Sunday, June 03, 2007

Convertible Counterpoint IX

It has been over a year since my last post on this subject, and this is simply an end note for the series which I will post a link to in the sidebar section that deals with Sergi Taneiev.

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:


Note that this law is permissive: It tells you what you may do. From this, we can deduce the the two secondary prohibitive laws:




Finally, from this triumvirate of immutable contrapuntal laws, we can deduce the final exceptional law:


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.


Blogger Octavio said...

The fourth in Fux's treatise is a dissonant interval, how harmonic overtone series explain this?

We could symmetrically say that counterpoint considerations based on harmonic series collapse because of this.

And yet there are mathematical reasons for not considering the fourth as a consonance.

Modulo octave, there are 12 intervals. There are 6 classic consonances (unison, minor and major thirds, perfect fifth adn minor and major sixth) and hence 6 dissonances. This is the only bipartition of the intervals that has only one affine transformation that swaps them. Additionally, this bipartition is maximally separated with respect to the distance of thirds (i. e., the minimum number of either minor or major thirds between to pitches modulo octave)

See "The Topos of Music" by G. Mazzola for details.

10:42 AM  
Blogger Hucbald said...

I suggest reading the book I cited above, Tenney's History of Consonance and Dissonance. Basically, the technical acoustic justification for treating the perfect fourth as a dissonance is that the interference pattern created has more in common with the dissonances of a major ninth and minor seventh than with the consonances of thirds and sixths. Of course, the ancients had no technical way to analyze sound, all they knew was that the perfect fourth MADE THE SYLLABLES OF THE SUNG TEXT DIFFICULT TO UNDERSTAND, so that's the practical justification that they used. For us moderns, the perfect fourth is a perfect consonance and that's that, so Taneiev's entire thesis crumbles into dust in that light, which is why I set up vertical and horizontal contrapuntal shifts mechanically on staves where I can use the modern laws I discovered as well as see the results. Anything that gets into mathematical formulas is really non-musical in that the musical mind just doesn't think that way. I made these discoveries while doing this post series, which is why I never finished it.



7:26 PM  
Blogger Octavio said...

But if a theory is meant to be consistent, it cannot support two contradictory claims. Perhaps I am missing something.

In fact, the response of the brain to the affine transformation I mentioned have been tested via deep EEG, so I think it is little bit bold to say that mathematical "formulas" are not present in the musical mind. Of course, not all the aspects of music come from mathematics, just as valuable mathematics does not come from elementary computations.

My "musical mind", for instance, tells me that the tritone is a consonance, and actually I discovered a counterpoint scheme that uses it as a consonance (along with the unison and the major third).

Nevertheless, I accept that the theories you have used have proven effective for you to write beautiful music. So, congratulations!

11:39 PM  
Blogger Hucbald said...

1] The point I made that you missed is that the perfect fourth is now a fully perfect consonance. So, it is consistent now, whereas it was not in the era of strict modal counterpoint, which is what Taneiev was writing about.

2] There may be a mathematically inclined musical mind somewhere in the world, but I've never met one. My suspicion is that most mathematically inclined people of above average intelligence end up in physics or related fields. Just a guess of course. Those musicians I have met who use strict mathematical schemes in their music have universally failed to create anything that I find musically convincing, regardless of how objective I attempt to be. In related study about Taneiev, I discovered that he sketched out his ideas for canons and other shifting counterpoint on staves the old fashioned way, which makes perfect sense to me. Another reason I didn't bother to continue my studies of him any further.

3] You can only rationalize the tritone as a consonance if you ignore the implications of the harmonic series. You are free to do that, of course.




3:24 PM  
Anonymous Anonymous said...

According to Jose Sottorio's book (Tone Spectrum?)... A fourth is considered dissonant in common practice harmony. While it IS an acoustic consonance, its 'harmoniousness' has conditions. If a fourth is presented in the bass, the lowest tone suggests a harmonic series, complete with perfect 5th. This clashes with the fourth (&its overtones)...


However, if the fourth has an octave underneith (as it might appear in middle and upper parts) it sounds harmonious because the bottom note is no longer setting up an incongruous series (perceptually)...


So according to him, the intuition of the ancients (on both sides of the argument) was correct. They were really arguing about acoustics verses context.

Great Blog.

9:42 AM  
Anonymous Anonymous said...

... just like to add, CP harmony is really about acoustic clarity. A major third is dissonant when it is played in the lower registers! Schenker also treated the fourth as a special case, just as Bach, Mozart, Haydn, Beethoven accepted that fourths were a special case in CP harmony. Artistically, however, they treated it how they must.

It is a strange that your Taneyev blog starts with "This is the greatest counterpoint book ever written" and ends with "Mechanically wrong, on so many levels." All because he treats the fourth as a dissonance?

Oh well, enjoyed it anyway.

9:55 AM  
Blogger Hucbald said...

It's not only that Taneiev - and strict style counterpoint in general - treats the fourth as a dissonance, but also that the formulaic methodology is an anti-intuitive way for most composers to work. The natural way is mechanical: Work the canons and vertical shifts out on staff paper (Or on a computer version of same, as I do, with all of the cut and paste and modal transposition possibilities).

As for the acoustical "problems" with the fourth, I understand them just fine - my source was Tenney's The History of Consonance and Dissonance - but the solution is to simply recognize the mechanical equivalence of the admittedly asymmetrical contrapuntal inversions: A fifth becomes a fourth, and that does not negate the validity of the combination.

Probably needless to say, but the issue of clarity is greatly ameliorated when writing instrumental music: Fourths really muck about with vocal formants, which makes understanding the text more difficult. This was actually the primary objection of the ancients before acoustics entered the picture.



11:01 AM  

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