Why Music Works: Chapter Five

The Harmonies of the Secondary Dominant Contextual Sub-System

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 Five:

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 will take us out of the diatonic harmonic world and into the chromatic realm as we discover the origins of the secondary dominant sub-system sonorities.

*****

CHAPTER FIVE:

As the intuition of composers began to exhibit a more complete understanding of the overtone sonority's implications, they began to employ it for effect targeting degrees other than the tonic as secondary dominants. This process started with the nearest secondary dominants - V(m7)/V and V(m7)/IV - and progressed roughly a step at a time through V(m7)/ii and V(m7)/vi until finally the most remote secondary dominant, V(d5m7)/iii was reached. As we shall see, that V(d5m7)/iii - functioning first as V(d5m7)/V in the minor mode - unleashed entirely new classes of secondary dominant sonorities, many of which have not been properly described until I began to organize and classify them about five years ago (Though I figured out that the French Augmented Sixth was a V(d5m7)/V in minor when I was a doctoral candidate circa 1995).

EXAMPLE 20:



Listen to Example 20

To turn a I(M7) chord into a V(m7)/IV secondary dominant, all that is required is to lower the seventh by a semitone from a major seventh to a minor seventh, and to change a ii(m7) chord into a V(m7)/V secondary dominant, all that is required is to raise the third by a semitone from a minor third to a major third. This second formula - raising the third from minor to major - also works for V(m7)/ii and V(m7)/vi, but a funny thing happens if you apply it to the vii(d5m7) chord: You end up with a V(d5m7)/iii.

Though I have never been able to nail this down with any degree of certainty, since the common practice minor mode was based on Alpha 6, - the Aeolian mode, where the vii(d5m7) from Ionian is the ii(d5m7) chord - it seems most likely that the historical origins for this sonority were as a V(d5m7)/V - or v - in minor.

Since in the evolution of western art music counterpoint preceded harmony, the augmented sixth effect targeting the dominant degree that results from the major third above a diminished fifth was already known, so the orientation of the V(d5m7)/V when it first appeared was in second inversion. This is the so-called classic French Augmented Sixth sonority. Unfortunately, this historical baggage combined with a ridiculous and utterly non-descriptive name relegated the V(d5m7) sonority to the level of an obscure and difficult to understand curiosity. The same is true to an even greater degree with the so-called German Augmented Sixth sonority, because it actually has the intervallic structure of an overtone chord with enharmonic notation. As I shall demonstrate in this chapter, both of these sonorities are just altered secondary dominants, and their historically limited use is unfortunate and was unnecessary: Understanding this chapter will give any composer vastly increased sonic resources.

OBSERVATIONS:

1. This is example ten, which was example seven contextualized, with secondary dominants added.

All of these are the standard secondary dominants except for the V(d5m7)/iii. That chord could also be made into a standard secondary dominant by raising the fifth a half-step, but for the genesis of the other secondary dominant sonorities, I left it in its most natural state.

2. The V(d5m7)/iii is the point of origin for the altered dominant that generated Gamma Prime.

Remember, Gamma Prime is best described as a Phrygian mode with a major sixth and a major seventh, and Phrygian is the Alpha 3 mode here.

3. The V(d5m7)/iii is also the point of origin for the so-called "French Augmented Sixth" sonority.

4. The traditional so-called "French Augmented Sixth" chord is just a V(d5m7) in second inversion.

5. Historically, the V(d5m7)2nd probably first appeared in minor, where the ii(d5m7) became V(d5m7)2nd/V.

As you've certainly guessed by now if you've followed this series from the beginning, I have attempted to keep as much terminology and symbology from traditional classical and jazz theory as possible to make these concepts accessible to anybody trained in those disciplines, only modifying them and adding to them enough to properly describe the musical phenomena I'm defining. One set of modifications to the symbology is that I've replaced any arcane symbols with what can easily be found on a QWERTY keyboard, and another has been to at long last eliminate the figured bass formulations from inversions, as that old nomenclature is ponderous and confusing, even to me sometimes: It is much easier to understand a French sonority as V(d5m7)2nd - the 2nd meaning second inversion - than as a V(4/2/b).

6. The V(d5m7)2nd is just a naturally occuring altered dominant, available on any degree that can support a secondary dominant, and in any inversion.

We will see this in example twenty-one.

7. The diminished fifth in the V(d5m7) is an active leaning tone, replacing the passive perfect fifth in the overtone sonority.

8. Adding this additional active tone increases both the tension, and the resolution effect.

Once the intuition of composers lead them to discover that they could increase the resolutional desire of the overtone sonority by replacing a passive tone with another active tone, several new sonorities were created. In this case, the resolutional impetus is at 150% of normal, whereas later examples will completely double it.

9. To avoid parallel octaves, the root of a chord must always be a passive tone.

That is the case with the traditional French chord, but the upcoming fully-diminished seventh and German chords have no real root present in them, as we shall see.

*****

EXAMPLE 21:



Listen to Example 21

This is example twenty with secondary V(d5m7) on every degree until the final cadence, so as you can see, there are far more options for employing these chords than any of the traditional composers ever realized.

OBSERVATIONS:

1. This is example twenty with diminished fifths added to the secondary dominants.

2. A V(d5m7) can reside on any degree that can host a secondary dominant.

3. The traditional "French" terminology does not properly describe the function - or the origin - of these chords.

4. The traditional "French" terminology limits these chords to only one of four possible orientations, the second inversion.

5. The traditional "French" terminology is ridiculous, and must be abandoned.

*****

EXAMPLE 22:



Listen to Example 22

This is example twenty with minor ninths added to the secondary dominants.

