is to look at and listen to the isolated root motion and transformation types.

*****

CHAPTER ELEVEN:

EXAMPLE 58:



Listen to Example 58

Here we have the progressions and regressions in Eta Prime, and you may find the effects other-worldly, as I do, which is what lead me to classify these contextual systems as alien: They are very, very far removed from the native systems, and even more foreign than the exotic systems.

EXAMPLE 59:



Listen to Example 59

These relatively smoother half-progressions and half-regressions can't do much to mitigate the strange harmonic effects that Eta Prime contains.

EXAMPLE 60:



Listen to Example 60

One of the problems with the augmented seconds is that - since there are two now - the listener is more likely to perceive them as minor thirds. That's pretty apparent in these super-progressions and super-regressions, where they are exposed in the bass.

*****

EXAMPLE 61:



Listen to Example 61

The Theta system is even stranger, as in addition to diminished thirds - which the listener is likely to perceive as major seconds - and the augmented seconds in the scale, we now have augmented thirds in the harmonies as well, which the listener will probably understand as perfect fourths. These distortions of native system realities can put listeners adrift, which is a nice way to affect them if that's the composer's desire. These systemic distortions of native reality are more compelling than the old techniques used in the serial systems of the century just passed, because a native system harmonic continuity can be progressively morphed into these alternate harmonic realities, and still retain their recognizability, as we will see in chapter twelve. One of the main criticisms of the various atonal methods is the, "any note could be replaced by any other note" feeling that listeners get, which is not an issue at all with systemic modifications from native, to exotic, and finally alien diatonic systems.

EXAMPLE 62:



Listen to Example 62

The funky thirds are quite apparent with the third movements in the bass with these half-progressions and half-regressions, but the connection to modal reality, though tenuous, is never entirely broken. This is far more effective than simply emerging listeners into total chaos, which they usually object to (And rightly so, in most instances). The exceptions, as always, involve situations such as film and stage vehicles, where there are extra-musical contextual defining elements at work. But for absolute music, especially, the alternate contextual morphologies available with exotic and alien systems are a great way to evoke the uncanny while still anchoring the listener to a modal locus.

EXAMPLE 63:



Listen to Example 63

The bizarre nature of Theta Prime is nicely exposed by the super-progressions and super-regressions.

*****

EXAMPLE 64:



Listen to Example 64

The minor tonic of Iota Prime adds even more darkness to the character of this alien system.

EXAMPLE 65:



Listen to Example 65

By the way, if you have been listening to all of the examples as we have progressed through the nine diatonic contextual systems, your brain is being hacked by them: Since almost nobody other than myself has ever listened to them systematically like this, I can tell you that this exposure will alter your conception of what musical reality can be. This is all leading up to the next chapter, where we will take an harmonic Mobius loop and morph it through all nine systems. If you have a musically sensitive mind, this will be very enlightening for you.

EXAMPLE 66:



Listen to Example 66

And so, all nine of the diatonic contextual systems have now been presented: Alpha, Beta, and Gamma for the three native systems; Delta, Epsilon, and Zeta for the three exotic systems; and finally Eta, Theta, and Iota for the alien systems. The total of sixty-three modal contexts and sub-contexts provide many more sonic resources than the old common practice guys were ever aware of, and even more than more modern jazz composers ever conceived of, all in a simple and intuitive system that can be applied to any diatonic theme a composer comes up with.

There are systems outside of the diatonic realm of course - which I have briefly alluded to previously - and we will begin to look at those in chapter thirteen.



I like that photo, even though she bears an uncanny resemblance to my ex-wife.

Read More

Why Music Works: Chapter Ten

The Alien Diatonic Contextual Systems: Eta, Theta, and Iota

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 Ten:

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. After the secondary dominants, 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.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. Last time, in chapter nine, I demonstrated a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

Chapters ten and eleven will be devoted to the alien diatonic contextual systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and with these chapters we will complete all nine master diatonic contextual systems and the total of sixty-three independent and dependent diatonic modes. Today's chapter ten will look at the genesis and structure of the alien systems.

