Secondary Dominants and Harmonic Canons
I have decided to leave counterpoint behind for the time being (Sort of), as I am currently involved with several harmonic-approach compositions but no purely contrapuntal ones. As with all obsessive/compulsive types, once I get involved in a particular project, I'm in it to stay until it's taken it's full course.
As mentioned earlier, the Natural Harmonic Overtone Series defines for us the primordial falling fifth V7 to I harmonic progression through it's first seven partials, which spell out a dominant seventh chord. Throughout the late Renaissance and early Baroque eras, this progression was extended to secondary degrees of the key until the entire pantheon of Secondary Dominant Sevenths was completed. The way this works is that in any root progression in a key that is Progressive in nature, and that involves degrees other than V and I, the first chord in the progression can be changed from it's natural state - either a minor seventh or a major seventh - to a dominant seventh chord so that it targets the secondary degree in the same manner as the V7 to I progression does.
In the top example above, I have written a progression that allows for all of the secondary dominants to be displayed. By progressing directly from the tonic to the seventh degree, I have set up for a series of falling fifth Progressive Root Motions that will involve each of the seven degrees of the key.
The top example shows the triadic Circular Transformations. After the Strong Decending root motion from the I to the vii diminished triad that kicks off the progression with it's associated clockwise transformation, all subsequent root motions are of the falling fifth (Or, rising fourth) Progressive type. Note that in the Progressive root motions all of the Circular Transformations are counterclockwise (1 becomes 5, 5 becomes 3, and 3 becomes the root), the triads share a single common tone, and that the voices rise in overall pitch from chord to chord. In the final bar there is a Strong Ascending root motion from the IV to the V to turn the progression around on itself, and it's transformation is counterclockwise opposite of the Strong Decending progression from the first measure to the second. This is an example of a very unbalanced chord progression: Since there are not a variety of root motions that alternate between clockwise and counterclockwise transformations, subsequent repititions of the progression will soon lead to problems of range if an all-triad environment is maintained.
In the second example of this progression, I have presented it in an all-seventh chord diatonic environment. Note that in this example the Progressive root motions transform in a crosswise manner (1 becomes 5, 5 becomes 1; 3 becomes 7, and 7 becomes 3), there are two common tones between adjacent chords, and that the voices fall in overall pitch throughout the progression. In an unbalanced progression involving a majority of Progressive root motions (Which includes the overwhelming majority of progressions one encounters in Western music), potential range problems can be avioded by alternating between periods of triadic textures and seventh chord textures. The importance of having this understanding should be obvious because, after all, true freedom is total control (Besides tending toward the obsessive/compulsive, I am also a "control freak", as my ex-wife will freely attest to).
In the third version of the progression, I have changed the diatonic chords to dominant sevenths in the second half of each measure to demonstrate the most natural transformation type for the resulting chain of Secondary Dominants. The seventh degree requires that the third and the fifth be raised - unless of course you want the sound of a dominant seventh with a diminished fifth (Which is a very cool sonority, not to mention that it is the origin of the so-called French Sixth) in order to get the resulting V7 of iii. The Natural tendency of the raised third degree of a secondary dominant is for it to resolve to the root of the target chord. This results in the target chord being a triad with the root doubled. Now, the other possibility is for the secondary dominant's raised third degree to "resolve" down chromatically to become the minor seventh of the target chord as we saw in the Minuet by J.S. Bach that I analyzed earlier. Here, I want to use the more natural arrangement to demonstrate the transformation type that is most often used or encountered, and also to fulfill a compositional goal that I have for this progression.
Since the target chord becomes a triad with a doubled root momentarily before it continues on to become another secondary dominant seventh chord, the usual crosswise transformation is delayed. I have indicated this with perentheses around the arrow going to the seventh degree in the transformation analysis symbol. If the less-usual decending chromatic "resolution" of the raised third degree of the secondary dominant was used, the transformation would be the same as the diatonic version, and no parenthesis would be needed. Another thing to note in these natural transformations of secondary dominants is that there is only a single common tone between the adjacent chords. This serves in conjunction with the secondary leading tone to give the progressions a stronger effect than the diatonic versions.
When the progression reaches the V and the I, I have changed the triads on these degrees from major to minor to keep the established pattern going. There is also another reason for doing this, which will become apparent in a minute.
Since there are two tones that change in each measure, I could get a constant quarter note surface rhythm if I put the progression in 3/4 time and changed first one note, and then the other; and since the progression is mostly a series of falling fifths, I decided to use the rhythmic resultant of interference of the interval of the perfect fifth to achieve this, which is the resultant of 3:2= 2+1+1+2.
