Why Music Works: Chapter Thirteen
Genesis of the Extra-Diatonic Contextual Systems: Kappa, Lambda, and Mu
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 Thirteen:
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. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.
After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we now turn to the extra-diatonic contextual systems of Kappa, Lambda, and Mu. Extra-diatonic systems are those that lie between the seven note diatonic systems and the fully chromatic twelve note systems. I have mentioned one of these nonatonic - nine note - systems previously, and that is melodic minor, which will start our examples off in the present chapter.
*****
CHAPTER THIRTEEN:
EXAMPLE 69:
Listen to Example 69
As I mentioned way back in an early chapter, the Kappa contextual system is easily internalized as a bi-modal combination of Alpha 6 - the Aeolian mode - and Beta Prime, which is Aeolian with a major sixth and seventh. Traditionally, Beta Prime has been used as the ascending form, and Alpha 6 for the descending. These conventions were not, however, scrupulously followed in common practice music, as a descending line over a IV(m7) or V(m7) harmony required the ascending form and its attendant raised degrees.
That leads to the true way that nature tells us this system is generated, which begins with an overtone sonority resolving to a minor/minor seventh tonic, as we see in the first progressive resolution above. This yields fully seven of the nine required notes all on its own, since we are no longer hemmed in by the diatonic system's paradigm of retaining the inflection of notes that appeared in a previous harmony: The B-flat minor seventh over the tonic is now perfectly acceptable (And it sounds less hotly dissonant than the minor/major seventh as well).
With the seven note diatonic systems, all that was required was a single additional progressive resolution to the subdominant, and the system was completely generated. With Kappa Prime, however, the resolution to IV(m7) only adds the eighth note - A-natural - so an additional third progressive resolution to the secondary subdominant of bVII(m7) is needed to get the A-flat. At this point, the astute reader will realize that there is no reason to stop the progressive resolutions there, and that fully chromatic systems are also implied by nature's resolutional paradigm. We will examine the manifold possibilities of chromatic contextual systems at a later point.
*****
EXAMPLE 70:
NOTE: The example below has an error in it, as the primary subdominant should be an overtone chord. Unfortunately, I didn't discover that until I had uploaded everything, so I'll have to fix that for the final book. One of the points of doing this series of posts is to iron that stuff out.
Listen to Example 70
The Lambda contextual system is commonly heard in Flemenco music, and the simplest way for traditionally trained musicians to internalize it is as a bi-modal combination of Alpha 3 - the Phrygian mode - and Alpha 5 which is Mixolydian. In vernacular usage, the v(d5m7) dominant stand-in is seldom heard, but the bII(M7) subdominant function harmony is (Usually as a triad). Though nominally a decatonic - ten note - system, the singers and soloists - usually guitarists - adhere pretty regularly to the Phrygian mode for vocals and improvisation.
For the first progressive resolution here we have the previously mentioned v(d5m7) - which is native to Alpha 3/Phrygian - resolving to an overtone sonority on the tonic degree. This yields six of the ten required notes, and the further resolution to the IV(m7) subdominant - which is usually replaced by the Phrygian-native minor/minor seventh in this idiom - gets the system to nine notes. So once again, a further resolution into the secondary subdominant realm is required to pick up the final note, which is the D-natural that is native to Alpha 5/Mixolydian. As I stated earlier, in the idiomatic traditional employment of this system, the vast majority of the melodic elements adhere to the diatonic Alpha 3/Phrygian mode.
The acoustical reason that this bi-modal combination works is because the dissonant overtone sonority can support a wide variety of upper structure tones, such as the minor ninth, augmented ninth/minor third, and the minor thirteenth that Phrygian adds to Mixolydian here. This same ability of the overtone sonority to support a wide variety of tones in the upper structures is also responsible for our final Mu contextual system.
*****
EXAMPLE 71:
Listen to Example 71
The Mu contextual system is the modern king of all of the extra-diatonic systems, as it virtually defined the sound of the twentieth century through it's ubiquitous employment in blues music, which evolutionarily lead to jazz, R&B, and rock and roll. In fact, I start all of my students out with the blues - even if they wish to study classical music - for this very reason: You can't understand the music of the twentieth century without a solid foundation in the blues.
For the traditionally trained theorists among us, the easiest way to pocket an understanding of blues tonality is as a tri-modal combination of Alpha 2 - the Dorian mode - Alpha 5/Mixolydian, and Alpha Prime. In the idiomatic vernacular, what you see above is exactly what you get: Overtone sonorities on all three cardinal degrees. In fact, in the most basic blues and blues based R&B and rock and roll music, these are the only three chords you ever hear. By the time swing and bebop guys like Charlie Parker got ahold of the blues, however, it was elevated into another musical art form entirely, with many harmonic excursions within its brief twelve measure form.
Obviously, the overtone sonorities on the dominant, tonic, and subdominant correspond to Ionian, Mixolydian, and Dorian respectively. In idiomatic practice, blues, rhythm and blues, blues based rock, and blues based jazz singers and soloists employ a multi-layered combination of modalities, which consist of a minor pentatonic skeleton, the fleshed out diatonic Dorian, and then a virtually fully clothed chromatic sub-system of passing and approach notes for the bling. Blues is something a beginner can learn to play in a week, and then explore for a lifetime, which is the primary aspect of its charm and enduring appeal.
