Musical Implications of the Harmonic Overtone Series: Chapter X
Formal Implications of the Series
NOTE: I have been going back and revising some of the earlier posts to correct misspellings, awkward grammar, and to get rid of some of the more overwrought sentences. I've been trying to clear up some of the more opaque sections with these revisions as well. In doing this, I have renumbered the chapters, so the chapter numbers in the examples won't match up after chapter four now. I'm revising all of this in the original Encore example file - and I'm using consistent Roman numeral/Arabic numeral logic in it as well - so the final revision which will go into Word and end up as a PDF file will be far better sorted out. I think that after I'm done with the current version, I'll only be a couple of revisions away from having this in a publishable form.
One of the nice things about going over this material multiple times is that each go-round I gain new insights I didn't have previously. So, not only did I extrapolate some melodic implications from the series this time around, but I also got a toehold on some formal implications. This chapter, as it stands now, will be little more than a place holder, as I just started making some connections on this topic since the last time I looked at the rhythmic implications of the series.
Form is really a meta-rhythmic aspect of music - rhythm on a broad temporal scale - so it relates back to the resultants of interference that are derived from the intervallic ratios in the series. Since I am unaware of any composers who have applied this type of logic to form in any other way than intuitively, there is not much material to draw on. Nonetheless, just two quick examples will show that formal elements can be related to the rhythmic resultants of the series' ratios in more than one way.
One of the most ancient musical forms - if not the most ancient - is the simple binary form in 4/4 time with four measure phrases which I have illustrated in Example 1: Note that the local time signature is expanded into meta-rhythmic phrase lengths. The first of these phrases is repeated, and the second phrase brings the piece (Or, the verse) to a conclusion. Obviously, the duple-related time relates to the 2:1 ratio of the octave interval, but the 2*A + B proportions of the form itself do as well. Since the octave is the first interval in the series, it is no surprise that this form was intuited over a thousand years ago.
If you take this form and vary the repeated A-section, you start to move up the series: In that instance you are moving toward an eight measure section - not just a literally repeated four bar phrase - followed by a four measure section. This follows the 2 + 1 antecedent part of the resultant of interference that the perfect fifth's 3:2 ratio gives. Now, if you repeat this entire structure - A8 + B4: A8 + B4 - and then add a bridge to it followed by a final A-section, you get two antecedents of the perfect fifths resultant followed by the consequent: Many songs follow this form or something very close to it.
From this you can see that the possibility of applying the series-derived resultants to formal proportions offers a lot of possibilities which have only casually and intuitively been explored by composers of the past. I really find the resultant's potential application to the meta-rhythmic aspects of form to be far more compelling than their potential use as surface rhythm embellishments, and it certainly offers a more series-derived approach to form than the old Golden Mean method (Which is really nothing more than using one possible ratio all of the time and on every level).
In Example 2 I have presented the underlying skeleton of the twelve bar blues form: Though countless embellishments have been made to this form over the years - from busy Bebop arangements to rousing Rockabilly renditions - this is the irreducible essence of it, and it demonstrates some very interesting characteristics.
If we do a simple measure count for the three cardinal harmonies, we find that the tonic overtone chord appears in 6.0 measures total, the subdominant overtone chord appears in 4.5 measures total, and the dominant overtone chord appears in 1.5 measures total. Multiplication by ten gets rid of the decimal points and yeilds a ratio of 60:45:15, and since all of the terms are divisible by 15 this yeilds 4:3:1. The 4:3 ratio is the series' perfect fourth, of course, and not only is the blues' root motion continuity dominated by fourths and biased toward the fourth degree, but the ratio multiplied gives the number of measures at twelve. Intuition is a powerful force.
This points out another possible meta-rhythmic application for ratios in the series: Durational proportions for tonic, dominant, and subdominant harmonies; durational proportions for secondary key-regions within a piece, &c. The possibilities are mind boggling. But then, my mind is easily boggled.
This concludes the musical subjects addressed by implication in the harmonic overtone series for now. Though many more musical details can be extraplated from the series - the countless totality of them in fact - all of the main aspects of music have now been addressed at least in part. The final section will knit all of these subjects together by looking at some simple musical examples.
What would Mr. Spock estimate the odds for this situation to be?