Tuesday, September 14, 2010

Why Music Works: Chapter Two

Analysis of the Modal Sub-Contexts of the Alpha, Beta, and Gamma 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 Two:

In chapter one, we looked at the structure of the harmonic series and found that it is what theorists call, for good reason, a dominant seventh chord: A major triad with a minor seventh. After removing the fundamental generator and its superfluous perfect twelfth, I identified the basic musical forces that reside within the tritone - the leading tone and leaning tone impetuses - and demonstrated how these forces imbue the overtone sonority with a desire for resolution. We found that with the five part texture of pure harmony, the resolution creates a delayed or interrupted crosswise transformation in the upper stratum, and with a single further diatonic resolution the Alpha Prime contextual system is created. Then we saw how with a resolution to a minor triad the Beta Prime diatonic contextual system results, and finally, how by starting from a V(d5m7) - the point of origin for the ridiculously so-called French Augmented Sixth sonority - that the Gamma Prime contextual system results.

Here in chapter two, we will look at the independent and dependent sub-systems of Alpha, Beta, and Gamma. Furthermore, I will demonstrate the primacy of Alpha Prime, the comparatively limited nature of Beta, and the outright flawed character of Gamma.




On the top system, I have presented all of the harmonies of Alpha Prime in standard notation. Underneath, I have put the analysis symbols in the format I will be using. A capital Roman numeral will represent a major triad, and a small case Roman numeral will be assumed to be a minor triad, unless there is a further indication for the fifth, as is the case for vii(d5m7) in the final measure, where the fifth is diminished. The parenthetical indicators are for alterations to the triad, descriptions of the seventh, and also any upper structure tones that may be present: M, m, A, d represent Major, minor, Augmented, and diminished, respectively. This results in more detail than is often presented: For example, what is usually called a V7 in most other systems is a V(m7) here. I prefer this more perfect logic and level of detail. Finally, under the chord analysis symbols is the tone/semitone pattern for Alpha Prime: 2, 2, 1, 2, 2, 2, 1. Obviously, this is the traditional major mode specifically called Ionian.

When I was coming up, diagrams almost precisely like example 7a were given at or near the beginning of harmonic theory. This really does a disservice to the student, as it can lead to the incorrect notion that scales generate harmony, when the truth - as I demonstrated in chapter one - is the other way around: Harmonic continuities generate the scales or modes.

With this in mind, I have demonstrated how the displacement modes of Alpha Prime are best represented by their less perfect resolutional paradigms that imitate the original: Alpha independent sub-context 2: Dorian is defined by v(m7), i(m7), and IV(m7); Alpha independent sub-context 3: Phrygian is defined by v(d5m7), i(m7), and iv(m7); Alpha independent sub-context 4: Lydian is defined by V(M7), I(M7), and #iv(d5m7); Alpha independent sub-context 5: Mixolydian is defined by v(m7), I(m7), and IV(M7); Alpha independent sub-context 6: Aeolian is defined by v(m7), i(m7), and iv(m7); and finally, the Alpha dependent sub-context 7: Locrian is defined by bV(M7), i(d5m7), and iv(m7). The Locrian mode is the only dependent sub-context in the Alpha system because it does not contain a perfect fifth, and so it has no proper dominant function harmony, and the i(d5m7) is not stable enough to provide a proper conclusion. In a chromatic versus a diatonic context, it is perfectly possible to target a (d5m7) harmony with an overtone sonority - or any other a perfect fifth above - so it is primarily the dissonant nature of the sonority that renders it unable to function on its own. Within a larger independent context, of course, Locrian effects are a perfectly fine resource.


Listen to Example 7B

The Alpha System allows for all seven harmonies to be put into a progressive order. As we shall see later, the construction of the Beta and Gamma systems do not allow for this. Here is one factor that explains the primacy of Alpha. Additionally, all of the diatonic degrees from vii(d5m7) up to and including the tonic can carry secondary dominant harmonies, as I shall demonstrate later. Furthermore, from the primary subdominant of IV(M7) on, the secondary subdominants can continue the cycle into the chromatic realm. Ultimately, a barber pole loop consisting of the primary subdominant and secondary subdominants leading away from the tonic into the chromatic realm will prepare for the most remote of the secondary dominants leading back to the primary dominant and then the tonic. That will be the ultimate musical proof, whereas this is the initial.


1. All master contexts will have functional dominant, tonic, and subdominant harmonies.

2. All independent sub-contexts will have functional dominant and tonic harmonies.

3. A non-functional subdominant harmony does not destroy the independence of a sub-contextual mode.

4. A dominant harmony with a diminished fifth is functional.

5. Only one sub-contextual mode of the Alpha System is contextually dependent: The Locrian mode.

6. A string of harmonies in progressive order creates a double harmonic canon in the upper stratum.

This isn't glaringly obvious without some elaboration of the parts, but - within the upper stratum - the soprano voice follows the tenor by a measure at the fourth above, and the alto follows the bass at the same distance and interval. This is such a cool phenomenon, that I'll devote an entire future chapter to it.

7. A string of harmonies in progressive order lowers the two strata.

As you can see, as the voices progressively transform, they get lower. This is what I call musical gravity, and it is the same phenomenon that Heinrich Schenker discovered, but he never figured out exactly what he was looking at: He was seeing an artifact of music that has a preponderance of progressive harmonic relationships in it. Since progressive root motions are statistically the most common type in western art music, you'll get the 3, 2, 1's; 5, 4, 3, 2, 1's &c. It's no big deal, really, and I can't think of too many musical exercises more futile than Schenkerian analysis: While mildly interesting, it's laborious and doesn't really teach the composer anything of much practical value.

