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.
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.
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