Sunday, February 11, 2007

Quadrant Rotation of Entire Musical Continuities

When I wrote the Fuga: Reductio ad Absurdum studies a while back - I've since changed the titles to Irreducible Fugues - I realized that one could use Schillinger's concept of quadrant rotation to get four versions of any musical continuity so long as the thematic material was set up rhythmically to allow for it. In the Invertible Canon for Wind Choir of a couple of posts back I demonstrated inverting an entire continuity, but the themes involved with that piece were not set up with the backwards versions in mind, so the other two possibilities - retrograde and inverted retrograde - do not work so well.

Like many musicians, I suppose, I had always considered retrogrades to be pretty much out of bounds except for parlor tricks - even Bach's crab canons sound forced to me - and themes played backwards usually sound goofy and are notoriously difficult to recognize. Well, those little irreducible fugue exercises made me realize why that is so: The rhythms are not properly set up to sound natural and be recognizable in the reverse direction.

I experimented with symmetrical rhythms - reversing them would obviously be no problem - but they lack a certain degree of interest. What I wanted was a head and tail fugue subject that made a canon and would work in all four positions. What I found the key to be (At least, one of the keys) is slowing down the rhythm at the end of the tail figure. Not as slow as the head, but at least slower than the fastest note values (Or, shortest note values).

The following example is just an experiment - I write stuff like this all the time (It's sort of like doing compositional workouts) - and I don't know if it will ever get beyond this stage, but I thought it interesting enough to share. As with all of the fugue subjects I write, this has a premise: The theme accelerates so that in the fourth measure there is simultaneous 2:1, 3:1, and 4:1 counterpoint happening. This rhythmic peculiarity also aids in recognizing the retrograde forms of the continuity.

Here is the canon in its original form:

As you can see, the subject starts out with a typical do, sol head figure in half notes which is then followed by an ascending chromatic tetrachord in quarters. The third measure then has quarter note triplets in an ascending diatonic tetrachord, followed by eighths descending chromatically to the final le, sol in quarters to end the subject.

Remember: I use the irreducible set of contrapuntal laws derived from implications of the overtone series:

1) Parallel Perfect Consonances are Not Allowed.

2) Parallel Imperfect Consonances are Allowed.

3) Parallel Dissonances are Not Allowed.

4) Unequal Parallels are Allowed.

5) Contrary Stepwise Motion Justifies Any Intervallic Succession (Simultaneous and non-simultaneous cross-relations are allowed).

This means the sound is quite dissonant and unusual, and the more I do this sort of thing, the more I like the language I'm developing here: I first used it in the Perpetual Canon for String Choir and it has continued to develop from there.

Here is the intervallically strict inversion of the canon:

Note that I have raised the third degree of the mode to maintain intervallic strictness. If I had not done that, there would have been parallel perfect fourths into the second half of the third measure of this version: The unequal parallel of the augmented fourth to the perfect fourth is perfectly acceptable though. This becomes unequal fifths in the fourth measure where the melodic fragments are inverted.

Now for the retrograde:

Notice how the quarter notes launch into the theme smoothly. Eighths right off the bat would not have worked so well, which is why most fugue subjects don't reverse well. The gradual slowing of the theme also leads nicely to the cadence at the end.

And, the inverted retrograde:

If you want to hear these, they are now on my Downloads Page in both PDF and MP3 formats.

I think I may link these together into a double crab canon for my next exercise.

It's in the 70's here today, so I think I'll go out and get some vitimin D in my system.


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