In the fourth example of the progression, I have opened up the chord voicings to avoid unisoni and to make it easier to visualize. Starting with the top voice in the third measure, you can see the rhythmic resultant of interference applied where 2+1+1+2 becomes a half note plus a quarter note, plus another quarter note, plus a final half note. You can also see this figure in the top voice of the left hand (Assuming we're dealing with an organ piece here) starting in the second measure. These rhythms dovetail to give the overall surface rhythm of constant quarter notes. But there is more to what I have done here. Much more.
In the final example, I have removed the preperatory setup voices involving the I to vii diminished minor seventh progression to reveal that what I have composed through this series of exercises is an accompanied harmonic double canon at the fifth below (Or, at the fourth above). I have arranged the voices so that the two most obviously canonic ones are on the top, and they follow at one measure at the fourth above. The two less obviously canonic voices are underneath and at the fifth below. Since these voices enter on the roots of their respective chords and start a series of decending fifths, I have withheld the constant root accompaniment stratum until the third measure and allowed it to continue with the series of descending fifths in this orchestral sketch. And, this is actually the second section of the introduction to a composition of mine called Theorem of Pythagoras for string choir. I chose the title, of course, because the Pythagorean tuning system's stack of fifths tuning scheme is reflected in the falling fifth root motion here, and I chose the string choir because they can more closely approximate the just fifths in that tuning scheme. Add to that my employment of the rhythmic resultant of the interval of a perfect fifth, and all the details of the composition relate to the perfect fifth. All_of_them.
This method of composing harmonic canons is by no means new, though my employment of rhythmic resultants of interference is relatively unexplored. J.S. Bach used this device constantly in his 371 Four-Part Chorales, in his figuration preludes for various instruments (Where the canons are so well hidden that it takes quite a bit of work to extract them), as well as in many of the episodes in his fugues. Progressions constructed using this device are infinitely more compelling than progressions with similar but casual formations, and any time you have a series of repeating root progression patterns you have the opportunity to compose harmonic canons! These progressions elicit in the listener a sense of wonder and sublimnity, whether they are consciously aware of the canons or not. In harmonic composition, there is no more artful or elegant solution. Period. Exclamation point. That this art has been lost, ignored, not utilized, and never taught is a tragedy of epic proportions, in my estimation, because the technological understanding of harmonic progression mechanics has never been greater than it is now, and by employing ninth chords or other sonorities with extended upper structures many new and unused harmonic canons remain never before used in all of music history.
As mentioned earlier, the Natural Harmonic Overtone Series defines for us the primordial falling fifth V7 to I harmonic progression through it's first seven partials, which spell out a dominant seventh chord. Throughout the late Renaissance and early Baroque eras, this progression was extended to secondary degrees of the key until the entire pantheon of Secondary Dominant Sevenths was completed. The way this works is that in any root progression in a key that is Progressive in nature, and that involves degrees other than V and I, the first chord in the progression can be changed from it's natural state - either a minor seventh or a major seventh - to a dominant seventh chord so that it targets the secondary degree in the same manner as the V7 to I progression does.
In the top example above, I have written a progression that allows for all of the secondary dominants to be displayed. By progressing directly from the tonic to the seventh degree, I have set up for a series of falling fifth Progressive Root Motions that will involve each of the seven degrees of the key.
The top example shows the triadic Circular Transformations. After the Strong Decending root motion from the I to the vii diminished triad that kicks off the progression with it's associated clockwise transformation, all subsequent root motions are of the falling fifth (Or, rising fourth) Progressive type. Note that in the Progressive root motions all of the Circular Transformations are counterclockwise (1 becomes 5, 5 becomes 3, and 3 becomes the root), the triads share a single common tone, and that the voices rise in overall pitch from chord to chord. In the final bar there is a Strong Ascending root motion from the IV to the V to turn the progression around on itself, and it's transformation is counterclockwise opposite of the Strong Decending progression from the first measure to the second. This is an example of a very unbalanced chord progression: Since there are not a variety of root motions that alternate between clockwise and counterclockwise transformations, subsequent repititions of the progression will soon lead to problems of range if an all-triad environment is maintained.
In the second example of this progression, I have presented it in an all-seventh chord diatonic environment. Note that in this example the Progressive root motions transform in a crosswise manner (1 becomes 5, 5 becomes 1; 3 becomes 7, and 7 becomes 3), there are two common tones between adjacent chords, and that the voices fall in overall pitch throughout the progression. In an unbalanced progression involving a majority of Progressive root motions (Which includes the overwhelming majority of progressions one encounters in Western music), potential range problems can be avioded by alternating between periods of triadic textures and seventh chord textures. The importance of having this understanding should be obvious because, after all, true freedom is total control (Besides tending toward the obsessive/compulsive, I am also a "control freak", as my ex-wife will freely attest to).