Another set of secondary dominant function sonorities that are not taught properly are the so-called secondary fully-diminished seventh chords. What is usually called the root of these sonorities is actually a leading tone, so it's active and can't be the real root. The real root is always a major third below the leading tone, and so it's missing if all you are presented with is the symmetrical structure that consists of nothing but minor thirds (It's still OK to describe it as a fully-diminished seventh in that situation, so long as you realize that's just describing the structure, and not the function). The true function of the secondary fully-diminished seventh chord is as a secondary dominant with a minor ninth and no root: (Root), M3rd, P5th, m7th, and m9th. Since the root has to be a passive tone, the 9th in the transformational stratum is replacing the root with an active leaning tone. When looked at this way, the normal delayed crosswise transformation that secondary dominants make is not changed: 9 > 5, 5 > R, 7 > 3, and 3 > 7 after the resolutional interruption.

As with the V(d5m7) - the sonority formerly known as French - there are now three active tones in the upper stratum instead of two: The fifth is still passive... except for in the case of the V(d5m7m9)/iii that starts things off here. That sonority is the one traditionally described as a German Augmented Sixth, and all four voices in the transformational stratum there are active: (Root), M3rd (leading tone), d5th (leaning tone 1), m7th (leaning tone 2), and m9th (leaning tone 3).

OBSERVATIONS:

1. This is example twenty with minor ninths added to the secondary dominants.

2. The V(d5m7m9)/iii is the point of origin for the so-called "German Augmented Sixth" sonority.

3. The traditional so-called "German Augmented Sixth" sonority is just a V(d5m7m9) without a root, with the diminished fifth under the leading tone (To get the augmented sixth interval instead of the diminished third heard here), and the minor ninth in the bass (To lead into a second inversion sonority, and so avoid the parallel perfect fifths that result from the normal transformation of this chord, as we have here).

4. Historically, the V(d5m7m9/0) probably appeared first in minor, where the ii(d5m7) became V(d5m7m9/0)/V.

5. The V(d5m7m9/0) has the same intervallic structure as an overtone chord, except it is spelled enharmonically.

This has caused tons of confusion about the nature and function of this sonority. Basically, if you notate the D-sharp in the second measure above enharmonically as an E-flat, the transformational stratum is an F(m7)3rd chord. That coincidence - happy though it may be - has no bearing whatsoever on the functions of the notes in the chord: The F isn't a passive root, it's an active diminished fifth.

Nonetheless, generations of jazz musicians have been taught that errant way of looking at the chord through the so-called "substitute secondary dominant" theory: I know, because I was one of them. In that theory, to cite a single example, the V(m7)/I in C - a G(m7) sonority - can be replaced by a subV(m7)/I - which is a Db(m7) chord. Though expedient and simple - and certainly superior to the German terminology - this just isn't the way in which the overtone sonority implies that these chords are generated. The classical notation is correct, but its description is useless, while the jazz notation is incorrect, but at least its terminology is useful.

6. None of the notes in the traditional spelling can be the real root, however, because all of them are active.

7. The V(m7m9) chords also often appear without roots as so-called secondary fully-diminished seventh chords.

8. (If you're keeping up, you know this is not correct above: The notated third in a fully-diminished seventh is a passive 5th in the V(m7m9) chord from which it comes. - Geo) The notated root in a fully-diminished seventh chord is an active leading tone, so it can't be the real root.

9. A secondary fully-diminished seventh chord is properly understood as a V(m7m9/0): the root is simply missing.

10. Since real roots must be passive, minor ninths replace roots with active leaning tones.

11. With the minor ninth as a root substitute, the interrupted crosswise transformation is normal.

As with the V(d5m7) sonorities, the V(d5m7m9) chords can live on any degree that can carry a secondary dominant too.

*****

EXAMPLE 23:



Listen to Example 23

OBSERVATIONS:

1. This is example twenty with diminished fifths and minor ninths added to the secondary dominants.

2. A V(d5m7m9) can reside on any degree that can host a secondary dominant.

3. The traditional "German" nomenclature does not properly describe the function of these chords.

4. The traditional "German" nomenclature limits these chords to only one or two of four possible inversions.

5. The traditional "German" nomenclature is ridiculous, and must be abandoned.

6. Parallel perfect fifths result from the transformations of these chords: This is normal.

That is why there was often a so-called I(6/4) chord between the V(d5m7m9/0) and the tonic triad in common practice music; to avoid the parallel perfect fifth.

All of these secondary dominant types can also transform directly. While this maintains surface tension by always presenting a seventh chord, it sounds slippery and strange because the chromatically inflected leading tones are thwarted, and return to their non-inflected diatonic state instead of resolving. For me, it's an effect best used sparingly, but the rest of the examples in this chapter are 20-23 above with direct transformations.

*****

EXAMPLE 24:



Listen to Example 24

OBSERVATIONS:

1. This is example twenty with direct transformations of the secondary dominants.

2. Direct transformations are less natural sounding, as the leading tones are not resolved.

3. Direct transformations maintain greater surface tension, since a seventh chord is always sounding.

4. Choosing between directs or interrupts comes down to the effect/affect desired.

*****

EXAMPLE 25:



Listen to Example 25

OBSERVATIONS:

1. This is example twenty-one with the V(d5m7) chords transforming directly.

*****

EXAMPLE 26:



Listen to Example 26

OBSERVATIONS:

1. This is example twenty-two with the V(m7m9) chords transforming directly.

EXAMPLE 27:



Listen to Example 27

OBSERVATIONS:

1. This is example twenty-three with the V(d5m7m9) chords transforming directly.

I have managed to work all of the secondary dominant types into this systematic presentation with the exception of the so-called Augmented Seventh chords, where the passive perfect fifth is replaced by a minor sixth. I have that worked out, but not in this example set, so we'll now go on to secondary subdominants.

