Musical Implications of the Harmonic Overtone Series: Chapter VII
Contrapuntal Implications of the Series
I have added a sidebar section for this series now, as the intial post will soon disappear off of the bottom of the front page, and I do not wish to display more posts per page due to bandwidth considerations. In order to get the title to fit on a single line, I had to use "Musical Relativity Theory," which I thought was kind of humorous. But then, I'm easily amused.
Historically, counterpoint evolved before harmony did, but the contrapuntal desires that tone combinations have originate in the passive-tone/active-tone dichotomy found in the series. This is where the mechanics of melody originate as well, and I am now going to have a tenth chapter which addresses melody, but everything contrapuntal and melodic finds its origin and impetus in harmonic aspects of the series, so it actually makes more sense to present these subjects in the reverse order from their historical order of appearance.
Furthermore, harmony is actually a far more complex system than counterpoint is, despite how counterintuitive that may seem: The problem is that counterpoint, as traditionally taught, is filled with burdensome rule-sets which never explain the underlying logic beneath the concept.
The reason that these rule-sets evolved was because they were based on deduced practices of composers, and not on any underlying scientific logic. So, the rule-set that is traditionally taught as Sixteenth-Century Counterpoint is simply the rule-set which describes the practices of the Giovanni Pierluigi da Palestrina school, and the rule-set that is traditionally taught as Eighteenth-Century Counterpoint is just the rule-set which describes the practices of the Johann Sebastian Bach school.
When I was teaching myself counterpoint, I bought just about every book on counterpoint ever printed in the English language, and studied my way through them all. Books by Benjamin, Fux, Gauldin, Gedalge, Jeppesen, Kennan, Mann, Norden, Piston, Shillinger, Soderlund, Taneiev, and Zarlino, which is not a complete list; only the ones I've kept around. Though virtually all of these authors offered some insight or other that I found useful, none of them ever got behind the descriptioins of the processes to the actual origin of the mechanics.
As I began to compose counterpoint for myself, I began to discard - one by one - all of the rules which I found to be either useless or just plain wrong. Useless rules are - to me - prohibitions which only describe a particular aspect of a composer's style, and not what the series implies is and isn't permissable. Rules that are wrong are simply prohibitions which the series implies are not.
What I was discovering was that, just as traditional harmonic voice leading was a simplified countrapuntal system, traditional tonal counterpoint was only a more intricate one: Traditional tonal counterpoint and traditional harmony are differentiated by the preponderance of contrapuntal effects versus harmonic effects, and that is really all there is to it.
Once the true nature of harmonic transformational logic is revealed and understood, adding contrapuntal effects to the system - and going back and forth between harmonic and contrapuntal effects - becomes quite simple. Writing melodies from this perspective is also easier, as I'll show in chapter ten.
As I'm sure is the case with many artists, I have a very asymmetrical set of talents and abilities. Though I always test in the top one percent in abstract reasoning, my verbal scores are only slightly above average, and my numerical abilities are actually below the fiftieth percentile. This causes me no end of frustration, as formulaic systems such as those presented by Schillinger and Taneiev - especially Taniev, who I think I could greatly benefit from if I could only wrap my brain around his formulas - are, quite simply, beyond my ability to grasp. I only bring this up because it is with much trepidation that I present anything involving numbers for fear of turning off similarly inclined musician-readers.
Ratios, however, I can visualize internally - and I'm betting most musicians can do this as well - so I am going to go over the series and its ratios at this point, because counterpoint simply cannot be understood properly without doing this. Nither can rhythm, which I'll demonstrate in the next chapter.
On the top system and the first half of the second system, in Example 1, the harmonic overtone series is presented to the twelfth harmonic, which is where the first semitone appears. There are several reasons for this, only one of which is to distill out the three immutable laws of counterpoint. If you simply number the partials starting with the fundamental as "1" (Versus numbering the harmonics starting at "1" with the fundamental being "0"), the series actually works all of the ratios out for you, as can be seen.
Within the first seven partials, all adjacent intervals and their inversions are consonances, and beyond the seventh partial, all adjacent intervals and their inversions are dissonances.
Among the consonances there are two types: Perfect and imperfect. Dissonances are imperfect by definition.
It is important to reiterate at this point that the contrapuntal impetus is contained within the overtone chord and its passive-tone/active-tone dichotomy, as per Example 2.
The series explains for us exactly why the consonances fall into two categories. Example 3 has the consonances and their octave displacement inversions presented along with their ratios to demonstrate this. The perfect octave has the ratio of 2:1 and its octave displacement (A double octave, actually) yeilds the same 2:1 ratio. The ratio of 2:1 is what is called a super-particular ratio, which means that the two terms differ by one: 2 - 1= 1. Even I understand aritmetic that simple.
The perfect fifth at 3:2 inverts to the perfect fourth of 4:3, and both of those ratios are also super-particulars.
CONSONANCES WHICH REMAIN SUPER-PARTICULAR RATIOS WHEN INVERTED ARE PERFECT.
The major third at 5:4 is a super-particular ratio (5 - 4= 1), but when inverted it becomes a minor sixth, which at 8:5 differs by three (8 - 5= 3), and so is not a super-particular. Likewise, the minor third at 6:5 is a super-particular (6 - 5= 1), but in inversion it becomes a major sixth, which at 5:3 differs by two (5 - 3= 2), and so is not a super-particular.
