I abandoned this project because I reached a critical mass point with my understanding of contrapuntal laws as defined by the implications present in the harmonic overtone series. You will find those defined in the Musical Relativity series Chapter VII, but briefly:
The series defines for itself what are perfect and imperfect consonances through the octave inversion principle. Perfect consonances remain super-particular ratios when inverted (In a super-particular ratio the terms differ by one). So, since the 2:1 octave remains a 2:1 octave when the voices are inverted, it is a perfect consonance. Likewise, the perfect fifth at 3:2 inverts to a perfect fourth at 4:3: Both ratios are super-particulars.
Imperfect consonances are only super-particulars in one orientation: The 5:4 major third inverts to an 8:5 minor sixth, and the 6:5 minor third inverts to a 5:3 major sixth.
Once I understood this, Taneiev's system totally collapsed, since he treats the fourth as a dissonance. There are acoustic reasons for this peculiarity which I won't go into here, but the best discussion of the fourth ever is in James Tenney's book A History of Consonance and Dissonance.
Then, I realized that almost all of the other restrictions in counterpoint were rules based upon the styles of a particular school, and not laws based on what the overtone series suggests is possible. Most of those rules were simply based on personal or group taste.
You'll have to follow the link above to get the full rationale, but basically there is one supreme law of counterpoint:
1] ONLY IMPERFECT CONSONANCES MAY MOVE TOGETHER IN PARALLEL STEPWISE MOTION.
Note that this law is permissive: It tells you what you may do. From this, we can deduce the the two secondary prohibitive laws:
2] PERFECT CONSONANCES MAY NOT MOVE TOGETHER IN PARALLEL STEPWISE MOTION.
and...
3] DISSONANCES MAY NOT MOVE TOGETHER IN PARALLEL STEPWISE MOTION.
Finally, from this triumvirate of immutable contrapuntal laws, we can deduce the final exceptional law:
4] PERFECT OR IMPERFECT CONSONANCES MAY MOVE IN PARALLEL STEPWISE MOTION INTO DISSONANT AUGMENTED OR DIMINISHED VERSIONS OF THEMSELVES AND VICE VERSA.
So, for example, the progression P5, d5, P5 is allowed, as is the progression m3, d3, m3, and all other variations on that scheme.
When you view counterpoint in the light of this level of understanding, there are so few restrictions that the possibilities increase exponentially over what Taneiev defines as "The Rules of Simple Counterpoint." Since all further convertible technology is based on those rules, I'm no longer interested in his system. It might seem a bit brash to say that I've outgrown the need for it, but I have.
If you wish to write an original combination that yeilds derivatives, all you have to do is go through the process mechanically on musical staves where you can present all of the possibilities you want simultaneously. This is how I work when I write counter-subjects to fugue subjects, for example. And the subjects themselves I compose in canon mechanically as well. I'm absolutely certain that this is how Palestrina, Zarlino, and Bach came up with their derivatives that Taneiev uses as examples (Though some of this stuff can be done in one's head after a certain amount of experience, obviously).
So, there really is no need for formulas in music at all. Ever.
Now, it could be possible to take Taneiev's system and modify it starting off with different basic rules for simple counterpoint, but, in my personal opinion, that would be an exercise in futility.
I love Taneiev's music, by the way - his Fourth Symphony I put on par with any by Brahms - and I have read quite a bit about the man, who was by all accounts a broadly talented genius: He spoke Esperanto, for example. However, by all accounts I've found, when he composed, he started by writing out canons... mechanically.
Mechanically wrong, on so many levels.
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