Read More

Why Music Works: Chapter Four

Root Motion and Voice Transformation Types in the Beta and Gamma 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 Four:

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 is dedicated to examining how Beta Prime and Gamma Prime compare to Alpha, using the same musical proof formats.

*****

CHAPTER FOUR:

EXAMPLE 14:



Listen to Example 14

Example fourteen is the same as example eleven, but with the inflections necessary to put it into the Beta Prime system. Obviously, these are end-contextualized, and are comparing the progressive resolutions to their opposite regressive versions.

The first of these root motions - the one from vii(d5m7) to bIII(A5M7) - is labelled PA5, for Progressive Augmented Fifth (This could also conceivably be labelled a Pd4 for Progressive Diminished Fourth, but since falling fifths are the most natural progressive root motions, I've decided to stick with fifths here). [As you'll see, my terminology eventually evolved into calling these quadra-tones. - Geo] This is a surprising and uncanny effect, because of both the root motion and the structure of the target harmony. Immediately following that - into the vi(d5m7) - we get a Progressive Tritone, which is labelled Ptt in the analysis. This root motion can actually be found in the Alpha system when moving from IV(M7) to vii(d5m7), but since the original intent of this example was to put all seven harmonies of Alpha into normal progressive order, we have not seen that yet.

After these initial histrionics - a very nice resource to affect the listener into the realm of the uncanny - the rest of the progressive relationships are relatively normal.

Since the second system is essentially the first system in reverse, the strangeness occurs near the end there. When I created this example back in 2008, I had still not completely nailed down the way I wanted to treat the progressive and regressive augmented fifths, so there is an Rd4 in the analysis, but the final version will say Rqt there for regressive quadra-tone. Note that the Ptt in Alpha is IV(M7) progressing to vii(d5m7) while the Rtt there is moving from vii(d5m7) to IV(M7): Both root motions are by tritone, but one is progressive and the other is regressive. The same thing applies in PA5 versus RA5.

OBSERVATIONS:

1. Overtone chord progressions imply falling fifths, hence, Progressive Augmented Fifth, Crosswise.

2. Overtone chord regressions imply rising fifths, hence, Regressive Augmented Fifth, Crosswise.

3. Progressive and regressive tritone root motions are perfectly usable in proper context.

4. Progressive and regressive augmented fifth root motions are perfectly usable in proper context.

5. Since augmented and diminished fifth root motions yield uncanny effects, employ accordingly.

6. The tonic minor/major seventh also evokes the uncanny.

7. The lowered mediant degree augmented/major seventh sonority also evokes the uncanny.

8. the harmonies and root motions possible in the Beta System provide new expressive resources.

*****

EXAMPLE 15:



Listen to Example 15

Example fifteen is the same as example twelve, but inflected to put it into Beta Prime.

OBSERVATIONS:

1. Beta system .5P's and .5R's contain less dramatically uncanny effects than the P's and R's did.

2. This is because the descending and ascending thirds smooth out the tritones and quadra-tones.

*****

EXAMPLE 16:



Listen to Example 16

This is example thirteen inflected to put it into Beta Prime

OBSERVATIONS:

1. Beta Prime SP's and SR's are again less uncanny than the system's P's and R's are.

In this case, all root motion and transformation is stepwise, hence the smoothness.

*****

EXAMPLE 17:



Listen to Example 17

Now we are ready to use example eleven/fourteen, example twelve/fifteen, and example thirteen/sixteen inflected into the Gamma System. By this point in creating these musical proofs, my terminology for the augmented fifth progressions and regressions had evolved to the point of referring to them as quadra-tones, which compares better to the tritone analysis symbols: Pqt and Rqt respectively, to fit in better with Ptt and Rtt. by this point, you should understand the analysis symbols well enough that the proofs become self-explanatory.

OBSERVATIONS:

1. Gamma System progressions and regressions contain even more uncanny sounding effects.

2. The areas of alternating quadra-tone and tritone root motions sound particularly sinister.

Obviously, these are gnarly sonic resources.

*****

EXAMPLE 18:



Listen to Example 18

Here are the half-progressions and half-regressions in Gamma. Note that I neglected to parenthetically denote the diminished thirds for the vii(d3d5m7) sonorities here. They are still notated properly - and so they sound correct - so that is just an oversight on my part.

OBSERVATIONS:

1. Gamma System .5P's and .5R's again contain fewer dramatically uncanny effects then the system's P's and R's do.

2. This is again, as before, because the .5P's and .5R's smooth out the tritones and quadra-tones.

*****

EXAMPLE 19:



Listen to Example 19

OBSERVATIONS:

1. Gamma Prime SP's and SR's are again less dramatically uncanny sounding than the system's P's and R's are.

2. Again, the fact that all root motion and transformation is stepwise aids smoothness.

3. Nonetheless, the Gamma System is filled with unsettling harmonic effects, which is an effective and affective resource.

Now that we have seen and heard the basic resources of the normal diatonic Alpha, Beta and Gamma contextual systems, it is time to enter the chromatic realm with secondary dominants and secondary subdominants as derived from the Alpha system.

Read More

Why Music Works: Chapter Three

Root Motion and Voice Transformation Types in the Alpha Contextual System

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 Three:

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.

Here we will look at the contextualization of Alpha Prime, the various different root progressions types it can exhibit, their various transformation types, and through this we will also begin to peer into the world of musical affect and effect.