In the third version of the progression, I have changed the diatonic chords to dominant sevenths in the second half of each measure to demonstrate the most natural transformation type for the resulting chain of Secondary Dominants. The seventh degree requires that the third and the fifth be raised - unless of course you want the sound of a dominant seventh with a diminished fifth (Which is a very cool sonority, not to mention that it is the origin of the so-called French Sixth) in order to get the resulting V7 of iii. The Natural tendency of the raised third degree of a secondary dominant is for it to resolve to the root of the target chord. This results in the target chord being a triad with the root doubled. Now, the other possibility is for the secondary dominant's raised third degree to "resolve" down chromatically to become the minor seventh of the target chord as we saw in the Minuet by J.S. Bach that I analyzed earlier. Here, I want to use the more natural arrangement to demonstrate the transformation type that is most often used or encountered, and also to fulfill a compositional goal that I have for this progression.
Since the target chord becomes a triad with a doubled root momentarily before it continues on to become another secondary dominant seventh chord, the usual crosswise transformation is delayed. I have indicated this with perentheses around the arrow going to the seventh degree in the transformation analysis symbol. If the less-usual decending chromatic "resolution" of the raised third degree of the secondary dominant was used, the transformation would be the same as the diatonic version, and no parenthesis would be needed. Another thing to note in these natural transformations of secondary dominants is that there is only a single common tone between the adjacent chords. This serves in conjunction with the secondary leading tone to give the progressions a stronger effect than the diatonic versions.
When the progression reaches the V and the I, I have changed the triads on these degrees from major to minor to keep the established pattern going. There is also another reason for doing this, which will become apparent in a minute.
Since there are two tones that change in each measure, I could get a constant quarter note surface rhythm if I put the progression in 3/4 time and changed first one note, and then the other; and since the progression is mostly a series of falling fifths, I decided to use the rhythmic resultant of interference of the interval of the perfect fifth to achieve this, which is the resultant of 3:2= 2+1+1+2.
In the fourth example of the progression, I have opened up the chord voicings to avoid unisoni and to make it easier to visualize. Starting with the top voice in the third measure, you can see the rhythmic resultant of interference applied where 2+1+1+2 becomes a half note plus a quarter note, plus another quarter note, plus a final half note. You can also see this figure in the top voice of the left hand (Assuming we're dealing with an organ piece here) starting in the second measure. These rhythms dovetail to give the overall surface rhythm of constant quarter notes. But there is more to what I have done here. Much more.
In the final example, I have removed the preperatory setup voices involving the I to vii diminished minor seventh progression to reveal that what I have composed through this series of exercises is an accompanied harmonic double canon at the fifth below (Or, at the fourth above). I have arranged the voices so that the two most obviously canonic ones are on the top, and they follow at one measure at the fourth above. The two less obviously canonic voices are underneath and at the fifth below. Since these voices enter on the roots of their respective chords and start a series of decending fifths, I have withheld the constant root accompaniment stratum until the third measure and allowed it to continue with the series of descending fifths in this orchestral sketch. And, this is actually the second section of the introduction to a composition of mine called Theorem of Pythagoras for string choir. I chose the title, of course, because the Pythagorean tuning system's stack of fifths tuning scheme is reflected in the falling fifth root motion here, and I chose the string choir because they can more closely approximate the just fifths in that tuning scheme. Add to that my employment of the rhythmic resultant of the interval of a perfect fifth, and all the details of the composition relate to the perfect fifth. All_of_them.
This method of composing harmonic canons is by no means new, though my employment of rhythmic resultants of interference is relatively unexplored. J.S. Bach used this device constantly in his 371 Four-Part Chorales, in his figuration preludes for various instruments (Where the canons are so well hidden that it takes quite a bit of work to extract them), as well as in many of the episodes in his fugues. Progressions constructed using this device are infinitely more compelling than progressions with similar but casual formations, and any time you have a series of repeating root progression patterns you have the opportunity to compose harmonic canons! These progressions elicit in the listener a sense of wonder and sublimnity, whether they are consciously aware of the canons or not. In harmonic composition, there is no more artful or elegant solution. Period. Exclamation point. That this art has been lost, ignored, not utilized, and never taught is a tragedy of epic proportions, in my estimation, because the technological understanding of harmonic progression mechanics has never been greater than it is now, and by employing ninth chords or other sonorities with extended upper structures many new and unused harmonic canons remain never before used in all of music history.
posted by Hucbald at 8:54 AM
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