Why Music Works: Chapter Fifteen
The Integrated Chromatic Contextual Systems: Chi, Psi, and Omega
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 Fifteen:
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 then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. The penultimate chapter in this section of WMW, chapter fourteen, examined just versus equal temperament, and so now in fifteen - which is the final chapter in this section on harmonic aspects of music - we will put the secondary dominant and secondary subdominant sub-systems together into the integrated chromatic contextual systems of Chi, Psi, and Omega. I skipped to the end of the Greek alphabet here because there may be some other systems to add in the future (But I got every one that can be generated by the overtone resolutional paradigm, so far as I know).
*****
CHAPTER FIFTEEN:
EXAMPLE 73:
Listen to Example 73
Starting on the tonic of Alpha Prime, the resolutional paradigm takes us immediately to the primary subdominant, and then into the secondary subdominant contextual sub-system at the next resolution to bVII(M7), which is an Alpha 4/Lydian sonority. Further Lydian sonorities follow as we traverse the secondary subdominants through bIII(M7), bVI(M7), bII(M7), and finally bV(M7), which is an enharmonic sharped fourth degree. The next enharmonic progressive resolution to V(m7)/iii puts us into the secondary dominant contextual sub-system, and we traverse V(m7)/vi, V(m7)/ii, and V(m7)/V before arriving at the primary dominant preparation for the final progressive resolution to the tonic. So, the secondary subdominant contextual sub-system acts as a huge subdominant preparation for the most remote of the secondary dominants, which in turn act as a gargantuan dominant preparation for the return to the tonic: For centuries this was reduced to the kernel of I, IV, V, I, with the super-progression from IV to V literally short circuiting what the overtone series implied was the ultimate outcome for a major locus integrated chromatic contextual system.
Note that the V(m7)/iii does not have a diminished fifth when we follow the resolutional paradigm through the secondary subdominant sub-system: The paradigm rules that the root of the targeting chord is retained as the fifth of the target chord, so the G-flat is sustained as the enharmonic equivalent of F-sharp into the B major/minor seventh, eliminating the diminished fifth that Alpha Prime produces.
*****
EXAMPLE 74:
Listen to Example 74
If we perform the same exercise with pure minor, which is the Alpha 6 or Aeolian mode, instead of a series of major/major sevenths acting as subdominants, we get a series of minor/minor sevenths. Again, the secondary subdominants are a different species of harmony than the tonic, as they are Alpha 2/Dorian sonorities.
Paleo-theorists handled the minor modes differently with respect to where the secondary dominants reside, but through the examples here we see that the only real difference between major and minor is what gender the primary dominant resolves to. I resolved to a final major tonic to accentuate this, which is the ancient Tierce de Picardie practice that goes back as far as composers have been resolving to triads instead of open fifths: Musical intuition is a powerful force.
Note again that the transit through the secondary subdominant sub-system eliminates the diminished fifth in the Alpha Prime version of the V(m7)/iii.
*****
EXAMPLE 75:
Listen to Example 75
The Beta Prime contextual system showed us that a subdominant could be an overtone chord too, and with that we end up with a full chromatic cycle of overtone sonorities. Omega is a good name for this system, as, "beyond here there be dragons" so to speak. It is easy to get totally lost as to where you are without following the music or at least listening closely, and this will bring up the ultimate implication of the overtone sonorities resolutional paradigm in the following example.
I again ended with a Picardy third to imply that the tonic of this system could also be major.
*****
This is what is inescapably ingrained into us by the overtone series resolutional paradigm: An endless series of overtone sonorities resolving to major triads, which in turn acquire minor sevenths to become new overtone chords and continue the progressive cycle. If you have ever lived anywhere there are maple trees, it is an auditory analogue to watching a maple seed fall, spiraling to the ground with its gyroscopic spinning: I can land on a rock or on the ground, just as any of the following overtone sonorities can alight on a major or a minor tonic.
I have only provided a two octave cycle, but subconsciously this pattern fills the entire spectrum of human hearing.
EXAMPLE 76A:
EXAMPLE 76B:
Listen to Example 76
Sorry for the Bizarro World piano sound font, but I neglected to properly assign the PC88 Marcato Strings to that example. Arg.
*****
There are many, many possibilities for fully integrated contextual systems - which I think I'll put into an appendix in the final book - but I just want to demonstrate one that has all twenty-four major and minor tonics as well as the twelve overtone chords. Each tonic starts minor, becomes major, then acquires a major seventh, and finally a minor seventh to become an overtone sonority. Obviously - if you've been following to this point - this creates a strict double canon.
EXAMPLE 77:
Listen to Example 77
This could be further adorned with diminished fifths and minor ninths of course, but I really like this one.
That brings to a conclusion the section of WMW that deals with harmony. There are sections to follow on counterpoint, rhythm, and form, but I don't know when I'll get to them, because I have other projects I need to get to right now. In fact, I've been anxious to finish this up because of some things I'm getting to the critical mass point with, the primary one of which is my programming of the Yamaha FS1R FM/Formant Sequence digital synthesizer.