*****

CHAPTER THREE:

EXAMPLE 10:



Listen to Example 10

Example ten is the contextualization of example seven, which was simply the ordering of the seven harmonies present in the Alpha System into a progressive order. To contextualize example seven into Alpha Prime, all that is required is to put a tonic harmony at the beginning, and a V(m7) P_ I at the end. The root motion form I(M7) to vii(d5m7) at the beginning - down by step - is called a Super-Regression, which gets the symbol SR in the analysis, and the voices transform in a clockwise - > - manner: R > 3 > 5 > 7 > R. I will present these in detail later in this chapter.

After that beginning contextualization, the harmonic continuity is just like example seven until measure eight, where the IV(M7) moves up into a V(m7) before the ending progressive resolution. The subdominant to dominant root motion - up by step - is the opposite of that at the beginning, and is called a Super-Progression. it gets SP in the analysis, and the voices transform in a counter-clockwise - < - manner: R > 7 > 5 > 3 > R. Opposite root motion types will always have opposite transformation types if they are circular and not crosswise.

Note also that Super-Progressions and Super-Regressions can produce parallel perfect fifths, as they do here into measure two, and into the final measure: This is normal. The biggest problem with traditional voice leading as it has been historically taught is that it is an amalgam of harmony and counterpoint, and not harmony isolated into its pure state, as we have here. In pure harmonic transformations, parallel perfect fifths sometimes result, and they are not a problem. In fact, it is the most organic and natural way of things, as the smooth transformational logic proves. Once you know this, the centuries of agonizing over whether to allow parallel perfect fifths in homophonic music becomes positively funny.

Another thing to notice is that in these root motions that are not Progressive, the transformations are direct, with no interruption as the Progressive root motions produce. The fact is, Progressive root motions can support direct transformations too, which I have demonstrated with the same continuity rendered that way on the second system: All of the Progressive transformations are direct until the final resolution to the tonic triad. Remembering that "P_+" is a Progressive Interrupted Crosswise Transformation, that becomes simply "P+" for Progressive Crosswise Transformation (The "P_" at the end is simply Progressive Interrupted Transformation, which is what you'll always see at perfect endings).

OBSERVATIONS:

1. In root motions other than progressive, transformations are direct, with no interruptions.

2. Progressive root motions may also support direct transformations.

3. The instant the overtone chord resolution varies, the realm of musical affect and effect is entered.

Affect and effect are nominally breached with all non-dominant progressive root motions that we've seen so far in the diatonic systems. One overtone sonority transforming to another through the chromatic system would be the default pure natural succession, and we'll look at that later. But really, when you allow for direct transformations over progressive root motions and especially non-progressive root motions, that is where the manifold harmonic effects that can effect the listener arise, and those are what we will be looking at for the rest of this chapter and the next.

4. Super-Regressions transform in a clockwise, circular manner: R > 3 > 5 > 7 > R.

5. Super-Progressions transform in a counterclockwise, circular manner: R > 7 > 5 > 3 > R.

6. Opposite root motions will always have opposite transformational directions, unless crosswise.

7. Direct transformations maintain surface tension by always presenting complete seventh chords.

This is one of the best features of direct transformations, as continuously interrupting the transformations sounds like a series of final-type resolutions, even if they are less perfect modal variants of the primary overtone sonority's resolution. Deft use of direct versus interrupted resolutions is one of a composer's basic resources for producing expressive effect.

8. musical contextualization is provided by beginnings and endings, or at least endings.

9. Omitting a contextualizing beginning can be an effective resource for affecting the listener.

Even in these examples, the beginning tonic harmony is a seventh chord, and so a stable context is not initially provided. In the following examples we will see how it is really the ending that provides a pure musical contextual definition. Obviously, starting the listener out in a foreign land, so to speak, and bringing them home can be a great musical plot device. This can be done by providing no initial context, or a false one.

10. Super-progressions and super-regressions can result in parallel perfect fifths, which is normal.

It is humorous, in retrospect, to view the conniptions some composers went to in order to avoid this effect in eras past.

*****

EXAMPLE 11:



Listen to Example 11

In these examples, I have only contextualized the endings to demonstrate observation nine above. Here, we also begin to look in detail at the types of root motion other than Progressive. The opposite of a progressive root motion is regressive, and it gets an "R" in the analysis (Many theorists have called these retrogressions in the past, but I prefer the simple yin/yang of two tri-syllabic words). This is what I have presented on the second system. There is also the root motion of a descending third into the penultimate measure of the second system, which is called a half-progression, and it gets .5P in the analysis. We'll see these in isolation in example twelve.

OBSERVATIONS:

1. Progressions move the transformational stratum lower.

2. Regressions move the transformational stratum higher.

3. Half-progressions and half-regressions result in three common tones between harmonies.

4. Progressions and regressions result in two common tones between harmonies.

5. Super-progressions and super-regressions result in one common tone between harmonies.

Composers need to know this, because the number of common tones between harmonies - note I'm referring to the transformational stratum - is what gives the effect of smoothness versus abruptness in the various root motion types.

6. The leading tone cannot be treated as a real root in a final resolution to a tonic triad.

7. The leading tone can be treated as a real root in an intermediate resolution to a tonic seventh chord.

If we were to attempt to move from the vii(d5m7) to I at the end of the second system, parallel octaves would result because both leading tones would move to the root of the tonic. While parallel fifths are fine in transformations that produce them in the upper stratum, parallel octaves are not between the two strata, for the simple reason that two voices are transforming the same; 7 > 1. Therefore, the leading tone cannot be treated as a real root in a final resolution (In an intermediate super-progression, of course, the upper stratum leading tone is held as a common tone, so it is 1 > 7). Discovering this was a much bigger deal than I initially thought, as it allowed me to figure out so-called secondary diminished seventh chords and also "German" augmented sixths, neither of which contain a real root, because all four tones are active: The root must be a passive tone.

