MM V: Harmonic Progression Mechanics
As I have alluded to before, the quantum mechanics of counterpoint and the quantum mechanics of harmonic progressions are two different things. Unfortunately, very few musicians and music educators understand this. As a result, the teaching of these two subjects is surrounded with nearly cabalistic obfuscation and therefore needless difficulty for the student or the aspiring composer. In reducing the largely taste-based rules of counterpoint to a small set of underlying irreducible laws, I have made it much easier on myself vis-a-vis developing an individual contrapuntal style. By jettisoning all of the old rules that are based on taste - or the style of a particular composer or compositional school - and using just the underlying irreducible laws of contrapuntal motion, I can allow my own taste to operate within a contrapuntal environment that offers the broadest possible range of choices, and thereby develop those principles that are only a matter of taste for myself using my own judgement. This is, in fact, the basic set of operations that are required to develop an individual style. The mechanics of harmonic motion is a much simpler issue, and therefore easier to distill out to irreducible principles, than contrapuntal laws are, and so I've decided to introduce the essential concepts at this time.
First, it must be understood that there are different types of root progressions and that these various root progressions have different kinds of effects that can be described fairly easily. Only one traditional theorist, Arnold Schoenberg, has ever made an attempt to enumerate these root progressions and describe them - so far as I am aware - in his book The Structural Functions of Harmony. I have borrowed his idea but revised some of the terminology to better reflect how I think of these various root motions and their effects. Furthermore, I have developed abbreviations for these progression types that can be added to any harmonic analysis so that the root progressions can be visualized. Being able to visualize the root progressions is very helpful because through visualization patterns can be more readily recognized, and it is precisely these patterns that give a chord progression it's cohesiveness and drive. So far as I know, I am the only one who currently teaches the writing of harmonic progressions through pattern making, and all of my harmony-based compositions are written using this technique. I never encountered this approach in either my undergraduate or my post graduate studies, which included taking all the courses required for a DMA in traditional composition. That the writing of chord progressions isn't taught this way is a shame, because then what makes a good progression remains in the realm of mystery and writing them is a hit-or-miss game that depends on the writer's intuition instead of his logical abilities.
The primordial root progression is the falling perfect fifth (Or, rising perfect fourth). Statistically, it is the most common root progression in all of Western music: Traditional, jazz, folk, and popular. As with all things musical, there is a logical reason for this that nature presents to us. If you look back at the Natural Harmonic Overtone Series, you will readily see that the first seven partials spell out a dominant seventh chord. As we know from the V7 - I progression, this sonority wants to have it's root fall a perfect fifth (Or, rise a perfect fourth) to the tonic at cadence points. This dominant progression is often replicated and extended to other degrees of the key. In total, you could have a chain that involves all seven of the diatonic chords in this arrangement: vii dim - iii - vi - ii - V - I - IV - (V - I). The chords on the degrees of this progression chain were often changed from their diatonic versions to dominant seventh chords to get a stronger effect that more closely resembles the V7 - I that nature defines for us. The resultant chords are called secondary dominant sevenths: V7/iii - V7/vi - V7/ii - V7/V - V7 - V7/IV - IV (V7 - I). This most pervasive of all root progressions I simply call a Progressive Root Motion, and it gets the abbreviation "P" in the analysis. The full list is:
1) Root ascends by step: Strong Ascending root motion= "S+"
2) Root decends by step: Strong Decending root motion= "S-"
3) Root ascends by third: Mild Ascending root motion= "M+"
4) Root decends by third: Mild Decending root motion= "M-"
5) Root ascends by perfect fourth: Progressive root motion= "P"
6) Root decends by perfect fourth: Retrogressive root motion= "R"
7) Root ascends by augmented fourth: Tri-Tone as Progressive root motion= "TT~P"
8) Root decends by augmented fourth: Tri-Tone as Retrogressive root motion= "TT~R"
Obviously, the corresponding intervallic inversions are labelled the same (Seconds= sevenths, thirds= sixths, etc.)
Next, the concept of circular transformation of the chord member tones must be understood, as this is the quantum mechanics of chord progression. For this we will need the following example.
Above I have written two versions of the same chord progression; one with triads and the other with seventh chords. The bass line is just a constant root to make it easier to see the root progressions. This chord progression has all of the different diatonic root progression types except for the decending augmented fourth (TT~R). If you look below the "C:" at the beginning of the top example's analysis, you will see 1, 3, and 5 arranged in a circle. This is the basis of the concept of circular transformations: between two triads in any diatonic root progression, there are only two possibilities; clockwise transformation or counterclockwise transformation. This concept is one of Joseph Schillinger's which I have enthusiastically embraced because it cuts through all the contrapuntal pollution inherant in the traditional method for teaching this subject and gets to the crux of the issue.