*****

CHAPTER TEN:

EXAMPLE 52:



Listen to Example 52

The Eta Prime master context is generated by a V(d5m7) altered overtone sonority resolving to a major seventh tonic chord, and then on to a minor/major seventh on the subdominant degree. This yields two augmented seconds: One between the minor second and the major third in the lower tetrachord, and the other between the minor sixth and the leading tone in the upper tetrachord. I've heard this scale referred to as double harmonic major, but there are several double harmonic major modes in the exotic systems, so - since the tonic is a major seventh and the fourth degree is perfect - it is more precise to call it Ionian minor second, minor sixth.

Eta 4 deserves a mention here, as I first encountered this mode as double harmonic minor while back at Berklee in the early 80's. Since then I've also heard it called an Arabian scale or the snake charmer scale. It's actually pretty cool.

EXAMPLE 53:



Listen to Example 53

Theta Prime is generated by a V(d5M7) on the dominant degree - and now you can see why these systems become alien: There is no primary tritone in some of the dominant stand-in sonorities - which resolves again to a major seventh tonic, but now the chord on the raised subdominant degree is the very strange #iv(d3d5d7) sonority. Since the fourth degree is now augmented, that makes this a Lydian minor second, minor sixth (Which is a different species of, "double harmonic major").

EXAMPLE 54:



Listen to Example 54

For Iota Prime, we are now resolving the V(d5M7) sonority into a minor/major seventh tonic, while the raised fourth degree still supports a #iv(d3d5d7) chord. As you can probably guess, there are going to be some very strange effects within these alien systems.

*****

EXAMPLE 55:



Listen to Example 55 (The example is the harmonized scale only).

Eta prime has the rare feature of being an intervalic palindrome, as it's intervals read the same forwards or backwards: 1, 3, 1, 2, 1, 3, 1. In the alpha system, this honor goes to Alpha 2, which is the Dorian mode. Eta 2, Eta 3, and Eta 4 are nominally independent since there is an harmony on the unaltered dominant degree and a functional tonic triad, however they are not simple to establish independently in practice, but it can be done. The other three sub-contexts are obviously dependent.

EXAMPLE 56:



Listen to Example 56 (The example is the harmonized scale only).

I neglected to put the intervals in example fifty-six, but they go; 1, 3, 2, 1, 1, 3, 1. There is only one independent sub-context in the Theta system - Theta 3 (Mistakenly labeled dependent) - because of a new phenomenon that destroys a tonic triad: an augmented third in the case of Theta 2 and a diminished third in the case of Theta 7. Obviously, this is a difficult system to work in, even with the independent modal sub-contexts.

EXAMPLE 57:



Listen to Example 57 (The example is the harmonized scale only).

There is a good case to be made for calling Iota 6 the prime here, as it has an actual altered dominant as well as a natural fourth degree. If you'll notice, though, I organized all of the diatonic systems around C-natural in such a way as to start with the fewest accidentals, and add from there. In this case, Eta Prime only required a D-flat and an A-flat, Theta Prime added the F-sharp to that, and Iota Prime here got the additional E-flat. Since I just recently worked these alien forms out, I may rethink my organization before the final version.

I again forgot to put the intervals under the harmonized scale, but it's; 1, 2, 3, 1, 1, 3, 1.

Next time we'll break out the musical proofs to look at and listen to the root progressions and transformations in the alien systems.

Read More

Why Music Works: Chapter Nine

Harmonic Canons and Musical Escher Morphs

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. After the secondary dominants, 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.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.

Now, in chapter nine, I will demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

*****

CHAPTER NINE:

EXAMPLE 49:



Listen to Example 49

On the top system we have the end-contextualized diatonic direct transforming progressive root motion example that we first saw way back in example seven when we initially arranged the harmonies of Alpha Prime in progressive order. As I pointed out when we contextualized that continuity for example ten, an artifact of the constant progressive root motions is an harmonic canon; specifically, a double canon at the fourth above. This means that the harmonic series, progressing most naturally, produces canon: It is an entirely natural phenomenon.