8. Progressions (Not regressions as the observation mistakenly says) result in an incremental decrease of intensity, akin to musical gravity.

9. Regressions result in an incremental increase of intensity, akin to musical anti-gravity.

The phenomena of musical gravity and anti-gravity are quite real, as these examples demonstrate, and along with musical gravity comes a decrease of intensity as the pitch level of the transformational stratum lowers, while musical anti-gravity - or propulsion - brings with it an increase in perceived intensity as the pitch level of the transformational stratum rises. These are also effects the composer must be aware of.

10. Context can be nebulous or even missing at the beginning, so long as it is present at the end.

Again, unless an extra-musical context is provided, such as in a film score, where the scene creates the context.

*****

EXAMPLE 12:



Listen to Example 12

Now were ready to look at the isolated half-progressions and half-regressions. As you can see and hear, these root motion types are very smooth sounding due to all of the common tones involved.

OBSERVATIONS:

1. It takes two half-progressions to move the bass as far as one progression.

2. Two half-progressions transform the voices exactly the same as one progression.

3. It takes two half-regressions to move the bass as far as one regression.

4. Two half-regressions transform the voices exactly the same as one regression.

I did not give the root motion types arbitrary names. Rather, starting with the resolution of the overtone sonority as a normative progression, I compared all other types to it, and named them logically: The opposite of a progression is a regression, so half of a progression or a regression is exactly that.

5. Half-progressions are musical gravity moving at half speed.

6. Half-regressions are musical anti-gravity moving at half speed.

7. Direct octaves occur between the bass and a transforming voice in half-progressions: This is normal.

Half-regressions do not have this feature, because the bass moves into a voice that is tied in the upper stratum. This is important to note, as in a half-progression, the root of each new chord is a new note that did not exist in the previous harmony, while in a half-regression the bass is not a new note, but one already established in the previous harmony. This is one of the features that produces the different effects between the two root motion types.

8. Half-progressions transform clockwise, and half-regressions transform counter-clockwise.

Missing above, but I'm going to recreate these in Sibelius anyway for the final version, I think.

*****

EXAMPLE 13:



Listen to Example 13

Finally for this chapter, here are the super-progressions and super-regressions. Despite the stepwise smoothness of the bass, these root motion types sound quite abrupt because there is only a single common tone between the adjacent harmonies.

OBSERVATIONS:

1. Two super-progressions move the bass as far as three progressions.

2. Two super-progressions transform the voices down as far as three progressions.

3. Two super-regressions move the bass as far as three regressions.

4. Two super-regressions transform the voices up as far as three regressions.

5. Super-progressions are musical gravity moving down at 1.5 times normal speed.

6. Super-regressions are musical anti-gravity moving up at 1.5 times normal speed.

For 5 & 6, this despite the direction of the bass line.

7. Super-progressive transformations can result in parallel perfect fifths: This is normal.

8. Super-regressive transformations can result in parallel perfect fifths: This is normal.

9. Super progressions transform counterclockwise, super-regressions transform clockwise.

We have now seen all of the root motion types in the Alpha Prime system with the exception of the tritone root motion that occurs between IV(M7) and vii(d5m7). This root motion type will be encountered in Beta Prime and Gamma Prime - where they are unavoidable if we follow the pattern of the examples presented in this chapter - so we will examine those, and more, in chapter four.

Read More

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.

*****

CHAPTER TWO:

EXAMPLE 7A:



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.

EXAMPLE 7B:



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.

OBSERVATIONS:

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.

*****

EXAMPLE 8A:



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.

EXAMPLE 8B:



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.

OBSERVATIONS:

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.

*****

EXAMPLE 9A:



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.

EXAMPLE 9B



Listen to Example 9B

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

OBSERVATIONS:

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.

*****

Read More

Why Music Works: Chapter One

How the Harmonic Series Generates the Three Diatonic Contextual Systems: Alpha, Beta, and Gamma

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 One:

My foremost intention with this monograph is to, at long last, describe in a comprehensive way the basic musical forces that exist, and to present them and their resultants in a way that will be of use to composers. These forces all spring from a tension that is inherent in the harmonic series, which desires a resolution. Allowing or thwarting this desire for resolution is what allows music to express things.

While many twentieth-century theorists attempted to describe this through mathematics, their efforts failed because of one simple oversight: The overwhelming majority of possessors of musical minds are not mathematically inclined. Our minds generally work in terms of shapes and sounds: If we can visualize a thing or hear a thing in our minds, we can freely manipulate it to our heart's content. Numbers don't look or sound like anything to us, so they might as well not exist in our world (With the exception of proportions and draw-out geometry). Eventually, this made me return to the beginnings of scientific thought, which was actually called natural philosophy. Sir Isaac Newton described himself as a natural philosopher. Music theory has never even reached the Newtonian stage of evolution because the fundamental forces have never been defined and demonstrated in a rigorous and logical way.

A natural philosopher looks at the God-given constant in a thing, and extrapolates its implications out to describe the observable phenomena (Or hearable phenomena, in this case). For music, that constant is the harmonic overtone series, and so I have used musical proofs based on the harmonic series instead of mathematical proofs to demonstrate all of the resolution possibilities present, as well as all of the manifold contextual results that it can produce.