Between the I and the V chord we have Retrogressive root motion and the circular transformation is clockwise: I have indicated this with an arrow to the right below the "R". In a clockwise transformation the 1 becomes 3, the 3 becomes 5, and the 5 becomes 1. This is the natural way that this particular root progression transforms it's tones when triads are involved. Note that there is a single common tone between the triads. Both progressive and retrogressive root motions share this single common tone feature, but retrogressions sound like they are going "backwards", which is why they got that name.
From the V to the vi there is a Strong Ascending root progression and the transformation is counterclockwise, as the arrow to the left below the "S+" indicates. In strong root motion there are no common tones between the adjecent triads, which is why the effect of the root motion of a second is so powerful or abrupt.
Then as the vi moves to the IV we get a Mild Descending root progression with a clockwise transformation. There are two common tones between mildly connected chords, which is why the effect of these root motions is so soft.
The Tritone root motion that divides the progression into halves is in an "as progressive" arrangement between the IV and the vii diminished triad, and it has a counterclockwise transformation. If it were in an "as retrogressive" arrangement between the vii diminished and the IV, the transformation would be clockwise. Like the progressive and retrogressive root motions, the adjacent triads share a single common tone.
The second half of the progression is a mirror image of the first half: Progressive versus retrogressive, strong decending versus strong ascending, and mild ascending versus mild decending. Note that the transformations are mirrored as well but the numbers of common tones (Or, the lack of them) remains the same. Note also that the pattern of the root motions is easy to visualize with the abbreviations added to the analysis. This construction of a chord progression into two halves with different root progression patterns allows the individual halves to be isolated and used in subsequent variations of the chord progression and that device lends a greater cohesiveness to the whole of the resultant composition.
The second example is the same progression with seventh chords instead of triads. Since there are four chord tones to work with, there is always at least one common tone regardless of the root progression type. There is also a new transformation type, which is Crosswise Transformation, and it is indicated by the symbol with arrows at each end and an up arrow in the middle. In a crosswise transformation the 1 becomes 5, 5 becomes 1, 3 becomes 7, and 7 becomes 3: The chord member tones simply exchange positions or functions. Finally, note that between the iii minor seventh and the ii minor seventh there is a parallel perfect fourth that results from the transformation. In a different inversion of these chords, a parallel perfect fifth would result: Parallel perfect fifths that result from transformations between seventh chords involved in strong root motions are perfectly natural and are not a problem.
We will return to this subject at a later date and treat it in more depth, but I wanted to put this primer here to emphasize the fundamental difference between the countrapuntal and harmonic approaches. This should give you a grester appreciation for the isolated and non-polluted laws of counterpoint I am distilling out.
First, it must be understood that there are different types of root progressions and that these various root progressions have different kinds of effects that can be described fairly easily. Only one traditional theorist, Arnold Schoenberg, has ever made an attempt to enumerate these root progressions and describe them - so far as I am aware - in his book The Structural Functions of Harmony. I have borrowed his idea but revised some of the terminology to better reflect how I think of these various root motions and their effects. Furthermore, I have developed abbreviations for these progression types that can be added to any harmonic analysis so that the root progressions can be visualized. Being able to visualize the root progressions is very helpful because through visualization patterns can be more readily recognized, and it is precisely these patterns that give a chord progression it's cohesiveness and drive. So far as I know, I am the only one who currently teaches the writing of harmonic progressions through pattern making, and all of my harmony-based compositions are written using this technique. I never encountered this approach in either my undergraduate or my post graduate studies, which included taking all the courses required for a DMA in traditional composition. That the writing of chord progressions isn't taught this way is a shame, because then what makes a good progression remains in the realm of mystery and writing them is a hit-or-miss game that depends on the writer's intuition instead of his logical abilities.
