Monday, January 12, 2009

Another Canon in Inversion

To explore symmetrical inversion further:

Yesterday's canon in inversion required each hand to be in a different key/scale. As I pointed out, this is a feature of strict intervallic inversion. When you invert this way (using the tonic as the axis of symmetry), major inverts to phrygian, mixolydian inverts to aeolian, lydian inverts to locrian and vice versa.

Oh, I didn't mention dorian...

This is a special case. Dorian inverts to itself or, to phrase it a different way: dorian is symmetrical around the tonic. See the examples.

The last example shows the chords that I arpeggiated as measures 12-15.

                          (Click on image to enlarge)

The violin is the dux until measure 10. I have the cello lead after that. The strict canon at 2 beats lasts until measure 15. From there to the end I use strict mirroring. Technically, this ending is still a canon in inversion -- but the time interval has changed to "zero."

Harmonic control was fairly easy using dorian in inversion.

"D" inverts to itself (tonic).

"A" and "G" invert to each other -- so I was able to use a seventh interval as a dominant for the middle section (starting in measure 10).

"B" and "F" invert to each other -- I exploited this by coming to this tritone at the end of the strict canon. At that point, I moved in contrary motion -- first to the seventh A-G (functioning as dominant and then, with more rhythmic interest, converging on the tonic at the end. The "stinger" is a Dsus4 chord -- symmetrical around the middle D axis.

                                      (Click on image to enlarge)

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