*****

CHAPTER ONE:

The overtone series is a harmonic system, so music is a harmonic system. If we look at the evolution of western art music, the harmonic series underpinnings were known from ancient Greek writings, but the implications of those writings were only discovered bit by bit by starting at the beginning, which was the unaccompanied melody of Gregorian plainchant. From plainchant, an admittedly oversimplified synopsis would proceed to organum, fauxbourdon, polyphony, traditional homophony, and then jazz, which was the real end result in the twentieth century: The seventh chords were finally ubiquitously used as tonics, including the overtone sonority itself in blues music.

Where the problem arose with both so-called traditional music theory and the succeeding jazz music theory was that the elements were never separated into their pure states: Pure harmony and pure counterpoint. By the time Joseph Schillinger properly described pure harmony, the "classical" guys had jumped the tracks and gone off into the desert of desiccation and musical nihilism known as - to cite but a single representative monicker - atonality (Though admittedly, some good did come of this for film music, but that's not what they intended). Meanwhile, even the most prominent Schillinger student of all, George Gershwin, never really exhibited in his works the revelation that was pure harmony: In other words, like all jazz cats, he didn't give a rip about the transformational voice leading that Schillinger revealed.

Finally, Schillinger himself never coupled his pure harmony with the overtone sonority, so he missed a crucial link, and that's why many of his ideas stray into the weeds of pure speculation. Admittedly, The System was a hodge-podge put together in haste by some of his students after his sudden and untimely death, so there is the possibility - regardless of how remote - that he did in fact make this connection. In any case, I made the connection independently due to the accidents of fortunate circumstance combined with my natural curiosity, and so here we are.

As mentioned previously, the natural philosopher's approach is to begin at the beginning, which in the case of music is with the harmonic overtone series, that every natural sound carries with it (A computer generated pure sine wave, which is an artificial phenomenon, would be an exception to this). In nature, the recognizable timbre of a sound is defined by the attack transients, formants, amplitudes, envelopes, and the phase relationships between the partials in the overtone series, which are individually pure sinusoidal periodicities.

Unfortunately, the pure waveform soundfont I found does not work in iTunes for some reason - it works fine in Encore though - so we're stuck with an organ sound that has relatively few harmonics present in it. If you play the overtone series in 12√2 equal temperament, this is what you get (I will address the simple logic of twelve-tone equal temperament much later in this book).

EXAMPLE 1:



Listen to Example 1

Though TTET does not render the mathematically perfect ratios of overtones on a vibrating string or in a vibrating column of air - which themselves vary minutely, especially on a string - it is quite close enough: A mathematically perfect overtone series sounds virtually the same, as many years of programming digital synthesizers taught me. To put a finer point on it, this 12√2 rendering of the series sounds like a so-called dominant seventh chord, as did the perfect ratio renderings I've created in years past with instruments such as the Synclavier. So please, spare me the, "TTET destroyed tonality" nonsense: The ratios in TTET may be irrational, but the concept isn't - 1/3 is an irrational number too, but a child can understand the idea of a third part of something. Likewise, 1/12 of an octave is a perfectly rational concept, even if the resulting ratios are expressed by irrational numbers.

Under the system are the various eras of western art music development - Plainchant, Organum, Fauxbourdon, Polyphony, Homophony, and Jazz - which roughly correspond to composers coming to understand ever higher partials in the harmonic series. That this understanding was primarily intuitive and not analytical need not concern us, as in musical composition - and performance - it has usually been intuition that lead the way, with theorists describing practice later.

Next I have labelled the partials 1-8, which I will call P1 through P8. It is useful to draw a distinction between partials and harmonics, as the fundamental generator of the series is not a harmonic: Only the overtones P2 and on are harmonics. By labeling the partials starting with the fundamental as P1, the series works out for us the pure intervallic ratios, which are found on the third, fourth and fifth lines.

Though self explanatory, 2:1 is the ratio for the inviolable perfect octave, 3:2 is the ratio for the just perfect fifth, 4:3 is a just perfect fourth, 5:4 is a just major third, 6:5 is the large minor third, 7:6 is the small minor third, and 8:7 is the first of the just major seconds. Line four shows the ratios of 5:3 for the major sixth and 8:5 for the minor sixth, and finally we have 7:4 for the minor seventh and 7:5 for the tritone on the bottom line.

OBSERVATIONS:

1. All adjacent intervals and their inversions from P1 through P7 are consonances.

2. All adjacent intervals and their inversions beyond P7 are dissonances.

Neither of these two observations should require any comment.

3. Consonances that remain super-particular ratios when inverted are considered perfect.

4. Consonances that are not super-particular ratios when inverted are considered imperfect.

A super-particular ratio is a ratio in which the terms differ by 1: All adjacent ratios for the consonances from P1 to P7 are super-particular; 2 - 1= 1, 3 - 2= 1, &c. Only those consonant intervals that remain super-particular when inverted are considered perfect, however: The 2:1 octave is unchanged during an inversion - which technically requires two octaves of displacement to avoid the unison - and the 3:2 perfect fifth inverts to a 4:3 perfect fourth, which are both super-particular: 3 - 2= 1 and 4 - 3= 1.

With the major and minor thirds, however, we get 8:5 for the minor sixth and 5:3 for the major sixth, which are not super-particular: 8 - 5= 3 and 5 - 3= 2.