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 Thirteen:
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. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.
After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we now turn to the extra-diatonic contextual systems of Kappa, Lambda, and Mu. Extra-diatonic systems are those that lie between the seven note diatonic systems and the fully chromatic twelve note systems. I have mentioned one of these nonatonic - nine note - systems previously, and that is melodic minor, which will start our examples off in the present chapter.
CHAPTER THIRTEEN:
EXAMPLE 69:
Listen to Example 69
As I mentioned way back in an early chapter, the Kappa contextual system is easily internalized as a bi-modal combination of Alpha 6 - the Aeolian mode - and Beta Prime, which is Aeolian with a major sixth and seventh. Traditionally, Beta Prime has been used as the ascending form, and Alpha 6 for the descending. These conventions were not, however, scrupulously followed in common practice music, as a descending line over a IV(m7) or V(m7) harmony required the ascending form and its attendant raised degrees.
That leads to the true way that nature tells us this system is generated, which begins with an overtone sonority resolving to a minor/minor seventh tonic, as we see in the first progressive resolution above. This yields fully seven of the nine required notes all on its own, since we are no longer hemmed in by the diatonic system's paradigm of retaining the inflection of notes that appeared in a previous harmony: The B-flat minor seventh over the tonic is now perfectly acceptable (And it sounds less hotly dissonant than the minor/major seventh as well).
With the seven note diatonic systems, all that was required was a single additional progressive resolution to the subdominant, and the system was completely generated. With Kappa Prime, however, the resolution to IV(m7) only adds the eighth note - A-natural - so an additional third progressive resolution to the secondary subdominant of bVII(m7) is needed to get the A-flat. At this point, the astute reader will realize that there is no reason to stop the progressive resolutions there, and that fully chromatic systems are also implied by nature's resolutional paradigm. We will examine the manifold possibilities of chromatic contextual systems at a later point.
EXAMPLE 70:
NOTE: The example below has an error in it, as the primary subdominant should be an overtone chord. Unfortunately, I didn't discover that until I had uploaded everything, so I'll have to fix that for the final book. One of the points of doing this series of posts is to iron that stuff out.
Listen to Example 70
The Lambda contextual system is commonly heard in Flemenco music, and the simplest way for traditionally trained musicians to internalize it is as a bi-modal combination of Alpha 3 - the Phrygian mode - and Alpha 5 which is Mixolydian. In vernacular usage, the v(d5m7) dominant stand-in is seldom heard, but the bII(M7) subdominant function harmony is (Usually as a triad). Though nominally a decatonic - ten note - system, the singers and soloists - usually guitarists - adhere pretty regularly to the Phrygian mode for vocals and improvisation.
For the first progressive resolution here we have the previously mentioned v(d5m7) - which is native to Alpha 3/Phrygian - resolving to an overtone sonority on the tonic degree. This yields six of the ten required notes, and the further resolution to the IV(m7) subdominant - which is usually replaced by the Phrygian-native minor/minor seventh in this idiom - gets the system to nine notes. So once again, a further resolution into the secondary subdominant realm is required to pick up the final note, which is the D-natural that is native to Alpha 5/Mixolydian. As I stated earlier, in the idiomatic traditional employment of this system, the vast majority of the melodic elements adhere to the diatonic Alpha 3/Phrygian mode.
The acoustical reason that this bi-modal combination works is because the dissonant overtone sonority can support a wide variety of upper structure tones, such as the minor ninth, augmented ninth/minor third, and the minor thirteenth that Phrygian adds to Mixolydian here. This same ability of the overtone sonority to support a wide variety of tones in the upper structures is also responsible for our final Mu contextual system.
EXAMPLE 71:
Listen to Example 71
The Mu contextual system is the modern king of all of the extra-diatonic systems, as it virtually defined the sound of the twentieth century through it's ubiquitous employment in blues music, which evolutionarily lead to jazz, R&B, and rock and roll. In fact, I start all of my students out with the blues - even if they wish to study classical music - for this very reason: You can't understand the music of the twentieth century without a solid foundation in the blues.
For the traditionally trained theorists among us, the easiest way to pocket an understanding of blues tonality is as a tri-modal combination of Alpha 2 - the Dorian mode - Alpha 5/Mixolydian, and Alpha Prime. In the idiomatic vernacular, what you see above is exactly what you get: Overtone sonorities on all three cardinal degrees. In fact, in the most basic blues and blues based R&B and rock and roll music, these are the only three chords you ever hear. By the time swing and bebop guys like Charlie Parker got ahold of the blues, however, it was elevated into another musical art form entirely, with many harmonic excursions within its brief twelve measure form.
Obviously, the overtone sonorities on the dominant, tonic, and subdominant correspond to Ionian, Mixolydian, and Dorian respectively. In idiomatic practice, blues, rhythm and blues, blues based rock, and blues based jazz singers and soloists employ a multi-layered combination of modalities, which consist of a minor pentatonic skeleton, the fleshed out diatonic Dorian, and then a virtually fully clothed chromatic sub-system of passing and approach notes for the bling. Blues is something a beginner can learn to play in a week, and then explore for a lifetime, which is the primary aspect of its charm and enduring appeal.
posted by Hucbald at 9:27 AM
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