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.
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.
posted by Hucbald at 6:27 PM
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