Sunday, October 10, 2010

Why Music Works: Chapter Twelve

Harmonic Mobius Loops, Harmonic Palindromes, and Comparative Morphology of the Diatonic Contextual Systems

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter Eleven:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

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

Now in chapter twelve, we return to the phenomenon I mentioned in chapter nine called harmonic Mobius loops, and the closely related phenomenon of harmonic palindromes.




Listen to Example 67

An harmonic Mobius loop is an harmonic continuity in which the root motion types are balanced out so that the ending voicing leads back into the beginning voicing. Almost all composers, working intuitively, will create continuities in which the voices descend over time, as the natural, ingrained tendency that intuitive understanding of the overtone sonority's resolutional desire imparts, is to use more progressive type motions: Progressions, half-progressions, and super-progressions. These progressive types of root motion are statistically more common in all forms of harmonic music for this reason, and that is what lead Schenker to devise his theories, which he didn't understand were only observations of an artifact of music that naturally contained a preponderance of these progressive type root motions. So, it takes both understanding and discipline to create harmonic Mobius loops, as the natural tendency is against creating them.

The easiest way to construct a harmonic Mobius loop is actually to create a harmonic palindrome, because the forward and reverse root motions will balance each other out so that the ending voicing leads back to the beginning voicing, as is desired. I have demonstrated this in the first part of example sixty-seven above.

To begin construction of one of these symmetrical Mobius loops - another way to describe an harmonic palindrome - is to put the tonic harmony in the first measure and the dominant harmony in the last measure. Since the dominant to tonic resolution is a progression, that means measure two has to be a regression back to the dominant. After that, the next three root motions/harmonies are up to the composer. Obviously, this limits the possibilities, especially in a single eight measure phrase, as we're working with here. I chose to go vi(m7), IV(M7), to ii(m7), as that is one way to yield the maximum possible five different harmonies in eight measures.

Breaking away from the symmetry imposed by harmonic palindromes can actually yield more interesting harmonic Mobius loops, as I have demonstrated in the second part of example sixty-seven. All I did here, obviously, was to lower the harmonies a step within the diatonic system for measures two, three, and four: The final four measures are the same. The reason this works here is because the initial progressive root motion is now balanced out by a super-regressive root motion into measure five: Note that the voicing in the fifth measure of each example is identical. This also has the additional benefit of presenting six of the diatonic harmonies instead of only five.

To present a comparative morphology of the nine primary diatonic systems, however, I will be required to present all seven diatonic harmonies in the Mobius loop. Therefore, neither of these two examples will suffice. I also would like to present as many different root motion types as possible - ideally all eight of them: Progression, regression, half-progression, half-regression, super-progression, super-regression, progressive tritone, and regressive tritone - so this will call for a more artful approach.


NOTE: I will present the audio example link at the end of the variation set.

My final harmonic Mobius loop starts out as the first part of sixty-seven did, but to get a progressive tritone in, I went from IV(M7) to vii(d5m7) from measure four into measure five. I then reversed the progression types in the first four measures, as you can see, in an attempt to balance things out and also to get all seven harmonies in: Progression answers regression, super-regression answers super-progression, and half-regression answers half-progression. I did manage to get all seven harmonies in - which was the primary goal - and I got seven out of eight root motion types in (Only the regressive tritone is missing) - which was the secondary goal, but I did not quite manage to balance the root motion types out, as there is still a preponderance of progressive types if we super-progress into the dominant in measure eight. If I went directly from the subdominant to the tonic, it would work, but that does not produce the desired dominant resolution, and if I went from the subdominant directly to the dominant, the voicings would miss their loop by a single inversion (7, 1, 3, 5 instead of the required 1,3,5, 7). So, I had to add another half-regression to make the loop happen. Doing this creates a nice turnaround figure that will alert the listener that a new variation and diatonic contextual system is forthcoming - and the final measure is a rhythmically diminished retrograde of the second three measures - so that makes this harmonic Mobius loop ideal for presenting a comparative morphology of the nine diatonic contextual systems.

Since this is a systematic run-through of only the nine master contexts, there are actually many nice possibilities within the sub-contexts not presented here. Presenting even only the independent sub-contexts, however, would lead to a huge and ponderous example: This is more than sufficient to alert the musically aware listener of the manifold new resources available, many of which remain almost totally unexplored.

I presented Alpha Prime twice here to allow the listener to get the lay of the land. I will not hold the reader's hand through this with a detailed explanation of every effect, as I assume that those for whom I created this will possess at least the basic intellectual curiosity to see that everything is in the analysis on the example pages.


This page presents the other two Native Diatonic Systems with Beta Prime and Gamma Prime. Since the only difference now is the presence of E-flat, there are both familiar and peculiar effects present. The last quarter note of measure twenty-four carries a V(d5m7) sonority, as this is the altered dominant belonging to the Gamma Prime contextual system: I anticipate the upcoming systems by leading into them with their dominants and dominant stand-ins throughout the variation set.

Gamma Prime, then, adds a D-flat to the E-flat already present, so the harmonic effects are more strange.


Page three gets us into the Exotic Diatonic Contextual Systems with Delta Prime, so our tonic is again a major/major seventh sonority. A-flat is the only disturbance to Alpha Prime here, so again there are both familiar and unfamiliar effects present. Epsilon Prime adds an E-flat, though, so we're back to a minor/major seventh tonic, and there are more unusual effects present.


Zeta Prime is the last of the Exotic Diatonic Contextual Systems, and it adds a D-flat to the A-flat and E-flat previously present: Now, all of the harmonies are strange compared to Alpha Prime.

With Eta Prime, we enter the Alien Diatonic Contextual Systems, and though the tonic is again a major/major seventh, the presence of the two augmented seconds produces copious amounts of uncanniness. The appearance of the V(d5M7) dominant stand-in - with it's F-sharp - really destabilizes everything, though.


The addition of the F-sharp with Theta Prime means that the final measure of the Mobius structure now has double chromatic approach tones to the dominant stand-in, which is quite nice. It's even better into a real dominant, as we see and hear in the final measure on the page where, finished with the nine contextual systems, we prepare to return to Alpha Prime. That brings up a point: Not only can you explore the sub-contexts as you please, but also nothing is stopping you from mixing and matching the contexts and sub-contexts to get whatever effects you desire on whatever degree you desire them. The possibilities available through a complete understanding of the diatonic contextual systems are truly staggering, and as I mentioned, many of them remain virtually unexplored, yet easily accessible.


Listen to Example 68

And with that, we return to Alpha Prime, where I again provide two statements of the Mobius loop to reorient yourselves, and also an ending, which finally allows the B-natural to resolve up to C-natural.

After listening to the example, you can hear that, no matter how bizarre the harmonic effects get, it is not only still possible to recognize the theme, it is impossible not to recognize it. This is one area where the so-called atonal composers of the twentieth century utterly failed: Variations on atonal constructs are virtually impossible to recognize, let alone follow, for the lay listener. Even for those of us with acutely well trained ears, it isn't exactly a satisfying experience, never mind fun.

The presence of a viable contextual system, no matter how warped, solves this problem completely, and as I said, the sub-contexts not presented here contain many more points of interest.


Post a Comment

<< Home