5. The only non-adjacent dissonance with a single skip within the first seven partials is the tritone.

6. The only other non-adjacent dissonance within the first seven partials is the minor seventh itself.

The ratios of 3:1, 4:2, 5:3, and 6:4 are all consonances: Perfect twelfth, perfect octave, major sixth, and perfect fifth. Only 7:5 is a dissonance, it being the tritone, and then 8:6 is another perfect fourth. The minor seventh requires two harmonics to be skipped from 7:4, and it's the only other dissonance from P1 to P7 (And of course, it converts to the 8:7 dissonant major second).

7. The harmonic series is complete at P7; P8 is only present to yield the minor sixth harmonic ratio.

As I said at the beginning of this chapter, "The overtone series is a harmonic system, so music is a harmonic system." Once the adjacent intervals in the series become dissonances at 8:7, the harmonic part of the series is over and the melodic part of the series begins. Sure, inversions of harmonic structures can yield seconds, but a root position harmony in close position will only contain adjacent thirds. While ninths can replace doubled roots in the basic harmonic musical system - and sixths can replace fifths - elevenths and thirteenths are actually acquired from harmony with two transformational strata, which is a subject for much, much later (Transformational stratum one is Root, 3rd, 5th, and 7th, while transformational stratum two is 9th, 11th, 13th, and Root: A ponderous system outside of the basic harmonic nature of music, but one useful for certain effects within a larger musical or extra-musical context). Theorists who claim that twentieth century music was an exploration of the harmonic series beyond P7 are in error.

8. Since the harmonic series contains a dissonant tritone, it is inherently unstable.

The life forces of music are in the tritone - the leading tone and leaning tone impetuses - and the inherent instability of the overtone sonority - a dominant seventh, remember - is what provides music with its possibility for forward motion. This is one reason that wild-eyed criticisms of 12√2 temperament are misguided: Since the harmonic series is inherently dissonant, why would a stable just tuning be an advantage, or even particularly desirable? Since just ratios are only possible with computers, voices and other variable pitch instruments like strings, it is just silly to criticize TTET, which is a perfectly equitable solution for fixed pitch idioms. Later, we will look at just how small the deviations from just are with TTET.

*****

The first thing that the harmonic series from P1 to P7 does for us is to define the format for pure harmony: Since we are dealing with a harmonic system, all we have to do is eliminate the non-harmonic fundamental generator and the superfluous twelfth that it produces to get this. The resulting structure is Root, Root, third, fifth, and seventh, which is the pattern for pure harmony.

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. Only the root is doubled - or trebled after a resolution - in pure harmony.

4. Though the transformational stratum can be in any close position inversion, all harmonies are root position in pure harmony due to the constant-root bass part.

*****

EXAMPLE 2:



Listen to Example 2

The four different notes of the overtone sonority are divided into two pairs: The root and perfect fifth are passive tones, while the tritone involving the major third and the minor seventh consists of the two active tones.

OBSERVATIONS:

1. The root is the foundation of the overtone sonority.

2. Together with the root, the perfect fifth provides context for the dissonant tritone.

Since there are twelve pitch classes in the chromatic system and the tritone involves two of them, there are only six tritones possible. Since there are twelve possible overtone sonorities, this means that each tritone is shared between two possible roots. In this case, if the tritone between B-natural and F-natural is enharmonically notated as C-flat to F-natural, it can belong to the overtone sonority with the root on D-flat and the perfect fifth of A-flat. That means that the two possible roots for each tritone are also in a tritone relationship with each other, as well as the two possible perfect fifths. When you change the context of a tritone with the other perfect fifth a tritone away, you also reverse the functions of the notes involved in the tritone. In this case, the B-natural leading tone becomes a C-flat leaning tone, and the F-natural leaning tone becomes an F-natural leading tone.

This is how jazz theorists justify their concept of substitute secondary dominant harmonies, by the way, though we will see later that this isn't really valid according to the implications of the series.

3. The root and perfect fifth are passive tones, neither desiring to rise or fall.

4. The major third and minor seventh are active tones, desiring to resolve their shared dissonance.

5. The major third is a leading tone, and it desires to rise by a semitone.

6. The minor seventh is a leaning tone, and it desires to fall by either a semitone or a tone.

7. The perfect fifth may rise or fall by a tone as the tritone resolves to a major target.

8. The root may remain stationary or fall a perfect fifth when the tritone resolves.

9. In order for the target sonority to be in root position, the lower root must fall by a perfect fifth.

10. In order for the target sonority to be complete, the upper root must remain stationary.

11. In order to avoid doubling a potential active tone in the target sonority, the perfect fifth must fall by a tone.

*****

When we follow the above observations, the following primordial resolution of the overtone chord is produced.

EXAMPLE 3:



Listen to Example 3

This is how you say, "The End" in music: The overtone sonority with a doubled root resolves to a targeted major triad with a trebled root. If we were to wish a continuation, the unison C-natural in the transformational stratum would have to dissolve with one of the C's going down to a seventh - either B-natural or B-flat.

The falling perfect fifth/rising perfect fourth root motion is called a Progressive root motion, and it will get a capital P in the analysis. When we get to more elaborate examples, they will not be called, "chord progressions" because a progression is this specific type of root motion; rather they will be referred to as harmonic continuities.

Though interrupted by the doubled root in the upper stratum, if one of the C's moved down into a seventh, this would be what is called an interrupted or delayed crosswise transformation, as you can see from the diagram in between the staves: The root and fifth exchange functions, and the seventh and third also exchange functions after the third's resolution into the unison. This would be a P_+ in the analysis, which reads, Progressive _Interrupted +Crosswise transformation. Uninterrupted crosswise transformations, though less than perfectly natural, can also be used when that effect is desired, as I shall demonstrate later.

