Sunday, September 24, 2006

Harmonic Implications of the Overtone Series, Part IV

Last time I got to the point where I demonstrated that it is harmonic implications of the overtone series which predict canon. In this installment I want to pursue that a bit more.

Anytime you create a harmonic sequence which has either the same root motion type, or a pattern of root motion combinations of the same types, and you are using the proper transformations in your voice leading, you are creating harmonic canons, whether you are aware of it or not. The only exception to this is if the voice transformation types do not create a complete cycle: For instance, if they simply alternate back and forth between clockwise and counterclockwise transformations, as I shall demonstrate.



Here is our first continuity example again, but this time I have extracted the very simple diatonic harmonic canon on the second system. This may not look or sound much like a canon, but it can be dressed up to make the canon more noticable as well as more perfect.

On the third system I have added secondary leading tones and made sure that all of the targets are minor triads to make the canon strict and more apparent. Then, in the fourth system, I added a further series of alterations to get augmented triad sonorities.

Down on the fifth system, I changed the time signature to 3/4 and ended up with a fairly interesting little harmonic canon which ends with a deceptive motion to the relative minor. Finally, on the bottom staves, I combined these versions to get a dovetailing harmonic canon in which the interest mounts as it progresses by beginning diatonically, then adding the secondary leading tones, and finally the augmented fifths. It also overshoots the key and ends with a funky and vaguely medieval-sounding cadence.

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Same example in a tetradic texture: First I extracted the simple and un-obvious diatonic double canon, then I added secondary dominants, then the French-derived secondary dominants with diminished fifths, and finally, I combined all of the versions to create a dovetailing double canon which increases in complexity and interest as it goes along.

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Here is an example which will demonstrate a couple of different things: First, that a repeating pattern of two different root motion types wil create harmonic canons, and second, if these root motion types only alternate between clockwise and counterclockwise transformations, the canon will be incomplete.

The root progression pattern is made up of a half-progression followed by a progression, and the triadic and tetradic versions are presented on the first two staves.

As you can see, the circular transformations in the triadic texture only cycle back and forth between clockwise and counterclockwise motion. That means the chord tones do not get to play all of the roles (Root, third, and fifth), so what you end up with is a two-part canon with a free voice, as shown on the fourth system, instead of a true three voice canon.

The tetradic version does not suffer from this "defect" however, and if you continue the original root motion pattern in strict intervals, it transcends the diatonic system in a very nice harmonic double canon which goes all the way through the chromatic system. Note that, as with the bass line in the final example of the previous post, this results in a twelve-tone row, but one which has inherent harmonic functionality. The harmonic series implies not only integrated chromatic tonality, but so-called tone rows, or serial systems as well! The caviat is, of course, that these tone rows will be harmonically functional, or rather, viable.

And some people still believe music is possible entirely without a harmonic or modal context. Well, it isn't. I can only demonstrate the truth, I can't comprehend it for you.

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Yes, my date is on the way. Ta, ta.

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