Reductio ad Absurdum: The Series

One of the things I stress to my students is that they should write a lot of miniatures, and preferrably several series of miniatures. It is especially useful if the series can fulfill dual purposes, and be both technical studies for the guitar and compositional studies as well. An added bonus - and one of my personal requirements for any studies that I write - is if the resulting series of studies are "good enough" to be worthy of performance.

Over the years I have written almost ten series of miniatures, and these little sets of studies invariably lead to increased technical mastery of the guitar, increased compositional technique, and - perhaps most importantly - larger and more significant works. I'm not sure where this series will lead compositionally, but I do know all of these pieces will end up in my performance set.

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As I mentioned previously, this set deals with dualities: The duality between the major and minor modes, the duality between triple and duple time, and the duality between 4-3 syncopes and 2-3 syncopes. The resulting twelve pieces are only about thirty seconds in performance time, and they are all on the pitch level of A (Which is the only place these four-voice contrapuntal studies will work on the guitar due to fingering considerations).

The set with the first two variables is ordered as follows:

01) Modus et Tempus Perfectum, or A major in triple time.

02) Modus et Tempus Imperfectum, or A minor and duple time.

03) Modus Imperfectum et Tempus Perfectum, or A minor in triple time.

04) Modus Perfectum et Tempus Imperfectum, or A major in duple time.

As you can see, the time alternates between triple and duple, but the mode has two minor versions sandwiched between the majors. This allows for the beginning and end of the series to be in the major mode, though the opening tempus is perfect, and the closing tempus is imperfect. This is cool, because the imperfect versions - in both modus and tempus - are the more "interesting" ones, as you will see.

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So, here are the four basic templates:









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Once this is worked out, the next step is to apply the syncopation chains. Please note that I have not added the left hand fingerings to these yet, and I may have to add a rest or two if I can't maintain the purity of the music due to fingering considerations. I was fortunate that the basic major and minor fingerings did not require any compromises to the purity of the musical conception, but adding the syncopes may end that string of luck. This is one of the nice things about this project: My idea of fingering possibilities on the guitar - already at a high level of understanding after thirty years of playing - will take another step forward.

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Here are the four versions with the 4-3 syncopation chains added:









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In the second and eleventh measures of the above versions, you'll notice that the shorter rhytmic value comes first in the leading voice: I did this to get the 7-6 syncopes in measures eight and seventeen (The second of which is, obviously, a multiple suspension), the resulting resolution and rhythm of which provides for a better end to the phrases. Just setting that up taught me that a rhythmic point of resistance near the launch of a melodic trajectory is a compelling device: You hardly ever hear this in western music (Or, at least, I can't recall any examples of it at the moment).

Note here that the counterpoint is generating the rhythm, and the palindromic nature of that rhythm - combined with the palindromic canon(s) - creates a solid inevitability to the overall effect. Though simple, these rhythms are compelling precicely because they are being generated by the music: The rhythm is organic on a very basic and inescapable level.

Remembering that I said that the imperfect (2/4) versions were "more interesting": The 3/4 versions simply launch into a surface rhythm of constant quarter notes, and dissolve back out of that; but the 2/4 versions create a more complex cumulative rhythmic palindrome, which creates a superior rhythmic cadential effect.

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Here are the four original versions with the 2-3 syncopes added:









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I have more hope that the 4-3 syncopes will be more easily executable on the guitar, as seconds are notoriously problematic. However, I am dealing with a lot of open strings, so I could yet be pleasantly surprised. The next step will to be to play through the versions with the syncopations to see.

As I said, I plan to add these - or at least some of them - to my set. Since I have my set organized into suites, and since these suites begin in A minor and follow the circle of thirds, these little mini-fugues will act as interludes and an axial ground to my set. The prelude to interlude (fuga) group for my set from beginning to intermission will look like this:

01) Suite in A minor/Reductio in A major

02) Suite in C major/Reductio in A minor

03) Suite in E minor/Reductio in A minor

04) Suite in G major/Reductio in A major

05) Suite in B minor/Reductio in A major

06) Suite in D major/Reductio in A minor

07) Suite in F-sharp minor/Reductio in A minor

08) Suite in A major/Reductio in A major

Cool, huh? I love it when a plan falls together. After my break the set continues on to B major, so I could get all twelve in if they happen to work out, but the A minor to A major segment is really nice with this organizational device thrown in.

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This fractal is not a Mandelbrot set, but its duality fit in with today's theme nicely.



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How about some variations on this theme?