The second system shows the extracted canon, which is still diatonic, and so it doesn't draw much attention to itself. If, however, we begin to embellish the diatonic version with secondary dominants and make all of the target chords minor, we get the strict canon on the third system. Penultimately, we can further adorn the canon with secondary V(d5m7) sonorities, as we have on the fourth system. Now it's very obviously a double canon. Finally, if we dovetail all of these versions together - diatonic, secondary dominant, and secondary V(d5m7) - we end up with the Musical Escher Morph on the fifth system.

I call these Musical Escher Morphs for reasons that should be obvious: They are a musical analog to this:



One harmonic form transforms into another over successive modulations of the root motion pattern. Realize that setting up this pure harmony version of the Musical Escher Morph is just the first step on the path to creating a final composition. Through further elaboration - which would take us into the realm of melody, and so is beyond the scope of this section of WMW - we could end up with something akin to Pachelbel's Canon in D, but much more modern and technologically proficient.

This repeating single-interval root motion is just the most basic kind of succession that creates harmonic canon as an artifact, however. Repeating root motion patterns of two intervals - like the one in Pachelbel's canon - also produce harmonic canons as artifacts.

*****

EXAMPLE 50:



Listen to Example 50

Here, on the top system, I have constructed a direct transforming harmonic continuity that consists of an half-progression alternating with a progression through diatonic Alpha Prime. The extracted diatonic canon on the second system reveals it as a four-part canon at the second above - not a double canon as before - but the continuity actually ends before the full canon is complete. When you have more than one root motion type, the transformations can allow each voice to play every part in the harmonies - root, third, fifth, and seventh - and so true four-voice canons can result.

The incompleteness of the diatonic canon coupled with the two intervals - descending minor third and ascending perfect fourth - presents me with the opportunity to create a two-interval twelve tone row for the bass part, and that creates the Musical Escher Morph on systems four and five (Sorry for the double bar line in the middle of that; just noticed). Since I introduced first secondary dominants and then secondary V(d5m7) chords in that one, there are interrupted crosswise transformations now, and so the four-part canon is at the unison. Again, this is just the skeleton of what the final canon could become through melodic elaboration, and yes, I plan to use this in a larger composition at some point. It's really quite wonderful.

*****

EXAMPLE 51:



Listen to Example 51

Justly, the most famous of all harmonic canons is Pachelbel's Canon in D, and the original continuity that Johann wrote is on the top system. This is a continuity of two root motion types as well, it being a regression followed by a super-progression. However, in the diatonic version the first super-progression is up by whole step, and the second is by half step. Now, Pachelbel composed this in the years just before J.S. Bach was born, so he didn't know anything about pure harmony, but it does tell us that the intuition of composers had figured out that repeating patterns in the bass could support canons as far back as three-hundred-twenty-five years ago. That's pretty amazing.

So, on the second system, I have converted Pachelbel's continuity into modern pure transformational harmony. This reveals to us that the underlying canon is at the sixth above - or third below - as we see on system three. If you are familiar with Pachelbel's version - and who isn't - you'll remember lots of parallel thirds and sixths, so at some level he figured this out too.

On the bottom two systems I have extended Pachelbel's continuity by making the original root progressions strict: Regression followed by super-progression of a whole step (Again, sorry for the double bar line there; I'll have to fix that for the final examples). This does not create a twelve tone row, as there are only eight pitch classes in the bass line, but it does create a direct modulation a tritone away - way gnarly - and that creates a very special kind of Musical Escher Morph that is also an Harmonic Mobius Loop.

Escher himself did a famous Mobius strip with ants.



I think I first saw that when I was about ten. Quite fascinating.

In an Harmonic Mobius Loop the root motion types are equalized so that musical gravity and anti-gravity are balanced out, and the end of the transformation runs back into the beginning. As you can see by comparing the first measure on the fourth system with the first measure on the fifth system - both over a I(M7) sonority - that is the case here. I break the strictness of the root motion pattern to bring the piece to an end, but I could just as easily repeat the first eight measures ad infinitum without the transformational stratum moving up or down at all. Yes, that's the plan for the final composition this will be in, as I plan to use this too.

We will look at Harmonic Mobius Loops more in chapter eleven, but chapter ten will present the three alien diatonic contextual systems of Eta, Theta, and Iota.

*****



Now that is an awesome redhead.

Read More

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.

*****

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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.

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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.

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