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.
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.
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