Harmonic Canons with Two Root Progression Types
In the course of musical exploration, on occasion one learns from unsuccessful experiments as much, if not more, than from the purely successful ones. Today's example will embody that truism. I had in mind to write triadic and tetradic harmonic canons over a root progression pattern that had two progression types in it. For this exercise I chose a pattern that consisted of a mild decending root motion followed by a progressive root motion. Part of this choice was a result of the fact that the patterns I have employed thus far have had an overall decending trajectory in the constant root bass line, and this combination offered a seemingly refreshing ascending pattern for the lower stratum. There was also the fact that the second progressive motion offered the oportunity to employ some of the secondary dominant sonorities I have described to this point.
Well, once I discovered harmonic canons, I naturally ass-u-me'd that any repeating root progression pattern would offer the oportunity to write them. I was only partly correct in that deduction: In a triadic texture, if the two root progression types alternate between clockwise transformations and counterclockwise transformations, the voices simply cycle back and forth between one chord function and another, and so there is literally no canon, strictly speaking, to be extracted that involves all three of the voices. This was... ah... an eye opener. This most likely also applys to tetradic upper stratum textures, as seems logical in retrospect, but I'll hold my tongue and refrain from commenting difinitively until I have had the oportunity to come to terms with the situation more fully. Fool me once...
On the top staff of the above example, we can see where the "problem" arises. The circular transformation types strictly alternate between clockwise and counterclockwise, so in the lowest triadic voice the 5 becomes 1, then it becomes 5, and then it becomes 1 again: That voice never has the third of any of the triads. The other two voices above rise in a series of parallel sixths exchanging 3 and 5, and 1 and 3 respectively. Now, this means that only two of the voices can really be arranged as canonic (And that in a less than perfectly satisfactory manner), instead of all three. This is a situation similar to the one where the double canon arose in the tetradic texture with all crosswise transformations in association with a constant root bass that only moved progressively, but in this situation one of the voices must be the odd man out, it not being given an echo in the triadic texture. I chose the two upper voices that share the third to make this quasi-canon, but either one of the upper two voices could be used against the lowest voice if so desired. I managed to create a series of chromatically parallel sixths in the upper two voices to work with, so I chose them. I still don't have my brain 100% wrapped around this subject, which is why I will continue to post on it until I'm satisfied that I have it under control.
On the second staff, it can be seen that no similar problem arises in the tetradic texture. Here, all four voices get all four of the canonic elements, so an eight bar canonic voice is created that follows the leader at a two bar distance. Applying intervalic strictness to these diatonic frameworks will take us to some interesting places for both the triadic and the tetradic harmonic canons, as we shall see, but the seventh chords take us all the way around the musical universe.
On the third stave I have made the triadic canon strict and added secondary dominant triads to it, as per previous examples. Here, however, the tonic triad must be changed to the minor gender, and the more remote root progressions must be chromatically altered to keep the pattern going. This creates a very unusual effect where the sense of tonality is lost, but the sense of direction isn't. I like it very much, and without this experiment, I would not have thought of going this route. If we analyze it in detail, we can see that the root progression is a falling minor third followed by a rising perfect fourth. This works fine diatonically from C to A, and from D to B, but from E to C, a chromatic adjustment must be made to avoid the decending major third. If you look at the analysis, I am targeting a diminished triad on F-sharp with a secondary dominant sonority on C-sharp. "You can't do that", except for the fact that it sounds righteous in this context, of course.
I used the syncope device to get a weird canon of alternating major and minor sixths (Or augmented fifths) in the triadic "canon" on the third staff, and with the momentary minor major seventh chords, it sounds highly exotic. Though I failed to create a canon proper as I set out to in this part of this experiment, I sure learned a few things and came up with some completely unique and unexplored sonic effects through it. I don't mind at all when that happens.
The bottom two staves show the extracted tetradic harmonic canon with added secondary dominant sevenths and secondary dominant seventh diminished fifth sonorities. I had to keep the canon going for so long to give the final trailing voice the oportunity to present the entire canonic voice. Note that since all four voices share all four of the canonic elements, diminished tenths occur in between the augmented sixths. This actually sounds fine in this context, which surprised me, as in two voice counterpoint on solo guitar I have been squeemish about employing the diminished tenth octave displacement inversion. This would qualify as one of those things that makes ya go, "hmmmm". Note also that with the chromatic alterations to the constant root bass, it is now a twelve tone row that begins to repeat at the upper octave C before it is interrupted by the final cadential figure. Because of this excursion through all of the most remote regions fo the key, any sense of a tonic center is entirely forgotten, as I mentioned previously concerning the triadic example. Here, though, the listener really ends up lost in the Sahara! Overall, I find it to be a magnificent effect worthy of employment. Finally, I should have given another bar for the final cadence, as it sounds overly abrupt, but I was running out of space on the page, and photo hosting is expensive! (Yeah, that was a hint).
