Musical Implications of the Harmonic Overtone Series: Chapter V
The Series' Prediction of Canon
As I mentioned in chapter four, the series, quite by itself when left to its own devices, will create canons. There are two circumstances required for this to take place: The root progression pattern has to be all of a single type or it must be a repeating pattern of two or more types, and the upper texture must follow proper transformational logic or at least a logic of all the same kind. If this is done casually within the diatonic system, non-strict diatonic canonic voices will result in the upper transformational stratum, but if the original established intervallic relationships are scrupulously followed in subsequent iterations, strict canons will result.
There are a couple of exceptions to note: In a three voice texture, if two root motion types are selected which result in alternating clockwise and counterclockwise transformations, then the chord tones will not get to play all of the roles within the triad (Root, third, and fifth). This will result in an incomplete canon in which two voices follow properly, but the third voice is "free" so to speak. Likewise, in a four voice texture, if the single or multiple root motion types don't allow for the voices to play all of the tetradic roles, then a four-voice canon will not result: But a double canon will. The double harmonic canon is what is actually implied in the series, but I will display a true four voice canon with two root motion types as well.
In Example I I have presented the old reliable initial proof in the triadic version. In Example II I extracted the three voice diatonic canon out of it. Obviously, this canon does not draw any attention to itself, but as we embelish it, it will.
By Example III, where I introduce secondary dominant triads into the mix, the canonic nature of the transformational stratum begins to become noticeable. Then, in Example IV, I introduce a nice altered dominant sonority available in triadic textures: The augmented triad. This second of the symmetrical harmonic structures - being a loop of major thirds - also generates a hexa-phonic whole tone scale just as the French and Italian-derived secondary dominants do.
Finally, in Example V, I dovetail the diatonic, secondary dominant, and augmented triad versions together to produce a three voice canon which increases in harmonic and melodic interest over the course of the phrase.
On this page of examples, I have presented the same continuity example with tetradic upper strata. Since all of the transformations are crosswise, one pair if voices alternates between the roots and fifths, while the other pair alternates between the thirds and sevenths (With the intermediate triads with doubled root). This results in a double canon instead of a four voice canon. So, the series-implied falling fifth root motions create a double canon, as you can see.
The process is the same as with the triadic examples: Example VI is the basic continuity, Example VII is the extracted diatonic double canon, Example VIII adds secondary dominant sevenths, Example IX adds French-derived V(d5m7) chords, and finally, Example X combines aspects of all of the previous versions into a dovetailing strict double canon.
In Example XI we have a root progression pattern of two types: A half-progression followed by a progression. This is the same pattern I used in the second integrated modality proof, but this version is diatonic and has a triadic upper stratum. Example XII adds secondary dominant triads, and Example XIII is only a two voice canon with a third "free" voice because the alternating clockwise and counterclockwise transformations do not allow all of the voices to take all of the chordal degrees.
Example XIV is the same root motion pattern with a tetradic upper stratum, and as you can see, it does not suffer from the same flaw. In fact, due to the clockwise/crosswise alternations, this will result in a true four voice canon since all four of the voices get to play all four of the chordal functions.
Here in Example XV we have another version of the second integrated modality proof which displays the fact that it is a true four voice harmonic canon (I labelled it a double harmonic canon, which it is, but is also a true four voice canon).
There are harmonic canons present - but well hidden - in some of Bach's preludes and chorale harmonizations, so the technology is not new by any means. I'm virtually certain that Bach was aware of these since he was such a transcendental contrapuntal master, but since these canons can be incidental to producing repeating root motion patterns, the very remote possibility does exist that some of them were simply artifacts of his harmonic compositional process. So far as I know, neither Bach or any of his students ever mentioned them pedagogically. Probably the most famous harmonic canon in all of music history is the Pachelbel Canon in D, which uses a I, V, vi, iii, IV, I, ii(6/5), V progression, which has alternating regressive and super-progressive root motions right until the end, and that was written a generation before Bach's time (Five years before Bach's birth, if memory serves), so I'm fairly certain this technique was well known in the north German school. I find it rather odd that I have never come across any mention of it within that period's context, but then I don't read German.
We need not stop here with melodic developments either, as we could create a long, drawn out Pachelbelian variation set from these progressions, but the proofs which I wanted to present are complete with this example: The point is that the series works out canon on its own if it is allowed to realize its resolutional desires in a cyclically repeating manner.
She could probably get me to eat more fruits and vegatables.