The Natural Laws of Pure Harmony:

A slightly elaborated list. I'm still figuring out how best to list, word, and present these observations.

1. Pure harmony consists of five total voices.

2. These five voices are divided into a four-part, close-position transformational stratum above a constant-root bass part.

3. All chords are in root position in pure harmony.

4. The upper stratum consists of complete seventh chords, or triads with a doubled root.

5. The upper stratum transforms in a crosswise or circular manner, depending upon the root motion type.

Number five is a bit of a peek ahead, as I will show the other root motion types at a later point.



This example of the Beta Contextual System is in the same format I presented for Alpha. The top system is a simple lineal presentation of the harmonies, and then the six displacement modes are on the following three systems. Alpha Prime - the Ionian mode - is considered the model for comparison, so any deviations from the pattern established there are represented in the analysis symbols. For example, the third degree of Alpha Prime is major, so the harmony residing on the minor third degree of Beta is described as a bIII(A5M7): the small case "b" standing in for the flat symbol. This convention will be followed throughout.

The proper way to name Beta Prime is as a Dorian mode with a major seventh. Independent sub-context 2, therefore, is described as a Phrygian mode with a major sixth. The third sub-context in Beta, Lydian augmented fifth, is a dependent sub-context, because the augmented triad cannot function as a tonic, and therefore there is also no dominant or progressive relationship between #v(d5m7) and the tonic.

Beta 4 is again an independent sub-context, and it is properly described as a Mixolydian mode with an augmented fourth. As I mentioned previously, this is the actual scale created by the harmonic series to P11.

Because the tonic harmony for Beta 5 is an overtone sonority, it also has to be compared to the Mixolydian mode, which means it is a Mixolydian with a minor sixth. Likewise, Beta 6 & 7 have (d5m7) chords on the tonic degree, so they are best compared to the Locrian mode: Locrian major second in the case of Beta 5, and Locrian diminished fourth in the case of Beta 6: Of course, both Beta 5 & 6 are dependent sub-contexts.


Listen to Example 8B

NOTE: I had to use separate example scores, so yes, the first harmony in 8B starts with a triad moving into a seventh. I just noticed that, sorry.

Whereas all seven harmonies in Alpha could be arranged in a progressive order, here in Beta only five of them line up that way.


1. An overtone chord can be a functional subdominant.

2. A minor.major seventh can be a functional dominant.

3. The mode created by the harmonic series to P11 would not allow the overtone chord to function as a dominant.

This means that, as far as music is concerned, the harmonic series is complete at P7.

4. The Beta System has only three independent sub-contexts.

5. Fully three Beta System sub-contexts are dependent on outside contextual definition.

6. The primacy of the Alpha System is demonstrated by the fact that all seven of its harmonies can be ordered in progressive relationships.

7. Traditional so-called melodic minor is a bi-modal combination of Alpha 6 and Beta Prime.



Here we have the Gamma contextual system, which is the last of the normal diatonic systems generated by the harmonic system, normal being defined as systems consisting of two semitones, and five tones. Gamma Prime is best described as a Phrygian mode with a major sixth and major seventh since the minor second degree is what distinguishes Phrygian from the other minor modes in Alpha. The Gamma system is interesting and difficult to navigate because only one of its sub-contexts, Gamma 4, is independent: All the rest are dependent. Gamma 7 is particularly bizarre, because the tonic triad has a diminished third, a the rest of the scale contains diminished fourth, and a diminished fifth.


Listen to Example 9B

Due to this structure, only three of the harmonies in the Gamma system can be arranged in a progressive order.


1. The Gamma System has only one independent sub-context.

2. Fully five of the Gamma System's sub-contexts are dependent on outside contextual definition.

3. Only three Gamma System harmonies occur in progressive order.

4. Due to 1-3, the Gamma system is the antithesis of the Alpha system, and the Beta system lies in between.

5. The Gamma System ammounts to a hexatonic whole tone scale with one of the tones filled in.

6. The augmented fifth/minor seventh on bIII - commonly called an augmented seventh chord - is a byproduct of the genesis of the Gamma System.

7. Both of the altered dominants on bIII and V can be used in non-diatonic contexts.

More on this later.

8. Between the Alpha, Beta, and Gamma Systems, all normal diatonic resources are present.

9. The combined diatonic contextual resources available - independent and dependent - totals 21 modes.

10. Twelve of these modes are independent, while 9 are contextually dependent.

11. Three additional contextual systems are possible allowing for one augmented second.

I've worked these out, and will present them later.

12. A further three contextual systems are possible allowing for two augmented seconds.

I thought I had worked these out, but I can't find them, so I may be mistaken here. In any event, this is a topic for much later. The next chapter will be looking at and listening to the various root motions in the Alpha system in context.



Blogger A Wolf said...


Before I get into reading these extensive postings, can you clarify something?

I haven't read your old postings from the sidebar. This basically replaces that stuff? Yes?

10:18 PM  
Blogger Hucbald said...

Yes. The concepts have finally evolved to the point that I have the schema worked out for the most logical order of presentation.

As usual, considering my middling verbal skills, it's the verbiage I'm working out now.

That said, the sidebar posts are still quite informative, I think.


6:42 AM  
Anonymous Justin said...

This series is so great. It's Schillinger with context. Beautiful!

1:21 PM  

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