OBSERVATIONS:

1. This resolution does not yield all seven tones of the diatonic system.

2. This resolution - or similar less perfect versions - does, however, yield all six tones of the ancient hexaphonic Church modes.

The modern Ionian mode implied here was not really a common Church mode, but as we shall see, modal displacements of this formula will produce modes commonly used back then. It is also useful to note that when ending resolutions first appeared, they were primitive versions of this one.

3. In order for the target chord to become a complete seventh chord, the doubled root in the transformational stratum must fall.

4. The doubled root may fall either a semitone or a tone.

5. If the doubled root falls by a semitone, the potential for a diatonic system will be possible.

6. If the doubled root falls by a tone, the potential for a diatonic system will be destroyed.

7. If the doubled root falls by a tone, another overtone sonority will be created.

The rule for creating diatonic systems through this resolutional paradigm is to retain the inflection of the notes present in the preceding chords. Here, for example, one would have the doubled root descend to B-natural, because that note is present in the preceding dominant harmony. Allowing for chromaticism with this paradigm will lead to various integrated modalities; again, we'll see this at a later point.

8. As the diagram shows, 1 becomes 5, 5 becomes 1, 7 becomes 3, and 3 becomes 7 after the interruption of the resolution.

9. Therefore, progressive resolution of the overtone sonority creates an interrupted crosswise transformation.

10. A single additional progressive resolution would complete a diatonic system.

*****

If we allow for that dissolution of the unison C so that the former major third descends to B-natural - becoming a major seventh - and add an additional progressive resolution, the Alpha Contextual System is produced.

EXAMPLE 4:



Listen to Example 4

The Alpha Contextual System has as Alpha Prime the traditional major or Ionian mode. Within this contextual system are the additional sub-contexts known as the displacement modes of Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian.

OBSERVATIONS:

1. With two progressive resolutions from the overtone sonority, the diatonic system is complete.

The only note missing in the single cycle resolution was A-natural, which now appears as the major third of the subdominant harmony.

2. All three possible harmonic functions - dominant, tonic and subdominant - are also now defined.

3. The progressive resolution from the tonic to the subdominant is a less perfect form of progressive resolution.

4. Additional less-than-perfect progressive resolutions will be found in the modal sub-contexts.

5. There exists an additional diatonic Beta Contextual System with resolution to a minor triad.

6. The Alpha Prime tonic scale is 2, 2, 1, 2, 2, 2, 1: The semitones are separated by two tones.

*****

If we start again with the formula and have the initial resolution to a minor tonic, the Beta Contextual System is produced.

EXAMPLE 5:



Listen to Example 5

The Beta Prime mode is like a Dorian mode from the Alpha System with a raised seventh degree. In common practice minor key music, this system was combined with the Aeolian mode to produce the nonatonic so-called melodic minor scale: Roughly speaking, his is the ascending version of that system, and Aeolian was used as the descending form (Though those conventions weren't always adhered to).

OBSERVATIONS:

1. The tonic seventh chord in this system is a highly dissonant minor triad with a major seventh.

This means the third mode here has an augmented triad, of course.

2. The subdominant chord in this system is another overtone sonority.

3. The sub-contextual subdominant mode is the scale that the overtone series creates to P11.

The overtone scale is best described as a Mixolydian mode with a raised fourth degree, not, "Lydian flat seven" as some jazz theorists describe it: The home harmony is a dominant seventh, not a major seventh.

4. There exists an additional Gamma Contextual System with resolution from a dominant harmony containing a diminished fifth.

The point of origin for the V(d5m7) chord - and the so called French augmented sixth - is from the V/V in minor, where the chord on the second degree is a ii(d5m7) before the third is raised to make it a secondary dominant. In Alpha Prime that sonority appears as the remote V(d5m7)/iii harmony. I will demonstrate this when we get to the secondary dominant harmonies, but here I'm just demonstrating the three possible diatonic contextual systems that contain two semitones and five tones.

EXAMPLE 6:



Listen to Example 6

The Gamma Prime scale created through this resolution process is best described as a Phrygian mode with the sixth and seventh degrees raised, since the minor second degree is the distinguishing characteristic of the Phrygian mode.

OBSERVATIONS:

1. Diminishing the fifth of the dominant chord makes that fifth into an active leaning tone.

As I just mentioned, this V(d5m7) has a natural origin in the Alpha System. When the fifth was D-natural, it was a passive tone that could theoretically rise or fall during the resolution, so long as the composer is prepared to deal with doubling a potential active tone. Now, the D-flat is a third active tone in the dominant harmony that desires to resolve down by semitone to the new root. This additional impetus increases the resolutional desire of the dominant harmony.

2. The tonic seventh chord is again a highly dissonant minor/major seventh.

3. The subdominant chord is again another overtone sonority.

4. The Gamma Prime tonic scale is, 1, 2, 2, 2, 2, 2, 1: The semitones are not separated.

5. All 21 possible diatonic modes consisting of five tones and two semitones have now been generated.

I will present these in detail in chapter two.

*****

What lead me to the idea of pure musical contextual systems was a phenomenon I noticed with so-called atonal works: They were completely unsatisfying - unlistenable, actually - in a purely musical concert context, but when used in a stage play or a film score, they could become quite effective. What I finally realized is that the play or the film provided an extra-musical context in which these pieces could be effective. Likewise, episodes of atonality within a larger purely musical context that is based on any one of the 21 diatonic modes that are independent sub-contexts - those with major or minor triads as tonics, and not diminished or augmented triads - can also be effective. When music provides its own context, it has to be based on an independent musical contextual system or sub-system, regardless of any arguments to the contrary.

Read More