Well, once I discovered harmonic canons, I naturally ass-u-me'd that any repeating root progression pattern would offer the oportunity to write them. I was only partly correct in that deduction: In a triadic texture, if the two root progression types alternate between clockwise transformations and counterclockwise transformations, the voices simply cycle back and forth between one chord function and another, and so there is literally no canon, strictly speaking, to be extracted that involves all three of the voices. This was... ah... an eye opener. This most likely also applys to tetradic upper stratum textures, as seems logical in retrospect, but I'll hold my tongue and refrain from commenting difinitively until I have had the oportunity to come to terms with the situation more fully. Fool me once...
On the top staff of the above example, we can see where the "problem" arises. The circular transformation types strictly alternate between clockwise and counterclockwise, so in the lowest triadic voice the 5 becomes 1, then it becomes 5, and then it becomes 1 again: That voice never has the third of any of the triads. The other two voices above rise in a series of parallel sixths exchanging 3 and 5, and 1 and 3 respectively. Now, this means that only two of the voices can really be arranged as canonic (And that in a less than perfectly satisfactory manner), instead of all three. This is a situation similar to the one where the double canon arose in the tetradic texture with all crosswise transformations in association with a constant root bass that only moved progressively, but in this situation one of the voices must be the odd man out, it not being given an echo in the triadic texture. I chose the two upper voices that share the third to make this quasi-canon, but either one of the upper two voices could be used against the lowest voice if so desired. I managed to create a series of chromatically parallel sixths in the upper two voices to work with, so I chose them. I still don't have my brain 100% wrapped around this subject, which is why I will continue to post on it until I'm satisfied that I have it under control.
On the second staff, it can be seen that no similar problem arises in the tetradic texture. Here, all four voices get all four of the canonic elements, so an eight bar canonic voice is created that follows the leader at a two bar distance. Applying intervalic strictness to these diatonic frameworks will take us to some interesting places for both the triadic and the tetradic harmonic canons, as we shall see, but the seventh chords take us all the way around the musical universe.
On the third stave I have made the triadic canon strict and added secondary dominant triads to it, as per previous examples. Here, however, the tonic triad must be changed to the minor gender, and the more remote root progressions must be chromatically altered to keep the pattern going. This creates a very unusual effect where the sense of tonality is lost, but the sense of direction isn't. I like it very much, and without this experiment, I would not have thought of going this route. If we analyze it in detail, we can see that the root progression is a falling minor third followed by a rising perfect fourth. This works fine diatonically from C to A, and from D to B, but from E to C, a chromatic adjustment must be made to avoid the decending major third. If you look at the analysis, I am targeting a diminished triad on F-sharp with a secondary dominant sonority on C-sharp. "You can't do that", except for the fact that it sounds righteous in this context, of course.
I used the syncope device to get a weird canon of alternating major and minor sixths (Or augmented fifths) in the triadic "canon" on the third staff, and with the momentary minor major seventh chords, it sounds highly exotic. Though I failed to create a canon proper as I set out to in this part of this experiment, I sure learned a few things and came up with some completely unique and unexplored sonic effects through it. I don't mind at all when that happens.
The bottom two staves show the extracted tetradic harmonic canon with added secondary dominant sevenths and secondary dominant seventh diminished fifth sonorities. I had to keep the canon going for so long to give the final trailing voice the oportunity to present the entire canonic voice. Note that since all four voices share all four of the canonic elements, diminished tenths occur in between the augmented sixths. This actually sounds fine in this context, which surprised me, as in two voice counterpoint on solo guitar I have been squeemish about employing the diminished tenth octave displacement inversion. This would qualify as one of those things that makes ya go, "hmmmm". Note also that with the chromatic alterations to the constant root bass, it is now a twelve tone row that begins to repeat at the upper octave C before it is interrupted by the final cadential figure. Because of this excursion through all of the most remote regions fo the key, any sense of a tonic center is entirely forgotten, as I mentioned previously concerning the triadic example. Here, though, the listener really ends up lost in the Sahara! Overall, I find it to be a magnificent effect worthy of employment. Finally, I should have given another bar for the final cadence, as it sounds overly abrupt, but I was running out of space on the page, and photo hosting is expensive! (Yeah, that was a hint).
posted by Hucbald at 4:25 AM
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