The primordial root progression is the falling perfect fifth (Or, rising perfect fourth). Statistically, it is the most common root progression in all of Western music: Traditional, jazz, folk, and popular. As with all things musical, there is a logical reason for this that nature presents to us. If you look back at the Natural Harmonic Overtone Series, you will readily see that the first seven partials spell out a dominant seventh chord. As we know from the V7 - I progression, this sonority wants to have it's root fall a perfect fifth (Or, rise a perfect fourth) to the tonic at cadence points. This dominant progression is often replicated and extended to other degrees of the key. In total, you could have a chain that involves all seven of the diatonic chords in this arrangement: vii dim - iii - vi - ii - V - I - IV - (V - I). The chords on the degrees of this progression chain were often changed from their diatonic versions to dominant seventh chords to get a stronger effect that more closely resembles the V7 - I that nature defines for us. The resultant chords are called secondary dominant sevenths: V7/iii - V7/vi - V7/ii - V7/V - V7 - V7/IV - IV (V7 - I). This most pervasive of all root progressions I simply call a Progressive Root Motion, and it gets the abbreviation "P" in the analysis. The full list is:
1) Root ascends by step: Strong Ascending root motion= "S+"
2) Root decends by step: Strong Decending root motion= "S-"
3) Root ascends by third: Mild Ascending root motion= "M+"
4) Root decends by third: Mild Decending root motion= "M-"
5) Root ascends by perfect fourth: Progressive root motion= "P"
6) Root decends by perfect fourth: Retrogressive root motion= "R"
7) Root ascends by augmented fourth: Tri-Tone as Progressive root motion= "TT~P"
8) Root decends by augmented fourth: Tri-Tone as Retrogressive root motion= "TT~R"
Obviously, the corresponding intervallic inversions are labelled the same (Seconds= sevenths, thirds= sixths, etc.)
Next, the concept of circular transformation of the chord member tones must be understood, as this is the quantum mechanics of chord progression. For this we will need the following example.
Above I have written two versions of the same chord progression; one with triads and the other with seventh chords. The bass line is just a constant root to make it easier to see the root progressions. This chord progression has all of the different diatonic root progression types except for the decending augmented fourth (TT~R). If you look below the "C:" at the beginning of the top example's analysis, you will see 1, 3, and 5 arranged in a circle. This is the basis of the concept of circular transformations: between two triads in any diatonic root progression, there are only two possibilities; clockwise transformation or counterclockwise transformation. This concept is one of Joseph Schillinger's which I have enthusiastically embraced because it cuts through all the contrapuntal pollution inherant in the traditional method for teaching this subject and gets to the crux of the issue.
Between the I and the V chord we have Retrogressive root motion and the circular transformation is clockwise: I have indicated this with an arrow to the right below the "R". In a clockwise transformation the 1 becomes 3, the 3 becomes 5, and the 5 becomes 1. This is the natural way that this particular root progression transforms it's tones when triads are involved. Note that there is a single common tone between the triads. Both progressive and retrogressive root motions share this single common tone feature, but retrogressions sound like they are going "backwards", which is why they got that name.
From the V to the vi there is a Strong Ascending root progression and the transformation is counterclockwise, as the arrow to the left below the "S+" indicates. In strong root motion there are no common tones between the adjecent triads, which is why the effect of the root motion of a second is so powerful or abrupt.
Then as the vi moves to the IV we get a Mild Descending root progression with a clockwise transformation. There are two common tones between mildly connected chords, which is why the effect of these root motions is so soft.
The Tritone root motion that divides the progression into halves is in an "as progressive" arrangement between the IV and the vii diminished triad, and it has a counterclockwise transformation. If it were in an "as retrogressive" arrangement between the vii diminished and the IV, the transformation would be clockwise. Like the progressive and retrogressive root motions, the adjacent triads share a single common tone.
The second half of the progression is a mirror image of the first half: Progressive versus retrogressive, strong decending versus strong ascending, and mild ascending versus mild decending. Note that the transformations are mirrored as well but the numbers of common tones (Or, the lack of them) remains the same. Note also that the pattern of the root motions is easy to visualize with the abbreviations added to the analysis. This construction of a chord progression into two halves with different root progression patterns allows the individual halves to be isolated and used in subsequent variations of the chord progression and that device lends a greater cohesiveness to the whole of the resultant composition.
The second example is the same progression with seventh chords instead of triads. Since there are four chord tones to work with, there is always at least one common tone regardless of the root progression type. There is also a new transformation type, which is Crosswise Transformation, and it is indicated by the symbol with arrows at each end and an up arrow in the middle. In a crosswise transformation the 1 becomes 5, 5 becomes 1, 3 becomes 7, and 7 becomes 3: The chord member tones simply exchange positions or functions. Finally, note that between the iii minor seventh and the ii minor seventh there is a parallel perfect fourth that results from the transformation. In a different inversion of these chords, a parallel perfect fifth would result: Parallel perfect fifths that result from transformations between seventh chords involved in strong root motions are perfectly natural and are not a problem.
We will return to this subject at a later date and treat it in more depth, but I wanted to put this primer here to emphasize the fundamental difference between the countrapuntal and harmonic approaches. This should give you a grester appreciation for the isolated and non-polluted laws of counterpoint I am distilling out.
posted by Hucbald at 1:55 AM
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