Stay tuned.
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 Fifteen:
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 then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. The penultimate chapter in this section of WMW, chapter fourteen, examined just versus equal temperament, and so now in fifteen - which is the final chapter in this section on harmonic aspects of music - we will put the secondary dominant and secondary subdominant sub-systems together into the integrated chromatic contextual systems of Chi, Psi, and Omega. I skipped to the end of the Greek alphabet here because there may be some other systems to add in the future (But I got every one that can be generated by the overtone resolutional paradigm, so far as I know).
CHAPTER FIFTEEN:
EXAMPLE 73:
Listen to Example 73
Starting on the tonic of Alpha Prime, the resolutional paradigm takes us immediately to the primary subdominant, and then into the secondary subdominant contextual sub-system at the next resolution to bVII(M7), which is an Alpha 4/Lydian sonority. Further Lydian sonorities follow as we traverse the secondary subdominants through bIII(M7), bVI(M7), bII(M7), and finally bV(M7), which is an enharmonic sharped fourth degree. The next enharmonic progressive resolution to V(m7)/iii puts us into the secondary dominant contextual sub-system, and we traverse V(m7)/vi, V(m7)/ii, and V(m7)/V before arriving at the primary dominant preparation for the final progressive resolution to the tonic. So, the secondary subdominant contextual sub-system acts as a huge subdominant preparation for the most remote of the secondary dominants, which in turn act as a gargantuan dominant preparation for the return to the tonic: For centuries this was reduced to the kernel of I, IV, V, I, with the super-progression from IV to V literally short circuiting what the overtone series implied was the ultimate outcome for a major locus integrated chromatic contextual system.
Note that the V(m7)/iii does not have a diminished fifth when we follow the resolutional paradigm through the secondary subdominant sub-system: The paradigm rules that the root of the targeting chord is retained as the fifth of the target chord, so the G-flat is sustained as the enharmonic equivalent of F-sharp into the B major/minor seventh, eliminating the diminished fifth that Alpha Prime produces.
EXAMPLE 74:
Listen to Example 74
If we perform the same exercise with pure minor, which is the Alpha 6 or Aeolian mode, instead of a series of major/major sevenths acting as subdominants, we get a series of minor/minor sevenths. Again, the secondary subdominants are a different species of harmony than the tonic, as they are Alpha 2/Dorian sonorities.
Paleo-theorists handled the minor modes differently with respect to where the secondary dominants reside, but through the examples here we see that the only real difference between major and minor is what gender the primary dominant resolves to. I resolved to a final major tonic to accentuate this, which is the ancient Tierce de Picardie practice that goes back as far as composers have been resolving to triads instead of open fifths: Musical intuition is a powerful force.
Note again that the transit through the secondary subdominant sub-system eliminates the diminished fifth in the Alpha Prime version of the V(m7)/iii.
EXAMPLE 75:
Listen to Example 75
The Beta Prime contextual system showed us that a subdominant could be an overtone chord too, and with that we end up with a full chromatic cycle of overtone sonorities. Omega is a good name for this system, as, "beyond here there be dragons" so to speak. It is easy to get totally lost as to where you are without following the music or at least listening closely, and this will bring up the ultimate implication of the overtone sonorities resolutional paradigm in the following example.
I again ended with a Picardy third to imply that the tonic of this system could also be major.
This is what is inescapably ingrained into us by the overtone series resolutional paradigm: An endless series of overtone sonorities resolving to major triads, which in turn acquire minor sevenths to become new overtone chords and continue the progressive cycle. If you have ever lived anywhere there are maple trees, it is an auditory analogue to watching a maple seed fall, spiraling to the ground with its gyroscopic spinning: I can land on a rock or on the ground, just as any of the following overtone sonorities can alight on a major or a minor tonic.
I have only provided a two octave cycle, but subconsciously this pattern fills the entire spectrum of human hearing.
EXAMPLE 76A:
EXAMPLE 76B:
Listen to Example 76
Sorry for the Bizarro World piano sound font, but I neglected to properly assign the PC88 Marcato Strings to that example. Arg.
There are many, many possibilities for fully integrated contextual systems - which I think I'll put into an appendix in the final book - but I just want to demonstrate one that has all twenty-four major and minor tonics as well as the twelve overtone chords. Each tonic starts minor, becomes major, then acquires a major seventh, and finally a minor seventh to become an overtone sonority. Obviously - if you've been following to this point - this creates a strict double canon.
EXAMPLE 77:
Listen to Example 77
This could be further adorned with diminished fifths and minor ninths of course, but I really like this one.
That brings to a conclusion the section of WMW that deals with harmony. There are sections to follow on counterpoint, rhythm, and form, but I don't know when I'll get to them, because I have other projects I need to get to right now. In fact, I've been anxious to finish this up because of some things I'm getting to the critical mass point with, the primary one of which is my programming of the Yamaha FS1R FM/Formant Sequence digital synthesizer.
Stay tuned.
posted by Hucbald at 5:27 PM
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