CONSONANCES WHICH ARE SUPER-PARTICULAR IN ONLY ONE INVERSION ARE IMPERFECT.
Understanding this leads us to the three immutable laws of contrapuntal motion:
1) Perfect consonances may not move together in parallel stepwise motion.
2) Imperfect consonances may move together in parallel stepwise motion.
3) Dissonances may not move together in parallel stepwise motion.
This really is all there is to counterpoint: Everything else can be extrapolated out between these three laws of contrapuntal motion, harmonic progression dynamics, and melodic theory, as I shall demonstrate.
Some may question the prohibition against parallel perfect fourths, since so-called "simple counterpoint" allows for these. There is no problem with simple counterpoint, per se but it is not pure counterpoint. As I mentioned previously, traditional harmony and tonal counterpoint both contain aspects of each - and the dividing line between highly decorated harmony and very plain tonal counterpoint is positively impossible to draw - so what is called simple counterpoint is just using the harmonic transformational logic which produces as artifacts parallel perfect fifths and parallel perfect fourths, though the parallel perfect fifths are quite exceptional (But there is a famous "hidden" one which Bach hides with a rest in Contrapunctus I from Die Kunst der Fuge, so they do appear from time to time).
Obviously, if you want to write counterpoint which inverts at the octave, parallel perfect fourths must be avoided, and therein lies the proof: All technically correct pure counterpoint will invert at the octave with no resulting forbidden parallels.
The series also gives us the logic for why this is so. Obviously, parallel octaves sound almost exactly like a single line, and this is because two overtone series which are generated by tones an octave apart blend so perfectly. In fact, the upper tone in the octave relationship is positively merged into the first and most powerful overtone in the series generated by the lower. If the idea is to combine two melodies and have them each maintain an independent nature, parallel octaves must be scrupulously avoided.
The relationship between the two series generated by tones a perfect fifth apart is quite similar: The first harmonic of the upper tone is absorbed into the second harmonic of the lower, and so they blend all too perfectly. The squishy rules allowing for parallel perfect fourths in so-called simple counterpoint aise because the perfect fifth's inversion, the perfect fourth, does not suffer the blending problems of the octave and fifth relationships. However, since counterpoint's intervallic logic is built upon the octave inversion principle, and fourths invert to fifths, these parallelisms must technically be avioded in pure counterpoint.
There is another factor to consider here, however, and that is the fact that all octaves are by definition perfect in any series-implied system: There is an absolute zero percent chance for variety with parallel octaves. Parallel fifths and fourths are similar in that they are all perfect in the diatonic system except for one: So, there is only an 14.3% chance for variety with parallel fifths and fourths. In that one place, however, parallel fifths and fourths are allowable, which leads us to Example 4.
The first two measures of Example 4 demonstrte the point at which parallel fifths and fourths are OK due to the intermediary A4/d5 interval. The rest of the example demonstrates DINO's: Dissonances In Name Only. The augmented second is as a sonoroty the same as a minor third, so it is no problem to move in and out of it in parallel. The same is true for the diminished seventh, which is in reality a major sixth; the diminished fourth, which is in reality a major third; the augmented fifth, which is actually a minor sixth, and so on. These situations become more common as the chromaticism of the counterpoint increases, naturally.
Finally, I want to return to the annotated series to point out a few things which relate back to the series' implications for equal temperament. There are two differing minor thirds at 6:5 and 7:6, then there are three major seconds at 8:7, 9:8, and 10:9, finally, there are several minor seconds from 12:11 - which borders on being a whole step in size - up to and beyond the 15:14 minor second which is used in 7-limit just tuning and the 16:15 minor second which is used in 5-limit just tuning.
You can say you like or prefer a particular temperament scheme if you want - I enjoy one called Kirnberger III for Baroque harpsichord music, for instance - but you must admit that there is no more logical system implied by the series than equal temperament. It solves so much so simply and allows for unrestricted compositional freedom within the implications of the series.
Another purely contrapuntal factor that the series implies and defines is the delayed resolution principle demonstrated in Example 5. It is a purely contrapuntal factor because in pure harmony all rsolutions are, by definition, immediately concurrent with the root motion. In this contrapuntal effect, the transformational stratum is "suspended" for a time before the resolution is allowed to proceed. Over the primordial resolution, this results in the delayed resolutions of 4-3, 9-8, and 7-8, as you can see.
Where these delayed resolutions imply a sequence of parallel imperfect consonances, they can be chained together, as I have shown in Example 6 and Example 7. Obviously, the second example has much more interest as counterpoint than the simple parallel thirds, which are in fact implying a purely harmonic effect.
Our final examples for today, numbers eight and nine, are a simple proof to expose a hidden shortcoming with traditional counterpoint pedagogy: The fact that 5-6 ascending syncopation chains are actually allowable. Virtually all teachers of counterpoint would forbid such a thing, but at the heart of it is what is being implied: As can be seen from Example 8, it is a chain of parallel sixths which is being implied, not a series of parallel fifths. Generations of contrapuntists have refrained from using this resource as well, and for no reason. It sounds perfectly fine too.
What is incorrect are series of 9-8's, 6-5's, and 5-4's, as they imply parallel perfect consonances. Likewise, 8-7's imply parallel dissonances: When in doubt, work it out... as I have done in these axamples.
It's time for "Highly Unlikely Embarrassments" I guess.