Sunday, September 26, 2010

Why Music Works: Chapter Seven

The Exotic Diatonic Contextual Systems: Delta, Epsilon, and Zeta

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 Seven:

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. Previously, 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.

At this point, in chapter seven, we will go back a bit, in a way, by looking at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and then in chapter eight we'll look in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create.



The Delta diatonic contextual system is created by an overtone sonority resolving to a major tonic, and then to a minor subdominant. Since the rules for diatonic resolutional genesis call for retaining the inflection of notes that appear in previous harmonies, that means that the subdominant minor triad carries a hotly dissonant major seventh (m3, P5, and M7 create an augmented triad in this chord).


Since resolutional genesis is now a familiar concept, audio examples will be saved for the progressive orientation examples from here on out.


1. Augmented seconds are perfectly acceptable in harmonic transformations.

2. A minor, major-seventh chord makes a perfectly acceptable subdominant harmony.

3. Delta Prime is often called, "harmonic major" but Ionian minor sixth is more descriptively accurate.

4. The Delta Prime tonic scale is, 2, 2, 1, 2, 1, 3, 1: Two whole steps are adjacent at the beginning of the mode.


The Epsilon diatonic contextual system is created by an overtone sonority resolving to a minor tonic, and then onto a minor subdominant.



1. Epsilon Prime is often called, "harmonic minor" but Aeolian major seventh is more descriptively accurate.

2. The Epsilon Prime tonic scale is, 2, 1, 2, 2, 1, 3, 1: Two whole steps are adjacent in the middle of the mode.


The Zeta diatonic contextual system is created by an overtone chord with a diminished fifth resolving to a minor tonic, and then into a minor subdominant.



1. Zeta prime is often called, "Phrygian harmonic" but Phrygian major seventh is more descriptively accurate.

2. The Zeta Prime scale is, 1, 2, 2, 2, 1, 3, 1: The three whole steps are adjacent.


Here are the displacement modes of the Delta contextual system.


Four harmonies of the Delta system can be arranged in progressive order.


Listen to Example 37B

As you can hear, the augmented second in the final transformation does not sound overly strange.


1. Delta or Epsilon can be considered as a point of origin for the V(m7m9) - vii(d5d7) sonorities.

Plainly, these harmonies are an artifact created through the genesis of these systems.

2. As with Beta, the Delta system has three independent and three dependent sub-contexts.

3. A m(d5d7) sonority can function perfectly well as a dominant with progressive root motion.

That would be as in Delta 3 above.


Here are the displacement modes of the Epsilon contextual system.


Four harmonies of the Epsilon system can be arranged in progressive order.


Listen to Example 38B

If you were into 80's and 90's virtuoso rock guitar, you might even detect a little of the feel of the background for some of Yngwie Malmsteen's stuff in that, as I do. He was very fond of Epsilon Prime aka "harmonic minor."


1. Like Beta and Delta, Epsilon has three independent and three dependent sub-contexts.


Here are the displacement modes of the Zeta contextual system.


Only three harmonies of the Zeta system can be arranged in progressive order.


Listen to Example 39B


1. Delta, Epsilon, and Zeta each have three independent and three dependent sub-contexts.

2. The Delta, Epsilon, and Zeta systems add an additional 21 modes to the 21 that resulted from the genesis of the Alpha, Beta and Gamma systems.

So, we're up to a total of 42 diatonic modes now: 21 normal modes and 21 exotic modes.

3. Three additional contextual systems are available with two augmented seconds.

As I mentioned previously, I thought I had worked these out, but I can't locate them now, so I'm not positive; there may be only two more. In any event, there will be a brief intermission as I create some more examples, because this is the end of the examples I created back in 2008 before I moved from Alpine to San Antonio. When we do continue, it will be to look at and listen to the various root motion and transformation types that Delta, Epsilon, and Zeta exhibit, using the same proofs we used for Alpha, Beta, and Gamma.

Since this post will put chapter one off of the home page, there is now a section in the sidebar for this series.

It's all drama with some girls.


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